Abstract
The existence of uncountably many positive solutions and Mann iterative approximations for a nonlinear three-dimensional difference system are proved by using the Banach fixed point theorem. Four illustrative examples are also provided.
1. Introduction
In recent years, the oscillation, nonoscillation, asymptotic behavior, existence and multiplicity of solutions, bounded solutions, unbounded solutions, positive solutions, and nonoscillatory solutions for some two- and three-dimensional difference systems have been studied by many authors and a significant number of important results have been found; see [1–12] and the references therein.
In order to solve the problem that the Picard iteration fails to converge under some conditions, Mann [13] introduced a modified iteration, which is now called Mann iterative scheme. It is well known that the Mann iterative schemes are often used in the fields of nonlinear differential equations, nonlinear equations, nonlinear mappings, optimization, variational inequalities, nonlinear analysis, and so forth.
Note that the difference systems in [1–12] are as follows: where , , and are nonnegative sequences, and is a real sequence with : where , , and are real sequences with , and is a nonnegative sequence: where , , , , , for , and , , , with However, to the best of our knowledge, there exists no result in the literature dealing with the following nonlinear three-dimensional difference system: which is abbreviated as, for convenience, where , , , , , and , with
The main purpose of this paper is to study solvability and convergence of the Mann iterative schemes for the system (6). Sufficient conditions for the existence of uncountably many positive solutions of the system (6) and convergence of the Mann iterative schemes relative to these positive solutions are provided by utilizing the Banach fixed point theorem. Four illustrative examples are given.
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, as Let denote the Banach space of all sequences in with norm and put It is easy to see that for each , is a nonempty closed convex subset of the Banach space with norm for each .
By a solution of the system (6), we mean a three-dimensional sequence with a positive integer such that (6) holds for all .
Let be a nonempty convex subset of a Banach space and let be a mapping. For any given , the sequence defined by where is a sequence in with certain condition, is called the Mann iterative scheme.
Lemma 1. Let be a nonnegative sequence and .(i)If , then (ii)If , then
Proof. It is easy to see that which yields (12). Note that where denotes the largest integral number not exceeding . Hence (13) holds. This completes the proof.
3. Uncountably Many Positive Solutions and Mann Iterative Schemes
Our main results are as follows.
Theorem 2. Assume that there exist constants , and nonnegative sequences and satisfying
Then one has the following.
(a) For any , there exist and + such that, for each , the Mann iterative sequence generated by the scheme
converges to a positive solution of the system (6) with for each and has the following error estimate:
where is an arbitrary sequence in such that
(b) The system (6) has uncountably many positive solutions in .
Proof. Firstly, we prove that (a) holds. Put . It follows from (19) that there exist and satisfying
where
Define mappings and byfor each .
Now we assert that
Using (18), (25), and (26), we get that, for any and ,
which implies (28). Thus (29) follows from (27) and (28).
Next we claim that
for all , . In view of (17), (23), (26), and (27), we deduce that, for any , , and ,which yield (31) and (32). Clearly, (29) and (32) ensure that is a contraction mapping in and it has a unique fixed point , that is,
which guarantee that
which together with (16) means that is a positive solution of the system (6) in . Observe that (18) and (19) give that, for each ,
which implies that
By means of (20), (26), and (31), we infer that
which imply that
that is, (21) holds. It follows from (21) and (22) that .
Secondly, we show that holds. Let , with . As in the proof of , we infer similarly that, for each , there exist constants , and mappings and satisfying (23)(32), where , , , , , , , , , , , and are replaced by , , , , , , , , , , , and , respectively, and the contraction mapping has a unique fixed point , which is a positive solution of the system (6); that is,
On account of (17), (23), and (40), we conclude thatwhich means that
which gives that
that is, . Hence the system (6) possesses uncountably many positive solutions in . This completes the proof.
Theorem 3. Assume that there exist constants , and nonnegative sequences , , and satisfying (17), (18), and (22) as follows:
Then one has the following.
(a) For any , there exist and such that, for each , the Mann iterative sequence generated by the schemeconverges to a positive solution of the system (6) with for each and has the error estimate (21).
(b) The system (6) has uncountably many positive solutions in .
Proof. Firstly, we prove that (a) holds. Set . It follows from (45) that there exist and satisfying
where
Define mappings and by (27) andfor each . It follows from (18), (49), and (50) that, for any and ,
which means (28). It is easy to see that (27) and (28) yield (29). By virtue of (17), (47), and (50), we know that for any , and ,which yield (31) and (32). Thus (29) and (32) guarantee that is a contraction mapping in and it has a unique fixed point , that is,
which mean that
which implies that
which together with (44) ensures that is a positive solution of the system (6) in . Obviously, (18) and (45) yield that
which means that
In light of (31), (46), and (50), we get that
which yield that
that is, (21) holds. It follows from (21) and (22) that .
Secondly, we show that (b) holds. Put , with . Similar to the proof of , we conclude that, for each , there exist constants , , , , + and mappings and satisfying (27)(32) and (47)(50), where , , , , , , , , , , , and are replaced by , , , , ,, , , , , , and , respectively, and the contraction mapping has a unique fixed point , which is a positive solution of the system (6); that is,
Using (17), (47), and (60), we infer that, for each and ,which implies that
which gives that
that is, . Hence the system (6) possesses uncountably many positive solutions in . This completes the proof.
Theorem 4. Assume that there exist constants , and nonnegative sequences , , and satisfying (17), (18), and (22) as follows:
Then one has the following.
(a) For any , there exist and such that, for each , the Mann iterative sequence generated by the schemeconverges to a positive solution of the system (6) with for each and has the error estimate (21).
(b) The system (6) has uncountably many positive solutions in .
Proof. Firstly, we show that (a) holds. Let . Observe that (65) implies that there exist and satisfying
where
Define mappings and by (27) andfor each . It follows from (18), (69), and (70) that, for any ,
which mean (28). Consequently, (29) follows from (27) and (28). By virtue of (17), (67), and (70), we infer that, for any , hh , and ,
which yield (31) and (32). Thus (29) and (32) ensure that is a contraction mapping in and it has a unique fixed point , that is,
which yields that
which implies that
which together with (64) means that is a positive solution of the system (6) in . Note that (18) and (65) give that, for each ,
which means that
Making use of (31), (66), and (70), we get that
which yield that
that is, (21) holds. It follows from (21) and (22) that .
Secondly, we show that (b) holds. Let , with . Similarly we infer that, for each , there exist constants , and mappings and satisfying (27)(32) and (67)(70), where , , , , , , , , , , , and are replaced by , , , , , , , , , , , and , respectively, and the contraction mapping has a unique fixed point , which is a positive solution of the system (6); that is,
Using (17), (67), (70), and (80), we know that, for each ,
which implies that
which gives that
that is, . Hence the system (6) possesses uncountably many positive solutions in . This completes the proof.
Theorem 5. Assume that there exist constants , , and nonnegative sequences , , and satisfying (17), (18), (22), (65), and
Then one has the following.
(a) For any , there exist and such that, for each , the Mann iterative sequence generated by (66) converges to a positive solution of the system (6) with for each and has the error estimate (21).
(b) The system (6) has uncountably many positive solutions in .
Proof. Let . It follows from (65) that there exist and satisfying (67) and Let , , , and be defined by (27) and (70), respectively. It follows from (18), (70), and (85) that, for any , which lead to (28). The rest of the proof is similar to that of Theorem 4 and is omitted. This completes the proof.
Similar to the proofs of Theorems 25, we have the following results and omit their proofs.
Theorem 6. Assume that there exist constants , and nonnegative sequences , , and for each satisfying (17), (18), and (22) as follows:
Then one has the following.
(a) For any , there exist and such that, for each , the Mann iterative sequence generated by the schemeconverges to a positive solution of the system (6) with
and has the error estimate (21).
(b) The system (6) has uncountably many positive solutions in .
Theorem 7. Assume that there exist constants , , and nonnegative sequences , , and for each satisfying (17), (18), (22), (88), and
Then one has the following.
(a) For any × , there exist and such that, for each , the Mann iterative sequence generated by the scheme (89)(91) converges to a positive solution of the system (6) with
and has the error estimate (21).
(b) The system (6) has uncountably many positive solutions in .
Theorem 8. Assume that there exist constants , , and nonnegative sequences , , and for each satisfying (17), (18), and (22) as follows:
Then one has the following.
(a) For any , , there exist and such that, for each , the Mann iterative sequence generated by the scheme (89), (91), andconverges to a positive solution of the system (6) with
and has the error estimate (21).
(b) The system (6) has uncountably many positive solutions in .
Theorem 9. Assume that there exist constants , , and nonnegative sequences , , and for each satisfying (17), (18), and (22) as follows:
Then one has the following.
(a) For any , , there exist and such that, for each , the Mann iterative sequence generated by the scheme (91), (96), andconverges to a positive solution of the system (6) with
and has the error estimate (21).
(b) The system (6) has uncountably many positive solutions in .
4. Examples
Now we suggest four examples to explain the main results in Section 3.
Example 1. Consider the nonlinear three-dimensional difference system where and are fixed. Let , , , , , , , , , and It is clear that (16)(18) are satisfied. Note that Lemma 1 implies that which yield that (19) holds. Thus Theorem 2 implies that the system (102) possesses uncountably many positive solutions in .
Example 2. Consider the nonlinear three-dimensional difference system where and are fixed. Let , , , , , , , , , and Obviously, (17) and (18) are satisfied. Note that Lemma 1 yields that which implies that (45) holds. Thus Theorem 3 implies that the system (105) possesses uncountably many positive solutions in .
Example 3. Consider the nonlinear three-dimensional difference system where and are fixed. Let , , , , , , , , , , , , and It is easy to see that (17), (18), and (64) are satisfied. Note that Lemma 1 yields that which implies that (65) holds. Thus Theorem 4 implies that the system (108) possesses uncountably many positive solutions in .
Example 4. Consider the nonlinear three-dimensional difference system where and are fixed. Let , , , , , , , , , , , , and Clearly, (17), (18), and (84) hold. Observe that Lemma 1 means that that is, (65) holds. Thus Theorem 5 implies that the system (111) possesses uncountably many positive solutions in .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors are indebted to the referee for carefully reading the paper and for making useful comments and suggestions. This research was supported by the Science Research Foundation of Educational Department of Liaoning Province (L2012380).