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Research Article | Open Access
On Positive Solutions of a Fourth Order Nonlinear Neutral Delay Difference Equation
The existence results of uncountably many bounded positive solutions for a fourth order nonlinear neutral delay difference equation are proved by means of the Krasnoselskii’s fixed point theorem and Schauder’s fixed point theorem. A few examples are included.
In the past few decades, the researchers [1–31] and others studied oscillation, asymptotic behavior, and solvability for a lot of second and third order nonlinear difference equations, some of which are as follows:
By employing a few famous tools in nonlinear analysis including the nonlinear alternative of Leray-Schauder type, Banach's fixed point theorem, Schauder's fixed point theorem, Krasnoselskii's fixed point theorem, coincidence degree theory and critical point theory, the authors [3, 4, 9, 13–16, 18, 25, 27, 28] and others proved the existence of nonoscillatory solutions, uncountably many bounded nonoscillatory solutions and periodic solutions for the difference equations above, where Lipschitz conditions were used in [14, 16]. Recently, the authors  used the Krasnoselskii's fixed point theorem to obtain -asymptotic stability results about the zero solution for a very general first order nonlinear neutral differential equation with functional delay.
However, to our knowledge, no one studied the following fourth order nonlinear neutral delay difference equation: where , , and are real sequences with for and with The purpose of this paper is to fill this gap in the literature and to study solvability of (2). Under certain conditions, we prove the existence of uncountably many bounded positive solutions of (2) by means of the Krasnoselskii's fixed point theorem and Schauder's fixed point theorem, respectively. Nine examples are included.
This paper is organized as follows. In Section 2 we present some notations, definitions, and lemmas. In Section 3 we establish nine sufficient conditions which guarantee the existence of uncountably many bounded positive solutions of (2) by using fixed point theorems and new techniques. In Section 4 we give nine examples to illustrate the effectiveness and applications of the results presented in Section 3.
Throughout this paper, we assume that and stand for the sets of all integers, positive integers, and nonnegative integers, respectively: denotes the forward different operator defined by and for . Let denote the Banach space of all bounded sequences on with norm: Obviously, , , and , are closed bounded and convex subsets of for any .
Definition 1 (see ). A subset of is said to be uniformly Cauchy (or equi-Cauchy) if for every there exists a positive integer such that whenever for any .
Lemma 2 (see ). Each bounded and uniformly Cauchy subset of is relatively compact.
Lemma 3 (Krasnoselskii's fixed point theorem). Let be a Banach space, let be a bounded closed convex subset of , and let be mappings from into such that for every pair . If is a contraction and is completely continuous, then the equation has a solution in .
Lemma 4 (Schauder's fixed point theorem). Let be a nonempty closed convex subset of a Banach space , continuous, and relatively compact. Then has at least one fixed point in .
Lemma 5. Let , , , , and be nonnegative sequences. If then where denotes the integer part of number .
Proof. Notice that That is, (9) holds. This completes the proof.
3. Existence of Uncountably Many Bounded Positive Solutions
Now we study the existence of uncountably many bounded positive solutions for (2) by using the Krasnoselskii's fixed point theorem and Schauder's fixed point theorem, respectively.
Theorem 6. Assume that there exist constants , , , and with and and nonnegative sequences , , , and satisfying Then (2) possesses uncountably many bounded positive solutions in .
Proof. Set . It follows from (13) that there exists sufficiently large such that
Define two mappings and : by
for each .
Now we prove that In view of (11), (12), and (14)–(16), we conclude that for any , , and , which yield that (17) hold
In order to prove that is completely continuous in , we have to show that is continuous in and is relatively compact. Suppose that is an arbitrary sequence in and with , where for each and . With the help of (12), (13), , and the continuity of , and , we know that for given , there exist , and with satisfying where By virtue of (16), (19), and (20), we get that which means that is continuous in .
Next we prove that is uniformly Cauchy. It follows from (13) that for given , there exists satisfying Using (16) and (22), we know that for any and , which yields that is uniformly Cauchy. It follows from Lemma 2 that is relatively compact. Consequently Lemma 3 guarantees that there exists satisfying , which together with (15) and (16) gives that which yields that which means that that is, is a bounded positive solution of (2) in .
Finally we prove that (2) has uncountably many bounded positive solutions in . Let and . For each , we conclude similarly that there exist a positive integer and two mappings and satisfying (14)–(16), where and are replaced by and , respectively, and has a fixed point , which is a bounded positive solution (2) in ; that is,
Equation (13) ensures that there exists some satisfying Combining (11), (12), (28), and (29), we deduce that for any , which implies that That is, . Thus (2) has uncountably many bounded positive solutions in . This completes the proof.
Proof. Let . It follows from (13) that there exists sufficiently large such that