Abstract

The existence, multiplicity, and properties of positive solutions for a third order nonlinear neutral delay difference equation are discussed. Six examples are given to illustrate the results presented in this paper.

1. Introduction and Preliminaries

Recently, some researchers used the Reccati transformation techniques, fixed point theorems, and iterative algorithms to study the oscillation, nonoscillation, asymptotic properties, and solvability for linear and nonlinear third order difference equations and systems; see, for example, [1–6] and the references therein. In particular, Saker [4], Andruch-Sobilo and Migda [1], and Grace and Hamedani [2] discussed the oscillation for the following third order difference equations: Making use of the Schauder fixed point theorem, Banach fixed point theorem, and Mann iterative schemes, Yan and Liu [5] and Liu et al. [3], respectively, proved the existence of a bounded nonoscillatory solution for the third order difference equation: and the existence of positive solutions and convergence of the Mann iterative schemes for the following third order nonlinear neutral delay difference equation:

However, the following third order nonlinear neutral delay difference equation: where , , , and are real sequences, and with has not been studied. The purpose of this paper is to study solvability of (4). By utilizing the Krasnoselskii fixed point theorem, Schauder fixed point theorem and some new techniques, we establish the existence, multiplicity, and properties of positive solutions of (4). Six examples are constructed to illuminate our results.

Throughout this paper, we assume that is the forward difference operator defined by , , , and and denote the sets of nonnegative integers and positive integers, respectively, Let denote the Banach space of all real sequences with norm Let , , , and be positive constants, , , , and be real sequences with Put It is easy to see that is a bounded closed and convex subset of .

By a solution of (4), we mean a sequence with a positive integer such that (4) holds for all .

The following lemmas play important roles in this paper.

Lemma 1 (see [6]). A bounded and uniformly Cauchy subset is relatively compact.

Lemma 2 (Krasnoselskii fixed point theorem). Let be a Banach space, a bounded closed convex subset of , and mappings such that for every pair . If is a contraction and is completely continuous, then the equation has a solution in .

Lemma 3 (Schauder fixed point theorem). Let be a nonempty closed convex subset of a Banach space and a continuous mapping such that is a relatively compact subset of . Then has at least one fixed point in .

Lemma 4. Let and . Then(i);(ii);(iii).

Proof. (i) Let denote the largest integer number not exceeding . It is clear that
(ii) It follows from (i) that
(iii) It follows from (ii) that This completes the proof.

2. Existence of Positive Solutions

Now we discuss the existence, multiplicity, and properties of positive solutions of (4) under various conditions on the sequence .

Theorem 5. Assume that there exist a constant and three nonnegative sequences , , and satisfying Then(i)equation (4) possesses a positive solution with (ii)equation (4) possesses uncountably many positive solutions in .

Proof. (i) Let . It follows from (16) and (17) that there exists satisfying Define two mappings and by for any .
Now we show that Using (14), (15), and (20)–(22), we get that for any which yield the fact that (23)–(25) hold.
Next we show that is completely continuous. Let and with Using (16), (17), (27), and the continuity of , , and , we know that for given , there exist , , and with satisfying Combining (15), (22), (28), and (29), we infer that which implies that is continuous in .
It follows from (15), (22), and (28) that for any and which means that is uniformly Cauchy, which together with (25) and Lemma 1 yields that is relatively compact. Hence is completely continuous in . Thus (14), (24), and Lemma 2 ensure that the mapping has a fixed point , which together with (21) and (22) implies that which gives that that is, is a positive solution of (4). It follows from (15)–(17), and (32) that which yield that that is, (18) and (19) hold.
(ii) Let and . As in the proof of (i), we infer that for each , there exist a constant and two mappings and satisfying (20)–(22), where and are replaced by and , respectively, and possesses a fixed point , which is a positive solution of (4); that is, Note that (16) and (17) imply that there exists satisfying which together with (15) and (36) means that for any which implies that which yields that . Therefore (4) possesses uncountably many positive solutions in . This completes the proof.