Solvability of a Higher-Order Nonlinear Neutral Delay Difference Equation
© Min Liu and Zhenyu Guo. 2010
Received: 19 March 2010
Accepted: 5 September 2010
Published: 13 September 2010
The existence of bounded nonoscillatory solutions of a higher-order nonlinear neutral delay difference equation , , where , , , and are integers, and are real sequences, , and is a mapping, is studied. Some sufficient conditions for the existence of bounded nonoscillatory solutions of this equation are established by using Schauder fixed point theorem and Krasnoselskii fixed point theorem and expatiated through seven theorems according to the range of value of the sequence . Moreover, these sufficient conditions guarantee that this equation has not only one bounded nonoscillatory solution but also uncountably many bounded nonoscillatory solutions.
1. Introduction and Preliminaries
where , , , and are integers, and are real sequences, , and is a mapping. Clearly, difference equations (1.1)–(1.10) are special cases of (1.11). By using Schauder fixed point theorem and Krasnoselskii fixed point theorem, the existence of bounded nonoscillatory solutions of (1.11) is established.
Lemma 1.1 (Schauder fixed point theorem).
Lemma 1.2 (Krasnoselskii fixed point theorem).
Let be a bounded closed convex subset of a Banach space , and let satisfy for each . If is a contraction mapping and is a completely continuous mapping, then the equation has at least one solution in .
The forward difference is defined as usual, that is, . The higher-order difference for a positive integer is defined as , . Throughout this paper, assume that , and stand for the sets of all positive integers and integers, respectively, , , , , and denotes the set of real sequences defined on the set of positive integers lager than where any individual sequence is bounded with respect to the usual supremum norm for . It is well known that is a Banach space under the supremum norm. A subset of a Banach space is relatively compact if every sequence in has a subsequence converging to an element of .
Definition 1.3 (see ).
Lemma 1.4 (discrete Arzela-Ascoli's theorem ).
By a solution of (1.11), we mean a sequence with a positive integer such that (1.11) is satisfied for all . As is customary, a solution of (1.11) is said to be oscillatory about zero, or simply oscillatory, if the terms of the sequence are neither eventually all positive nor eventually all negative. Otherwise, the solution is called nonoscillatory.
2. Existence of Nonoscillatory Solutions
In this section, a few sufficient conditions of the existence of bounded nonoscillatory solutions of (1.11) are given.
The rest of the proof is similar to that in Theorem 2.1. This completes the proof.
Similar to the proof of Theorem 2.5, we have the following theorem.
Similar to the proof of Theorem 2.7, we have
Similar to Remark 2.2, we can also prove that the conditions of Theorems 2.3–2.8 ensure that (1.11) has not only one bounded nonoscillatory solution but also uncountably many bounded nonoscillatory solutions.
In this section, two examples are presented to illustrate the advantage of the above results.
Choose and . It is easy to verify that the conditions of Theorem 2.1 are satisfied. Therefore Theorem 2.1 ensures that (3.1) has a nonoscillatory solution in . However, the results in [3, 4, 6, 8–17] are not applicable for (3.1).
Choose and . It can be verified that the assumptions of Theorem 2.5 are fulfilled. It follows from Theorem 2.5 that (3.2) has a nonoscillatory solution in . However, the results in [3, 4, 6, 8–17] are unapplicable for (3.2).
The authors are grateful to the editor and the referee for their kind help, careful reading and editing, valuable comments and suggestions.
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