Existence of positive periodic solutions for nonlinear neutral dynamic equations with variable coefficients on a time scale

Bol. Soc. Paran. Mat. c SPM –ISSN-2175-1188 on line SPM: www.spm.uem.br/bspm (3s.) v. 36 2 (2018): 185–198. ISSN-00378712 in press doi:10.5269/bspm.v36i2.31030 Existence of Positive Periodic Solutions for Nonlinear Neutral Dynamic Equations with Variable Coefficients on a Time Scale Abdelouaheb Ardjouni, Ahcene Djoudi abstract: Let T be a periodic time scale. The purpose of this paper is to use Krasnosel’skiı̆’s fixed point theorem to prove the existence of positive periodic solutions for nonlinear neutral dynamic equations with variable coefficients on a time scale. We invert these equations to construct a sum of a contraction and a compact map which is suitable for applying the Krasnosel’skiı̆’s theorem. The results obtained here extend the work of Candan [11]. Key Words: Positive periodic solutions, nonlinear neutral dynamic equations, fixed point theorem, time scales Contents 1 Introduction 185 2 Preliminaries 186 3 Existence of positive periodic solutions in the case |c (t)| > 1 190 4 Existence of positive periodic solutions in the case |c (n)| < 1 194 1. Introduction Let T be a periodic time scale such that 0 ∈ T. In this paper, we are interested in the analysis of qualitative theory of positive periodic solutions of dynamic equations. Motivated by the papers [1]– [7], [10]– [20] and the references therein, we consider the following two kinds of nonlinear neutral dynamic equations with variable coefficients (x (t) − c (t) x (t − τ ))△ = −a (t) xσ (t) + f (t, x (t − τ )) , (1.1) where x△ is the △-derivative on T (see [8]). Throughout this paper we assume that τ = mω if T has period ω and τ is fixed if T = R. Our purpose here is to use the Krasnosel’skiı̆’s fixed point theorem to show the existence of positive periodic solutions on time scales for equation (1.1). To reach our desired end we have to transform (1.1) into integral equation written as a sum of two mapping; one is a contraction and the other is compact. After that, we use Krasnosel’skiı̆’s fixed point theorem, to show the existence of a positive periodic solution for equation 2010 Mathematics Subject Classification: Primary 34K13, 34A34; Secondary 34K30, 34L30 Submitted February 20, 2016. Published July 23, 2016 185 Typeset by BSP style. M c Soc. Paran. de Mat. 186 A. Ardjouni, A. Djoudi (1.1). In the special case T = R, in [11] we show that (1.1) have a positive periodic solution by using Krasnosel’skiı̆’s fixed point theorem. The organization of this paper is as follows. In Section 2, we present some preliminary material that we will need through the remainder of the paper. We will state some facts about the exponential function on a time scale as well as the Krasnosel’skiı̆’s fixed point theorem. For details on Krasnosel’skiı̆’s theorem we refer the reader to [21]. Also, we present the inversion of (1.1), and we give the Green’s functions of (1.1), which play an important role in this paper. In Section 3 and Section 4, we present our main results on existence of positive periodic solutions of (1.1). The results presented in this paper extend the main results in [11]. 2. Preliminaries A time scale is an arbitrary nonempty closed subset of real numbers. The study of dynamic equations on time scales is a fairly new subject, and research in this area is rapidly growing (see [1], [3], [5]– [9], [16], [17] and papers therein). The theory of dynamic equations unifies the theories of differential equations and difference equations. We suppose that the reader is familiar with the basic concepts concerning the calculus on time scales for dynamic equations. Otherwise one can find in Bohner and Peterson books [8] and [9] most of the material needed to read this paper. We start by giving some definitions necessary for our work. The notion of periodic time scales is introduced in Atici et al. [6] and Kaufmann and Raffoul [16]. The following two definitions are borrowed from [6] and [16]. Definition 2.1. We say that a time scale T is periodic if there exists a ω > 0 such that if t ∈ T then t ± ω ∈ T. For T 6= R, the smallest positive ω is called the period of the time scale. Below are examples of periodic time scales taken from [16]. Example 2.2. [∞The following time scales are periodic. [2 (i − 1) h, 2ih] , h > 0 has period ω = 2h. (1) T = i=−∞ (2) T = hZ has period ω = h. (3) T = R. (4) T = {t = k − q m : k ∈ Z, m ∈ N0 } where, 0 < q < 1 has period ω = 1. Remark 2.3 ( [16]). All periodic time scales are unbounded above and below. Definition 2.4. Let T 6= R be a periodic time scales with period ω. We say that the function f : T → R is periodic with period T if there exists a natural number n such that T = nω, f (t ± T ) = f (t) for all t ∈ T and T is the smallest number such that f (t ± T ) = f (t). If T = R, we say that f is periodic with period T > 0 if T is the smallest positive number such that f (t ± T ) = f (t) for all t ∈ T. Remark 2.5 ( [16]). If T is a periodic time scale with period ω, then σ (t ± nω) = σ (t)±nω. Consequently, the graininess function µ satisfies µ (t ± nω) = σ (t ± nω)− (t ± nω) = σ (t) − t = µ (t) and so, is a periodic function with period ω. Nonlinear Neutral Dynamic Equations 187 Our first two theorems concern the composition of two functions. The first theorem is the chain rule on time scales ( [8], Theorem 1.93). e := Theorem 2.6 (Chain Rule). Assume ν : T → R is strictly increasing and T e e → R. If ν △ (t) and ω △ (ν (t)) exist for t ∈ Tk , then ν (T) is a time scale. Let ω : T   e △ (ω ◦ ν) = ω△ ◦ ν ν △ . In the sequel we will need to differentiate and integrate functions of the form f (t − r (t)) = f (ν (t)) where, ν (t) := t − r (t). Our second theorem is the substitution rule ( [8], Theorem 1.98). e := Theorem 2.7 (Substitution). Assume ν : T → R is strictly increasing and T ν (T) is a time scale. If f : T → R is rd-continuous function and ν is differentiable with rd-continuous derivative, then for a, b ∈ T, Z ν(b) Z b
e f ◦ ν −1 (s) △s. f (t) ν △ (t) △t = ν(a) a A function p : T → R is said to be regressive provided 1 + µ (t) p (t) 6= 0 for all t ∈ Tk . The set of all regressive rd-continuous function f : T → R is denoted by R while the set R+ = {f ∈ R : 1 + µ (t) f (t) > 0 for all t ∈ T}. Let p ∈ R and µ (t) 6= 0 for all t ∈ T. The exponential function on T is defined by Z t  1 ep (t, s) = exp Log (1 + µ (z) p (z)) △z . (2.1) s µ (z) It is well known that if p ∈ R+ , then ep (t, s) > 0 for all t ∈ T. Also, the exponential function y (t) = ep (t, s) is the solution to the initial value problem y △ = p (t) y, y (s) = 1. Other properties of the exponential function are given in the following lemma. Lemma 2.8 ( [8]). Let p, q ∈ R. Then (i) e0 (t, s) = 1 and ep (t, t) = 1; (ii) ep (σ (t) , s) = (1 + µ (t) p (t)) ep (t, s) ; 1 p (t) (iii) = e⊖p (t, s) , where ⊖p (t) = − ; ep (t, s) 1 + µ (t) p (t) 1 = e⊖p (s, t) ; (iv) ep (t, s) = ep (s, t) (v) ep (t, s) ep (s, r) = ep (t, r) ;  △ 1 p (t) (vi) e△ (., s) = pe (., s) and =− σ . p p ep (., s) ep (...truncated)