An algebraic study of Volterra integral equations and their operator linearity
aa r X i v : . [ m a t h . R A ] A ug AN ALGEBRAIC STUDY OF VOLTERRA INTEGRAL EQUATIONS AND THEIROPERATOR LINEARITY
LI GUO, RICHARD GUSTAVSON, AND YUNNAN LIA bstract . The algebraic study of special integral operators led to the notions of Rota-Baxter oper-ators and shu ffl e products which have found broad applications. This paper carries out an algebraicstudy for general integral equations and shows that there is a rich algebraic structure underlyingVolterra integral operators and the corresponding equations. First Volterra integral operators areshown to produce a matching twisted Rota-Baxter algebra satisfying twisted integration-by-partstype operator identities involving multiple operators. In order to provide a universal space to ex-press general integral equations, free operated algebras are then constructed in terms of bracketedwords and rooted trees with decorations on the vertices and edges. Further explicit constructionsof the free objects in the category of matching twisted Rota-Baxter algebras are obtained by atwisted and decorated generalization of the shu ffl e product, providing a universal space for separa-ble Volterra equations. As an application of these algebraic constructions, the operator linearity ofintegral equations with separable Volterra kernels is established. C ontents
1. Introduction 11.1. Background of integral operators and integral equations 21.2. Integral algebras and integral equations 31.3. Outline of the paper 52. Analytic and algebraic operators for integral equations 53. A general algebraic framework for integral equations 93.1. Background 93.2. Free operated algebras and decorated rooted trees 113.3. Free operated algebras in the relative context 153.4. Evaluations, solutions and equivalence of operator equations 194. Operator linearity of separable Volterra equations 204.1. Construction of free relative matching twisted Rota-Baxter algebras 214.2. Operator linearity of separable Volterra equations 26References 281. I ntroduction
This paper sets up an algebraic framework for integral equations by operated algebras andestablishes the operator linearity of separable Volterra integral equations by applying matchingtwisted Rota-Baxter algebras.
Date : August 18, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Integral equation, Volterra operator, Volterra equation, Rota-Baxter algebra, operatedalgebra, rooted trees, linearity of integral equation.
Background of integral operators and integral equations.
Analytic aspect.
Roughly speaking, an integral equation is an equation involving integraloperators of unknown functions. The study of integral equations was begun by Fredholm [10] andVolterra [31], and the two main branches of equations within the discipline, where both limits ofintegration are fixed and where one or both limits are allowed to vary, bear their respective names.In particular, Volterra equations are defined by the Volterra integral operators(1) P K ( f )( x ) : = P K , a ( f )( x ) : = Z xa K ( x , t ) f ( t ) dt , with kernels K ( x , t ).Integral equations and operators have seen many applications since their inception. Hilbert [18]used integral equations to begin the study of functional analysis. Integral operators appear whensolving boundary value problems through the use of Green’s functions, which are the kernels ofsuch operators [30]. Furthermore, there is an extensive literature on iterated path integrals anditerated integrals of di ff erential forms in di ff erential geometry [6] where they are called Chen inte-grals, and in multiple zeta values [19] where they are also called Drinfeld integrals or Kontsevichintegrals, as well as in quantum field theory [4]. See [32, 35] for the general theory of integralequations.1.1.2. Algebraic aspect.
The study of integral equations from an algebraic perspective was morerecent. One approach has been through the lens of Rota-Baxter algebras, in which a linear oper-ator P must satisfy the Rota-Baxter identity of weight λ for some fixed scalar λ (2) P ( f ) P ( g ) = P ( f P ( g )) + P ( P ( f ) g ) + λ P ( f g ) . When λ =
0, this can be taken as an abstraction of the integration by parts identity for the usualsingle-variable integral by letting P ( f )( x ) = R xa f ( t ) dt ; letting λ vary allows Rota-Baxter alge-bras to be applied in many areas outside of integral equations. Indeed Rota-Baxter algebras werefirst used by G. Baxter in fluctuation theory [1], and by G.-C. Rota in combinatorics [27]. Thealgebraic structure for the space of integral operators R xa f ( t ) dt using shu ffl e products was de-veloped from this perspective; see [16]. From another perspective, integro-di ff erential algebraswere developed to algebraically study boundary problems for linear ordinary di ff erential equa-tions [11, 17, 25].Other algebraic identities are known to hold for special integral operators. One distinguishedexample is the Reynolds operator identity(3) R ( f ) R ( g ) = R ( f R ( g )) + R ( R ( f ) g ) − R ( R ( f ) R ( g )) for all f , g ∈ A , satisfied by the Volterra integral operator P ( f )( x ) : = R xa e x − t f ( t ) dt which served as one of the firstexamples of the Reynolds operator originated from the famous work of O. Reynolds on turbulencetheory in fluid mechanics [23] and was interested to R. Birkho ff and G.-C. Rota [2, 26, 28].A recent example is the matching Rota-Baxter operators [14, 37], consisting of a family oflinear operators P ω , ω ∈ Ω , for a parameter set Ω satisfying the identities(4) P α ( f ) P β ( g ) = P α ( f P β ( g )) + P β ( P α ( f ) g ) + λ β P α ( f g ) for all f , g ∈ A , α, β ∈ Ω . They are satisfied by a family of integral operators P ω ( f )( x ) : = R xa h ω ( t ) f ( t ) dt for a parametricfamily of functions h ω ( t ).Despite all these developments, our understanding is still very limited on the algebraic aspectof integral operators and integral equations, in comparison with their di ff erential counterparts LGEBRAIC STUDY OF INTEGRAL EQUATIONS 3 which, through the pioneering work of Ritt and Kolchin, and contributions of many authors overthe recent decades, has evolved into a vast field of theory and applications [20, 22, 24].1.1.3. Di ff erentiation versus integration, algebraically. To get some idea on how to move for-ward with an algebraic study of integral operators and equations, we compare with the algebraicstudy of di ff erential operators and di ff erential equations.Evidently, the integral analysis is much more complicated than di ff erential analysis. To beginwith, while the derivation is a well-defined notion, there are a variety of meanings for an integra-tion. Taking the one variable case for example, the integral operator acting on a function f ( x ) canbe understood to be the indefinite integral (anti-derivative) R f ( t ) dt , the definite integral R ba f ( t ) dt or something in between such as R xa f ( t ) dt . This di ff erence is combined with considerations ofwhere to put the unknown functions and the nature of known functions.Next, the algebraic definition of a di ff erential operator is also long established, as the Leibnizrule or the product formula. But there is no good understanding of the algebraic characterizationof an integral. The identities in Eqs. (2) – (4) apply only for special integrals. It is critical touncover more such identities as we do with some of them in this paper, though one should notexpect to have any algebraic identities for an arbitrary integral operator.Furthermore, related to the previous point, the meaning of a di ff erential equation is also wellestablished, at least in their algebraic study, as the ring of di ff erential polynomials on a set of“di ff erentiable” variables, which are simply the polynomials on the given variables and theirdi ff erential derivations. The di ff erential polynomial algebras serve as the the universal spaceof di ff erential equations and thus the fundamental notion for di ff erential algebras. The ring ofdi ff erential polynomials is categorically characterized as the free di ff erential algebra, both in theabsolution sense (for di ff erential equations with constant coe ffi cients) and in the relative case (fordi ff erential equations with variable coe ffi cients).To emphasize, the last two points are closely interrelated because the algebraic characterizationof di ff erential operators determines the construction of free di ff erential algebras, which in turn,provides the precise meaning of di ff erential equations.Such algebraic characterizations are absent for integral operators except for some special casesas mentioned above. Thus we face the dilemma that the lack of understanding of algebraic char-acterizations of an integral operator prevents the construction of a suitable “integral” polynomialalgebra, without which one does not have a precise notion of an integral equation in which tostudy the integral operator, at least from an algebraic point of view.1.2. Integral algebras and integral equations.
The purpose of this paper is to address thesetwo important points and thus to give meanings to the notions of integral equations and integralalgebras, serving as the foundation of an algebraic study of integral equations, analogous to theirdi ff erential counterparts.More precisely, this paper provides(i) a precise framework to discuss integral equations in terms of bracketed words and dec-orated rooted trees. Indeed, the setup applies to equations of any linear operators. Thisleads to a notion of integral polynomials and hence, by setting the polynomials to zero,integral equations;(ii) a more amenable framework for integral equations with separable Volterra integral op-erators, through uncovering algebraic relations of such operators. As an application, weprove the operator linearity of separable Volterra integral equations. LI GUO, RICHARD GUSTAVSON, AND YUNNAN LI
Integral polynomials and integral equations.
An algebraic equation or a di ff erential equa-tion is defined to be the zeros of a polynomial or a di ff erential polynomial, respectively. From thisviewpoint, an integral equation should be an element of a suitably defined “integral polynomialalgebra”. To find this “integral polynomial algebra,” we note that the polynomial algebra k [ Y ] isthe free commutative algebra on a set Y , while the di ff erential polynomial algebra A { Y } is the freecommutative di ff erential algebra on the set Y . Thus this sought-after “integral polynomial alge-bra” should be the free commutative “integral” algebra on a set Y of unknown functions. Thuswe should find a suitable notion of integral algebra which, in contrast to di ff erential algebras, canhave di ff erent definitions depending on the nature of the integral operators and might not have anyalgebraic characterizations. To be versatile, we begin by imposing no conditions on the integraloperators except its linearity, in which case, such an algebra is called an operated algebra, definedin [15] but can be traced back to A.G. Kurosh [21] who called it an Ω -algebra. The constructionof free (noncommutative) operated algebras generated by sets was obtained in [15] in terms ofbracketed words, Motzkin paths and rooted trees.We adapt the construction of free operated algebras by bracketed words to the commutativecase, and then give their interpretations in terms of trees in analog to the noncommutative casebut utilizing typed decorated trees, that is, rooted trees whose vertices and edges are both deco-rated, arising from the recent study of renormalization [5, 9]. This setup will provide a generalframework for operator equations and, in particular, integral equations, with constant coe ffi cients.In order to deal with integral equations with variable coe ffi cients, we further generalize the con-struction of free commutative operated algebras on sets to those with coe ffi cients in a preassignedoperated algebra, in particular in a “coe ffi cient algebra” of functions equipped with integral oper-ators. This free object provides the formal and rigorous context for operator equations. Then anyintegral equation, regarded as a special case of an operator equation, comes from an element ofthe thus-defined free commutative operated algebra with coe ffi cients in an algebra with integraloperators.When one or a family of linear operators satisfy certain operator identities, an element in thecorresponding operator polynomial algebra can descend to a smaller algebra with more concisedescription, similar to the usual di ff erential polynomials. Such a descent is made precise by anequivalence relation of operator polynomials.1.2.2. Separable Volterra integral equations: from Reynolds operators to twisted Rota-Baxteroperators.
With the general notions of an operator equation and integral equation established,we next focus on equations in which the integral operators are the Volterra integral operatorsas in Eq. (1) in particular separable Volterra integral operators, for which the kernel K ( x , t ) isseparable. A special case is when K ( x , t ) = e x − t when the operator satisfies the Reynolds identityin Eq. (3). We explore similar identities that can be satisfied by other separable Volterra operatorsand discovered that they indeed satisfy a more general Reynolds identity which we call a weightedReynolds identity. However, their study posed challenges because of the self-cyclic nature of theirdefining equation like in Eq. (3). Unexpectedly, in our e ff ort to construct such free objects bymeans of a completion process through inverse limits, new operator identities emerged, leadingto the notion of twisted Rota-Baxter operators. To be sure, such transition is possible only in therelative context, that is, when the coe ffi cient algebra of continuous functions carries a family ofseparable Volterra operators.These twisted Rota-Baxter operators make the construction of their free objects much simpler.So just to show the end product, this paper leaves out the discussions of Reynolds operators. LGEBRAIC STUDY OF INTEGRAL EQUATIONS 5
Outline of the paper.
The main body of the paper is divided into three sections. We startwith giving an elementary discussion on the analytic and algebraic operators in integral equations,before moving to an abstract and general level to define integral equations. We finally apply thegeneral setup back to separable Volterra integral equations and establish their operator linearity,which is a weaker form of linearity.In Section 2, we give an overview of the analytic Volterra operators and equations and showthey have the algebraic property that, if the kernel is separable , i.e. is a product of single-variable functions, then the algebra of continuous functions equipped with a family { P ω | ω ∈ Ω } of separable Volterra operators is a matching Rota-Baxter algebra twisted by invertible elements τ ω (Theorem 2.9), that is, the Volterra integral operators P ω , ω ∈ Ω , satisfy identities of the form P α ( f ) P β ( g ) = τ α P β (cid:16) τ − α P α ( f ) g (cid:17) + τ β P α (cid:16) τ − β f P β ( g ) (cid:17) for all f , g ∈ C ( I ) , α, β ∈ Ω . Their relationship with the Reynolds operator is briefly discussed.In Section 3 we develop an algebraic structure for defining arbitrary integral equations, indeedarbitrary operator equations, in the context of free operated algebras of [15, 21]. The algebraicstructure we develop, called the free relative Ω -operated k-algebra over an algebra C , canbe described using bracketed words as in Theorem 3.10 or rooted trees as in Corollary 3.11.While the bracketed word setting is more precise, using rooted trees allows for a more intuitiveinterpretation as integral equations. Indeed, setting C = A [ Y ] for some function space A and Ω a set of integral operators, the free relative Ω -operated algebra is thus the integral polynomialalgebra with integral operator set Ω , coe ffi cient function space A , and variable functions Y . Thisallows us to not only rigorously define solutions to algebraic integral equations, but also describewhen two such equations are equivalent to each other; see Definitions 3.12(iii) and 3.16.In the final section of the paper we apply this abstract approach to the concrete setting ofVolterra integral equations with separable kernels. Applying the twisted Rota-Baxter relation sat-isfied by the Volterra operators allows us to obtain another construction of the free objects (The-orem 4.5). As an application, we show in Theorem 4.6 that any integral equation with separablekernels is equivalent to one that is operator linear. Examples are given throughout to illustratehow our algebraic results apply to specific Volterra equations. Notations.
In this paper, we fix a ground field k of characteristic 0. All the structures under dis-cussion, including vector spaces, algebras and tensor products, are taken over k unless otherwisespecified. Likewise, all algebras are assumed to be commutative unitary k -algebras.2. A nalytic and algebraic operators for integral equations In this section we give a first discussion of the relationship between analytically defined op-erators from integrals and algebraically defined operators by algebraic operator identities, beforetheir general study in later sections. We show that a Volterra integral operator is a Rota-Baxteroperator for only phantom kernels. For separable kernels, it can be realized by a more generaloperator that we introduce, called twisted Rota-Baxter operators.
Definition 2.1.
Let I ⊆ R be an open interval and let C ( I ) be the algebra of continuous functionson I . (i) A Volterra operator is a linear operator P K : = P K , a : C ( I ) → C ( I ) , P K ( f )( x ) : = Z xa K ( x , t ) f ( t ) dt , for some a ∈ I and kernel K ( x , t ) ∈ C ( I ); LI GUO, RICHARD GUSTAVSON, AND YUNNAN LI (ii) A kernel K and the corresponding Volterra operator are called separable if it can bedecomposed as K ( x , t ) = k ( x ) h ( t ) for some functions k and h in C ( I );(iii) A kernel K and the corresponding Volterra operator are called phantom if it is a functionof only the variable of integration (i.e. the “dummy” variable). It is the special case of aseparable kernel when k ( x ) is a constant;(iv) An integral equation is called separable (resp. phantom ) if all the integral operators inquestion are separable (resp. phantom) and share the same lower limit.By imposing suitable convergent conditions on C ( I ) at the endpoints of I , we can also take a to be one of the endpoints of I , allowing us to discuss the Thomas-Fermi equation below. Example 2.2. (i) Volterra’s population model for a species in a closed system [34] describesthe population u ( t ) of a species when exposed to both crowding and toxicity; it can bewritten as an integral equation of the form(5) u ( t ) = u + a Z t u ( x ) dx − b Z t u ( x ) dx − c Z t u ( x ) Z x u ( y ) dy dx , where a , b , and c are the birth rate, crowding coe ffi cient, and toxicity coe ffi cient, respec-tively, and u is the initial population. Here there is one integral operator(6) P : f ( t ) Z t f ( x ) dx . (ii) The Thomas-Fermi equation [33] describes the potential y ( x ) of an atom in terms of theradius x , and can be written as an integral equation of the form(7) y ( x ) = + Bx + Z x Z t s − / y ( s ) / ds dt , where B is a known parameter. In this setting, there are two integral operators(8) P : f ( x ) Z x f ( t ) dt , P : f ( x ) Z x t − / f ( t ) dt . Definition 2.3.
An integral equation is called operator linear if it does not contain any productsof Volterra integral operators.Note the di ff erence between an equation being operator linear and linear in the usual sense,namely the unknown function only appears once to the first power in every term of the equation.For example, Eq. (5) is not linear since it contains u , but it is operator linear since there is noproduct of integral operators. The two integrals in the last term are nested into each other, notmultiplied. Similarly, a term yP ( y ) is operator linear, but not linear. To the contrary, the equation(9) f ( x ) + (cid:16) Z x (sin t )( f ( t ) − g ( t )) dt (cid:17)(cid:16) Z x exp( t ) g ( t ) dt (cid:17) = LGEBRAIC STUDY OF INTEGRAL EQUATIONS 7
Definition 2.4. (i) For any λ ∈ k , a Rota-Baxter algebra of weight λ is a k -algebra A together with a linear operator P : A → A satisfying P ( f ) P ( g ) = P ( f P ( g )) + P ( P ( f ) g ) + λ P ( f g ) for all f , g ∈ A . (ii) [37] Fix a non-empty set Ω and let λ Ω : = ( λ ω | ω ∈ Ω ) ⊆ k . An Ω -matching Rota-Baxter algebra of weight λ Ω , or simply a matching Rota-Baxter algebra is a k -algebra A together with a family of linear operators P Ω : = ( P ω | ω ∈ Ω ), where P ω : A → A forall ω ∈ Ω , satisfying(10) P α ( f ) P β ( g ) = P α ( f P β ( g )) + P β ( P α ( f ) g ) + λ β P α ( f g ) for all f , g ∈ A , α, β ∈ Ω . If λ Ω = ( λ ), then we say that the matching Rota-Baxter algebra has weight λ .Note that any Rota-Baxter algebra is also a matching Rota-Baxter algebra by letting Ω containa single element.We first consider the simple case K ( x , t ) = K ( t ), that is, K is a phantom kernel. In this case P K is essentially the classical integral operator I ( f )( x ) : = R xa f ( t ) dt and the Rota-Baxter identity ofweight zero holds:(11) I ( f ) I ( g ) = I ( f I ( g )) + I ( I ( f ) g ) for all f , g ∈ C ( I ) . In fact, as K ∈ C ( I ) varies, the family of operators P K form a matching Rota-Baxter algebra inDefinition 2.4.(ii). More precisely, the following result holds. Proposition 2.5.
For any K , H ∈ C ( I ) , P K and P H satisfy the matching Rota-Baxter identityP K ( f ) P H ( g ) = P K ( f P H ( g )) + P H ( P K ( f ) g ) for all f , g ∈ C ( I ) , making ( C ( I ) , ( P K ) K ∈ C ( I ) ) a matching Rota-Baxter algebra of weight .Proof. We note that P K ( f ) = I ( K f ) and P H ( g ) = I ( Hg ) for the integral operator I ( f )( x ) : = R xa f ( t ) dt . Since I is a Rota-Baxter operator of weight zero, we obtain P K ( f ) P H ( g ) = I ( K f ) I ( Hg ) = I ( K f I ( Hg )) + I ( I ( K f ) Hg ) = P K ( f P H ( g )) + P H ( P K ( f ) g ) , as needed. (cid:3) We now consider the case when the kernel K ( x , t ) is indeed a function of x in addition to t . Here, the Volterra integral operator P K is no longer a Rota-Baxter operator. As a simpleexample, let K ( x , t ) = x and f = g =
1. Then the linear operator P K on C ( R ) with a = P K ( f )( x ) P K ( g )( x ) = x which does not agree with P K ( f P K ( g ))( x ) + P K ( P K ( f ) g )( x ) = x . In general, a Volterra operator with separable kernel K ( x , t ) = k ( x ) h ( t ) is a Rota-Baxter operatoronly in very specific circumstances, as shown in Corollary 2.11. Nevertheless, the operator cansatisfy a twisted Rota-Baxter identity. Definition 2.6. (i) For any λ ∈ k , an algebra R with a linear operator P and a specificinvertible element τ ∈ R is called a twisted Rota-Baxter algebra of weight λ with twist τ if(12) P ( x ) P ( y ) = τ P (cid:16) τ − P ( x ) y (cid:17) + τ P (cid:16) τ − xP ( y ) (cid:17) + λτ P (cid:16) τ − xy (cid:17) for all x , y ∈ R . (ii) Fix a non-empty set Ω and let λ Ω : = ( λ ω | ω ∈ Ω ) ⊆ k . An Ω -matching twisted Rota-Baxter algebra of weight λ Ω with twist τ Ω , or simply a matching twisted Rota-Baxteralgebra is a k -algebra R together with a family of linear operators P Ω : = ( P ω | ω ∈ Ω ), LI GUO, RICHARD GUSTAVSON, AND YUNNAN LI where P ω : R → R for all ω ∈ Ω , and a parametric family τ Ω : = ( τ ω | ω ∈ Ω ) of invertibleelements of R satisfying(13) P α ( x ) P β ( y ) = τ α P β (cid:16) τ − α P α ( x ) y (cid:17) + τ β P α (cid:16) τ − β xP β ( y ) (cid:17) + λ β τ α P β (cid:16) τ − α xy (cid:17) for all x , y ∈ R , α, β ∈ Ω . If τ Ω = ( τ ) (resp. λ Ω = ( λ )), then we say that the matchingtwisted Rota-Baxter algebra has twist τ (resp. weight λ ).For Ω -matching twisted Rota-Baxter algebras ( R , P Ω , τ Ω ) and ( R ′ , P ′ Ω , τ ′ Ω ), an algebra homo-morphism ϕ : R → R ′ is called an Ω -matching twisted Rota-Baxter algebra homomorphism if ϕ ( τ ω ) = τ ′ ω and ϕ P ω = P ′ ω ϕ for all ω ∈ Ω .The next result shows that a matching twisted Rota-Baxter operator is equivalent to a matchingRota-Baxter operator. Proposition 2.7.
Let P Ω be a family of linear operators on an algebra A and let τ Ω ⊆ A ofinvertible elements and λ Ω : = ( λ ω | ω ∈ Ω ) ⊆ k . Then ( A , P Ω ) is a matching twisted Rota-Baxteralgebra of weight λ Ω with twist τ Ω if and only if ( A , ˇ P Ω ) is a matching Rota-Baxter algebra ofweight λ Ω , where ˇ P ω : = τ − ω P ω τ ω , ω ∈ Ω .Proof. The proposition follows from P α ( x ) P β ( y ) − τ α P β ( τ − α P α ( x ) y ) − τ β P α ( τ − β xP β ( y )) − λ β τ α P β ( τ − α xy ) = τ α τ β (cid:16) ˇ P α ( τ − α x ) ˇ P β ( τ − β y ) − ˇ P β (cid:0) ˇ P α ( τ − α x )( τ − β y ) (cid:1) − ˇ P α (cid:0) ( τ − α x ) ˇ P β ( τ − β y ) (cid:1) − λ β ˇ P β (( τ − α x )( τ − β y )) (cid:17) . (cid:3) Convention.
With our application to integral equations in mind, we will only consider the caseof weight zero in the rest of the paper. So for a twisted Rota-Baxter operator, we always meanone with weight zero.We give more identities for a matching twisted Rota-Baxter algebra (of weight 0) for later use.
Lemma 2.8.
Let ( A , P Ω , τ Ω ) be an Ω -matching twisted Rota-Baxter algebra. Then for ˇ P ω,ω ′ : = τ − ω ′ P ω τ ω ′ , ω, ω ′ ∈ Ω , we haveP α ( x ) ˇ P β,γ ( y ) − τ α ˇ P β,γ ( τ − α P α ( x ) y ) − τ β ˇ P α,γ ( τ − β x ˇ P β,γ ( y )) = , (14) ˇ P α,η ( x ) ˇ P β,γ ( y ) − τ α τ − η ˇ P β,γ ( τ − α τ η ˇ P α,η ( x ) y ) − τ β τ − γ ˇ P α,η ( τ − β τ γ x ˇ P β,γ ( y )) = , (15) for any x , y ∈ A and α, β, η, γ ∈ Ω .Proof. To prove Eq. (14), we multiply τ γ to its left hand side and then use Eq. (13) to derive τ γ P α ( x ) ˇ P β,γ ( y ) − τ γ τ α ˇ P β,γ ( τ − α P α ( x ) y ) − τ γ τ β ˇ P α,γ ( τ − β x ˇ P β,γ ( y )) = P α ( x ) P β ( τ γ y ) − τ α P β ( τ − α P α ( x )( τ γ y )) − τ β P α ( τ − β xP β ( τ γ y )) = . For Eq. (15), we also use Eq. (13) to see thatˇ P α,η ( x ) ˇ P β,γ ( y ) − τ α τ − η ˇ P β,γ ( τ − α τ η ˇ P α,η ( x ) y ) − τ β τ − γ ˇ P α,η ( τ − β τ γ x ˇ P β,γ ( y )) = τ − η τ − γ (cid:16) P α ( τ η x ) P β ( τ γ y ) − τ α P β ( τ − α P α ( τ η x )( τ γ y )) − τ β P α ( τ − β ( τ η x ) P β ( τ γ y )) (cid:17) = . (cid:3) As the main application of matching twisted Rota-Baxter algebras we have
Theorem 2.9.
Let I ⊆ R be an open interval and let A = C ( I ) be the algebra of continuousfunctions on I. Let K ω ( x , t ) = k ω ( x ) h ω ( t ) ∈ C ( I ) , ω ∈ Ω be a family of separable kernels with allk ω ( x ) free of zeros. Let a ∈ I and let P K ω ( f )( x ) : = R xa K ω ( x , t ) f ( t ) dt , ω ∈ Ω , be the correspondingVolterra operators on A. Denote τ ω : = k ω ( x ) k ω ( a ) , ω ∈ Ω . Then ( A , P K Ω , τ Ω ) is an Ω -matching twistedRota-Baxter algebra. LGEBRAIC STUDY OF INTEGRAL EQUATIONS 9
Proof.
By assumption, the functions τ ω = k ω ( x ) k ω ( a ) , ω ∈ Ω , are invertible on I . To verify that( C ( R ) , P K Ω , τ Ω ) is an Ω -matching twisted Rota-Baxter algebra, we just need to check Eq. (13)as follows. For any f , g ∈ C ( I ) and α, β ∈ Ω , τ α P K β (cid:16) τ − α (cid:0) P K α ( f ) g (cid:1)(cid:17) + τ β P K α (cid:16) τ − β (cid:0) f P K β ( g ) (cid:1)(cid:17) = k α ( x ) k α ( a ) k β ( x ) Z xa h β ( t ) k α ( a ) k α ( t ) g ( t ) k α ( t ) Z ta h α ( u ) f ( u ) du ! dt ! + k β ( x ) k β ( a ) k α ( x ) Z xa h α ( t ) k β ( a ) k β ( t ) f ( t ) k β ( t ) Z ta h β ( u ) g ( u ) du ! dt ! = k α ( x ) k β ( x ) Z xa h β ( t ) g ( t ) dt Z ta h α ( u ) f ( u ) du + Z xa h α ( t ) f ( t ) dt Z ta h β ( u ) g ( u ) du ! = k α ( x ) k β ( x ) Z xa h α ( t ) f ( t ) dt ! Z xa h β ( t ) g ( t ) dt ! = P K α ( f ) P K β ( g ) , where the third equality is due to integration by parts. (cid:3) Corollary 2.10.
With the notations in Theorem 2.9, if Ω is a singleton, then ( C ( I ) , P K Ω ) is atwisted Rota-Baxter algebra with twist τ : = k ( x ) / k ( a ) . Corollary 2.11.
With the notations in Theorem 2.9, a Volterra operator P K ω , ω ∈ Ω is a Rota-Baxter operator if k ω ( x ) is a constant function, that is, the kernel K ω is phantom. When all kernelsK ω , ω ∈ Ω , are phantom, ( C ( I ) , P K Ω ) is a matching Rota-Baxter algebra.Proof. According to Theorem 2.9, P K ω is a twisted Rota-Baxter operator with τ ω = k ω ( x ) / k ω ( a ),which is a Rota-Baxter operator if τ ω =
1. This holds if and only if k ω ( x ) = k ω ( a ) is a constant.In this Eq. (13) becomes Eq. (10). (cid:3) Example 2.12. (i) For K ( x , t ) = e − x + t = e t / e x . The operator P K : C ( R ) → C ( R ) satisfiesthe Reynolds identity in Eq. (3). See [36] for further details on the identity. On the otherhand, for τ = k ( x ) / k ( a ) = e a − x , we obtain a twisted Rota-Baxter operator.(ii) For K ( x , t ) = x , taking τ = k ( x ) / k ( a ) = x / a , we obtain a twisted Rota-Baxter operator.3. A general algebraic framework for integral equations We now introduce a framework to express integral equations, in particular the Fredholm equa-tions and Volterra equations. This framework will on one hand provide a uniform and precisecontext for studying integral equations that are more general than their existing forms. On theother hand, this notion will also bring the algebraic methods and perspectives into the study ofintegral equations and lay the foundation for their symbolic computations. For some related liter-ature, see [3, 11, 14, 17, 25].3.1.
Background.
We will first express integral equations in terms of elements of a suitableoperated algebra [15, 16]. In a general and loose language, an algebraic integral equation is theannihilation of an “integral” algebraic expression Φ ( Y , P Ω , A ) consisting of several ingredientsand restrictions [35]:(i) an algebra of variable functions, including a set Y of unknown functions to be solvedfrom the integral equation;(ii) a set P Ω : = ( P ω | ω ∈ Ω ) of integral operators in various forms, set apart by (a) the lower or upper limits (each being fixed or variable, and in the later case, inde-pendent variables or intermediate variables);(b) the kernels for the integral operators, as functions in both the dummy variables ofthe integrals and the independent variables for the integral equation;Special cases are the Volterra operators and the Fredholm operators;(iii) a set A of “free term” or coe ffi cient functions which can appear both inside and outsideof the integrals. Some of them can be constant, treated as parameters.(iv) These ingredients are put together by the algebraic operations together with the action ofthe integral operators. Remark 3.1.
We note a subtle di ff erence in the meanings of algebraicity of an operator equation,as discussed in Item (i). If the unknown functions are polynomials of the unknown variables Y ,then the equation is called algebraic , while if other functions of Y can appear, the equation iscalled operator algebraic . In the following integral equations, the first one is algebraic while thesecond one is operator linear since y / is not polynomial in y . But the second equation is stilloperator algebraic since the operations for the integrals are algebraic (and in composition).We put Example 2.2 into this framework. Example 3.2. (i)
Volterra’s population model . Here Y = { u } , there is one integral operator P defined by Eq. (6) and the set A can be taken to be the set of all constants. ThenVolterra’s population model in Eq. (5) becomes u = u + aP ( u ) − bP ( u ) − cP ( uP ( u )) . (ii) Thomas-Fermi equation . In this setting, Y = { y } , there are two Volterra operators P and P in Eq. (8) and the set A can be taken to be the ring of polynomials R [ x ]. Then theThomas-Fermi equation (7) becomes(16) y = + Bx + P ( P ( y / )) . To inspire a precise general framework for integral equations, let us first recall how we formu-late algebraic equations. An algebraic equation consists of several ingredients:(i) a set X of variables;(ii) a set of “free term” elements from a prefixed k -algebra A over the ground field k .(iii) The ingredients can be put together by the algebraic operations.As is well-known, the general form of an algebraic equation is an element in the polynomialalgebra A [ X ].Another motivation which is more closely related to our study is di ff erential equations, con-sisting of ingredients and conditions:(i) a set Y of unknown functions;(ii) a set d Ω : = ( d ω | ω ∈ Ω ) of di ff erential operators;(iii) a set of “free term” or coe ffi cient functions from a di ff erential algebra ( A , d , Ω ) where d , Ω : = ( d ,ω | ω ∈ Ω ).(iv) The ingredients can be put together by the algebraic operations, together with the di ff er-ential operators.Note that here A carries its own commuting derivations d ,ω , ω ∈ Ω , called an Ω -ring in dif-ferential algebra [20, 29] where ∆ instead of Ω is used. Elements in A are called the coe ffi cientfunctions because they are the coe ffi cients in the di ff erential equation. A coe ffi cient function isnot necessarily a constant function (defined to be the ones in k ) for the derivations d ,ω or d ω . LGEBRAIC STUDY OF INTEGRAL EQUATIONS 11
In di ff erential algebra [20, 29], a di ff erential equation with coe ffi cients in an Ω -ring ( A , d , Ω ) isan element in the di ff erential polynomial algebra A { Y } : = A Ω { Y } which is simply the polynomialalgebra(17) A Ω { Y } : = A [ ∆ Ω Y ] , where ∆ Ω Y : = Y × C ( Ω ) . Here C ( Ω ) is the commutative monoid over Ω : C ( Ω ) : = Y ω ∈ Ω ω n ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n ω ∈ N , ω ∈ Ω , with finitely many n ω positive representing the iterations from Ω . An element τ = Q ω ∈ Ω ω n ω of C ( Ω ) is identified with the map n = n τ : Ω → N , ω n ω , with finite support. For ω ∈ Ω , the derivation d ω on A Ω { Y } first restricts to d ,ω on A and furthersends ( y , τ ) to ( y , ωτ ), that is, sending n τ to n τ,ω : Ω → N , α ( n ( α ) + , α = ω, n ( α ) , otherwise.These two conditions determine d ω uniquely since A [ ∆ Ω Y ] = A ⊗ k k [ ∆ Ω Y ] as an algebra for which d applies to a pure tensor a ⊗ f = a f by the Leibniz rule. See [20, 22, 24] for more details.For example, with Y = { y } and Ω = { d } both singletons, A { Y } = A { y } = A [ y ( n ) | n ≥
0] is thespace for di ff erential equations with one derivation d and one unknown function y .While this gives precise definitions of algebraic equations and (algebraic) di ff erential equa-tions, there does not appear to be a general notion of integral equations, even though in broadterms an integral equation is an equation involving integral operators.Based on the previous discussion, as an analog of an algebraic or a di ff erential equation, anintegral equation should be an element of a suitably defined “integral polynomial algebra” whichshould be a free object in a suitable category of integral algebras. Such integral algebras woulddepend on the integral operators involved. Since an arbitrary integral operator is not known tosatisfy any algebraic relations, in order to serve the general purpose, we impose only linearity onthe operators, which leads us to an operated algebra [15].First in Section 3.2 we give two constructions of free commutative operated algebras, one bybracketed words and one by typed decorated trees, that is, rooted trees whose vertices and edgesare both decorated. This setup will provide a general framework for operator equations and, inparticular, integral equations, with constant coe ffi cients.In order to deal with integral equations with variable coe ffi cients, we consider a relative versionof operated algebras in Section 3.3, in the sense of operated algebras with an operated homomor-phism from a given operated algebra, in practice a “coe ffi cient algebra” of functions equippedwith integral operators. The free objects in this category of relative operated algebras provide theformal and rigorous context for operator equations with variable coe ffi cients.Of course such an interpretation is not our goal by itself, but rather the first step in understand-ing integral equations from an algebraic viewpoint. Applications will be given in Section 4.3.2. Free operated algebras and decorated rooted trees.
We construct free operated algebrasfirst by bracketed works and then by rooted trees with decorations on both the vertices and theedges.
Construction of free operated algebras by bracketed elements.
We start with some generaldefinitions and notations.
Definition 3.3.
Let Ω be a nonempty set. An Ω -operated k-module (resp. Ω -operated k-algebra ) is a k -module (resp. k -algebra) V together with k -linear maps P ω : V → V , ω ∈ Ω .Denote P Ω : = ( P ω | ω ∈ Ω ). A homomorphism from an Ω -operated k -module (resp. k -algebra)( V , P Ω ) to an Ω -operated k -module (resp. k -algebra) ( U , Q Ω ) is a k -module (resp. k -algebra)homomorphism f : V → U such that f P ω = Q ω f , ω ∈ Ω .The class of Ω -operated algebras with homomorphisms between them forms the category of Ω -operated algebras.Let C be an algebra. We construct the free Ω -operated unitary algebra over C , by applyingbracketed words. See [15] for more details on bracketed words. Definition 3.4.
Let Ω be a nonempty set and C a k -algebra. The free Ω -operated k-algebra over C is an Ω -operated k -algebra ( F ( Ω , C ) , Π Ω ) together with a k -algebra homomorphism i C : C −→ F ( Ω , C ) satisfying the universal property that, for any Ω -operated k -algebra ( R , P Ω ) and k -algebra homomorphism f : C → R , there is a unique Ω -operated k -algebra homomorphism f : ( F ( Ω , C ) , Π Ω ) → ( R , P Ω ) such that f i C = f .In the special case when C is the symmetric algebra S ( V ) on a module V (resp. the polynomialalgebra k [ Y ] on a set Y ), we obtain the free Ω -operated algebra over the module V (resp. over theset Y ), where the algebra homomorphism i C is replaced by a linear map (resp. a set map). Recall S ( V ) = L n ≥ S n ( V ) where S n ( V ) is the n -th symmetric tensor power of V with the conventionthat S ( V ) = k . Denote S + ( V ) : = L n ≥ S n ( V ).Given an ω ∈ Ω and a module V , let ⌊ V ⌋ ω denote the module (cid:8) ⌊ v ⌋ ω | v ∈ V (cid:9) . So it is a copy of V but is linearly independent of V . We also assume that the modules ⌊ V ⌋ ω are linearly independentwhen ω varies in Ω . Another way to think of ⌊ V ⌋ ω is that ⌊ V ⌋ ω is a module together with a givenlinear bijection π ω : V → ⌊ V ⌋ ω for which we denote π ω ( v ) = ⌊ v ⌋ ω , hence the notation ⌊ V ⌋ ω . Inreality, ⌊ v ⌋ ω is meant to be the image of v under the action of a linear operator ⌊·⌋ ω . Note that bydefinition, we have the linearity ⌊ au + bv ⌋ ω = a ⌊ u ⌋ ω + b ⌊ v ⌋ ω for all a , b ∈ k , u , v ∈ V . Denote ⌊ V ⌋ Ω : = M ω ∈ Ω ⌊ V ⌋ ω . With the notations in Definition 3.4, we now construct the free Ω -operated algebra over analgebra C . The construction is modified after the one for free Ω -operated algebras in [15], withthe commutativity condition further imposed, as well as the condition on the generating set beingrelaxed to a generating algebra. See the remark after Definition 3.4.We will obtain the free object as the limit of a direct system { i n , n + : S ( C ) n → S ( C ) n + } ∞ n = of algebras S n : = S ( C ) n , n ≥
0, where the transition morphisms i n , n + are natural injective algebrahomomorphisms.For the initial step of n =
0, we define S : = C . LGEBRAIC STUDY OF INTEGRAL EQUATIONS 13
We then define S : = S ( C ) : = C ⊗ S ( ⌊ C ⌋ Ω ) = C ⊕ ( C ⊗ S + ( ⌊ C ⌋ Ω )) , with the natural injective algebra homomorphism i , : S : = C ֒ → S = C ⊕ ( C ⊗ S + ( ⌊ C ⌋ Ω )) . For the inductive step, let n ≥ S k , k ≤ n , and injectivealgebra homomorphisms(18) i k − , k : S k − ֒ → S k , ≤ k ≤ n , have been defined. We then define(19) S n + : = S ( C ) n + : = C ⊗ S ( ⌊ S n ⌋ Ω ) = C ⊕ ( C ⊗ S + ( ⌊ S n ⌋ Ω )) . The natural injection i n − , n : S n − ֒ → S n in Eq. (18) induces the natural linear injection ⌊ S n − ⌋ Ω ֒ → ⌊ S n ⌋ Ω and thus an algebra injection i n , n + : S n : = C ⊗ S (cid:0) ⌊ S n − ⌋ Ω (cid:1) ֒ → C ⊗ S (cid:0) ⌊ S n ⌋ Ω (cid:1) = : S n + . This completes the inductive construction of the direct system. Finally we define the directlimit of algebras(20) S ( Ω , C ) : = lim −→ S n = [ n ≥ S n . As a direct limit of algebras, S ( Ω , C ) is an algebra.For ω ∈ Ω , define a linear map ∆ ω : S ( Ω , C ) → S ( Ω , C ) , u
7→ ⌊ u ⌋ ω . Thus the pair (cid:0) S ( Ω , C ) , ∆ Ω (cid:1) is an Ω -operated algebra. Adapting the proof of the free Ω -operated algebras in [15, Corollary 3.6], we can show that (cid:0) S ( Ω , C ) , ∆ Ω (cid:1) is the free Ω -operatedalgebra over C . The proof is also a simplified version of the proof for the relative case in Theo-rem 3.10. So the reader can also be referred there for details. Theorem 3.5.
Let Ω be a nonempty set and C a k -algebra. Let j C : C ֒ → S ( Ω , C ) be the naturalinjection. The pair ( S ( Ω , C ) , ∆ Ω ) together with j C is the free Ω -operated k -algebra over C. Realization as typed trees.
Now we apply typed rooted trees, with decorated vertices andedges, to give another construction of the free Ω -operated algebra S ( Ω , C ) just obtained by brack-eted words. The rooted tree construction has the advantage of being intuitive and non-recursive,giving an easy understanding of the free object. The bracketed word construction is more precise,and the recursive definition is convenient to be applied in proofs.See [5, 9, 12] for more details on typed rooted trees from the perspective of renormalization,algebra and combinatorics. See also [7] for a related treatment for the rooted tree integrals andsums, and their renormalization.Let T denote the set of nonplanar rooted trees. For a nonempty set Ω and an algebra C , let T ( Ω , C ) denote the set of the trees in T with their vertices decorated by elements of C and theiredges decorated by elements of Ω . Such trees are called vertex-edge decorated trees . Someexamples are r a rr ab α r ✁✁ rr ❆❆ ab c α β r rrr ✡✡❏❏ ab c d α β γ r ✁✁ rr ❆❆ r a α b β c γ d r rrr ✡✡❏❏ abc d αβ γ r ✁✁ rr ❆❆ r ✁✁❆❆ r a α b β c γ d δ e , a , b , c , d , e ∈ C , α, β, γ, δ ∈ Ω . Let T ( Ω , C ) denote the module spanned by T ( Ω , C ), allowing ( k -)linearity on decorations ofthe vertices.We next define the grafting product T ⊻ U of two ( Ω , C )-decorated trees T and U to beobtained by merging the roots of T and U into a common root shared by the branches of both thetrees T and U , and decorating the common root by the product of the decorations of the roots of T and U . For example r a ⊻ r rrr ✡✡❏❏ bc d e α β γ = r rrr ✡✡❏❏ abc d e α β γ , rr ab α ⊻ rr cd β = r ✁✁ rr ❆❆ acb d α β . For each ω ∈ Ω , define a linear operator Λ ω on T ( Ω , C ) to be the extension operator , sendinga decorated tree T in T ( Ω , C ) to a new tree Λ ω ( T ) by adding a new root connecting to the root of T , decorating the edge connecting the new root to the root of T by ω and decorating the new rootby 1. For example, Λ ω ( r a ) = rr a ω , Λ ω (cid:16) r ✁✁ rr ❆❆ ab c α β (cid:17) = r rrr ✡✡❏❏ ab c ωα β . Note that often in the literature the product of trees is given by concatenation and the unaryoperator is given by grafting. See for example Connes-Kreimer Hopf algebra of rooted trees[8] and the constrution of free noncommutative operated algebras [15]. Here the role of the twooperations is somewhat reversed. Thus we use the terms grafting product and extension operatorto emphasize the di ff erence.With the notions above, the pair (( T ( Ω , C ) , ⊻ ) , Λ Ω ) is an Ω -operated algebra.We next define a bijection η : S ( Ω , C ) → T ( Ω , C )by the universal property of S ( Ω , C ) as the free Ω -operated algebra over C . More specifically, η isdefined as a direct limit of maps η n on the direct system S n , n ≥
0, defining S ( Ω , C ) in Eq. (20).By showing that η is a bijection, we will establish T ( Ω , C ) as another construction of the free Ω -operated algebra on C .In fact, T ( Ω , C ) is also the direct limit of a direct system T n , n ≥
0, which equips T ( Ω , C ) witha filtered algebra structure. First define the height of a rooted tree to be the length of the longestpath from the root to the leafs. For n ≥
0, let T n denote the subset of T with height less thanor equal to n and let T n be the linear span of T n . From the definition of the product ⊻ , T n is asubalgebra. Further the operator Λ ω , ω ∈ Ω , sends T n to T n + .To obtain the bijection η : S ( Ω , C ) → T ( Ω , C ), we first define η : S = C → T , a r a , a ∈ C , evidently a bijective algebra homomorphism.Next for a given n ≥
0, assume that a bijective algebra homomorphism η n : S n → T n has beendefined. Then the bijection defines a linear bijection ⌊ η n ⌋ ω : ⌊ S n ⌋ ω → Λ ω ( T n ) , ⌊ u ⌋ ω Λ ω ( η n ( u )) , u ∈ S n and hence an algebra homomorphism η n + : S n + = C ⊗ S ( ⌊ S n ⌋ Ω ) → T n + . To be specific, consider the decomposition S n + = C ⊕ ( C ⊗ S + ( ⌊ S n ⌋ Ω )) and consider the twocases when u is in either of the direct summands. If u is in C , then define η n + ( u ) : = r u . If u is apure tensor in C ⊗ S + ( ⌊ S n ⌋ Ω ), then u = c ⊗ ( ⌊ u ⌋ ω · · · ⌊ u k ⌋ ω k ) , c ∈ C , u i ∈ S n , ω i ∈ Ω , ≤ i ≤ k , k ≥ , LGEBRAIC STUDY OF INTEGRAL EQUATIONS 15 with the multiplication in S + ( ⌊ S n ⌋ Ω ) suppressed. Then η n ( u i ) , ≤ i ≤ k , are trees in T n by theinductive hypothesis. We thus define(21) η n + ( u ) : = r c ⊻ Λ ω ( η n ( u )) ⊻ · · · ⊻ Λ ω k ( η n ( u k )) . By the definitions of the extension operator Λ ω i and the grafting product ⊻ , the new tree η n + ( u )is the grafting of the trees Λ ω i ( η n ( u i )) , ≤ i ≤ k , with c decorating the common root. On theother hand, any rooted tree in T n + is either r c or of the forms described above and thus has thefactorization in Eq. (21). Thus η n + is surjective and an inverse map of η n + can be defined byreversing the above process. This completes the inductive construction of η and its bijectivity.To summarize, we have Theorem 3.6.
The Ω -operated k -algebra (( T ( Ω , C ) , ⊻ ) , Λ Ω ) , together with natural imbeddingi C : C → T ( Ω , C ) , is the free Ω -operated k -algebra over C. Free operated algebras in the relative context.
General discussion.
The notion of an (associative) algebra is a relative notion of a ringin the sense that an algebra is a ring R together with a ring homomorphism k → R , calledthe structure map. A similar notion is fundamental in di ff erential algebra where a di ff erentialalgebra usually means a di ff erential ring ( R , D ) together with a di ff erential ring homomorphism( A , d ) → ( R , D ) from a base di ff erential ring ( A , d ).Since an integral equation also has its coe ffi cient functions equipped with preassigned integraloperators, we give a relative notion of operated algebras. Definition 3.7.
For a given Ω -operated k -algebra ( A , α Ω ), an ( A , α Ω ) -algebra is an Ω -operated k -algebra ( R , P Ω ) together with a homomorphism (called the structure map ) j R : ( A , α Ω ) −→ ( R , P Ω ) of Ω -operated k -algebras. A homomorphism f : ( R , P Ω ) → ( S , Q Ω ) of ( A , α Ω )-algebras isa homomorphism of Ω -operated k -algebras that is compatible with the structure maps: f j R = j S .We next construct the free objects in the category of ( A , α Ω )-algebras. Let B be a k -augmentedalgebra in the sense that there is a direct sum decomposition B = k ⊕ B + of k -subalgebras. In other words, B is the unitization of a k -algebra B + . Then C = A ⊕ ( A ⊗ B + )is an A -augmented algebra in the sense that there is a direct sum decomposition C = A ⊕ C + A . Definition 3.8.
Let Ω be a nonempty set. Let ( A , α Ω ) be an Ω -operated k -algebra and let B be a k -augmented algebra. The free Ω -operated k-algebra over B with coe ffi cients in ( A , α Ω )(or simply the free ( A , α Ω ) -algebra over B ) is an ( A , α Ω )-algebra ( F A ( Ω , B ) , Π Ω ) together withan algebra homomorphism i B : B −→ F A ( Ω , B ) satisfying the universal property that, for any( A , α Ω )-algebra ( R , P Ω ) and algebra homomorphism f : B → R , there is a unique ( A , α Ω )-algebrahomomorphism f : ( F A ( Ω , B ) , Π Ω ) → ( R , P Ω ) such that f i B = f . Remark 3.9.
As our primary interest, taking B to be k [ Y ] for a set Y of variable functions and re-placing the algebra homomorphism from B by a map from Y , we obtain the free ( A , α Ω )-algebrasover the set Y . However the more general B allows non-algebraic functions in the variable func-tions to be included as noted in Remark 3.1. See also Remark 3.13. Construction by bracketed words.
To construct the free ( A , α Ω )-algebra over a k -augmentedalgebra B , we denote C : = A ⊗ B and define a direct subsystem { I n | n ≥ } of S n = { S ( C ) n | n ≥ } . Each I n is an A -augmentedalgebra with I n = A ⊕ I + n .First take I : = S = C , I + : = C + A : = A ⊗ B + , and I : = C ⊗ S ( ⌊ I + ⌋ Ω ) = C ⊕ ( C ⊗ S + ( ⌊ I + ⌋ Ω )) = A ⊕ C + A ⊕ ( C ⊗ S + ( ⌊ I + ⌋ Ω )) , I + : = C + A ⊕ ( C ⊗ S + ( ⌊ I + ⌋ Ω )) . Then we have the natural inclusions I : = C ֒ → I , I + ֒ → I + . For the inductive step, let n ≥ I k , I + k , ≤ k ≤ n , with I k = A ⊕ I + k , and injective algebra homomorphisms(22) i n − , n : I n − → I n , i + n − , n : I + n − → I + n . We then define(23) I n + : = C ⊗ S ( ⌊ I + n ⌋ Ω ) = A ⊕ C + A ⊕ ( C ⊗ S + ( ⌊ I + n ⌋ Ω )) , I + n + : = C + A ⊕ ( C ⊗ S + ( ⌊ I + n ⌋ Ω )) . The natural injection i + n − , n : I + n − ֒ → I + n in Eq. (22) induces the natural injections ⌊ I + n − ⌋ Ω ֒ → ⌊ I + n ⌋ Ω and S ( ⌊ I + n − ⌋ Ω ) ֒ → S ( ⌊ I + n ⌋ Ω ) , which then induce i n , n + : I n : = C ⊗ S ( ⌊ I + n − ⌋ Ω ) ֒ → C ⊗ S (cid:0) ⌊ I + n ⌋ Ω (cid:1) = : I n + , i + n , n + : I + n : = C + A ⊕ (cid:0) C ⊗ S + ( ⌊ I + n − ⌋ Ω ) (cid:1) ֒ → C + A ⊕ (cid:0) C ⊗ S + ( ⌊ I + n ⌋ Ω ) (cid:1) = : I + n + . This completes the inductive construction of the direct systems. Finally we define the direct limitsof algebras I : = I ( Ω , A ⊗ B ) : = lim −→ I n = [ n ≥ I n , I + : = lim −→ I + n = [ n ≥ I + n . As a direct limit of k -algebras, I and I + are k -algebras.For each ω ∈ Ω , ⌊ I + n ⌋ ω is contained in I + n + . Also, taking direct limits on the two sides in theequality I n = A ⊕ I + n , we obtain the equality I = A ⊕ I + . Thus there is a natural k -linear map Π ω : I ( Ω , A ⊗ B ) → I ( Ω , A ⊗ B ) , u ( α ω ( u ) , u ∈ A , ⌊ u ⌋ ω , u ∈ I + , ω ∈ Ω . Consequently the pair (cid:0) I ( Ω , A ⊗ B ) , Π Ω (cid:1) is an Ω -operated algebra. Further, there is the obviousinclusion j I : ( A , α Ω ) ֒ → ( I ( Ω , A ⊗ B ) , Π Ω ) of Ω -operated algebras. Hence ( I ( Ω , A ⊗ B ) , Π Ω ) isan ( A , α Ω )-algebra.We prove that I ( Ω , A ⊗ B ) satisfies the desired universal property. LGEBRAIC STUDY OF INTEGRAL EQUATIONS 17
Theorem 3.10.
Let Ω be a nonempty set and let ( A , α Ω ) be an Ω -operated algebra. Let B be a k -augmented algebra and let i B : B ֒ → I ( Ω , A ⊗ B ) be the natural injection. The pair ( I ( Ω , A ⊗ B ) , Π Ω ) together with i B is the free ( A , α Ω ) -algebra over B in the sense of Definition 3.8.Proof. Let ( R , P Ω ) be an ( A , α Ω )-algebra with the structure map j R : ( A , α Ω ) → ( R , P Ω ) which isan Ω -operated algebra homomorphism. Let f : B → R be an algebra homomorphism. We willdefine the desired ( A , α Ω )-algebra homomorphism f : ( I ( Ω , A ⊗ B ) , Π Ω ) → ( R , P Ω )from a sequence of algebra homomorphisms f n : I n → R , n ≥ , that is compatible with the direct system { I n , i n , n + } n ≥ .Again taking C : = A ⊗ B , first define f = j R ⊗ f : I : = C = A ⊗ B → R . Then define ⌊ f ⌋ : ⌊ I + ⌋ ω = ⌊ C + A ⌋ ω → R by ⌊ f ⌋ ( ⌊ u ⌋ ω ) : = P ω ( f ( u )), which then together with f : C → R induces a unique algebrahomomorphism f : I = C ⊗ S ( ⌊ I + ⌋ Ω ) → R . We also have f (cid:12)(cid:12)(cid:12)(cid:12) I = f .Inductively, for n ≥
1, suppose that f i : I i → R , i ≤ n , have been defined such that f i (cid:12)(cid:12)(cid:12)(cid:12) I i − = f i − , i ≥
1. We then define ⌊ f n ⌋ : ⌊ I + n ⌋ Ω → R , ⌊ u ⌋ ω P ω ( f n ( u )) , u ∈ I + n , ω ∈ Ω . Then together with the map f : C → R , we obtain an algebra homomorphism f n + : I n + = C ⊗ S ( ⌊ I + n ⌋ Ω ) → R . For u ∈ I + n − , we have f n + ( ⌊ u ⌋ ω ) = ⌊ f n ⌋ ( ⌊ u ⌋ ω ) = P ω ( f n ( u )) = P ω ( f n − ( u )) = ⌊ f n − ⌋ ( ⌊ u ⌋ ω ) = f n ( ⌊ u ⌋ ω ) . It then follows that f n + (cid:12)(cid:12)(cid:12)(cid:12) I n = f n . Therefore, we obtain an algebra homomorphism f : = lim −→ f n : I ( Ω , A ⊗ B ) = lim −→ I n → R . To check that f is an ( A , α Ω )-algebra homomorphism, for a ∈ A , we have f j I ( a ) = f ( a ) = f ( a ) = j R ( a ) . To see that f is an operated algebra homomorphism, for u ∈ I + = S n ≥ I + n and hence u ∈ I + n forsome n ≥
0, by the definition of f = lim −→ f n , we have f Π ω ( u ) = f ( ⌊ u ⌋ ω ) = f n + ( ⌊ u ⌋ ω ) = ⌊ f n ⌋ ( ⌊ u ⌋ ω ) = P ω f n ( u ) = P ω f ( u ) . Further, for a ∈ A , we have f Π ω ( a ) = f ( α ω ( a )) = j R ( α ω ( a )) = P ω j R ( a ) = P ω f j I ( a ) = P ω f ( a ) . This proves the existence of f . Finally, from the construction of f , it is the only way to define an ( A , α Ω )-algebra homomor-phism such that f i B = f . This completes the proof. (cid:3) Construction by rooted nonplanar forests.
Again take C : = A ⊗ B for a k -augmentedalgebra B , in particular B = k [ Y ] for a set Y . We consider the submodule E ( Ω , C ) of T ( Ω , C )spanned by vertex-edge decorated rooted trees E ( Ω , C ) ⊆ T ( Ω , C ) for which the leaf vertices aredecorated by C + A : = A ⊗ B + . For the one-vertex tree, the vertex is not taken to be a leaf and hencecan be decorated by any element in C . Then E ( Ω , C ) = { r a | a ∈ A } ⊔ E + , where E + consists of r a with a ∈ C + A and trees of height at least one with leaf vertices decorated by C + A . Then E ( Ω , C ) isclosed under the grafting product ⊻ . Further, for ω ∈ Ω , define Γ ω on E ( Ω , C ) by Γ ω ( T ) = ( r α ω ( a ) , T = r a , a ∈ A , Λ ω ( T ) , T ∈ E + ( Ω , C ) . Then the bijective ( A , α Ω )-algebra homomorphism η : S ( Ω , C ) → T ( Ω , C ) restricts to a bijec-tion η : I ( Ω , C ) → E ( Ω , C ). Thus we conclude Corollary 3.11.
Let ( A , α Ω ) be an Ω -operated algebra and let B be a k -augmented algebra. Thepair (( E ( Ω , A ⊗ B ) , ⊻ ) , Γ Ω ) is the free ( A , α Ω ) -algebra over B. Applying Theorem 3.10 and Corollary 3.11 to our subject of study, any integral equation canbe regarded as an element of E ( Ω , C ) or I ( Ω , C ). Here Ω is the set of integral operators, includingthe integral limits and kernels. So for example f ( x ) R x K ( x , t ) f ( t ) dt is taken as one integraloperator; C = A ⊗ B is the algebra of coe ffi cient functions A , such as C ( R ), together with thevariable functions B . In particular, let Y be a set of variable functions. Then C = C ( R )[ Y ]. Byenlarging this C , we can also consider the case when transcendental functions of Y are involved.As the following examples illustrate, we have a three way dictionary among E ( Ω , C ), I ( Ω , C )and integral expressions which, when set to zero, become integral equations. r a rr ab α r ✁✁ rr ❆❆ ab c α β r rrr ✡✡❏❏ ab c d α β γ r ✁✁ rr ❆❆ r a α b β c γ d r rrr ✡✡❏❏ abc d αβ γ r ✁✁ rr ❆❆ r ✁✁❆❆ r a α b β c γ d δ ea a Π α ( b ) a Π α ( b ) Π β ( c ) a Π α ( b ) Π β ( c ) Π γ ( d ) a Π α ( b ) Π β (cid:16) c Π γ ( d ) (cid:17) a Π α (cid:16) b Π β ( c ) Π γ ( d ) (cid:17) a Π α ( b ) Π β (cid:16) c Π γ ( d ) Π δ ( e ) (cid:17) a a ( R α b ) a ( R α b )( R β c ) a ( R α b )( R β c )( R γ d ) a ( R α b ) R β (cid:16) c ( R γ d ) (cid:17) a R α (cid:16) b ( R β c )( R γ d ) (cid:17) a ( R α b ) R β (cid:16) c ( R γ d )( R δ e ) (cid:17) Here α, β, γ, δ are in Ω and a , b , c , d , e are in C and further in C + A when decorating leaf vertices.Now we can define the main notions of our study, including a formal definition of integralequations. Definition 3.12.
Let ( A , α Ω ) be an Ω -operated algebra and B be a k -augmented algebra.(i) The ( A , α Ω )-algebra I ( Ω , A ⊗ B ) (resp. E ( Ω , A ⊗ B )) is called the operated polynomialalgebra in words (resp. trees ) with operator set Ω , coe ffi cient algebra A and variablealgebra B . Elements (setting to zero) in I ( Ω , A ⊗ B ) and E ( Ω , A ⊗ B ) are called operatorequations in words (resp. trees).(ii) Let B = k [ Y ] and hence A ⊗ B = A [ Y ] where Y is a set. Then I ( Ω , A [ Y ]) and E ( Ω , A [ Y ])are called the operated polynomial algebra with operator set Ω , coe ffi cient algebra A and independent variables Y . Their elements (setting to zero) are called operatorequations with the same qualifications. LGEBRAIC STUDY OF INTEGRAL EQUATIONS 19 (iii) Further let A = C ( I ) for an open interval I ⊆ R , α Ω a set of integral operators on A and Y a set of variable functions. Then I ( Ω , C ( I )[ Y ]) and E ( Ω , C ( I )[ Y ]) are called the integralpolynomial algebra with integral operator set Ω , coe ffi cient function space C ( I ) andvariable functions Y . Their elements (setting to zero) are called integral equations withthe same qualifications. Remark 3.13.
As noted in Remark 3.9, taking B to be an algebra containing non-polynomialfunctions of the variable functions Y , we can include more general operator equations than when B is taken A [ Y ]. For example, the Thomas-Fermi equation in Eq. (7), rewritten in Eq. (16), is notin I ( Ω , C ( I )[ y ]), but is in I ( Ω , C ( I ) ⊗ B ) for B = R [ y , z ] / ( z − y ).As a prototype of operator equations and integral equations thus defined, taking the operatedalgebra ( A , α Ω ) in Item (ii) to be a di ff erential algebra, we arrive at a notion of di ff erential equa-tions. Definition 3.14.
Let ( A , d ) be a di ff erential algebra. In particular, let A be the di ff erential algebraof smooth functions on an open interval I ⊆ R with the usual derivation. The operated algebra I ( { d } , A [ Y ]) (resp. E ( { d } , A [ Y ])) is called the di ff erential polynomial algebra over Y . Itselements (setting to zero) are called di ff erential equations in Y .These notions of the di ff erential polynomial algebra and a di ff erential equation is apparentlymore general than the usual notions in Eq. (17) in the usual sense of di ff erential algebra [20, 22].However, as to be shown in Proposition 3.17, the two sets of notions are equivalent.3.4. Evaluations, solutions and equivalence of operator equations.
As defined in Defini-tion 3.12, for a given element φ ∈ I ( Ω , A ⊗ B ), the equation φ = φ in the operated algebra( A , α Ω ) (or its extension), when the operator Π ω on I ( Ω , A ⊗ B ) is taken to be the specific linearoperator α ω on A , ω ∈ Ω . By the universal property of I ( Ω , A ⊗ B ), an algebra homomorphism f B : B → A gives rise to a unique Ω -operated algebra homomorphism, called an evaluation map : f : = f B : = eva (cid:12)(cid:12)(cid:12) f B , Π Ω α Ω : I ( Ω , A ⊗ B ) → A , such that f = f i B . Of particular interest to us is when B = k [ Y ] is a polynomial algebra, so f B : B → A is uniquely determined by a map f Y : Y → A , denoted by Y f Y . Then the equation φ = f Y when Π ω are taken to be the operators α Ω precisely meanseva (cid:12)(cid:12)(cid:12) Y f Y , Π Ω α Ω ( φ ) = . Thus the operator equations φ = f Y is a solution form the kernel J α Ω , f Y of f , whichis an operated ideal of I ( Ω , A [ Y ]). For a given set of linear operators α Ω on A , as we run throughall the evaluations of Y , consider the intersection(24) J α Ω : = \ Y f Y J α Ω , f Y . Elements in J α Ω are precisely the (algebraic) operator identities satisfied by elements of A and theoperators α Ω . Thus we have Proposition 3.15. (i)
Elements in I ( Ω , A [ Y ]) \ J α Ω give operator equations that do not holdidentically in ( A , α Ω ) . (ii) Two operator equations φ = and ψ = for φ, ψ ∈ I ( Ω , A [ Y ]) have the same solutionset in ( A , α Ω ) if the operators di ff er by an element in J α Ω . These properties lead us to define
Definition 3.16.
Let ( A , α Ω ) be an operated algebra and Y be a set.(i) Nonzero elements of I ( Ω , A [ Y ]) / J α Ω are called reduced operator polynomials for ( A , α Ω ).(ii) Operated polynomials φ and ψ (or their corresponding equations) in I ( Ω , A [ Y ]) are called equivalent if φ − ψ is in J α Ω .Even though it is usually not practical to display a canonical basis of the quotient I ( Ω , α Ω ) / J α Ω ,it is possible to do so when the operated ideal J α Ω is replaced by an operated subideal generatedby defining identities of algebraically defined operators such as the di ff erential operator (in dif-ferential algebra) or the Rota-Baxter operator, as we will show next.We will consider the case when the operators α Ω are separable Volterra operators in the nextsection to obtain further properties of the integral polynomials and the corresponding integralequations, namely that any such integral equation is equivalent to an integral equation that isoperator linear (Definition 2.3).As a prototype of this case, we first treat di ff erential operators and di ff erential equations. Proposition 3.17.
Let ( A , α Ω ) be a di ff erential algebra and let Y be a set. Any di ff erential poly-nomial in the sense of Definition 3.12. (i) , that is, as an element of I ( { d } , A [ Y ]) , is equivalent to adi ff erential polynomial in the usual sense of di ff erential algebra and di ff erential equation, that is,as an element of A Ω { Y } in Eq. (17) .Proof. Let φ = φ ( Π Ω , A , Y ) in the free ( A , α Ω )-operated algebra ( I ( Ω , A [ Y ]) , Π Ω ). Consider thedefining relations of di ff erential operators in a di ff erential algebra:(25) Π α ( uv ) − Π α ( u ) v − u Π α ( v ) , for all u , v ∈ I ( Ω , A [ Y ]) , α ∈ Ω , in I ( Ω , A [ Y ]).Since α Ω are di ff erential operators on A , for any f : Y → A , the induced ( A , α Ω )-algebrahomomorphism f sends the expressions in Eq. (25) to zero. Thus the operated ideal I di ff generatedby these expressions is contained in the ideal J α Ω defined in Definition 3.16.By definition, the quotient ( A , α Ω )-algebra I ( Ω , A [ Y ]) / I di ff is the free ( A , α Ω )-di ff erential alge-bra. Thus by the fact that A Ω { Y } in Eq. (17) is also the free ( A , α Ω )-di ff erential algebra, we havethe natural isomorphism I ( Ω , A [ Y ]) / I di ff (cid:27) A Ω { Y } of di ff erential algebras. In fact, the isomorphism (or its inverse) is given linearly by sending adi ff erential monomial ( y , Q ω ∈ Ω ω n ω ) ∈ Y × C ( Ω ) in A Ω { Y } = A [ Y × C ( Ω )] to the correspondingelement Q ω ∈ Ω Π n ω ω ( y ) in I ( Ω , A [ Y ]). This means that, in particular, the element φ ( Π Ω , A , Y ) in I ( Ω , A [ Y ]) is congruent modulo I di ff to an element of A Ω { Y } . Since I di ff is contained in J α Ω , byDefinition 3.16, φ ( Π Ω , A , Y ) is equivalent to a di ff erential equation in I ( Ω , A [ Y ]) of the form givenin A Ω { Y } . (cid:3)
4. O perator linearity of separable V olterra equations In this section, we first construct free relative matching twisted Rota-Baxter algebras. We thenapply the construction to prove the operator linearity of separable Volterra equations.
LGEBRAIC STUDY OF INTEGRAL EQUATIONS 21
Construction of free relative matching twisted Rota-Baxter algebras.
We start with therelated notions. As the analog of an algebra over a ring in the context of matching twisted Rota-Baxter algebras, we give
Definition 4.1.
Fix an Ω -matching twisted Rota-Baxter algebra ( A , P Ω , τ Ω ). An ( A , P Ω , τ Ω ) -matching twisted Rota-Baxter algebra is an Ω -matching twisted Rota-Baxter algebra ( R , P ′ Ω , τ ′ Ω )together with a matching twisted Rota-Baxter algebra homomorphism ϕ : ( A , P Ω , τ Ω ) → ( R , P ′ Ω , τ ′ Ω ).We also use ( R , P ′ Ω , τ ′ Ω , ϕ ) to denote the data. For ( A , P Ω , τ Ω )-matching twisted Rota-Baxter al-gebras ( R , P , Ω , τ , Ω , ϕ ) and ( R , P , Ω , τ , Ω , ϕ ), a matching twisted Rota-Baxter algebra homo-morphism f : ( R , P , Ω , τ , Ω ) → ( R , P , Ω , τ , Ω ) is called an ( A , P Ω , τ Ω ) -matching twisted Rota-Baxter algebra homomorphism if ϕ = f ϕ .We will determine the free objects in the category of ( A , P Ω , τ Ω )-matching twisted Rota-Baxteralgebras, defined precisely as follows. Definition 4.2.
Given an Ω -matching twisted Rota-Baxter algebra ( A , P Ω , τ Ω ) and an augmented k -algebra B , the free ( A , P Ω , τ Ω ) -matching twisted Rota-Baxter algebra over B , denoted by( R A ( B ) , P B , τ B , ϕ B , j B ), is an ( A , P Ω , τ Ω )-matching twisted Rota-Baxter algebra ( R A ( B ) , P B , Ω , τ B , Ω , ϕ B )with a k -algebra homomorphism j B : B → R A ( B ) satisfying the following universal property.For any ( A , P Ω , τ Ω )-matching twisted Rota-Baxter algebra ( R , P ′ Ω , τ ′ Ω , ϕ ) and a k -algebra homo-morphism f : B → R , there exists a unique ( A , P Ω , τ Ω )-matching twisted Rota-Baxter algebrahomomorphism ¯ f : ( R A ( B ) , P B , Ω , τ B , Ω , ϕ B ) → ( R , P ′ Ω , τ ′ Ω , ϕ ) such that f = ¯ f j B .In the case when B is the polynomial algebra k [ Y ] on a set Y or the symmetric algebra S ( V )over a k -module V , we obtain the notions of a free ( A , P Ω , τ Ω )-matching twisted Rota-Baxteralgebra over Y or V .Let ( A , P Ω , τ Ω ) be a fixed Ω -matching twisted Rota-Baxter algebra. Let B = k ⊕ B + be anaugmented k -algebra. Denote A : = A ⊗ B , A + : = A ⊗ B + . Then A = A ⊕ A + .We next construct the free ( A , P Ω , τ Ω )-matching twisted Rota-Baxter algebra over B . See [12,13] for the case of matching Rota-Baxter algebras, that is, when τ Ω = (1 A ).For a k -module U , denote the colored tensor power U ⊗ Ω n : = k { u ⊗ ω u ⊗ ω u ⊗ ω · · · ⊗ ω n − u n − | u i ∈ U , ω i ∈ Ω , ≤ i ≤ n − } . Consider the submodule F TRB ( A , B ) : = M k ≥ A ⊗ Ω ( A + ) ⊗ Ω k ⊆ X Ω ( A ) : = M k ≥ A ⊗ Ω k . Denote ˇ P ω,ε : = τ − ε P ω τ ε , ω, ε ∈ Ω . First define a system of linear operators P B ,ω , ω ∈ Ω on F TRB ( A , B ) by(26) P B ,ω ( u ) : = P ω ( u ) , u ∈ A , n = , ˇ P ω,ω ( u ) ⊗ ω u ⊗ ω · · · ⊗ ω n u n − τ ω τ − ω ⊗ ω τ − ω τ ω ˇ P ω,ω ( u ) u ⊗ ω u ⊗ ω · · · ⊗ ω n u n , u ∈ A , n > , A ⊗ ω u ⊗ ω u ⊗ ω · · · ⊗ ω n u n , u ∈ A + . for any u = u ⊗ ω u ⊗ ω · · · ⊗ ω n u n ∈ A ⊗ Ω ( A + ) ⊗ Ω n , n ≥ Define a binary operation ⊛ : = ⊛ Ω : F TRB ( A , B ) ⊗ F TRB ( A , B ) → F TRB ( A , B ) , u ⊛ Ω v : = u v ⊗ ε v ′ , if m = , n ≥ , u v ⊗ ω u ′ , if m > , n = , u v τ ω ⊗ ε (cid:16)(cid:16) τ − ω ⊗ ω u ′ (cid:17) ⊛ Ω v ′ (cid:17) + u v τ ε ⊗ ω (cid:16) u ′ ⊛ Ω (cid:16) τ − ε ⊗ ε v ′ (cid:17)(cid:17) , if m , n > , (27) for any u = u ⊗ ω u ′ ∈ A ⊗ Ω ( A + ) ⊗ Ω m and v = v ⊗ ε v ′ ∈ A ⊗ Ω ( A + ) ⊗ Ω n with m , n ≥
0, and extendlinearly.
Proposition 4.3.
Let ( A , P Ω , τ Ω ) be an Ω -matching twisted Rota-Baxter algebra and B an aug-mented k -algebra. The triple (( F TRB ( A , B ) , ⊛ Ω ) , P B , Ω , τ Ω ) is an Ω -matching twisted Rota-Baxteralgebra.Proof. By Eq. (27), the binary operation ⊛ Ω is clearly commutative. We first prove the associa-tivity of ⊛ Ω . For this we only need to deal with the case when u = u ⊗ ω u ′ ∈ A ⊗ Ω ( A + ) ⊗ Ω m , v = v ⊗ ε v ′ ∈ A ⊗ Ω ( A + ) ⊗ Ω n and w = w ⊗ γ w ′ ∈ A ⊗ Ω ( A + ) ⊗ Ω l with m , n , l > ω, ε, γ ∈ Ω , sincethe other cases are clear. Then by induction on m + n + l we can use Eq. (27) to show that( u ⊛ Ω v ) ⊛ Ω w = (cid:16) u v τ ω ⊗ ε (cid:16)(cid:16) τ − ω ⊗ ω u ′ (cid:17) ⊛ Ω v ′ (cid:17) + u v τ ε ⊗ ω (cid:16) u ′ ⊛ Ω (cid:16) τ − ε ⊗ ε v ′ (cid:17)(cid:17)(cid:17) ⊛ Ω w = u v w τ ω τ ε ⊗ γ (cid:16)(cid:16) τ − ε ⊗ ε (cid:16)(cid:16) τ − ω ⊗ ω u ′ (cid:17) ⊛ Ω v ′ (cid:17) + τ − ω ⊗ ω (cid:16) u ′ ⊛ Ω (cid:16) τ − ε ⊗ ε v ′ (cid:17)(cid:17)(cid:17) ⊛ Ω w ′ (cid:17) + u v w τ ω τ γ ⊗ ε (cid:16)(cid:16)(cid:16) τ − ω ⊗ ω u ′ (cid:17) ⊛ Ω v ′ (cid:17) ⊛ Ω (cid:16) τ − γ ⊗ γ w ′ (cid:17)(cid:17) + u v w τ ε τ γ ⊗ ω (cid:16)(cid:16) u ′ ⊛ Ω (cid:16) τ − ε ⊗ ε v ′ (cid:17)(cid:17) ⊛ Ω (cid:16) τ − γ ⊗ γ w ′ (cid:17)(cid:17) = u v w τ ω τ ε ⊗ γ (cid:16)(cid:16) τ − ω ⊗ ω u ′ (cid:17) ⊛ Ω (cid:16)(cid:16) τ − ε ⊗ ε v ′ (cid:17) ⊛ Ω w ′ (cid:17)(cid:17) + u v w τ ω τ γ ⊗ ε (cid:16)(cid:16) τ − ω ⊗ ω u ′ (cid:17) ⊛ Ω (cid:16) v ′ ⊛ Ω (cid:16) τ − γ ⊗ γ w ′ (cid:17)(cid:17)(cid:17) + u v w τ ε τ γ ⊗ ω (cid:16) u ′ ⊛ Ω (cid:16)(cid:16) τ − ε ⊗ ε v ′ (cid:17) ⊛ Ω (cid:16) τ − γ ⊗ γ w ′ (cid:17)(cid:17)(cid:17) = u ⊛ Ω (cid:16) v w τ ε ⊗ γ (cid:16)(cid:16) τ − ε ⊗ ε v ′ (cid:17) ⊛ Ω w ′ (cid:17) + v w τ γ ⊗ ε (cid:16) v ′ ⊛ Ω (cid:16) τ − γ ⊗ γ w ′ (cid:17)(cid:17)(cid:17) = u ⊛ Ω ( v ⊛ Ω w ) , where the third equality uses the induction hypothesis.We next show that P B , Ω satisfies relation (13). Depending on how P B ,α ( u ) and P B ,β ( v ) aredefined in Eq. (26), we have the following six cases to check.(1) For u = a , v = b ∈ A , it is clear as P B , Ω (cid:12)(cid:12)(cid:12) A = P Ω satisfies relation (13).(2) For u = a ∈ A and v = v ⊗ ε v ′ ∈ A ⊗ Ω ( A + ) ⊗ Ω n with n >
0, we use Eq. (14) of P Ω to showrelation (13) as follows. τ α ⊛ Ω P B ,β (cid:16) τ − α ⊛ Ω P B ,α ( a ) ⊛ Ω v (cid:17) + τ β ⊛ Ω P B ,α (cid:16) τ − β ⊛ Ω a ⊛ Ω P B ,β ( v ) (cid:17) = τ α ⊛ Ω P B ,β (cid:16) τ − α P α ( a ) v ⊗ ε v ′ (cid:17) + τ β ⊛ Ω P B ,α (cid:16) τ − β a ⊛ Ω (cid:16) ˇ P β,ε ( v ) ⊗ ε v ′ − τ β τ − ε ⊗ ε (cid:16) τ − β τ ε ˇ P β,ε ( v ) ⊛ Ω v ′ (cid:17)(cid:17)(cid:17) = (cid:16) τ α ˇ P β,ε ( τ − α P α ( a ) v ) + τ β ˇ P α,ε ( τ − β a ˇ P β,ε ( v )) (cid:17) ⊗ ε v ′ − P α ( a ) τ β τ − ε ⊗ ε (cid:16) τ − β τ ε ˇ P β,ε ( v ) ⊛ Ω v ′ (cid:17) − τ α τ β τ − ε ⊗ ε (cid:16) τ − α τ − β τ ε (cid:16) P α ( a ) ˇ P β,ε ( v ) − τ α ˇ P β,ε ( τ − α P α ( a ) v ) − τ β ˇ P α,ε ( τ − β a ˇ P β,ε ( v )) (cid:17) ⊛ Ω v ′ (cid:17) = P α ( a ) ˇ P β,ε ( v ) ⊗ ε v ′ − P α ( a ) τ β τ − ε ⊗ ε (cid:16) τ − β τ ε ˇ P β,ε ( v ) ⊛ Ω v ′ (cid:17) LGEBRAIC STUDY OF INTEGRAL EQUATIONS 23 = P α ( a ) ⊛ Ω (cid:16) ˇ P β,ε ( v ) ⊗ ε v ′ − τ β τ − ε ⊗ ε (cid:16) τ − β τ ε ˇ P β,ε ( v ) ⊛ Ω v ′ (cid:17)(cid:17) = P B ,α ( a ) ⊛ Ω P B ,β ( v ) . (3) For u = a ∈ A and v ∈ ( A + ) ⊗ Ω n with n >
0, we use τ ε ˇ P ω,ε τ − ε = P ω to check that τ α ⊛ Ω P B ,β (cid:16) τ − α ⊛ Ω P B ,α ( a ) ⊛ Ω v (cid:17) + τ β ⊛ Ω P B ,α (cid:16) τ − β ⊛ Ω a ⊛ Ω P B ,β ( v ) (cid:17) = τ α ⊗ β (cid:16) τ − α P α ( a ) ⊛ Ω v (cid:17) + P α ( a ) ⊗ β v − τ α ⊗ β (cid:16) τ − α P α ( a ) ⊛ Ω v (cid:17) = P B ,α ( a ) ⊛ Ω P B ,β ( v ) . (4) For u = u ⊗ ω u ′ ∈ A ⊗ Ω ( A + ) ⊗ Ω m and v = v ⊗ ε v ′ ∈ A ⊗ Ω ( A + ) ⊗ Ω n with m , n >
0, Eqs. (26)and (27) of ⊛ Ω mean that τ α ⊛ Ω P B ,β (cid:16) τ − α ⊛ Ω P B ,α ( u ) ⊛ Ω v (cid:17) = τ α ⊛ Ω P B ,β (cid:16) ˇ P α,ω ( u ) v τ − α τ ω ⊗ ε (cid:16)(cid:16) τ − ω ⊗ ω u ′ (cid:17) ⊛ Ω v ′ (cid:17) + ˇ P α,ω ( u ) v τ − α τ ε ⊗ ω (cid:16) u ′ ⊛ Ω (cid:16) τ − ε ⊗ ε v ′ (cid:17)(cid:17)(cid:17) − τ α ⊛ Ω P B ,β (cid:16) v ⊗ ε (cid:16)(cid:16) τ − ω ⊗ ω (cid:16) τ − α τ ω ˇ P α,ω ( u ) ⊛ Ω u ′ (cid:17)(cid:17) ⊛ Ω v ′ (cid:17)(cid:17) − τ α ⊛ Ω P B ,β (cid:16) v τ − ω τ ε ⊗ ω (cid:16)(cid:16) τ − α τ ω ˇ P α,ω ( u ) ⊛ Ω u ′ (cid:17) ⊛ Ω (cid:16) τ − ε ⊗ ε v ′ (cid:17)(cid:17)(cid:17) = τ α ˇ P β,ε (cid:16) ˇ P α,ω ( u ) v τ − α τ ω (cid:17) ⊗ ε (cid:16)(cid:16) τ − ω ⊗ ω u ′ (cid:17) ⊛ Ω v ′ (cid:17) + τ α τ − ω τ ε ˇ P β,ε (cid:16) ˇ P α,ω ( u ) v τ − α τ ω (cid:17) ⊗ ω (cid:16) u ′ ⊛ Ω (cid:16) τ − ε ⊗ ε v ′ (cid:17)(cid:17) − τ α τ β τ − ε ⊗ ε (cid:16)(cid:16) τ − β τ − ω τ ε ˇ P β,ε (cid:16) ˇ P α,ω ( u ) v τ − α τ ω (cid:17) ⊗ ω u ′ (cid:17) ⊛ Ω v ′ (cid:17) − τ α τ β τ − ω ⊗ ω (cid:16) u ′ ⊛ Ω (cid:16) τ − β τ ω τ − ε ˇ P β,ω (cid:16) ˇ P α,ω ( u ) v τ − α τ ε (cid:17) ⊗ ε v ′ (cid:17)(cid:17) − τ α ˇ P β,ε ( v ) ⊗ ε (cid:16)(cid:16) τ − ω ⊗ ω (cid:16) τ − α τ ω ˇ P α,ω ( u ) ⊛ Ω u ′ (cid:17)(cid:17) ⊛ Ω v ′ (cid:17) − τ α ˇ P β,ω ( v τ − ω τ ε ) ⊗ ω (cid:16)(cid:16) τ − α τ ω ˇ P α,ω ( u ) ⊛ Ω u ′ (cid:17) ⊛ Ω (cid:16) τ − ε ⊗ ε v ′ (cid:17)(cid:17) + τ α τ β τ − ε ⊗ ε (cid:16)(cid:16) τ − β τ − ω τ ε ˇ P β,ε ( v ) ⊗ ω (cid:16) τ − α τ ω ˇ P α,ω ( u ) ⊛ Ω u ′ (cid:17)(cid:17) ⊛ Ω v ′ (cid:17) + τ α τ β τ − ω ⊗ ω (cid:16)(cid:16) τ − α τ − β ˇ P α,ω ( u ) ˇ P β,ε ( v ) ⊛ Ω u ′ (cid:17) ⊛ Ω (cid:0) τ ω ⊗ ε v ′ (cid:1)(cid:17) . Then using the commutativity of ⊛ Ω and Eq. (15) of P Ω further, we have τ α ⊛ Ω P B ,β (cid:16) τ − α ⊛ Ω P B ,α ( u ) ⊛ Ω v (cid:17) + τ β ⊛ Ω P B ,α (cid:16) τ − β ⊛ Ω u ⊛ Ω P B ,β ( v ) (cid:17) = (cid:16) ˇ P α,ω ( u ) ⊗ ω u ′ − τ α τ − ω ⊗ ω (cid:16) τ − α τ ω ˇ P α,ω ( u ) ⊛ Ω u ′ (cid:17)(cid:17) ⊛ Ω (cid:16) ˇ P β,ε ( v ) ⊗ ε v ′ − τ β τ − ε ⊗ ε (cid:16) τ − β τ ε ˇ P β,ε ( v ) ⊛ Ω v ′ (cid:17)(cid:17) = P B ,α ( u ) ⊛ Ω P B ,β ( v ) . (5) For u = u ⊗ ω u ′ ∈ A ⊗ Ω ( A + ) ⊗ Ω m and v ∈ ( A + ) ⊗ Ω n with m , n >
0, Eqs. (26) and (27) of ⊛ Ω imply that τ α ⊛ Ω P B ,β (cid:16) τ − α ⊛ Ω P B ,α ( u ) ⊛ Ω v (cid:17) + τ β ⊛ Ω P B ,α (cid:16) τ − β ⊛ Ω u ⊛ Ω P B ,β ( v ) (cid:17) = τ α ⊗ β (cid:16) τ − α ⊛ Ω P B ,α ( u ) ⊛ Ω v (cid:17) + τ β ⊛ Ω P B ,α (cid:16) τ − β ⊛ Ω (cid:16) u τ ω ⊗ β (cid:16)(cid:16) τ − ω ⊗ ω u ′ (cid:17) ⊛ Ω v (cid:17) + u τ β ⊗ ω (cid:16) u ′ ⊛ Ω (cid:16) τ − β ⊗ β v (cid:17)(cid:17)(cid:17)(cid:17) = ˇ P α,ω ( u ) τ ω ⊗ β (cid:16)(cid:16) τ − ω ⊗ ω u ′ (cid:17) ⊛ Ω v (cid:17) + ˇ P α,ω ( u ) τ β ⊗ ω (cid:16) u ′ ⊛ Ω (cid:16) τ − β ⊗ β v (cid:17)(cid:17) + τ α ⊗ β (cid:16) τ − α ⊛ Ω (cid:16) P B ,α ( u ) − ˇ P α,ω ( u ) ⊗ ω u ′ (cid:17) ⊛ Ω v (cid:17) − τ α τ β τ − ω ⊗ ω (cid:16)(cid:16) τ − α τ ω ˇ P α,ω ( u ) ⊛ Ω u ′ (cid:17) ⊛ Ω (cid:16) τ − β ⊗ β v (cid:17)(cid:17) = ˇ P α,ω ( u ) τ ω ⊗ β (cid:16)(cid:16) τ − ω ⊗ ω u ′ (cid:17) ⊛ Ω v (cid:17) + ˇ P α,ω ( u ) τ β ⊗ ω (cid:16) u ′ ⊛ Ω (cid:16) τ − β ⊗ β v (cid:17)(cid:17) − τ α ⊗ β (cid:16)(cid:16) τ − ω ⊗ ω (cid:16) τ − α τ ω ˇ P α,ω ( u ) ⊛ Ω u ′ (cid:17)(cid:17) ⊛ Ω v (cid:17) − τ α τ β τ − ω ⊗ ω (cid:16)(cid:16) τ − α τ ω ˇ P α,ω ( u ) ⊛ Ω u ′ (cid:17) ⊛ Ω (cid:16) τ − β ⊗ β v (cid:17)(cid:17) = (cid:16) ˇ P α,ω ( u ) ⊗ ω u ′ − τ α τ − ω ⊗ ω (cid:16) τ − α τ ω ˇ P α,ω ( u ) ⊛ Ω u ′ (cid:17)(cid:17) ⊛ Ω (1 A ⊗ β v ) = P B ,α ( u ) ⊛ Ω P B ,β ( v ) , where the second equality uses the identity τ β ˇ P α,β τ − β = P α = τ ω ˇ P α,ω τ − ω .(6) For u ∈ ( A + ) ⊗ Ω m , v ∈ ( A + ) ⊗ Ω n with m , n >
0, it follows from Eq. (27) of ⊛ Ω .Hence, F TRB ( A , B ) is a matching twisted Rota-Baxter algebra. (cid:3) We give an explicit formula for the product ⊛ Ω on F TRB ( A , B ) defined in Eq. (27). Equip themodule X Ω ( A ) : = M n ≥ A ⊗ Ω n = A ⊕ A ⊗ Ω ⊕ · · · with the (augmented) mixable shu ffl e product ⋄ Ω recursively defined by u ⋄ Ω A = A ⋄ Ω u = u , u ⋄ Ω v = u v ⊗ ω ( u ′ ⋄ Ω (1 A ⊗ ε v ′ )) + u v ⊗ ε ((1 A ⊗ ω u ′ ) ⋄ Ω v ′ ) , (28)for any u = u ⊗ ω · · · ⊗ ω m u m = u ⊗ ω u ′ ∈ A ⊗ Ω ( m + , v = v ⊗ ε · · · ⊗ ε n v n = v ⊗ ε v ′ ∈ A ⊗ Ω ( n + with m , n ≥ ⊗ γ w ⊗ γ · · · ⊗ γ m + n w m + n : = ⊗ ω u ⊗ ω · · · ⊗ ω m u m ⊗ ε v ⊗ ε · · · ⊗ ε n v n , that is, set w i : = ( u i , ≤ i ≤ m , v i − m , m + ≤ i ≤ m + n , γ i : = ( ω i , ≤ i ≤ m ,ε i − m , m + ≤ i ≤ m + n . Let S m , n : = n σ ∈ S m + n (cid:12)(cid:12)(cid:12) σ − (1) < · · · < σ − ( m ) , σ − ( m + < · · · < σ − ( m + n ) o be the set of ( m , n )-(un)shu ffl es. Then(29) u ⋄ Ω v = X σ ∈ S m , n u v ⊗ γ σ (1) w σ (1) ⊗ γ σ (2) · · · ⊗ γ σ ( m + n ) w σ ( m + n ) . We further introduce the twists into Eq. (29) to define u i Ω v : = X σ ∈ S m , n u v τ ω τ ε τ − γ σ (1) ⊗ γ σ (1) τ − γ σ (2) θ σ (1) w σ (1) ⊗ γ σ (2) · · · ⊗ γ σ ( m + n − τ − γ σ ( m + n ) θ σ ( m + n − w σ ( m + n − ⊗ γ σ ( m + n ) w σ ( m + n ) , (30)where θ i : = τ γ i + , i , m , m + n , A , i = m or m + n , for any i = , . . . , m + n .For example, when m = n =
2, we have u ⋄ Ω v = u v ⊗ ω u ⊗ ε v ⊗ ε v + u v ⊗ ε v ⊗ ω u ⊗ ε v + u v ⊗ ε v ⊗ ε v ⊗ ω u , u i Ω v = u v τ ε ⊗ ω τ − ε u ⊗ ε v ⊗ ε v + u v τ ω ⊗ ε τ − ω τ ε v ⊗ ω τ − ε u ⊗ ε v LGEBRAIC STUDY OF INTEGRAL EQUATIONS 25 + u v τ ω ⊗ ε v ⊗ ε τ − ω v ⊗ ω u . Proposition 4.4.
Given any u = u ⊗ ω · · · ⊗ ω m u m ∈ A ⊗ Ω ( A + ) ⊗ Ω m and v = v ⊗ ε · · · ⊗ ε n v n ∈ A ⊗ Ω ( A + ) ⊗ Ω n with m , n > , the product u ⊛ Ω v in Eq. (27) coincides with u i Ω v given by Eq. (30) .Proof. We will prove that u ⊛ Ω v = u i Ω v by induction on m + n when m , n >
0. If m , n =
1, bydefinition (27) it is clear that u ⊛ Ω v = u v τ ω ⊗ ε τ − ω v ⊗ ω u + u v τ ε ⊗ ω τ − ε u ⊗ ε v = u v τ ω τ ε τ − γ ⊗ γ τ − γ θ w ⊗ γ w + u v τ ω τ ε τ − γ ⊗ γ τ − γ θ w ⊗ γ w = u i Ω v . Now for m , n ≥
1, let u = u ⊗ ω u ′ ∈ A ⊗ Ω ( A + ) ⊗ Ω m and v = v ⊗ ε v ′ ∈ A ⊗ Ω ( A + ) ⊗ Ω n , thedefinition of i Ω implies that u i Ω v = u v τ ω ⊗ ε (cid:16)(cid:16) τ − ω ⊗ ω u ′ (cid:17) i Ω v ′ (cid:17) + u v τ ε ⊗ ω (cid:16) u ′ i Ω (cid:16) τ − ε ⊗ ε v ′ (cid:17)(cid:17) , with the convention that u i Ω v : = u v ⊗ ω u ′ and u i Ω v : = u v ⊗ ε v ′ . Hence, by the inductionhypothesis we have u ⊛ Ω v = u v τ ω ⊗ ε (cid:16)(cid:16) τ − ω ⊗ ω u ′ (cid:17) ⊛ Ω v ′ (cid:17) + u v τ ε ⊗ ω (cid:16) u ′ ⊛ Ω (cid:16) τ − ε ⊗ ε v ′ (cid:17)(cid:17) = u v τ ω ⊗ ε (cid:16)(cid:16) τ − ω ⊗ ω u ′ (cid:17) i Ω v ′ (cid:17) + u v τ ε ⊗ ω (cid:16) u ′ i Ω (cid:16) τ − ε ⊗ ε v ′ (cid:17)(cid:17) = u i Ω v . This completes the inductive proof of Eq. (30). (cid:3)
Let ϕ B : A → F TRB ( A , B ) , a a ; j B : B → F TRB ( A , B ) , u u be the natural injections. Theorem 4.5.
Let ( A , P Ω , τ Ω ) be a matching twisted Rota-Baxter algebra and B an augmented k -algebra. Then (cid:16) ( F TRB ( A , B ) , ⊛ Ω ) , P B , Ω , τ Ω , ϕ B , j B (cid:17) is the free ( A , P Ω , τ Ω ) -matching twisted Rota-Baxter algebra over B.Proof. First by Proposition 4.3, we know that F TRB ( A , B ) is an ( A , P Ω , τ Ω )-matching twisted Rota-Baxter algebra, as ϕ B is clearly a matching twisted Rota-Baxter algebra homomorphism.Next we check the universal property of F TRB ( A , B ). Given any ( A , P Ω , τ Ω )-matching twistedRota-Baxter algebra ( R , P ′ Ω , τ ′ Ω , ϕ ) and k -algebra homomorphism f : B → R , we first extend f tobe the following algebra homomorphismˇ f : A → R , av · · · v n ϕ ( a ) f ( v ) · · · f ( v n ) , a ∈ A , v , . . . , v n ∈ B . In particular, ˇ f ( τ ω ) = τ ′ ω , ω ∈ Ω . Then ˇ f induces the linear map ¯ f : F TRB ( A , B ) → R satisfying¯ f ( u ) = ˇ f ( u ) P ′ ω ( ˇ f ( u ) P ′ ω ( · · · P ′ ω n ( ˇ f ( u n )) · · · )) , for any u = u ⊗ ω · · · ⊗ ω n u n ∈ A ⊗ Ω ( A + ) ⊗ Ω n with n ≥
0. It is clear that f = ¯ f j B . Moreover, usingEq. (26), we check that ¯ f ( P B ,ω ( u )) = P ′ ω ( ¯ f ( u )) , ω ∈ Ω . The case when u ∈ A , n = u ∈ A + is easy to see. Now if u ∈ A , n >
0, then we check it as follows.¯ f ( P B ,ω ( u )) = ¯ f (cid:16) ˇ P ω,ω ( u ) ⊗ ω u ⊗ ω · · · ⊗ ω n u n − τ ω τ − ω ⊗ ω τ − ω τ ω ˇ P ω,ω ( u ) u ⊗ ω u ⊗ ω · · · ⊗ ω n u n (cid:17) = ˇ f ( ˇ P ω,ω ( u )) P ′ ω (cid:16) ˇ f ( u ) P ′ ω ( · · · P ′ ω n ( ˇ f ( u n )) · · · ) (cid:17) − τ ′ ω ( τ ′ ω ) − P ′ ω (cid:16) ( τ ′ ω ) − τ ′ ω ˇ f ( ˇ P ω,ω ( u ) u ) P ′ ω ( ˇ f ( u ) P ′ ω ( · · · P ′ ω n ( ˇ f ( u n )) · · · )) (cid:17) = ˇ P ′ ω,ω ( ϕ ( u )) P ′ ω (cid:16) ˇ f ( u ) P ′ ω ( · · · P ′ ω n ( ˇ f ( u n )) · · · ) (cid:17) − τ ′ ω ( τ ′ ω ) − P ′ ω (cid:16) ( τ ′ ω ) − τ ′ ω ˇ P ′ ω,ω ( ϕ ( u )) ˇ f ( u ) P ′ ω ( · · · P ′ ω n ( ˇ f ( u n )) · · · ) (cid:17) = P ′ ω (cid:16) ϕ ( u ) P ′ ω ( ˇ f ( u ) P ′ ω ( · · · P ′ ω n ( ˇ f ( u n )) · · · )) (cid:17) = P ′ ω ( ¯ f ( u )) , where the fourth equality is due to the relation (14) between ˇ P ′ ω,ω and P ′ ω . Hence, ¯ f P B ,ω = P ′ ω ¯ f .Also, ¯ f ϕ B = ϕ by definition.It remains to check that ¯ f is an algebra homomorphism and hence an ( A , P Ω , τ Ω )-matchingtwisted Rota-Baxter algebra homomorphism. That is,¯ f ( u ⊛ Ω v ) = ¯ f ( u ) ¯ f ( v ) for all u = u ⊗ ω u ′ ∈ A ⊗ Ω ( A + ) ⊗ Ω m , v = v ⊗ ε v ′ ∈ A ⊗ Ω ( A + ) ⊗ Ω n , m , n ≥ . It is clear when m = n = f . Otherwise, if m , n >
0, we can prove it byinduction on m + n as follows.¯ f ( u ⊛ Ω v ) = ¯ f (cid:16) u v τ ω P B ,ε (cid:16) τ − ω ⊛ Ω P B ,ω ( u ′ ) ⊛ Ω v ′ (cid:17) + u v τ ε P B ,ω (cid:16) τ − ε ⊛ Ω u ′ ⊛ Ω P B ,ε ( v ′ ) (cid:17)(cid:17) = ˇ f ( u v τ ω ) P ′ ε (cid:16) ¯ f (cid:16) τ − ω ⊛ Ω P B ,ω ( u ′ ) ⊛ Ω v ′ (cid:17)(cid:17) + ˇ f ( u v τ ε ) P ′ ω (cid:16) ¯ f (cid:16) τ − ε ⊛ Ω u ′ ⊛ Ω P B ,ε ( v ′ ) (cid:17)(cid:17) = ˇ f ( u ) ˇ f ( v ) (cid:16) τ ′ ω P ′ ε (cid:16) ( τ ′ ω ) − P ′ ω ( ¯ f ( u ′ )) ¯ f ( v ′ ) (cid:17) + τ ′ ε P ′ ω (cid:16) ( τ ′ ε ) − ¯ f ( u ′ ) P ′ ε ( ¯ f ( v ′ )) (cid:17)(cid:17) = ˇ f ( u ) P ′ ω ( ¯ f ( u ′ )) ˇ f ( v ) P ′ ε ( ¯ f ( v ′ )) = ¯ f ( u ) ¯ f ( v ) , where the third equality uses the induction hypothesis, and the fourth equality is due to the rela-tion (13) for P ′ Ω .Since the definition of ¯ f as a homomorphism of ( A , P Ω , τ Ω )-matching twisted Rota-Baxteralgebras is determined by its image on A , the universal property of F TRB ( A , B ) is verified. (cid:3) Operator linearity of separable Volterra equations.
We now apply the construction offree matching twisted Rota-Baxter algebras to obtain the operator linearity (Definition 2.3) ofintegral equations with separable Volterra operators.
Theorem 4.6.
Let I ⊆ R be an open interval and A : = C ( I ) . Let K Ω : = ( K ω ( x , t ) | ω ∈ Ω ) bea family of separable kernels K ω ( x , t ) = k ω ( x ) h ω ( t ) ∈ C ( I ) with k ω ( x ) zero free on I. Let P K Ω be the corresponding family of Volterra operators P K ω ( f ) : = R xa K ω ( x , t ) f ( t ) dt with a ∈ I and let ( A , P K Ω ) be the matching twisted Rota-Baxter algebra. For any given set Y, any element (andits corresponding integral equation) in ( I ( Ω , A [ Y ]) , Π Ω ) defined in Theorem 3.10 is equivalent toone that is operator linear and in which each of the (nested) operators acts on variable functions ( that is, in A [ Y ] + ) . As noted after Definition 3.16, this theorem shows that in order to study integral equationsof a Volterra operator with separable kernels, we only need to consider operator linear integralequations with these Volterra operators.
Proof.
By Theorem 2.9, the quadruple ( A , P K Ω , τ Ω ) : = (cid:0) C ( I ) , P K Ω , (cid:8) k ω ( x ) k ω ( a ) (cid:9) ω ∈ Ω (cid:1) is an Ω -matchingtwisted Rota-Baxter algebra. Let φ be in the free ( A , P K Ω )-operated algebra ( I ( Ω , A [ Y ]) , Π Ω ).Consider the defining relation of the Ω -matching twisted Rota-Baxter operator(31) Π α ( u ) Π β ( v ) − τ α Π β (cid:16) τ − α Π α ( u ) v (cid:17) − τ β Π α (cid:16) τ − β u Π β ( v ) (cid:17) , for all u , v ∈ I ( Ω , A [ Y ]) , α, β ∈ Ω , in I ( Ω , A [ Y ]). Since ( A , P K Ω , τ Ω ) is a matching twisted Rota-Baxter algebra, for any f : Y → A , the induced ( A , P K Ω , τ Ω )-matching twisted Rota-Baxter algebra homomorphism f sends the LGEBRAIC STUDY OF INTEGRAL EQUATIONS 27 expressions in Eq. (31) to zero. Thus the operated ideal I TRB generated by these expressions iscontained in the ideal J α Ω defined in Definition 3.16, where α Ω = P K Ω .By definition, the quotient ( A , P K Ω )-algebra I ( Ω , A [ Y ]) / I TRB is the free ( A , P K Ω , τ Ω )-matchingtwisted Rota-Baxter algebra. With the assumption on ( A , P K Ω , τ Ω ) in the theorem, by Theorem 4.5,this free object is also given by F TRB ( A , B ) = (cid:16) ( F TRB ( A , B ) , ⊛ Ω ) , P B , Ω , τ Ω , j B (cid:17) with B = R [ Y ]. Thuswe have the natural isomorphism I ( Ω , A [ Y ]) / I TRB (cid:27) F TRB ( A , R [ Y ])of ( A , P K Ω , τ Ω )-twisted Rota-Baxter algebras. This means that, in particular, the element φ ( Π Ω , A , Y )in I ( Ω , A [ Y ]) is congruent modulo I TRB to an element of F TRB ( A , R [ Y ]) : = L k ≥ A [ Y ] ⊗ Ω ( A [ Y ] + ) ⊗ Ω k and thus to an element that is operator linear and each of the (nested) operators acts on variablefunctions. Since I TRB ⊆ J α Ω , by Definition 3.16, φ ( Π Ω , A , Y ) is equivalent to an integral equationin I ( Ω , A [ Y ]) that is operator linear with the mentioned additional property. (cid:3) We next apply the theorem to some examples of Volterra equations, beginning with the ones inExample 2.2 that motivated our study.
Example 4.7. (i) We first note that the Volterra population model in Example 2.2 (i) isalready operator linear.(ii) The Thomas-Fermi equation from Example 2.2, Eq. (7) is already operator linear. Butthere is no variable function between the outer and inner integrals, as prescribed in The-orem 4.6 (by taking the independent variable to be z = y / ). Integration by parts gives Z x Z t s − / y ( s ) / ds dt = x Z x s − / y ( s ) / ds − Z x t / y ( t ) / dt , now operator linear with the desired form.As shown in Theorem 2.9, the Volterra integral operator P K ( f ) : = k ( x ) Z xa h ( t ) f ( t ) dt is a twisted Rota-Baxter operator by τ = k ( x ) k ( a ) , and thus satisfies the twisted Rota-Baxter identity P K ( f ) P K ( g ) = τ P K (cid:16) τ − ( P K ( f ) g + f P K ( g )) (cid:17) for any f ( x ) , g ( x ) ∈ C ( I ). We apply this identity in the following Volterra equations to obtainoperator linear forms. Naturally, these examples are guaranteed to work thanks to Theorem 4.6. Example 4.8. (i) Let the kernel be K ( x , t ) = e − x + t and a =
0, so τ = e − x . On C ( R ), theVolterra equation = Z x e − x + t f ( t ) Z t e − t + u g ( u ) du dt ! Z x e − x + t h ( t ) dt ! − Z x e − x + t f ( t ) Z t e − t + u h ( u ) du dt ! Z x e − x + t g ( t ) dt ! written in operator form is0 = P K ( f P K ( g )) P K ( h ) − P K ( f P K ( h )) P K ( g ) = τ P K ( τ − ( P K ( f P K ( g )) h + f P K ( g ) P K ( h ))) − τ P K ( τ − ( P K ( f P K ( h )) g + f P K ( h ) P K ( g ))) = τ P K ( τ − hP K ( f P K ( g ))) − τ P K ( τ − gP K ( f P K ( h ))) . Rewriting this back using the integral notations, we obtain = e − x Z x e − x + t e t h ( t ) Z t e − t + u f ( u ) Z u e − u + s g ( s ) ds ! du ! dt − e − x Z x e − x + t e t g ( t ) Z t e − t + u f ( u ) Z u e − u + s h ( s ) ds ! du ! dt . Observe that this equation is operator linear and each of the (nested) operators acts onthe variable functions f , g and h .(ii) Given kernel K ( x , t ) = xt with a =
1, we have τ = x . Thus on C ((1 , ∞ )), we can rewritethe equation f ( x ) = Z x xt f ( t ) g ( t ) dt ! Z x xt f ( t ) Z t tuh ( u ) du dt ! as f = P K ( f g ) P K ( f P K ( h )) = τ P K (cid:16) τ − ( P K ( f g ) f P K ( h ) + f gP K ( f P K ( h ))) (cid:17) = τ P K (cid:16) f P K (cid:16) τ − hP K ( f g ) (cid:17)(cid:17) + τ P K (cid:16) f P K (cid:16) τ − f gP K ( h ) (cid:17)(cid:17) + τ P K (cid:16) τ − f gP K ( f P K ( h )) (cid:17) Again, the last expression rewrites to integrals in operator linear forms and each of the(nested) operators acts on the variable functions f , g and h . Acknowledgments.
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