Existence of periodic or nonnegative periodic solutions for totally nonlinear neutral di ﬀ erential equations with in ﬁ nite delay

In this work, we investigate the existence of periodic or nonnega-tive periodic solutions for a totally nonlinear neutral di ﬀ erential equation with in ﬁ nite delay. In the process, we convert the given neutral di ﬀ erential equation into an equivalent integral equation. Then, we employ Krasnoselski ˘ı -Burton’s ﬁ xed point theorem to prove the existence of periodic or nonnegative periodic solutions. Two examples are provided to illustrate the obtained results.


Introduction
Delay differential equations have attracted considerable attention in mathematics during recent years since these equations have been showed to be valuable tools in the modeling of many phenomena in various fields of science, physics, chemistry and engineering, etc.In particular, problems concerning qualitative analysis such as periodicity, positivity and stability of solutions for delay differential equations have been studied extensively by many authors, see the references [1]- [15].In the current paper, we present sufficient conditions for the existence of periodic or nonnegative periodic solutions of the totally nonlinear neutral differential equation with infinite delay where a is a positive continuous function.The functions h, f : R → R are continuous, Q : R × R → R satisfying the Carathéodory condition.
The main purpose of this work is to use Krasnoselskiȋ-Burton's fixed point theorem (see [11]) to prove the existence of periodic or nonnegative periodic solutions for (1.1).During the process, we employ the variation of parameter formula and the integration by parts to transform (1.1) into an equivalent integral equation written as a sum of two mappings; one is a large contraction and the other is compact.After that, we use Krasnoselskiȋ-Burton's fixed point theorem, to prove the existence of periodic or nonnegative periodic solutions.Two examples are given to illustrate the obtained results.

Preliminaries
For T > 0 define where C (R, R) is the space of all real valued continuous functions.Then P T is a Banach space when it is endowed with the supremum norm Existence of periodic or nonnegative periodic solutions for... 1299 In this paper, we assume that a(t − T ) = a(t), D(t − T, s − T ) = D(t, s), τ (t − T ) = τ (t) ≥ τ * > 0, g(t − T ) = g(t) ≥ g * > 0. (2.1) with τ and g are continuously differentiable functions, τ * and g * are positive constants, a is a positive function and The function Q(t, x) is periodic in t of period T , that is Also, there is a positive constant E such that, The following lemma is fundamental to our results.Lemma 1. Suppose (2.1)-(2.3)hold.If x ∈ P T , then x is a solution of (1.1) if and only if Proof.Let x ∈ P T be a solution of (1.1).Rewrite (1.1) as Multiply both sides of the above equation by exp By dividing both sides of the above equation by exp ³ R t 0 a (u) du ´and using the fact that x(t) = x(t − T ), we obtain Integration by parts the second integral in the above expression, we get (2.8) Then substituting the result of (2.8) into (2.7) to obtain (2.5).The converse implication is easily obtained and the proof is complete.
Theorem 1 (Krasnoselskiȋ-Burton [11]).Let M be a closed bounded convex nonempty subset of a Banach space (B, k.k).Suppose that A and B map M into M such that (i) A is completely continuous, (ii) B is large contraction, (iii) x, y ∈ M, implies Ax + By ∈ M. Then there exists z ∈ M with z = Az + Bz.
Then the mapping H define by (2.6) is a large contraction on the set M.

Existence of periodic solutions
To apply Theorem 1, we need to define a Banach space B, a closed bounded convex subset M of B and construct two mappings; one is a completely continuous and the other is a large contraction.So, we let (B, k.k) = (P T , k.k) and with L ∈ (0, 1] and K > 0. M is a closed convex and bounded subset of P T .
Define a mapping S : Therefore, we express the above mapping as where A, B : P T → P T are given by Existence of periodic or nonnegative periodic solutions for... 1303 We will assume that the following conditions hold.
(H8) The function Q (t, x) is also supposed locally Lipschitz in t, i.e, there exists

Now, we need the following assumptions
where where where γ 1 , γ 2 , γ 3 , γ 4 and J are positive constants with J ≥ 3. Also, suppose that there are constants and Proof.
Since M is a uniformly bounded and equicontinuous subset of the space of continuous functions on the compact [0, T ], we can apply the Arzela-Ascoli theorem to confirm that M is a compact subset from this space.Also, since any continuous operator maps compact sets into compact sets, then to prove that A is a compact operator it suffices to prove that it is continuous.
We prove that A is continuous in the supremum norm, let ϕ n ∈ M where n is a positive integer such that ϕ n → ϕ as n → ∞.Then

Proof.
We have h : R → R is continuous on where L = √ 3/3 and K = 100.Doing straightforward computations, we obtain All hypotheses of Theorem 3 are fulfilled and so (3.22) has a 2π-periodic solution belonging to M. 2

Existence of nonnegative periodic solutions
In this section we obtain the existence of a nonnegative periodic solution of (1.1).By applying Theorem 1, we need to define a closed, convex, and bounded subset M of P T .So, let where L and K are positive constants.To simplify notation, we let It is easy to see that for all (t, u) Then, we obtain the existence of a nonnegative periodic solution of (1.1) by considering the two cases (1) F (t, x (t)) ≥ 0, ∀t ∈ [0, T ] , x ∈ M.

Proof.
For having Aϕ, Bϕ ∈ M, we show that 0 ≤ Aϕ, Bϕ ≤ L and (3.3).So, for any ϕ ∈ M, we have From Lemma 2, we see that That is Aϕ ∈ M. Now, let B defined by (3.4).So, for any ϕ ∈ M, we have and from Lemma 4, we see that Theorem 4. Suppose the hypotheses of Lemmas 3-5 hold.Then (1.1) has a nonnegative T -periodic solution x in the subset M.

Proof.
By Lemma 3, A is completely continuous.Also, from Lemma 4, the mapping B is a large contraction.By Lemma 5, A, B : M → M. Next, we show that if ϕ, ψ ∈ M, we have 0 On the other hand, we have Clearly, all the hypotheses of Krasnoselskiȋ-Burton's theorem are satisfied.Thus there exists a fixed point z ∈ M such that z = Az + Bz.By Lemma 1 this fixed point is a solution of (1.1) and the proof is complete. ´.
Proof.By Example 1, the mapping H (x) = x−x 3 is a large contraction on the set A simple calculation yields   All conditions of Theorem 4 hold and so (4.10) has a nonnegative 2πperiodic solution belonging to M. 2 In the case two, we substitute conditions (4.6)-(4.9)with the following conditions, respectively.We assume that there exist a negative constant c 3 such that Clearly, all the hypotheses of Krasnoselskiȋ-Burton's theorem are satisfied.Thus there exists a fixed point z ∈ M such that z = Az + Bz.By Lemma 1 this fixed point is a solution of (1.1) and the proof is complete.2

2 Theorem 3 .
Suppose the hypotheses of Lemmas 2-4 hold.Let M defined by (3.1), then (1.1) has a T -periodic solution in M.

2
Definition 1.The map P : [0, T ]×R n → R is said to satisfy Carathéodory conditions with respect to L 1 [0, T ] if the following conditions hold.(i) For each z ∈ R n , the mapping t 7 −→ P (t, z) is Lebesgue measurable.(ii) For almost all t ∈ [0, T ], the mapping t 7 −→ P (t, z) is continuous on R n .(iii) For each r > 0, there exists α r ∈ L 1 ([0, T ] , R) such that for almost all t ∈ [0, T ] and for all z such that |z| < r, we have |P (t, z)| ≤ α r (t).