Abstract
This paper studies the second-order nonlinear neutral delay difference equation , . By means of the Krasnoselskii and Schauder fixed point theorem and some new techniques, we get the existence results of uncountably many bounded nonoscillatory, positive, and negative solutions for the equation, respectively. Ten examples are given to illustrate the results presented in this paper.
1. Introduction
We are concerned with the second-order nonlinear neutral delay difference equation of the form where , , , and are real sequences with for each , , and with Note that a few special cases of (1.1) were studied in [1–9]. In particular, González and Jiménez-Melado [3] used a fixed-point theorem derived from the theory of measures of noncompactness to investigate the existence of solutions for the second-order difference equation By applying the Leray-Schauder nonlinear alternative theorem for condensing operators, Agarwal et al. [1] studied the existence of a nonoscillatory solution for the second-order neutral delay difference equation where . Using the Banach contraction principle, Cheng [5] discussed the existence of a positive solution for the second-order neutral delay difference equation with positive and negative coefficients where , Liu et al. [6] and Liu et al. [7] extended the results due to cheng [5] and got the existence of uncountably many bounded nonoscillatory solutions for (1.1) and the second-order nonlinear neutral delay difference equation with respect to , where is Lipschitz continuous, respectively.
The purpose of this paper is to establish the existence results of uncountably many bounded nonoscillatory, positive, and negative solutions, respectively, for (1.1) by using the Krasnoselskii fixed point theorem, Schauder fixed point theorem, and a few new techniques. The results obtained in this paper improve essentially the corresponding results in [5–7] by removing the Lipschitz continuity condition. Ten nontrivial examples are given to reveal the superiority and applications of our results.
2. Preliminaries
Throughout this paper, we assume that is the forward difference operator defined by , and , , and stand for the sets of all integers, positive integers, and nonnegative integers, respectively, Let denote the Banach space of all bounded sequences in with norm represent the closed ball centered at and with radius in .
By a solution of (1.1), we mean a sequence with a positive integer such that (1.1) is satisfied for all . As is customary, a solution of (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative. Otherwise, it is said to be nonoscillatory.
Lemma 2.1 ([2]). A bounded, uniformly Cauchy subset of is relatively compact.
Lemma 2.2 (Krasnoselskii fixed point theorem [10]). Let be a nonempty bounded closed convex subset of a Banach space and mappings such that for every pair . If is a contraction and is completely continuous, then has a solution in .
Lemma 2.3 (Schauder fixed point theorem [10]). 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 a fixed point in .
Lemma 2.4. Let , and be a nonnegative sequence. Then, Moreover, if then
Proof. For each , let stand for the largest integer not exceeding . It follows that Combining (2.6) and (2.7), we infer that (2.4) holds. Assume that In view of (2.4), we get that , which gives that . It follows that This completes the proof.
3. Existence of Uncountably Many Bounded Positive Solutions
Now, we use the Krasnoselskii fixed point theorem to prove the existence of uncountably many bounded nonoscillatory, positive, and negative solutions of (1.1) under various conditions relative to the sequence .
Theorem 3.1. Assume that there exist and two nonnegative sequences and satisfying Then, (1.1) has uncountably many bounded nonoscillatory solutions in .
Proof. Let . It follows from (3.3) that there exists satisfying
Define two mappings and by
for any .
Now, we assert that
It follows from (3.1), (3.2), and (3.4)–(3.6) that for any , and ,
which imply that (3.7)–(3.9) hold.
Next, we prove that is continuous and is uniformly Cauchy. It follows from (3.3) that for each , there exists satisfying
Let and satisfy that
In view of (3.12) and the continuity of and , we know that there exists such that
Combining (3.6), (3.11), and (3.13), we obtain that
which means that is continuous in .
In view of (3.6) and (3.11), we obtain that for any and
which implies that is uniformly Cauchy, which together with (3.9) and Lemma 2.1 yields that is relatively compact. Consequently, is completely continuous in . Thus, (3.7), (3.8), and Lemma 2.2 ensure that the mapping has a fixed point , which together with (3.5) and (3.6) implies that
which yields that
That is, is a bounded nonoscillatory solution of (1.1) in .
Let , and . Similarly, we can prove that for each , there exist a constant and two mappings and satisfying (3.4)–(3.6), where , and are replaced by , and , respectively, and has a fixed point , which is a bounded nonoscillatory solution of (1.1); that is,
Note that (3.3) implies that there exists satisfying
Using (3.2), (3.18), and (3.19), we get that for any ,
that is, . Therefore, (1.1) possesses uncountably many bounded nonoscillatory solutions in . This completes the proof.
Theorem 3.2. Assume that there exist , and two nonnegative sequences and satisfying (3.2), (3.3), and Then, (1.1) has uncountably many bounded solutions in .
The proof of Theorem 3.2 is analogous to that of Theorem 3.1 and hence is omitted.
Theorem 3.3. Assume that there exist , and two nonnegative sequences and satisfying (3.2), (3.3), and Then, (1.1) has uncountably many bounded solutions in .
Proof. Let . It follows from (3.3) that there exists satisfying
Define two mappings and by
for any .
Now, we assert that (3.7), (3.9), and the below
hold. It follows from (3.2) and (3.22)–(3.25) that for any , and ,
which imply (3.7), (3.9), and (3.26).
Next, we show that is continuous and is uniformly Cauchy. It follows from (3.3) that for each , there exists satisfying (3.11). Let and with (3.12). It follows from (3.12) and the continuity of and that there exists satisfying (3.13). In light of (3.11), (3.13), and (3.25), we deduce that
which yields that is continuous in .
Using (3.1) and (3.25), we get that for any and
which means that is uniformly Cauchy, which together with (3.9) and Lemma 2.1 yields that is relatively compact. Consequently, is completely continuous in . Thus, (3.22), (3.26), and Lemma 2.2 ensure that the mapping has a fixed point ; that is,
which gives that
That is, is a bounded solution of (1.1) in .
Let and . Similarly, we conclude that for each , there exist a constant and two mappings and satisfying (3.23)–(3.25), where , and are replaced by , and , respectively, and has a fixed point , which is a bounded solution of (1.1); that is,
for all and . Note that (3.3) implies that there exists satisfying (3.19). By means of (3.2), (3.19), and (3.32), we infer that for any ,
that is, . Therefore, (1.1) possesses uncountably many bounded solutions in . This completes the proof.
Similar to the proofs of Theorems 3.1 and 3.3, we have the following results.
Theorem 3.4. Assume that there exist and two nonnegative sequences and satisfying (3.2), (3.3), and Then, (1.1) has uncountably many bounded nonoscillatory solutions in .
Theorem 3.5. Assume that there exist and two nonnegative sequences and satisfying (3.2), (3.3), and Then, (1.1) has uncountably many bounded positive solutions in .
Theorem 3.6. Assume that there exist and two nonnegative sequences and satisfying (3.2), (3.3), and Then, (1.1) has uncountably many bounded negative solutions in .
Theorem 3.7. Assume that there exist and two negative sequences and satisfying (3.2), (3.3), and Then, (1.1) has uncountably many bounded nonoscillatory solutions in .
Theorem 3.8. Assume that there exist and two negative sequences and satisfying (3.2), (3.3), and Then, (1.1) has uncountably many bounded nonoscillatory solutions in .
Next, we investigate the existence of uncountably bounded nonoscillatory solutions for (1.1) with the help of the Schauder fixed point theorem under the conditions of .
Theorem 3.9. Assume that there exist and two nonnegative sequences and satisfying (3.2), (3.3), and Then, (1.1) has uncountably many bounded nonoscillatory solutions in .
Proof. Let . It follows from (3.3) that there exists satisfying
Define a mapping by
for any .
Now, we prove that
It follows from (3.2) and (3.39)–(3.41) that for any and ,
which imply (3.42).
Next, we prove that is continuous and is uniformly Cauchy. It follows from (3.3) that for each , there exists satisfying (3.11). Let and satisfying (3.12). It follows from (3.12) and the continuity of and that there exists satisfying (3.13). Combining (3.11), (3.13), and (3.41), we infer that
which implies that is continuous in .
By means of (3.11) and (3.41), we get that for any and
which means that is uniformly Cauchy, which together with (3.42) and Lemma 2.1 yields that is relatively compact. It follows from Lemma 2.3 that the mapping has a fixed point ; that is,
which give that
That is, is a bounded nonoscillatory solution of (1.1).
Let and . Similarly, we infer that for each , there exist a constant and a mapping satisfying (3.41), where are replaced by , and , respectively, and has a fixed point , which is a bounded nonoscillatory solution of (1.1); that is,
for . Note that (3.3) implies that there exists satisfying (3.19). Using (3.2), (3.19), and (3.48), we conclude that for any
which gives that . Therefore, (1.1) possesses uncountably many bounded nonoscillatory solutions in . This completes the proof.
Theorem 3.10. Assume that there exist and two nonnegative sequences and satisfying (3.2), Then, (1.1) has uncountably many bounded nonoscillatory solutions in .
Proof. Let . It follows from (3.41) that there exists satisfying
Define a mapping by
for any . It follows from (3.2), (3.52), and (3.53) that for any and
which imply (3.42).
Next, we show that is continuous and is uniformly Cauchy. It follows from (3.51) and Lemma 2.4 that for each , there exists satisfying
Let and satisfying (3.12). By means of (3.12) and the continuity of and , we deduce that there exists satisfying
In light of (3.2), (3.53)–(3.56) and Lemma 2.4, we conclude that
which implies that is continuous in .
By virtue of (3.53), (3.55), and Lemma 2.4, we get that for any and ,
which means that is uniformly Cauchy, which together with (3.42) and Lemma 2.1 yields that is relatively compact. It follows from Lemma 2.3 that the mapping has a fixed point ; that is,
which means that
That is, is a bounded nonoscillatory solution of (1.1) in .
Let and . Similarly, we conclude that for each , there exist a positive integer and a mapping satisfying (3.53), where are replaced by , and , respectively, and has a fixed point , which is a bounded nonoscillatory solution of (1.1); that is,
for . Note that (3.41) implies that there exists satisfying
which together with (3.2), (3.53), and (3.61) gives that
that is, . Therefore, (1.1) possesses uncountably many bounded nonoscillatory solutions in . This completes the proof.
Remark 3.11. Theorems 3.1 and 3.4–3.10 generalize Theorem??1 in [5] and Theorems 2.1–2.7 in [6, 7], respectively. The examples in Section 4 reveal that Theorems 3.1 and 3.4–3.10 extend authentically the corresponding results in [5–7].
4. Examples and Applications
Now, we construct ten examples to explain the advantage and applications of the results presented in Section 3. Note that Theorem??1 in [5] and Theorem ??2.1–2.7 in [6, 7] are invalid for Examples 4.1–4.10, respectively.
Example 4.1. Consider the second-order nonlinear neutral delay difference equation where and are fixed. Let , , , , , , and It is easy to show that (3.1)–(3.3) hold. It follows from Theorem 3.1 that (4.1) possesses uncountably many bounded nonoscillatory solutions in .
Example 4.2. Consider the second-order nonlinear neutral delay difference equation where and are fixed. Let , , , , , , and It is clear that (3.2), (3.3), and (3.21) hold. It follows from Theorem 3.2 that (4.3) possesses uncountably many bounded solutions in .
Example 4.3. Consider the second-order nonlinear neutral delay difference equation where and are fixed. Let , , , , , , , and It is easy to see that (3.2), (3.3), and (3.22) hold. It follows from Theorem 3.3 that (4.5) possesses uncountably many bounded solutions in .
Example 4.4. Consider the second-order nonlinear neutral delay difference equation where and are fixed. Let , , , , , , , and It is easy to show that (3.2), (3.3), and (3.34) hold. It follows from Theorem 3.4 that (4.7) has uncountably many bounded nonoscillatory solutions in .
Example 4.5. Consider the second-order nonlinear neutral delay difference equation where and are fixed. Let , , , , , , , and It is easy to show that (3.2), (3.3), and (3.35) hold. It follows from Theorem 3.5 that (4.9) has uncountably many bounded positive solutions in .
Example 4.6. Consider the second-order nonlinear neutral delay difference equation where and are fixed. Let , , , , , , , and It is easy to show that (3.2), (3.3), and (3.36) hold. It follows from Theorem 3.6 that (4.11) has uncountably many bounded negative solutions in .
Example 4.7. Consider the second-order nonlinear neutral delay difference equation where and are fixed. Let , , , , , , and It is clear (3.2), (3.3), and (3.37) hold. It follows from Theorem 3.7 that (4.13) has uncountably many bounded nonoscillatory solutions in .
Example 4.8. Consider the second-order nonlinear neutral delay difference equation where and are fixed. Let , , , , , , and It is clear (3.2), (3.3), and (3.38) hold. It follows from Theorem 3.8 that (4.15) has uncountably many bounded nonoscillatory solutions in .
Example 4.9. Consider the second-order nonlinear neutral delay difference equation where and are fixed. Let , , , , , and It is clear (3.2), (3.3), and (3.39) hold. It follows from Theorem 3.9 that (4.17) has uncountably many bounded nonoscillatory solutions in .
Example 4.10. Consider the second-order nonlinear neutral delay difference equation where and are fixed. Let , , , , , and It is clear (3.2), (3.50), and (3.51) hold. It follows from Theorem 3.10 that (4.19) possesses uncountably bounded nonoscillatory solutions in .
Acknowledgment
This research is financially supported by Changwon National University in 2011.