R-boundedness Approach to linear third differential equations in a UMD Space
aa r X i v : . [ m a t h . SP ] J un R-boundedness Approach to linear thirddifferential equations in a UMD Space
Bahloul Rachid Department of Mathematics, Faculty of Sciences and Technology,
Fez ,Morocco.
ABSTRACT
The aim of this work is to study the existence of a periodic solutions of third order differentialequations z ′′′ ( t ) = Az ( t ) + f ( t ) with the periodic condition x (0) = x (2 π ) , x ′ (0) = x ′ (2 π ) and x ′′ (0) = x ′′ (2 π ). Our approach is based on the R-boundedness and L p -multiplier of linearoperators. Keywords: differential equations, L p -multipliers. Contents R -bounded and L p -multiplier . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 (1.1)
44 Main result 75 Application 96 Conclusion 9
Motivated by the fact that functional differential equations arise in many areas of appliedmathematics, this type of equations has received much attention in recent years. In particular,the problem of existence of periodic solutions, has been considered by several authors. Werefer the readers to papers [[1], [6], [9], [14]] and the references listed therein for informationson this subject. n this work, we study the existence of periodic solutions for the following differentialequations z ′′′ ( t ) = Az ( t ) + f ( t ) x (0) = x (2 π ) , x ′ (0) = x ′ (2 π ) and x ′′ (0) = x ′′ (2 π ) . (1.1)where A : D ( A ) ⊆ X → X is a linear closed operator on Banach space ( X, k . k ) and α canbe any real number and f ∈ L p ( T , X ) for all p ≥ u ′ ( t ) = Au ( t ) + F ( u t ) + f ( t ) , (1.2)B´atkai et al. [5] obtained results on the hyperbolicity of delay equations using the theoryof operatorvalued Fourier multipliers. Bu [8] has studied C α -maximal regularity for theproblem (1.2) on R. Recently, Lizama [14] obtained necessary and sufficient conditions forthe first order delay equation (1.2) to have L p -maximal regularity using multiplier theoremson L p -( T ;X), and C α -maximal regularity of the corresponding equation on the real line hasbeen studied by Lizama and Poblete [15].Arendt [1] gave necessary and sufficient conditions for the existence of periodic solutions ofthe following evolution equation. ddt x ( t ) = Ax ( t ) + f ( t ) for t ∈ R , where A is a closed linear operator on an UMD-space Y .Hernan et al [9], studied the existence of periodic solutions for the class of linear abstractneutral functional differential equation described in the following form: ddt [ x ( t ) − Bx ( t − r )] = Ax ( t ) + G ( x t ) + f ( t ) for t ∈ R where A : D ( A ) → X and B : D ( B ) → X are closed linear operator such that D ( A ) ⊂ D ( B )and G ∈ B ( L p ([ − π, , X ); X ).Bahaj et al [3] studied the existence of periodic solution of second degenerate differentialequation described in the following form:( M x ) ′′ ( t ) + Ax ( t ) + G ( x t ) = f ( t ) for t ∈ R where A : D ( A ) → X and M : D ( M ) → X are closed linear operator such that D ( A ) ⊂ D ( M ) and G ∈ B ( L p ([ − π, , X ); X ).The organization of this work is as follows: In section 2, collects definitions and basicproperties of R-bounded, UMD space and Fourier multipliers, In section 3, we study thesufficient Conditions For the Periodic solutions of Eq. (1.1), In section 4, we establish theperiodic solution for the equation (1.1) of this work solely in terms of a property of R-boundedness for the sequence of operators − ik ( − ik + ( α − A ) − . We optain that thefollowing assertion are equivalent in UMD space :1) D A = d dt − A : H ,p ( T , X ) ∩ L p ( D ( A ) , X ) → L p ( T , X ) is an isomorphism . σ Z (∆) = φ, (cid:8) − ik ∆ − k (cid:9) k ∈ Z is R-bounded . ∀ f ∈ L p ( T , X ) there exists a unique 2 π -periodic strong L p -solution of Eq. (1.1) .
2n section 5, we propose an application. In section 6, we give the conclusion.
Let X be a Banach Space. Firstly, we denote By T the group defined as the quotient R / π Z . There is an identification between functions on T and 2 π -periodic functions on R .We consider the interval [0 , π ) as a model for T .Given 1 ≤ p < ∞ , we denote by L p ( T ; X ) the space of 2 π -periodic locally p -integrablefunctions from R into X , with the norm: k f k p := (cid:18)Z π k f ( t ) k p dt (cid:19) /p For f ∈ L p ( T ; X ), we denote by ˆ f ( k ), k ∈ Z the k -th Fourier coefficient of f that is definedby: ˆ f ( k ) = 12 π Z π e − ikt f ( t ) dt for k ∈ Z and t ∈ R . For 1 ≤ p < ∞ , the periodic vector-valued space is defined by H ,p ( T ; X ) = { u ∈ L p ( T , X ) : ∃ v ∈ L p ( T , X ) , ˆ v ( k ) = ik ˆ u ( k ) for all k ∈ Z } (2.1) Definition 2.1.
Let ε ∈ ]0 ,
1[ and 1 < p < ∞ . Define the operator H ε by:for all f ∈ L p ( R ; X ) ( H ε f )( t ) := 1 π Z ε< | s | < ǫ f ( t − s ) s ds if lim ε → H ε f := Hf exists in L p ( R ; X ) Then Hf is called the Hilbert transform of f on L p ( R , X ). Definition 2.2. [1]
A Banach space X is said to be UMD space if the Hilbert transform is bounded on L p ( R ; X )for all 1 < p < ∞ . R -bounded and L p -multiplier Let X and Y be Banach spaces. Then B ( X, Y ) denotes, the space of bounded linearoperators from X to Y.
Definition 2.3. [1]
A family of operators T = ( T j ) j ∈ N ∗ ⊂ B ( X, Y ) is called R -bounded ( Rademacherbounded or randomized bounded ), if there is a constant
C > p ∈ [1 , ∞ ) such that3or each n ∈ N, T j ∈ T , x j ∈ X and for all independent, symmetric, {− , } -valued randomvariables r j on a probability space (Ω , M, µ ) the inequality (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X j =1 r j T j x j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p (0 , Y ) ≤ C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X j =1 r j x j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p (0 , X ) is valid. The smallest C is called R -bounded of ( T j ) j ∈ N ∗ and it is denoted by R p ( T ). Definition 2.4. [1]
For 1 ≤ p < ∞ , a sequence { M k } k ∈ Z ⊂ B ( X, Y ) is said to be an L p -multiplier if for each f ∈ L p ( T , X ), there exists u ∈ L p ( T , Y ) such that ˆ u ( k ) = M k ˆ f ( k ) for all k ∈ Z . Proposition 2.1. [[ ] , P roposition . Let X be a Banach space and { M k } k ∈ Z be an L p -multiplier, where ≤ p < ∞ . Then theset { M k } k ∈ Z is R -bounded. Theorem 2.1. (Marcinkiewicz operator-valud multiplier Theorem) .Let X , Y be UMD spaces and { M k } k ∈ Z ⊂ B ( X, Y ) . If the sets { M k } k ∈ Z and { k ( M k +1 − M k ) } k ∈ Z are R -bounded, then { M k } k ∈ Z is an L p -multiplier for < p < ∞ . Theorem 2.2. ( F ejer ′ s theorem ) Let f ∈ L p ( T , X ) . Then f = lim n →∞ σ n ( f ) in L p ( T , X ) where σ n ( f ) := 1 n + 1 n X m =0 m X k = − m e k ˆ f ( k ) with e k ( t ) := e ikt . Lemma 2.3. [1] . Let f, g ∈ L p ( T ; X ) . If ˆ f ( k ) ∈ D ( A ) and A ˆ f ( k ) = ˆ g ( k ) for all k ∈ Z Then f ( t ) ∈ D ( A ) and Af ( t ) = g ( t ) for all t ∈ [0 , π ] . (1.1) In this section, we will give conditions which guarantee the periodic solution of the somesecond differential equation. We denote by H ,p ( T ; X ) = { u ∈ L p ( T , X ) : ∃ v ∈ L p ( T , X ) , ˆ v ( k ) = ik ˆ u ( k ) f or all k ∈ Z } H ,p ( T ; X ) = (cid:8) u ∈ L p ( T , X ) : ∃ v ∈ L p ( T , X ) , ˆ v ( k ) = − k ˆ u ( k ) f or all k ∈ Z (cid:9) H ,p ( T ; X ) = (cid:8) u ∈ L p ( T , X ) : ∃ v ∈ L p ( T , X ) , ˆ v ( k ) = − ik ˆ u ( k ) f or all k ∈ Z (cid:9) efinition 3.1. [14] For 1 ≤ p < ∞ , we say that a sequence { M k } k ∈ Z ⊂ B ( X, Y ) is an ( L p , H ,p )-multiplier, iffor each f ∈ L p ( T , X ) there exists u ∈ H ,p ( T , Y ) such that ˆ u ( k ) = M k ˆ f ( k ) for all k ∈ Z . Lemma 3.1. [1]
Let ≤ p < ∞ and ( M k ) k ∈ Z ⊂ B ( X ) ( B ( X ) is the set of all bounded linear operators from X to X ). Then the following assertions are equivalent:(i) ( M k ) k ∈ Z is an ( L p , H ,p )-multiplier.(ii) ( ikM k ) k ∈ Z is an ( L p , L p )-multiplier. We define D A := d dt − A ∆ k = ( − ik I + A ) and σ Z (∆) = { k ∈ Z : ∆ k is not bijective } We begin by establishing our concept of strong solution for Eq. (1.1)
Definition 3.2.
Let f ∈ L p ( T ; X ). A function x ∈ H ,p ( T ; X ) is said to be a 2 π -periodicstrong L p -solution of Eq.(1.1) if x ( t ) ∈ D ( A ) for all t ≥ Proposition 3.1.
Let A be a closed linear operator defined on an UMD space X . Supposethat σ Z (∆) = φ .Then the following assertions are equivalent : (i) (cid:0) − ik ( − ik I + A ) − (cid:1) k ∈ Z is an L p -multiplier for < p < ∞ (ii) (cid:0) − ik ( − ik I + A ) − (cid:1) k ∈ Z is R -bounded.Proof. (i) ⇒ (ii) As a consequence of Proposition (2.1) (ii) ⇒ (i) Define M k = − ik N k where N k = ( − ik I + A ) − . By Marcinkiewcz Theorem itis sufficient to prove that the set { k ( M k +1 − M k ) } k ∈ Z is R -bounded. Since k [ M k +1 − M k ] = k [ − i ( k + 1) ( − i ( k + 1) I + A ) − + ik ( − ik I + A ) − ]= k [ − i ( k + 1) N k +1 + ik N k ]= kN k +1 [ − i ( k + 1) (( − k I + A ))) + ik (( − i ( k + 1) I + A )] N k = kN k +1 [ − i ( k + 1) A + ik A )] N k = ikN k +1 [( k − ( k + 1) ) A ] N k = ikN k +1 [ − (3 k + 3 k + 1) A ] N k = ik N k +1 [ − (3 + 3 1 k + 1 k )]( I + ik N k )= k ( k + 1) M k +1 [ − (3 + 3 1 k + 1 k )]( I − M k )Since products and sums of R -bounded sequences is R -bounded [ [14] . Remark 2.2]. Thenthe proof is complete. 5 emma 3.2. Let ≤ p < ∞ . Suppose that σ Z (∆) = φ and D A is surjective. Then D A isbijective.Proof. We have D A is surjective the H ,p ( T , X ) to L p ( T , X ). Then ∀ f ∈ L p ( T , X ) ∃ z ∈ H ,p ( T , X ) such that D A z = f Suppose that there exists z and z such that D A z = f and D A z = f . then for z = z − z we have D A z = 0. Taking Fourier transform, we obtain that( − ik + A )ˆ z ( k ) = 0 , k ∈ Z . i.e ∆ k ˆ z ( k ) = 0It follows that ˆ z ( k ) = 0 for every k ∈ Z and therefore z = 0. Then z = z and D A isbijective. Theorem 3.3.
Let X be a Banach space. Suppose that the operator D A := d dt − A is anisomorphism of H ,p ( T , X ) onto L p ( T , X ) for ≤ p < ∞ . Then1. for every k ∈ Z the operator ∆ k = ( − ik I + A ) has bijective,2. (cid:8) − ik ∆ − k (cid:9) k ∈ Z is R -bounded. Before to give the proof of Theorem (3.3) , we need the following Lemma.
Lemma 3.4. if x ∈ Ker ∆ k , then e ikt x ∈ KerD A Proof. x ∈ Ker ∆ k ⇒ − ik x = Ax .Put z ( t ) = e ikt x , then z ′′′ ( t ) = ( ike ikt x ) ′′ = ( − k e ikt x ) ′ = − ik e ikt x = e ikt Ax = Az ( t ) ⇒ D A z ( t ) = 0 ⇒ e ikt x ∈ KerD A . Proof of Theorem (3.3) : 1) Let k ∈ Z and y ∈ X . Then for f ( t ) = e ikt y , there exists z ∈ H ,p ( T ; X ) such that: D A z ( t ) = f ( t )Taking Fourier transform, we deduce that:( − ik + A )ˆ z ( k ) = ˆ f ( k ) = y ⇒ ∆ k = ( − ik + A ) is surjective.Let u ∈ Ker ∆ k . By Lemma , we have e ikt u ∈ KerD A , then u = 0 and ( − ik + A ) isinjective. 6) Let f ∈ L p ( T ; X ). By hypothesis, there exists a unique z ∈ H ,p ( T , X ) such that D A z = f . Taking Fourier transforms, we deduce thatˆ z ( k ) = ( − ik + A ) − ˆ f ( k ) for all k ∈ Z . Hence − ik ˆ z ( k ) = − ik ( − ik + A ) − ˆ f ( k ) for all k ∈ Z Since z ∈ H ,p ( T ; X ) , then there exists v ∈ L p ( T ; X ) such thatˆ v ( k ) = − ik ˆ z ( k ) = − ik ( − k + A ) − ˆ f ( k ) . Then (cid:8) − ik ∆ − k (cid:9) k ∈ Z is an L p -multiplier and (cid:8) − ik ∆ − k (cid:9) k ∈ Z is R -bounded. Our main result in this section is to establish that the converse of Theorem , are true,provided X is an UMD space. Lemma 4.1.
Let ≤ p < ∞ and ( M k ) k ∈ Z ⊂ B ( X ) . Then the following assertions areequivalent:(i) ( M k ) k ∈ Z is an ( L p , H ,p )-multiplier.(ii) ( − k M k ) k ∈ Z is an ( L p , L p )-multiplier.Proof. We have( − k M k ) k ∈ Z is an ( L p , L p ) − multiplier ⇔ ik ( ikM k ) k ∈ Z is an ( L p , L p ) − multiplier ⇔ ( ikM k ) k ∈ Z is an ( L p , H ,p ) − multiplier (by Lemma3 . Lemma 4.2.
Let ≤ p < ∞ and ( M k ) k ∈ Z ⊂ B ( X ) . Then the following assertions areequivalent:(i) ( M k ) k ∈ Z is an ( L p , H ,p )-multiplier.(ii) ( − ik M k ) k ∈ Z is an ( L p , L p )-multiplier. Theorem 4.3.
Let X be an UMD space and A : D ( A ) ⊂ X → X be an closed linearoperator. Then the following assertions are equivalent for < p < ∞ . (1) The operator D A := d dt − A is an isomorphism of H ,p ( T , X ) ∩ L p ( D ( A ) , X ) onto L p ( T , X ) . (2) σ Z (∆) = φ and (cid:8) − ik ∆ − k (cid:9) k ∈ Z is R -bounded.Proof. ⇒
2) see Theorem (3.3) ⇐
2) Let f ∈ L p ( T ; X ) . Define ∆ k = ( − ik I + A ),By Lemma , the family (cid:8) − ik ∆ − k (cid:9) k ∈ Z is an L p -multiplier it is equivalent to 7he family (cid:8) ∆ − k (cid:9) k ∈ Z is an L p -multiplier that maps L p ( T ; X ) into H ,p ( T ; X ) (Lemma 4.2),namely there exists z ∈ H ,p ( T , X ) such thatˆ z ( k ) = ∆ − k ˆ f ( k ) = ( − ik I + A ) − ˆ f ( k ) (4.1)In particular, z ∈ L p ( T ; X ) and there exists v ∈ L p ( T ; X ) such that ˆ v ( k ) = − ik ˆ z ( k ) ByTheorem , we have z ( t ) = lim n → + ∞ n + 1 n X m =0 m X k = − m e ikt ˆ z ( k )Using now (4.1) we have: ( − ik I − A )ˆ z ( k ) = ˆ f ( k ) for all k ∈ Z . i.e \ ( D A z )( k ) = ˆ f ( k ) for all k ∈ Z . Since A is closed, then z ( t ) ∈ D ( A ) and D A z ( t ) = f ( t ) [Lemma ].Uniqueness, suppose that ∃ z , z : D A z ( t ) = f ( t ) and D A z ( t ) = f ( t ).Then z = z − z ∈ ker ( D A ) i.e d dt x ( t ) = Ax ( t ) . Taking Fourier transform, we deduce that:∆ k ˆ z ( k ) = 0 ⇒ ˆ z ( k ) = 0 ∀ k ∈ Z ⇒ z = 0. i.e z = z . Or D A is linear operator then D A isisomorphism. Corollary 4.4.
Let X be an UMD space and A : D ( A ) ⊂ X → X be an closed linearoperator. Then the following assertions are equivalent for < p < ∞ . (1) for every f ∈ L p ( T ; X ) there exists a unique π -periodic strong L p -solution of Eq. (1.1) . (2) σ Z (∆) = φ and (cid:8) − ik ∆ − k (cid:9) k ∈ Z is R -bounded.Proof. By theorem 4.3, we have(2) ⇔ D A : H ,p ( T , X ) ∩ L p ( D ( A ) , X ) → L p ( T , X ) is an isomorphism . ⇔ ∀ f ∈ L p ( T , X ) there exits a unique z ∈ H ,p ( T , X ) ∩ L p ( D ( A ) , X ) : D A z = f ⇔ ∀ f ∈ L p ( T , X ) there exits a unique z ∈ H ,p ( T , X ) ∩ L p ( D ( A ) , X ) : z = D − A f ⇔ ∀ f ∈ L p ( T , X )there exists a unique 2 π -periodic strong L p -solution of Eq. (1.1) . Application
To apply the provious result, we propose the following partial functional differential equation ∂ ∂t w ( t, x ) = ∂ ∂t w ( t, x ) + g ( t, x ) , x ∈ [0 , π ] , t ≥ ,w (0 , t ) = w ( π, t ) = 0 , t ≥ .w ( x, t ) = w ( x, t ) , x ∈ [0 , π ] , − r ≤ t ≤ . (5.1)Let X = C [0 , π ] = { u ∈ C ([0 , π ] , R ) : u (0) = u ( π ) = 0 } . we define the linear operator A : D ( A ) ⊂ X → X by Ay = y ′′ D ( A ) = { y ∈ C ([0 , π ] , R ) : y (0) = y ( π ) = 0 } . Put y ( t )( x ) = w ( t, x ) and f ( t )( x ) = g ( t, x )Thus, Eq. (5.1) takes the following abstract form d dt y ( t ) = Ay ( t ) + f ( t ) (5.2)It is well known that A is linear operator. Then by [[9], Section 3.7] (see also referencestherein), there exists a constant c > || ( − ik − A ) || ≤ c | k | Then sup k ∈ Z || k ( − ik − A ) || < + ∞ We deduce from Corollary (4.4) that the above periodic problem has L p -strong solution. we are obtained necessary and sufficient conditions to guarantee existence and uniqueness ofperiodic solutions to the equation z ′′′ ( t ) = Az ( t ) + f ( t ) in terms of either the R-boundednessof the modified resolvent operator determined by the equation. Our results are obtained inthe UMD spaces. References [1] W.Arend and S.Bu,
The operator-valued Marcinkiewicz multiplier theorem and maximalregularity, Math.Z. 240, (2002), 311-343.
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