A β-Sturm Liouville problem associated with the general quantum operator
aa r X i v : . [ m a t h . C A ] J a n A β − STURM LIOUVILLE PROBLEM ASSOCIATED WITHTHE GENERAL QUANTUM OPERATOR
J.L. CARDOSO
Abstract.
Let I ⊆ R be an interval and β : I → I a strictly increasingand continuous function with a unique fixed point s ∈ I which satisfies( s − t )( β ( t ) − t ) ≥ t ∈ I , where the equality holds only when t = s .The general quantum operator defined in 2015 by Hamza et al., D β [ f ]( t ) := f (cid:0) β ( t ) (cid:1) − f ( t ) β ( t ) − t if t = s and D β [ f ]( s ) := f ′ ( s ) if t = s , generalizes theJackson q -operator D q and also the Hahn ( q, ω )-operator, D q,ω .Regarding a β − Sturm Liouville eigenvalue problem associated with theabove operator D β , we construct the β − Lagrange’s identity, show that it isself-adjoint in L β ([ a, b ]) , and exhibit some properties for the correspondingeigenvalues and eigenfunctions. Introduction
The β − operator D β described in the abstract was introduced in [33] togetherwith the corresponding general quantum difference calculus. It generalizes the( q, ω ) − derivative operator (the Hahn’s quantum operator)(1.1) D q,ω [ f ]( x ) := f (cid:0) qx + ω (cid:1) − f ( x )( q − x + ω , which, in turn, generalizes both the Jackson q − derivative D q [ f ]( x ) := f ( qx ) − f ( x )( q − x , and the (forward difference) ω − derivative △ ω [ f ]( x ) := f ( x + ω ) − f ( x ) ω , where 0 < q < ω ≥ q, ω ) − integral Z ba f d q,ω := Z bω f d q,ω − Z aω f d q,ω , Date : January 11, 2021.2020
Mathematics Subject Classification.
Primary 39A70; Secondary 47B39, 39A12, 34B24.
Key words and phrases. general quantum operator, β -difference operator, β -derivative, β -integral, β -Sturm-Liouville problem.This research was partially supported by Portuguese Funds through the FCT - Funda¸c˜ao paraa Ciˆencia e a Tecnologia - within the Project UID/MAT/00013/2013. the Jackson q − integral Z ba f d q := (1 − q ) + ∞ X k =0 (cid:2) bf ( bq k ) − af ( aq k ) (cid:3) q k and the N¨orlund integral Z ba f ∆ ω := ω + ∞ X k =0 (cid:2) f ( b + kω ) − f ( a + kω ) (cid:3) . For more details over the q − integrals see, for example, [12].Those difference operators together with its inverse operators are very importantin mathematics investigation and in applications, with a large number of publica-tions and a variety of topics including, but not limited to, the quantum calculus[16], the quantum variational calculus [41, 25, 8, 42, 40], q − difference equationsproperties [6, 7, ? , 14], Sturm-Liouville problems [29, 15, 10, 13, 40, 14], Paley-Wiener results [27, 4, 26], Sampling theory [9, 37, 2, 26, 13, 11, 32], q − exponential,trigonometric, hyperbolic and other important families of functions associated withFourier expansions and corresponding properties connected and derived from itsorthogonality feature [39, 17, 5, 44, 3, 18, 38, 19, 23, 20, 21, 1, 22]. These and manyother subjects has attracted many researchers.In 2015, Hamza et al. [33] introduced a general quantum difference operator,the β − derivative, generalizing the Hahn’s quantum operator (for certain func-tions β ), and its inverse operator, the β − integral. Also in 2015 [34], β − H¨older, β − Minkowski, β − Gronwall, β − Bernoulli and β − Lyapunov inequalities were ex-hibited. In 2016, it was proved the existence and uniqueness of solutions of generalquantum difference equations [35]. Later, in [30], some new results on homogeneoussecond order linear general quantum difference equations were presented and, in[36], the exponential, trigonometric and hyperbolic functions were introduced. Thetheory of n th-order linear general quantum difference equations was developed in[31] while the general quantum variational calculus was build up in [24]. In [22],properties of the β -Lebesgue spaces were produced and, recently, in [43], a generalquantum Laplace transform was displayed and studied.All of these publications generalizes previous results and are directly related withthe above mentioned general quantum difference operator.The aim of this work is to obtain a self-adjoint formulation of a β -Sturm-Liouvilleproblem and to prove properties for the corresponding eigenvalues and eigenfunc-tions. Its construction follows ideas from [14] and from other previous publications.In section 2 we recall the definition of the β -difference operator and its inverseoperator, the β -integral, together with some of its properties. Section 3 is devotedto the β -Sturm-Liouville problem to be considered.The outcome of this work can be found in section 3. We believe that Lemmas3.1 and 3.2, Corollaries 3.3, 3.5, 3.8, 3.9 and 3.10, as well as Theorems 3.5 and 3.7are original. Subsection 3.4 is also new.2. The β − difference operator and the β − integral The β − difference operator. In the following, I ⊆ R will denote an intervaland β : I → I a strictly increasing and continuous function with a unique fixed − STURM LIOUVILLE PROBLEM 3 point s ∈ I satisfying(2.1) ( t − s )( β ( t ) − t ) ≤ t ∈ I , where the equality holds only when t = s .For each function f : I → K where K is either R or C , Hamza et al. [33]defined the general quantum difference operator(2.2) D β [ f ]( t ) := f (cid:0) β ( t ) (cid:1) − f ( t ) β ( t ) − t if t = s ,f ′ ( s ) if t = s . , provided that f ′ ( s ) exists. D β [ f ]( t ) is called the β − derivative of f at t ∈ I . If f ′ ( s ) exists then f is said to be β − differentiable on I .It is obvious that when β ( t ) = qt + ω one obtains the Hahn operator (1.1), beingthe fixed point given by s = ω − q .We remark that it is possible to replace the above condition (2.1) by ( t − s )( β ( t ) − t ) ≥ t ∈ I .An introduction to the β − calculus related with this general quantum differenceoperator can be found in [33].2.2. The β − integral. Defining β k ( t ) := ( β ◦ β ◦ . . . ◦ β | {z } k times )( t ) for t ∈ I and k ∈ N = N ∪ { } , with β ( t ) := t, we can consider the β − interval with extreme points a and b, [ a, b ] β := (cid:8) β n ( x ) | ( x, n ) ∈ { a, b } × N (cid:9) . Of course that a, b ∈ I ⇒ [ a, b ] β ⊂ I , whenever a and b are real numbers.The following proposition can be found in [33, Lemma 2.1, page 3]. Proposition A . The sequence of functions (cid:8) β k ( t ) (cid:9) k ∈ N converges uniformly tothe constant function ˆ β ( t ) := s on every compact interval J ⊂ I containing s .The quantum difference inverse operator, the β − integral, with a, b ∈ I, is definedby(2.3) Z ba f d β := Z bs f d β − Z as f d β where(2.4) Z xs f d β := + ∞ X k =0 (cid:16) β k ( x ) − β k +1 ( x ) (cid:17) f (cid:0) β k ( x ) (cid:1) . Thus,(2.5) Z ba f d β = + ∞ X k =0 (cid:16) β k ( b ) − β k +1 ( b ) (cid:17) f (cid:0) β k ( b ) (cid:1) − + ∞ X k =0 (cid:16) β k ( a ) − β k +1 ( a ) (cid:17) f (cid:0) β k ( a ) (cid:1) . If the infinite sum in the right side of (2.4) is convergent then we say that thefunction f is β − integrable in [ s , x ] . The β − integral in the left side of (2.3) is In fact, K = X can represent any Banach space [33, p.2] CARDOSO well defined provided that at least one of the β − integrals in the right side is finiteand we say that f is β − integrable in [ a, b ] if it is both β − integrable in [ s , a ]and in [ s , b ] . As important particular cases, one obtains the Jackson-Thomae-N¨orlund integralwhen β ( t ) = qt + ω with 0 < q < ω ≥
0. The Jackson q − integralcorresponds to ω = 0 in the previous case. Its fixed points are given by s = ω − q and s = 0 , respectively.2.3. Properties of the β − derivative and of the β − integral. We go back tothe the definition of the β − derivative operator (2.2).Notice that if f is differentiable at a point t ∈ I , thenlim β ( t ) → t D β [ f ]( t ) = f ′ ( t ) , hence D β is a beta − analogue of the standard derivative operator.The β − derivative satisfies properties which may be regarded as β − analogues ofthe corresponding properties for the usual derivative. For instance, the quantumoperator (2.2) is linear, i.e., D β [ αf + βg ]( t ) = αD β [ f ]( t ) + βD β [ g ]( t ) , where α and β are any real or complex numbers, and satisfies the following β − product rule: for t ∈ I, (2.6) D β [ f · g ]( t ) = D β [ f ]( t ) · g ( t ) + f (cid:0) β ( t ) (cid:1) · D β [ g ]( t )= D β [ g ]( t ) · f ( t ) + g (cid:0) β ( t ) (cid:1) · D β [ f ]( t )if f and g are β − differentiable in I . Also, f will be the constant function suchthat f ( t ) = f ( s ) for all t ∈ I whenever D β [ f ]( t ) = 0 for all t ∈ I . For these andother properties of the general quantum difference operator D β see [33]. Theseequalities hold for all t = s , and also for t = s whenever f ′ ( s ) and g ′ ( s ) exist.2.3.1. The fundamental theorem of β − calculus. The following statement of the β − analogue of the fundamental theorem of calculus can be found in [22]. Theorem B . Let β : I → I be a function satisfying the conditions describedin subsection 2.1. Fix a, b ∈ I and let f : I → K be a function such that D β [ f ] ∈ L β [ a, b ]. Then:(i) The equality Z ba D β [ f ] d β = (cid:20) f ( s ) − lim k → + ∞ f (cid:0) β k ( s ) (cid:1)(cid:21) bs = a holds, provided the involved limits exist.(ii) In addition, assuming that a < s < b, if f has a discontinuity of first kindat s then Z ba D β [ f ] d β = f ( b ) − f ( a ) − (cid:16) f ( s +0 ) − f ( s − ) (cid:17) . Of course, if f is continuous at s then Z ba D β [ f ] d β = f ( b ) − f ( a ) . − STURM LIOUVILLE PROBLEM 5
The β − integration by parts formula. Now we state the β − analogue of theintegration by parts formula [22]. Theorem C . Let β : I → I be a function satisfying the conditions described insubsection 2.1. Fix a, b ∈ I and two functions f : I → K and g : I → K . Then: Z ba f · D β [ g ] d β = (cid:20) ( f · g )( s ) − lim k → + ∞ ( f · g ) (cid:0) β k ( s ) (cid:1)(cid:21) bs = a − Z ba (cid:0) g ◦ β (cid:1) · D β [ f ] d β holds, provided f, g ∈ L β [ a, b ], D β [ f ] and D β [ g ] are bounded in [ a, b ] β , and thelimits exist.If, in addition, f and g are continuous at s and a < s < b, then Z ba f · D β [ g ] d β = h f · g i ba − Z ba (cid:0) g ◦ β (cid:1) · D β [ f ] d β . The spaces L pβ [ a, b ] and L pβ [ a, b ] . The space L pβ [ a, b ] . For a, b ∈ I , we will denote by L pβ [ a, b ] the set of func-tions f : I → C such that | f | p is β − integrable in [ a, b ], i.e., L pβ [ a, b ] = ( f : I → C (cid:12)(cid:12)(cid:12) Z ba | f | p d β < ∞ ) . We also set L ∞ β [ a, b ] = (cid:26) f : I → C (cid:12)(cid:12)(cid:12) sup k ∈ N n(cid:12)(cid:12) f (cid:0) β k ( a ) (cid:1)(cid:12)(cid:12) , (cid:12)(cid:12) f (cid:0) β k ( b ) (cid:1)(cid:12)(cid:12)o < ∞ (cid:27) . It was proved in [22, Corollary 3.4] that if a ≤ s ≤ b and 1 ≤ p ≤ ∞ , then theset L pβ [ a, b ], with the usual operations of addition of functions and multiplicationof a function by a number (real or complex), becomes a linear space over K .2.4.2. The space L pβ [ a, b ] . For f, g ∈ L pβ [ a, b ], writing f ∼ g when(2.7) f (cid:0) β k ( a ) (cid:1) = g (cid:0) β k ( a ) (cid:1) and f (cid:0) β k ( b ) (cid:1) = g (cid:0) β k ( b ) (cid:1) holds for all k = 0 , , , · · · , i.e., we say that f ∼ g if f = g in [ a, b ] β . Clearly, ∼ defines an equivalence relation in L pβ [ a, b ]. We will represent by L pβ [ a, b ] thecorresponding quotient set: L pβ [ a, b ] := L pβ [ a, b ] (cid:14) ∼ . Also in [22] it was proved the following theorems.
Theorem D . If a ≤ s ≤ b and 1 ≤ p ≤ ∞ then L pβ [ a, b ] is a normed linearspace over K with norm(2.8) k f k L pβ [ a,b ] := Z ba | f | p d β ! p if 1 ≤ p < ∞ ;sup k ∈ N n (cid:12)(cid:12) f (cid:0) β k ( a ) (cid:1)(cid:12)(cid:12) , (cid:12)(cid:12) f (cid:0) β k ( b ) (cid:1)(cid:12)(cid:12) o if p = ∞ . In the usual way, here f ∈ L pβ [ a, b ] denotes any representative of the correspon-dent class in L pβ [ a, b ] where it belongs. Notice that, in view of (2.5) and (2.7), thedefinition of the norm k f k L pβ [ a,b ] is independent of the chosen representative. Theorem E . If a ≤ s ≤ b and 1 ≤ p ≤ ∞ , then the following holds: CARDOSO (i) L pβ [ a, b ] , endowed with the norm (2.8), is a Banach space for 1 ≤ p ≤ ∞ ,which is separable if 1 ≤ p < ∞ and reflexive if 1 < p < ∞ .(ii) L β [ a, b ] is a Hilbert space with inner product(2.9) h f, g i β := Z ba f g d β , f, g ∈ L β [ a, b ] . The β -exponential and β -trigonometric functions. The following β -analogues of the exponential and trigonometric functions, as well as some of itsproperties, were introduced in [36].It is assumed that p : I → C is a continuous function at s . The β -exponentialfunctions are defined by(2.10) e p,β ( t ) = 1 ∞ Y k =0 h − p (cid:0) β k ( t ) (cid:1) (cid:16) β k ( t ) − β k +1 ( t ) (cid:17)i and(2.11) E p,β ( t ) = ∞ Y k =0 h p (cid:0) β k ( t ) (cid:1) (cid:16) β k ( t ) − β k +1 ( t ) (cid:17)i . We notice that, under the assumptions on the functions p and β , the infiniteproducts of (2.10) and of (2.11) are both convergent [36, page 28].On the other hand, the trigonometric functions are defined by(2.12) cos p,β ( t ) = e ip,β ( t ) + e − ip,β ( t )2 , sin p,β ( t ) = e ip,β ( t ) − e − ip,β ( t )2and(2.13) Cos p,β ( t ) = E ip,β ( t ) + E − ip,β ( t )2 , Sin p,β ( t ) = E ip,β ( t ) − E − ip,β ( t )2 . It is known that e p,β ( t ) = 1 E − p,β ( t ) , e p,β ( s ) = 1 , E p,β ( s ) = 1and(2.14) D β e p,β ( t ) = p ( t ) e p,β ( t ) , D β E p,β ( t ) = p ( t ) E p,β (cid:0) β ( t ) (cid:1) ,D β sin p,β ( t ) = p ( t ) cos p,β ( t ) , D β cos p,β ( t ) = − p ( t ) sin p,β ( t ) ,D β Sin p,β ( t ) = p ( t ) Cos p,β ( t ) , D β Cos p,β ( t ) = − p ( t ) Sin p,β ( t ) . The β − Sturm-Liouville problem
Simple formulation of the classical Sturm-Liouville problem.
The sim-plest formulation of the classical Sturm-Liouville problem with separate type con-ditions, is the following:(3.1) − y ′′ + ν ( t ) y = λy , −∞ < a ≤ t ≤ b < ∞ a y ( a ) + a y ′ ( a ) = 0 ,a y ( b ) + a y ′ ( b ) = 0 . , where ν ( · ) is a real valued continuous function on [ a, b ] , λ ∈ C and | a i | + | a i | 6 = 0 , i = 1 , − STURM LIOUVILLE PROBLEM 7 are only a countable number of real numbers (eigenvalues) λ < λ < λ < · · · ,with λ n → ∞ , such that (3.1) has nontrivial solutions ϕ , ϕ , ϕ · · · (eigenfunc-tions). Moreover, to each eigenvalue λ n , up to a multiplicative constant, therecorresponds only one solution ϕ n . Thus, the eigenvalues are geometrically simpleand they are also algebraically simple since they are simple zeros of the character-istic function associated with (3.1). Also, the set { ϕ n } ∞ n =0 is an orthonormal basisof L ( a, b ) . Regarding all of these aspects see [45].Annaby and Mansour [15, 16], after Exton [28, 29], developed a q − Sturm-Liouville theory in the regular setting. Later, together with Makharesh [14], theymake progress in the direction of a Sturm-Liouville theory in the regular setting,with separate-type boundary conditions, associated with the D q,ω operator (1.1).In the following, we will establish conditions to develop a similar Sturm-Liouvilletheory associated with the general quantum operator D β defined by (2.2).3.2. Further properties for the β -difference operator and the β -integral. In the next lemmas, b is a fixed real parameter. Lemma 3.1. Z bs f (cid:0) β ( t ) (cid:1) d β t = Z β ( b ) s f ( u ) (cid:0) D β β − (cid:1) ( u ) d β u .Proof. The proof is straightforward since, by definition, the right side Z β ( b ) s f ( u ) (cid:0) D β β − (cid:1) ( u ) d β u equals ∞ X k =0 f (cid:16) β k ( β ( b )) (cid:17) β − (cid:16) β k (cid:0) β ( b ) (cid:1)(cid:17) − β k (cid:0) β ( b ) (cid:1)(cid:17) β k (cid:0) β ( b ) (cid:1)(cid:17) − β (cid:16) β k (cid:0) β ( b ) (cid:1)(cid:17)(cid:17) (cid:16) β k ( b ) − β k +1 ( b ) (cid:17) ,which becomes, after simplification, ∞ X k =0 f (cid:16) β (cid:0) β k ( b ) (cid:1) (cid:17)(cid:16) β k ( b ) − β k +1 ( b ) (cid:17) = Z bs f (cid:0) β ( t ) (cid:1) d β t . (cid:3) In the following, in order to simplify the notation of the inner-product (2.9) wewill consider h · , · i rather than h · , · i β . Of course that when we write the interval( s , b ) we admit that s ≤ b . If b < s then one must replace that interval by( b, s ) . The corresponding proofs follow exactly the same steps. Lemma 3.2.
Let f, g ∈ L β (cid:0) s , b (cid:1) be both continuous functions at s . Then, for t ∈ (cid:0) s , b (cid:3) we have (i) ( D β f ) (cid:0) β − ( t ) (cid:1) = (cid:0) D β − f (cid:1) ( t ) ;(ii) h D β f , g i = f ( b ) g (cid:0) β − ( b ) (cid:1) − f ( s ) g (cid:0) s (cid:1) + h f , − D β β − D β − g i ;(iii) h− D β β − D β − f , g i = f (cid:0) s (cid:1) g (cid:0) s (cid:1) − f (cid:16) β − ( b ) (cid:17) g ( b ) + h f , D β g i . Proof.
The proof of (i) is trivial. Let’s prove (ii):By the β -integration by parts theorem (Theorem C) we have h D β f , g i = Z bs (cid:16) D β f (cid:17) ( t ) g ( t ) d β t = f ( b ) g ( b ) − f ( s ) g ( s ) − Z bs f (cid:0) β ( t ) (cid:1)(cid:16) D β g (cid:17) ( t ) d β t . CARDOSO
Considering u = β ( t ) then, by Lemma 3.1, one obtains h D β f , g i = f ( b ) g ( b ) − f ( s ) g ( s ) − Z β ( b ) s f ( u ) (cid:16) D β β − (cid:17) ( u ) (cid:16) D β g (cid:17) ( β − ( u )) d β u , which, by (i), becomes(3.2) h D β f , g i = f ( b ) g ( b ) − f ( s ) g ( s ) − Z β ( b ) s f ( u ) D β β − ( u ) D β − g ( u ) d β u. Writing (see (iv) of Lemma 3.5 in [33]) Z β ( b ) s f ( u ) D β β − ( u ) D β − g ( u ) d β u = Z bs f ( u ) D β β − ( u ) D β − g ( u ) d β u + Z β ( b ) b f ( u ) D β β − ( u ) D β − g ( u ) d β u . then, by Corollary 3.7 of [33], Z β ( b ) s f ( u ) D β β − ( u ) D β − g ( u ) d β u = Z bs f ( u ) D β β − ( u ) D β − g ( u ) d β u + (cid:0) β ( b ) − b (cid:1) f ( b ) D β β − ( b ) D β − g ( b ) . Introducing this last identity in (3.2) one gets h D β f , g i = f ( b ) g ( b ) − f ( s ) g ( s ) − Z bs f ( u ) D β β − ( u ) D β − g ( u ) d β u − (cid:0) β ( b ) − b (cid:1) f ( b ) D β β − ( b ) D β − g ( b ) . Since f ( b ) g ( b ) − f ( s ) g ( s ) − (cid:0) β ( b ) − b (cid:1) f ( b ) (cid:16) D β β − (cid:17) ( b ) (cid:16) D β − g (cid:17) ( b ) = f ( b ) g (cid:0) β − ( b ) (cid:1) − f ( s ) g (cid:0) s (cid:1) . we obtain h D β f , g i = f ( b ) g (cid:0) β − ( b ) (cid:1) − f ( s ) g (cid:0) s (cid:1) − h f , D β β − D β − g i , which proves (ii).Finally, (iii) is a consequence of (ii). (cid:3) Taking into account the β -inner-product (2.9), we thus have the following corollary.It is a direct consequence of (ii), Lemma 3.2, therefore its proof will be omitted. Corollary 3.3.
Let a ≤ s ≤ b and f, g ∈ L β ( a, b ) be both continuous functionsat s . Then, h D β f , g i = f ( b ) g (cid:0) β − ( b ) (cid:1) − f ( a ) g (cid:0) β − ( a ) (cid:1) + h f , − D β β − D β − g i . − STURM LIOUVILLE PROBLEM 9
The β -eigenvalue problem. Consider the following β -Sturm-Liouville prob-lem ( β -SLP) in L β ( s , b ) defined by the β -difference equation(3.3) ℓ β y ( t ) := − D β β − D β − D β y ( t ) + r ( t ) y ( t ) = λ y ( t ) , with s ≤ t ≤ b < ∞ , λ ∈ C , and the boundary conditions(3.4) ( a y ( s ) + a D β − y ( s ) = 0 b y ( b ) + b D β − y ( b ) = 0 . We also assume that r ( t ) is a real valued continuous function on [ s , b ] and | a | + | a | 6 = 0 = | b | + | b | .The operator ℓ β (3.3) satisfies the following β -Lagrange’s identity. Theorem 3.4. If y , z ∈ L β ( s , b ) then (3.5) Z bs (cid:20)(cid:16) ℓ β y ( t ) (cid:17) z ( t ) − y ( t ) (cid:16) ℓ β z ( t ) (cid:17)(cid:21) d β u = [ y, z ]( b ) − [ y, z ] ( s ) , where (3.6) [ y, z ]( t ) = y ( t ) D β − z ( t ) − z ( t ) D β − y ( t ) Proof.
Consider y , z ∈ L β ( s , b ) .On one hand, using (iii), Lemma 3.2, with f ( t ) = D β y ( t ) and g ( t ) = z ( t ) , we have h− D β β − D β − D β y, z i = − D β y (cid:0) β − ( b ) (cid:1) z ( b ) + D β y ( s ) z ( s ) + h D β y , D β z i . By (i) of Lemma 3.2 we then obtain(3.7) h− D β β − D β − D β y, z i = − D β − y ( b ) z ( b ) + D β − y ( s ) z ( s ) + h D β y, D β z i . On the other hand, using (ii), Lemma 3.2, with f ( t ) = y ( t ) and g ( t ) = D β z ( t ) ,we get h D β y , D β z i = y ( b ) D β z ( β − ( b )) − y ( s ) D β z ( s ) + h y, − D β β − D β − D β z i , which becomes, by (i) of Lemma 3.2,(3.8) h D β y , D β z i = y ( b ) D β − z ( b ) − y ( s ) D β − z ( s ) + h y, − D β β − D β − D β z i . Combining (3.7) with (3.8) it results h− D β β − D β − D β y , z i = y ( b ) (cid:0) D β − z (cid:1) ( b ) − (cid:0) D β − y (cid:1) ( b ) z ( b ) − y ( s ) (cid:0) D β − z (cid:1) ( s ) + D β − y ( s ) z ( s ) + h y, − D β β − D β − D β z i which is equivalent to(3.9) h− D β β − D β − D β y , z i = [ y, z ]( b ) − [ y, z ] ( s ) + h y, − D β β − D β − D β z i . Now we easily derive the β -Lagrange’s identity (3.5): using (3.9) and the fact that r ( t ) is real we have h ℓ β y , z i = h D β β − D β − D β y ( t ) + r ( t ) y ( t ) , z ( t ) i = h D β β − D β − D β y ( t ) , z ( t ) i + h r ( t ) y ( t ) , z ( t ) i = [ y, z ]( b ) − [ y, z ] ( s ) + h y, − D β β − D β − D β z ( t ) i + h y ( t ) , r ( t ) z ( t ) i = [ y, z ]( b ) − [ y, z ] ( s ) + h y, − D β β − D β − D β z ( t ) + r ( t ) z ( t ) i = [ y, z ]( b ) − [ y, z ] ( s ) + h y , ℓ β z i (cid:3) The following corollary follows now easily.
Corollary 3.5.
The β -Sturm Liouville eigenvalue problem (3.3- 3.4) is self-adjointin L β ( s , b ) (cid:0) i.e., ℓ β is self-adj. in (cid:8) y ∈ L β ( s , b ) : y satisfies (3.4) (cid:9)(cid:1) .Proof. Let y and z satisfy the boundary conditions (3.4).(i) If a = 0 then, from (3.6),[ y, z ] ( s ) = y ( s ) (cid:18) − a a z ( s ) (cid:19) + a a y ( s ) z ( s ) = 0 . (ii) If a = 0 then, since | a | + | a | 6 = 0 , one must have a = 0 which, by(3.4), implies that y ( s ) = 0 .By similar arguments it follows that z ( s ) = 0 . Thus, [ y, z ] ( s ) = 0 . Arguing as above one proves also that [ y, z ]( b ) = 0 thus, h ℓ β y , z i = h y , ℓ β z i . (cid:3) Remark . As a consequence of (3.9), under the boundary conditions (3.4), ofcourse we also have h− D β β − D β − D β y , z i = h y, − D β β − D β − D β z i . Theorem 3.7.
All eigenvalues of the problem (3.3)-(3.4) are real. Eigenfunctionscorresponding to different eigenvalues are orthogonal.Proof.
We separate the proof in two parts: (i) and (ii).(i) First we show that the eigenfunctions are real.Let λ be an eigenvalue and y ( t ) be a corresponding eigenfunction. Since ℓ β y ( t ) = λ y ( t ) , ℓ β y ( t ) = λ y ( t )and, by Corollary 3.5, Z bs ℓ β y ( t ) y ( t ) d β t = Z bs y ( t ) ℓ β y ( t ) d β t , then, λ Z bs | y ( t ) | d β t = λ Z bs | y ( t ) | d β t , − STURM LIOUVILLE PROBLEM 11 thus (cid:0) λ − λ (cid:1) Z bs | y ( t ) | d β t = 0 . Since y ( t ) is an eigenfunction then λ = λ .(ii) Finally, we show that eigenfunctions corresponding to different eigenvaluesare orthogonal.Let λ = λ be distinct eigenvalues with eigenfunctions φ and φ , respectively.From Corollary 3.5 and because the eigenvalues are real one have λ Z bs φ ( t ) φ ( t ) d β t = λ Z bs φ ( t ) φ ( t ) d β t . Since λ = λ the orthogonality follows. (cid:3) Now we generalize these results in the following way.If a ≤ s ≤ b then we may consider the β -SLP in L β ( a, b ) defined by the β -difference equation(3.10) ℓ a,bβ y ( t ) := − D β β − D β − D β y ( t ) + r ( t ) y ( t ) = λ y ( t ) , with −∞ < a ≤ s ≤ b < ∞ , λ ∈ C , and the boundary conditions(3.11) ( a y ( a ) + a D β − y ( a ) = 0 b y ( b ) + b D β − y ( b ) = 0 . We also assume that r ( t ) is a real valued continuous function on [ a, b ] and | a | + | a | 6 = 0 = | b | + | b | .This operator ℓ a,bβ defined by (3.10) -(3.11) satisfies the following Corollaries. Corollary 3.8. If y , z ∈ L β ( a, b ) then Z ba (cid:20)(cid:16) ℓ a,bβ y ( t ) (cid:17) z ( t ) − y ( t ) (cid:16) ℓ a,bβ z ( t ) (cid:17)(cid:21) d β u = [ y, z ]( b ) − [ y, z ] ( a ) , with [ y, z ] defined by (3.6).Proof. To prove this Corollary one follows the procedure of the proof of Theorem3.5 and make use of Corollary 3.3. (cid:3)
Corollary 3.9.
The β -Sturm Liouville eigenvalue problem (3.10- 3.11) is self-adjoint in L β ( a, b ) (cid:0) i.e., ℓ a,bβ is self-adj. in (cid:8) y ∈ L β ( a, b ) : y satisfies (3.11) (cid:9)(cid:1) . Corollary 3.10.
All eigenvalues of the problem (3.10)-(3.11) are real. Eigenfunc-tions corresponding to different eigenvalues are orthogonal.
A particular case.
In this subsection the function p satisfies the same condi-tions assumed in the beginning of subsection 2.5. We have the following Propositionrelative to the β -exponential and β -trigonometric functions (2.10)-(2.13). Proposition 3.11.
The following identities hold on I : D β − e p,β ( t ) = p (cid:0) β − ( t ) (cid:1) e p,β ( t ) , D β − E p,β ( t ) = p (cid:0) β − ( t ) (cid:1) E p,β (cid:0) β ( t ) (cid:1) ,D β − sin p,β ( t ) = p (cid:0) β − ( t ) (cid:1) cos p,β ( t ) , D β − cos p,β ( t ) = − p (cid:0) β − ( t ) (cid:1) sin p,β ( t ) ,D β − Sin p,β ( t ) = p (cid:0) β − ( t ) (cid:1) Cos p,β ( t ) ,D β − Cos p,β ( t ) = − p (cid:0) β − ( t ) (cid:1) Sin p,β ( t ) . Proof.
We prove the first identity. D β − e p,β ( t ) = e p,β ( t ) − e p,β (cid:0) β − ( t ) (cid:1) t − β − ( t ) == − − p ( β − t ) ) (cid:16) β − t ) − t (cid:17)(cid:0) t − β − ( t ) (cid:1) ∞ Y k =0 h − p (cid:0) β k ( t ) (cid:1) (cid:16) β k ( t ) − β k +1 ( t ) (cid:17)i = − p (cid:0) β − ( t ) (cid:1) (cid:16) β − ( t ) − t (cid:17) t − β − ( t ) e p,β ( t ) = p (cid:0) β − ( t ) (cid:1) e p,β ( t ) . (cid:3) In a similar way to the β -product rule (2.6) we have(3.12) D β − [ f · g ]( t ) = D β − [ f ]( t ) · g ( t ) + f (cid:0) β − ( t ) (cid:1) · D β − [ g ]( t )= D β − [ g ]( t ) · f ( t ) + g (cid:0) β − ( t ) (cid:1) · D β − [ f ]( t ) . Using the corresponding definitions and this latter formula we obtain the followingproperties.
Proposition 3.12.
The following identities hold on I : D β − D β e p,β ( t ) = (cid:2) p (cid:0) β − ( t ) (cid:1) + D β − p ( t ) (cid:3) e p,β ( t ) ,D β − D β E p,β ( t ) = p ( t ) p (cid:0) β − ( t ) (cid:1) E p,β ( t ) + D β − p ( t ) E p,β (cid:0) β ( t ) (cid:1) ,D β − D β sin p,β ( t ) = − p (cid:0) β − ( t ) (cid:1) sin p,β ( t ) + D β − p ( t ) cos p,β ( t ) ,D β − D β cos p,β ( t ) = − p (cid:0) β − ( t ) (cid:1) cos p,β ( t ) − D β − p ( t ) sin p,β ( t ) ,D β − D β Sin p,β ( t ) = − p (cid:0) β − ( t ) (cid:1) Sin p,β ( t ) + D β − p ( t ) Cos p,β ( t ) ,D β − D β Cos p,β ( t ) = − p (cid:0) β − ( t ) (cid:1) Cos p,β ( t ) − D β − p ( t ) Sin p,β ( t ) . Proof.
The proofs are straightforward. Just to illustrate it we give the proof of thesecond identity.By (2.14) and (3.12) we obtain D β − D β E p,β ( t ) = D β − h p ( t ) E p,β (cid:0) β ( t ) (cid:1)i = D β − p ( t ) E p,β (cid:0) β ( t ) (cid:1) + p (cid:0) β − ( t ) (cid:1) D β − E p,β (cid:0) β ( t ) (cid:1) , hence, by (i) of Lemma 3.2, D β − D β E p,β ( t ) = D β − p ( t ) E p,β (cid:0) β ( t ) (cid:1) + p (cid:0) β − ( t ) (cid:1) D β E p,β ( t )= p ( t ) p (cid:0) β − ( t ) (cid:1) E p,β ( t ) + D β − p ( t ) E p,β (cid:0) β ( t ) (cid:1) . (cid:3) For the particular case where p is the constant function p ( t ) = z , the followingCorollaries hold. − STURM LIOUVILLE PROBLEM 13
Corollary 3.13. If p is the constant function p ( t ) = z on I then the followingidentities hold: D β − D β e z,β ( t ) = z e z,β ( t ) , D β − D β E z,β ( t ) = z E z,β ( t ) ,D β − D β sin z,β ( t ) = − z sin z,β ( t ) , D β − D β cos z,β ( t ) = − z cos z,β ( t ) ,D β − D β Sin z,β ( t ) = − z Sin z,β ( t ) , D β − D β Cos z,β ( t ) = − z Cos z,β ( t ) . Corollary 3.14.
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