Existence of periodic solutions for two types of second-order nonlinear neutral integro-differential equations with infinite distributed mixed-delays

We consider two types of second-order neutral functional differential equations with infinite distributed delays and offer existence criteria for periodic solutions. During the process we invert the integro-differential equations into equivalent integral equations and derive suitable fixed point mappings. We show that these mappings fit into the framework of Schauder’s fixed point theorem so that periodic solutions are readily obtained.


Introduction
All biological systems and processes take time delays to complete.The delays can represent gestation times, incubation periods, or transport delays.In many cases time delays can be substantial such as gestation and maturation or can represent little lags such as acceleration and deceleration in physical processes.Therefore, it become natural to include time delay terms into the differential equations that model population dynamics.Such models are referred as delay differential equation models.Thus, it seems clear that ordinary differential models are, at best, approximations of real word problems.In the last fifty years, delay models are becoming more common, appearing in many branches of biological, economical and physical modeling (see [1]- [12], [15]- [18]).This is due to their advantage of combining a simple, intuitive derivation with a wide variety of possible behavior regimes and to the fact that such models operate on an infinite dimensional space consisting of continuous functions that accommodate high dimensional dynamics (see [15]- [16]).More recently investigators have given special attentions to the study of equations in which the delay occurs in the derivative of the state variable as well as in the independent variable, so called neutral differential equations (see, [1]- [10], [12], [13], [17]).As known by Hale [15], Hale and Lunel [16], neutral delay differential equations appear as models of electrical networks which contain lossless transmission lines.Such networks arise, for example, in high speed computers where lossless transmission lines are used to interconnect switching circuits.
Existence and periodicity of solutions of functional differential equations are of great interest in mathematics and its applications to the modeling of various practical problems and have been extensively studied in recent times (see [1]- [13], [17]- [18] and references therein).
The study on neutral functional differential equations is more intricate than ordinary delay differential equations.This is why the studies of periodic solutions for neutral differential equations are relatively less than those devoted to ordinary differential equation.Most of the investigations on neutral type equations are confined to first order neutral differential equations.Very recently, Wu and Wang (see [18]) discussed the second order neutral delay differential equation where λ is a positive parameter, δ and c are constants We consider the following two types of second-order neutral functional integro-differential equations with infinite distributed mixed-delays and Also we assume that f : − × R → R + , g : + × R → R + and h 1 , h 2 : R → R are uniformly continuous functions at x. Special cases of (1.1) and (1.2) have been considered and investigated by many authors.Particularly, W. Han and J. Ren in [17], Ardjouni et al. in [1], have, by choosing available operators and applying Krasnoselskii's fixed-point theorem, obtained sufficient conditions providing existence of periodic solutions to special cases of equations (1.1) and (1.2).
The main features of this exposition are the following.In first section we introduce some notations and lemmas and state some preliminary results needed in later sections.Then we give the Green's function of (1.1) and (1.2), which plays an important role in our investigation.Also, we present the inversions of (1.1) and (1.2) and Schauder's fixed point theorem.For details on Schauder's theorem we refer the reader to [14].In the last section, we present our main results on existence of periodic solutions of (1.1) and (1.2).

Preliminaries
For T > 0, let C T be the set of all continuous scalar functions x that are periodic in t with period T .Then, endowed with the supremum norm, and , and Throughout this section we let Also, in order to simplify notation, we define the function H by Clearly, H is a positive continuous function on R.

Lemma 2.1 ([13]
).The equation has a unique T -periodic solution where where 13]).The equation has a unique T -periodic solution The following lemma is essential for our results on existence of periodic solution of (1.1).
Lemma 2.5.If x ∈ C T , then x is a solution of equation (1.1) if and only if where P 1 is the map given by (2.1).
Proof.Let x ∈ C T be a solution of (1.1).Equation (1.1) can be rewritten as Taking Then, (2.2) is transformed into From Lemma 2.1, we have This yields Therefore, since Obviously It is clear that y (t) is the unique T -periodic solution of (2.3) for h ∈ C − T .
The following lemma is essential for our results on existence of periodic solution of (1.2).
Lemma 2.10.If x ∈ C T then x is a solution of equation (1.2) if and only if where P 2 is the map given by (2.4).
Proof.Let x ∈ C T be a solution of (1.2).Equation (1.2) can be rewritten as Taking From Lemma 2.6, we have This yields Obviously It is obvious that y (t) is the unique T -periodic solution of (2.6) for h ∈ C + T .

Periodic solutions
In this section we offer existence criteria for the periodic solutions of the second-order nonlinear neutral integro-differential equations with infinite distributed delay.Lastly in this section we state the Schauder fixed point theorem which enables us to prove the existence of periodic solutions to (1.1) and (1.2).For its proof we refer the reader to [14].This section is devoted to results concerning the condition (3.2) below.We already know, from Lemma (2.5), that the existence of periodic solutions for (1.1) is equivalent to the existence of solutions for the operator equation x = D 1 x (3.1), that is, the fixed points in C J T of D 1 .So we assume, for any J > 0, that there are continuous functions F J : − → R + and G J : Whenever necessary, we shall consider We define an operator D 1 on C T as follows, ϕ ∈ C T implies that We are now ready to state existence T −periodic solution criteria for (1.1)-(1.2).