ASYMPTOTIC BEHAVIOUR OF SOLUTIONS OF NONLINEAR DELAY DIFFERENCE EQUATIONS IN BANACH SPACES

We consider the second-order nonlinear difference equations of the form Δ(rn−1Δxn−1)


Introduction
Let C be the set of complex numbers and let R be the set of real numbers and l 1 (C) the space of complex-valued sequences (c n ) such that (1.1) Let (X, • ) be a complex (real) Banach space and l ∞ (X) denote the space of bounded sequences x = (x n ) in X with the norm With this norm l ∞ (X) is a Banach space.
In this paper, we are concerned with the difference equation in Banach space: where ∆is the forward difference operator, that is, and f : X → X.
By a solution of (1.3) we understand a sequence x = (x n ) in l ∞ (X) which satisfies (1.3).
The results obtained here generalize some results of M. Migda and J. Migda [9,10].In [10] the second-order difference equation of the form was considered.Authors give a condition when this equation has a solution, asymptotically equal to c, c ∈ R.

Main results
We give necessary and sufficient conditions for the existence of solutions.
Let f be the function from X to X, (p n ), (r n ) sequences of real numbers, and (h n ) a sequence in Banach space.
Let D be a nonempty, closed, convex, and bounded subset of Banach space.Our result will be proved by the following fixed point theorem.
Let F : D → D be a continuous mapping, which is condensing with respect to the measure of noncompactness α: (2.1) Then F has fixed point, where α is the Kuratowski's measure of noncompactness.
Theorem 2.2.Let V ⊂ C(N + ,X) be a family of functions.Then where α(V ) denotes the measure of noncompactness in C(N + ,X).
A theorem similar to Theorem 2.1 was proved by Arino et al. [2], see also [8,11], when f is weakly-weakly sequentially continuous, that is, if for each sequence (x n ), and instead of α we used β-weak measure of noncompactness.

A. Kisiolek and I. Kubiaczyk 2771
Then for every a,b ∈ R there exists a solution (x n ) of (1.3) which possesses the asymptotic behaviour Assume that j=n l j for n ∈ N. Define the operator T : D → K, where (2.8) For x ∈ D and n ∈ N we have (2.9) For n > m we have (2.10) So the operator T : D → K and the continuity of f , T is continuous.Now, we will prove that T satisfies condition (2.1) of Theorem 2.1.
For n > m we have (2.12) A. Kisiolek and I. Kubiaczyk 2773 Using the inequality we obtain that (2.15) This completes the proof of the theorem.
Because we can extend Theorem 2.1 as in [2,8], so similar as Theorem 2.3 we can prove the following theorem.
Theorem 2.4.Let f : X → X be the bounded and weakly-weakly continuous function. Let

.16)
and kt < 1. Moreover There exist many important examples of mappings which are weakly sequentially continuous but not weakly continuous.
The relationship between strong weak and weak sequential continuity for mappings is studied in [3].

.14) By Theorem 2 . 1 T
has a fixed point, and by the definition of T the solution x = (x n )

. 17 )Remark 2 . 5 .
Then for every a,b ∈ R there exists a solution (x n ) of (1.3) which possesses the asymptotic behaviour Observe that the class of continuous functions is different than the class of weakly-weakly sequentially continuous functions and weakly-weakly continuous functions.