Periodicity and stability of solutions for nonlinear
delay dynamic equations on time scales

Notation and preliminaries

Definition 1.1 (Metric space) A pair (X, d) is a metric space if X is a set and d : X × X → [0,∞) such that when y, z and u are in X then i) d(y, z) ≥ 0, d(y, y) = 0 and d(y, z) = 0 implies y = z, ii) d(y, z) = d(z, y), iii) d(y, z) ≤ d(y, u) + d(u, z). The metric space is complete if every Cauchy sequence in (X, d) has a limit in that space. A sequence {xn} ⊂ X is a Cauchy sequence if for each ε > 0 there exists N such that n, m > N imply d(xn, xm) < ε. Theorem 1.1 (Convergent sequence ) Every convergent sequence in a metric space is a Cauchy sequence. 7 Chapter 1. Fixed point theory, functional differential equations and stability 8 Definition 1.2 (Normed space) A vector space (X, +, .) is a normed space if for each x, y ∈ X there is a nonnegative real number kxk, called the norm of x, such that i) kxk = 0 if and only if x = 0, ii) kαxk = |α| kxkfor each α ∈ R, iii) kx + yk ≤ kxk + kyk. A normed space is a vector space and it a metric space with d (x, y) = kx − yk. But a vector space with a metric is not always a normed space. Definition 1.3 A normed space (X, k.k) is said to be complete if it is complete as a metric space (X, d), i.e., every Cauchy sequence is convergent in X. Definition 1.4 (Banach space) A Banach space is a complete normed space. Example 1.1 (a) Let X = R n , n > 1 be a linear space. Then R n is a normed space with the following norms kxk1 = Pn i=1 |xi | for all x = (x1, x2, …, xn) ∈ R n ; kxkp = Pn i=1 |xi | p 1 p for all x = (x1, x2, …, xn) ∈ R n and p ∈ (1,∞); kxk∞ = max 1≤i≤n |xi | for all x = (x1, x2, …, xn) ∈ R n . (b) With any of these norms, (R n , k.k) is a Banach space. It is complete because the real numbers are complete. Theorem 1.2 A closed subspace of a Banach space is a Banach space. Definition 1.5 (Compactness ) A subset M of a metric space (X; d) is compact if any sequence {xn} of M admits a subsequence with limit in M. M is relatively compact if every sequence of M admits a subsequence converging towards a limit belonging to X (i.e. M is compact). Lemma 1.1 A compact subset M of a metric space is closed and bounded. Proposition 1.1 The set M is compact if f it is relatively compact and closed. Proposition 1.2 Each relatively compact set is bounded. Definition 1.6 Let {fn} be a sequence of real valued functions with fn : [a, b] → R. a) {fn} is uniformly bounded on [a, b] if there exists M > 0 such that |fn (t)| ≤ M for all n and all t ∈ [a, b]. 1.1. Notation and preliminaries Chapter 1. Fixed point theory, functional differential equations and stability 9 b) {fn} is equicontinuous if for any ε > 0 there exists δ > 0 such that t1, t 2 ∈ [a, b] and |t1 − t2| < δ imply |fn (t1) − fn (t2)| < ε for all n. Theorem 1.3 (Ascoli-Arzela [14]) If {fn (t)} is a uniformly bounded and equicontinuous sequence of real functions on an interval [a, b], then there is a subsequence which converges uniformly on [a, b] to a continuous function. But here we manipulate function spaces defined on infinite t-intervals. So, for compactness we need an extension of the Arzel`a-Ascoli theorem. This extension is taken from ([19], Theorem 1.2.2 p. 20) and is as follows. Theorem 1.4 Let q : R + → R be a continuous function such that q (t) → 0 as t → ∞. If {fn} is an equicontinuous sequence of R m-valued functions on R + with |fn (t)| ≤ q (t) for t ∈ R +, then there is a subsequence that converges uniformly on R + to a continuous function f (t) with |f (t)| ≤ q (t) for t ∈ R +, where |·| denotes the Euclidean norm on R m. Definition 1.7 Let M be a subset of a Banach space X and T : M → X. If T is continuous and T (M) is contained in a compact subset of X, then T is a compact mapping

 Fixed point theorems

Definition 1.13 Let f be a mapping in the set M. we call fixed point of f any point x satisfying f (x) = x. If there exists such x, we say that f has a fixed point, which is equivalent to saying that the equation f (x) − x = 0 has a null solution. 1.2. Fixed point theorems Chapter 1. Fixed point theory, functional differential equations and stability 11 Theorem 1.5 (Brouwer Fixed Point Theorem (1912)) Suppose that M is a nonempty, convex, compact subset of R N where N > 1, and that f : M → M is a continuous mapping. Then f has a fixed point. Theorem 1.6 (Contraction Mapping Principle (1922) [14]) Let (X, d) a complete metric space and let P : X → X a contraction mapping. Then there is one and only one point z ∈ X with P z = z. Moreover z = lim zn where zn+1 = P zn and z1 chosen arbitrarily in X. Theorem 1.7 ([14]) Let (X, d) a compact nonempty metric space and let P : X → X. If d (P x, P y) < d (x, y), for x 6= y Then P has a fixed point. Theorem 1.8 ([14]) If (X, d) is a complete metric space and P : X → X is a αcontraction operator with fixed point x, then for any y ∈ X we have (a) d (x, y) ≤ d (y, P y)  (1 − α). (b) d (P n y, x) ≤ α nd (y, P y)  (1 − α). 

Krasnoselskii fixed point theorem

The fixed point theorem of Krasnoselskii is an hybrid result and is based on Banach and Schauder theorems. Firstly, we recall the theorem of Schauder Definition 1.14 ([53]) A topological space X has the fixed-point property if whenever P : X → X is continuous, then P has a fixed point. Theorem 1.9 (Schauder’s first fixed-point theorem (1930) [53]) Any compact convex nonempty subset M of a Banach space has the X fixed-point property. Theorem 1.10 (Schauder’s second fixed point theorem [53]) Let M be a nonempty closed convex bounded subset of a Banach space (X, k.k). Then every continuous compact mapping P : M → M has a fixed point. The fixed point theorem of Krasnoselskii is a combination of Banach theorem and that of Schauder. It was the object of several studies these last years and one meets it in several forms. In particular, the theorem of Krasnoselskii gives the existence and the stability of the solutions of the functional differential equations and the nonlinear integral equations with delay of mixed type. In 1955 Krasnoselskii (see [52], [53]) observed that in a good number of problems, the integration of a perturbed differential operator gives rise to a sum of two applications, a contraction and a compact application. It declares then, 

Fixed point theorems

Principle: the integration of a perturbed differential operator can produce a sum of two applications, a contraction and a compact operator. Consider the differential equation x 0 (t) = −a (t) x (t) − g (t, x). (1.1) We can transform this equation in another form while writing, formally x 0 (t) e R t 0 a(s)ds = −a (t) x (t) e R t 0 a(s)ds − g (t, x) e R t 0 a(s)ds , thus x 0 (t) e R t 0 a(s)ds + a (t) x (t) e R t 0 a(s)ds = −g (t, x) e R t 0 a(s)ds , or  x (t) e R t 0 a(s)ds0 = −g (t, x) e R t 0 a(s)ds , then integrating from t − T to t, we obtain Z t t−T  x (u) e R u 0 a(s)ds0 du = − Z t t−T g (u, x) e R u 0 a(s)dsdu, what gives x (t) = x (t − T) e − R t t−T a(s)ds − Z t t−T g (u, x) e − R t u a(s)dsdu. (1.2) If we suppose that e − R t t−T a(s)ds = α < 1 and if (M, k.k) is the Banach space of functions continuous and T-periodic ϕ : R → R, then the equation (1.2) can be written as ϕ (t) = (Bϕ) (t) + (Aϕ) (t) := (P ϕ) (t), where B is contraction provides that the constant α < 1 and A is compact mapping. This example shows the birth of the mapping (P ϕ) := (Bϕ) + (Aϕ) who is identified with a sum of a contraction and a compact mapping. Thus, the search of the solution for (1.2) requires an adequate theorem which applies to this hybrid operator P and who can conclude the existence for a fixed point which will be, in his turn, solution of the initial equation (1.1). Krasnoselskii found the solution by combining the two theorems of Banach and that of Schauder in one hybrid theorem which bears its name. In light, it establishes the following result ([15], [53]). Theorem 1.11 (Krasnoselskii (1955)) Let M be a closed bounded convex nonempty subset of a Banach space (X, k.k). Suppose that A and B map M into X such that (i) A is compact and continuous, (ii) B is a contraction mapping, 1.2. Fixed point theorems Chapter 1. Fixed point theory, functional differential equations and stability 13 (iii) x, y ∈ M, implies Ax + By ∈ M, Then there exists z ∈ M with z = Az + Bz. Note that if A = 0, the theorem become the theorem of Banach. If B = 0, then the theorem is not other than the theorem of Schauder. Proof. According to the condition (iii) we have k(I − B) x − (I − B) yk = k(x − y) − (Bx − By)k ≤ kx − yk + kBx − Byk ≤ kx − yk + α kx − yk = (1 + α) kx − yk , and k(I − B) x − (I − B) yk = k(x − y) − (Bx − By)k ≥ kx − yk − kBx − Byk ≥ kx − yk − α kx − yk = (1 − α) kx − yk . In short (1 − α) kx − yk ≤ k(I − B) x − (I − B) yk ≤ (1 + α) kx − yk . This inequality shows that (I − B) : M → (I − B)M is continuous and bijective. Thus, (I − B) −1 exist and is continuous. Let us pose U := (I − B) −1A. It is clear that U is compact mapping, because U is a composition of a continuous mapping with a compact. Under the theorem of Schauder, U has a fixed point, i.e. ∃z ∈ M such that (I − B) −1Az = z. This is equivalent to z = Az + Bz. 1.2.2 Krasnoselskii-Burton fixed point theorem In this many work on stability with the help of the technique of fixed point T.A. Burton ( [11]) observed that Krasnoselskii result can be more interesting in applications with certain changes and formulated the Theorem 1.13 below (see [11] for the proof). Burton ([11]) remarked that in certain problems the situation does not arise in contraction form. For example, if we consider the equation x 0 = −x 3 = −x + (x − x 3 ). It is proved in [11] that a large contraction defined on a bounded and complete metric space has a unique fixed point. 1.2. Fixed point theorems Chapter 1. Fixed point theory, functional differential equations and stability 14 Theorem 1.12 (Burton [11]) Let (X, d) be a complete metric space and P be a large contraction. Suppose there is an x ∈ X and an L > 0, such that (x, P nx) ≤ L for all n ≥ 1. Then P has a unique fixed point in X. Proof. Suppose there exist x ∈ X, consider {P nx}. If this is a Cauchy sequence then by the triangle inequality we have for m ≥ n d (P nx, P mx) ≤ d .