PERIODIC SOLUTIONS FOR SOME PARTIAL FUNCTIONAL DIFFERENTIAL EQUATIONS

We study the existence of a periodic solution for some partial functional differential equations. We assume that the linear part is nondensely defined and satisfies the Hille-Yosida condition. In the nonhomogeneous linear case, we prove the existence of a periodic solution under the existence of a bounded solution. In the nonlinear case, using a fixed-point theorem concerning set-valued maps, we establish the existence of a periodic solution.


Introduction
Consider the partial functional differential equation d dt x(t) = Ax(t) + L t,x t + G t,x t , for t ≥ 0, where A : D(A) ⊂ E → E is a nondensely defined linear operator on a Banach space E. Throughout this paper, we suppose that (H 1 ) A is a Hille-Yosida operator: there exist M 0 ≥ 1 and ω 0 ∈ R such that where ρ(A) is the resolvent set of A and R(λ,A) = (λ − A) −1 .C is the space of continuous functions from [−r,0] into E endowed with the uniform norm topology, and for every t ≥ 0, the history function x t ∈ C is defined by the existence, the regularity of solutions, and the local stability have been treated when A is nondensely defined and satisfies the Hille-Yosida condition.In this work, we will deal with the existence of periodic solutions of (1.1) when A satisfies the Hille-Yosida condition.The problem of finding periodic solutions is an important subject in the qualitative study of functional differential equations.The famous Massera's theorem on twodimensional periodic ordinary differential equations [11] explains the relationship between the boundedness of solutions and periodic solutions.In [15], using Browder's fixed-point theorem, it has been proved that if the solutions of an n-dimensional periodic ordinary differential equation are either uniformly bounded or uniformly ultimately bounded, then the system has a periodic solution.In [5], the existence of a periodic solution has been established under the existence of a bounded solution for some inhomogeneous, linear functional differential equation in infinite dimensional space.In [10], using Horn's fixed-point theorem, the existence of periodic solutions for functional differential equation with finite delay was established.Recently in [12], several criteria were obtained to ensure the existence and uniqueness of a periodic solution for some inhomogeneous linear partial functional differential equations with infinite delay.In [4], we developed some results dealing with the existence of a periodic solution for (1.1) when A generates a strongly continuous semigroup on E. In [7], it was established that the existence of bounded and ultimate bounded solutions of (1.1) implies the existence of periodic solutions.The approach that was used was based on Horn's fixed-point theorem.In this paper, we generalize the results obtained in [4,5,11] for (1.1), where the operator A is not necessarily densely defined but satisfies the Hille-Yosida condition.In Section 2, we prove the existence of periodic solutions in the nonhomogeneous linear case under the assumption that a bounded solution on R + exists.In Section 3, we study the nonlinear case; our approach makes use of a fixed-point theorem for set-valued maps to obtain sufficient conditions, ensuring the existence of a periodic solution for (1.1).Section 4 is devoted to an example.

Inhomogeneous linear case
Definition 2.1 [1,8].A continuous function It follows from the closedness of A that if x is an integral solution of (1.1), then x(t) ∈ D(A), for t ≥ 0. The following result dealing with the existence and the uniqueness of the integral solution was established.
Theorem 2.2 [1,8].Assume that (H 1 ) holds and G is Lipschitz with respect to the second argument.Then for all ϕ ∈ C such that ϕ(0) ∈ D(A), (1.1) has a unique integral solution on R + .Moreover, the integral solution depends continuously on the initial data.
Let A 0 be the part of A in D(A) given by (2.1) R. Benkhalti and K. Ezzinbi 11 Then, from [2], A 0 generates a strongly continuous semigroup (T 0 (t)) t≥0 on D(A).Moreover, from [13], if the integral solution of (1.1) exists, then it is given by this variation of constant formula where Consider the equation where f is continuous and ω-periodic in t, and suppose the hypothesis stated below.(H 2 ) The semigroup (T 0 (t)) t≥0 is compact on D(A), meaning that for t > 0, the operator T 0 (t) is compact on D(A).
Let u be the bounded integral solution of (2.3) on R + , then the following two lemmas are needed in the proof of Theorem 2.3.
Proof of Lemma 2.4.For simplicity, we equate F(t,ϕ) = L(t,ϕ) + f (t), and let ε > 0 and t > ε.Then, The compactness property of the semigroup (T 0 (t)) t≥0 and the boundedness of the solution u show that {T 0 (ε)u(t − ε) : t > ε} is relatively compact in E. Using the boundedness of B λ and F, there exists a positive constant a such that Hence, {u(t) : t ≥ 0} is relatively compact in E.
To show the uniform continuity of u, let t > τ > 0.Then, we have Now the range of u is relatively compact, so From the uniform continuity of u, we determine that {u t : t ≥ 0} is an equicontinuous family of functions on [−r,0]; moreover, the range of u is relatively compact.Hence, by Arzèla-Ascoli theorem, we determine that {u t : t ≥ 0} is relatively compact in C.
Lemma 2.5 [9].Let X be a Banach space, let Φ : X → X be a continuous linear operator, let y ∈ X be given, and let Θ : X → X be given by Θx = Φx + y.Suppose that there exists x 0 ∈ X such that {Θ n x 0 : n ∈ N} is relatively compact.Then Θ has a fixed point.

R. Benkhalti and K. Ezzinbi 13
Proof of Theorem 2.3.As usual, define the Poincaré map P(ϕ) = x ω (•,ϕ, f ) on the phase space C 0 = {ϕ ∈ C : ϕ(0) ∈ D(A)}, where x(•,ϕ, f ) is the integral solution of (2.3).Because of the uniqueness property, it is enough to show that P has a fixed point to get an ω-periodic solution of (2.3).Also, the uniqueness property of the solution with respect to ϕ allows the Poincaré map P to be decomposed as where x ω (•,ϕ,0) is the integral solution of (2.3) with f = 0, and x ω (•,0, f ) is the integral solution of (2.3) with ϕ = 0. Let u be the bounded solution of (2.3) on [0,+∞) and u 0 = ϕ.Then, by Lemma 2.4, is relatively compact in C 0 , and the mapping P has a fixed point in C 0 using Lemma 2.5.Hence, (2.3) has an ω-periodic solution.

Nonlinear case
Consider the nonlinear equation and assume the hypothesis stated below.
(H 3 ) G takes every bounded set into a bounded set.
Let B ω be the space of all continuous ω-periodic functions from R + into E, endowed with the uniform norm topology.Theorem 3.1.Assume that (H 1 ), (H 2 ), and (H 3 ) hold.Further, assume that there exists a positive ρ such that for any y ∈ S ρ = {v ∈ B ω : v ≤ ρ}, the equation has an ω-periodic integral solution in S ρ .Then, (3.1) has an integral ω-periodic solution on R + .
For the proof, we need the following definition and theorem.(ii) The map Ᏻ is called upper semicontinuous if Ᏻ −1 (D) is closed for all closed set D in M. Theorem 3.3 (see [16,Corollary 9.8]).Let Ᏻ : M → 2 M be a multivalued map, where M is a nonempty convex set in the Banach space X such that (i) the set Ᏻ(x) is nonempty, closed, and convex for all x ∈ M, (ii) the set Ᏻ(M) is relatively compact, (iii) the map Ᏻ : M → 2 M is upper semicontinuous.Then Ᏻ has a fixed point in the sense that there exists x ∈ M such that x ∈ Ᏻ(x).
Proof of Theorem 3.1.Define the set-valued mapping Ᏻ : S ρ → 2 Sρ , for y ∈ S ρ , by We will show that the mapping Ᏻ satisfies the conditions of Theorem 3.3.
(ii) Let x ∈ Ᏻ(S ρ ), then there exists y ∈ S ρ such that We first show that {x(t) : x ∈ Ᏻ(S ρ )} is relatively compact in E. Let t > 0 and ε > 0 such that t > ε.Then, From the boundedness of L, G and (H 2 ), we deduce that is relatively compact in E. On the other hand, for some positive constant b, we have Hence, {x(t) : x ∈ Ᏻ(S ρ )} is relatively compact in E, for every t > 0, and by periodicity, we also have that {x(0) : x ∈ Ᏻ(S ρ )} is relatively compact in E. For the equicontinuity, one has, for t > τ > 0, R. Benkhalti and K. Ezzinbi 15 The semigroup (T 0 (t)) t≥0 is compact, so (T 0 (t)) t≥0 is continuous in the uniform topology whenever t > 0. Hence, By (H 3 ), we deduce that for some positive constant c, Consequently, Similarly, one can also prove that lim Therefore, Ᏻ(S ρ ) is a family of uniformly bounded and equicontinuous ω-periodic functions.By the Arzèla-Ascoli theorem, we deduce that Ᏻ(S ρ ) is relatively compact in B ω .
(iii) To prove that Ᏻ is upper semicontinuous, it is enough to show that Ᏻ is closed.Let (y n ) n≥0 and (z n ) n≥0 be sequences, respectively, in S ρ and Ᏻ(S ρ ) such that Then, Letting n go to infinity and by a continuity argument, we obtain Hence, z ∈ Ᏻ(y), which implies that Ᏻ is closed.Now let D be a closed set in S ρ and take a sequence (x n ) n ⊂ Ᏻ −1 (D) such that x n → x as n → ∞.Since x n ∈ Ᏻ −1 (D), it follows that there exists y n ∈ D such that y n ∈ Ᏻ(x n ).Moreover, Ᏻ(S ρ ) is compact; thus, there exists a subsequence (y n ) n of (y n ) n such that y n → y as n → ∞.Therefore, Ᏻ is closed and it follows that y ∈ Ᏻ(x) and y ∈ Ᏻ −1 (D).Consequently, Ᏻ is upper semicontinuous.All the assumptions of Theorem 3.3 hold.Hence, there exists x ∈ S ρ such that x ∈ Ᏻ(x).Finally, x is an ω-periodic solution of (3.1) on R + .
To prove that (3.2) has an ω-periodic solution in S ρ , it suffices, by Theorem 2.3, to show that it has a solution which is bounded by ρ.
Proof.Let u be a bounded solution of (3.2) such that u 0 = ϕ.Following the proof of [9, Theorem 2.5], the map P has a fixed point which belongs to co{P n ϕ : n ≥ 0}, where co denotes the closure of the convex hull.Let ψ be the fixed point of P and x(•,ψ, f ) the associated integral solution; by virtue of the continuous dependence on the initial data, the solution x(•,ψ, f ) is also bounded by ρ.

Definition 3 . 2
(see[16, Definition 9.3]).Let Ᏻ : M → 2 M be a multivalued map, where M is a subset of a Banach space and 2 M is the power set of M.(i) For D ⊂ M, the inverse image Ᏻ −1 (D) is the set of all x ∈ M such that Ᏻ(x) ∩ D = ∅.