Abstract
The aim of this paper is to introduce some generalized contractions and prove certain new fixed point results for self-mappings satisfying these contractions in the setting of -metric space. As an application of our results, we investigate the problem of dynamic programming related to the multistage process which formulates the problems of computer programming and mathematical optimization. We also provide an example to support the validity of our main results.
1. Introduction
Because of utility and applications of metric fixed point theory in mathematics and related fields like social sciences, physical sciences, computer sciences, and engineering have grown in many directions. More general theorems have been introduced along with the provision of many useful tools to solve problems arising in several diverse areas of research.
Recently, Jlei and Samet [1] initiated a generalized metric space named as -metric space and showed a generalization of the Banach contraction principle. Meanwhile, researchers have picked keen interests in extending results in this generalized metric space; see for instance, [2–5]. In this paper, we define some generalized contractions and establish some results in the context of -metric spaces.
2. Preliminaries
Here, we record some requisites definitions and results for the purpose of the next sections.
Definition 1 (see [1]). Let be the class of functions satisfying these assertions:
,
Example 2. If are defined as. (1)(2)(3)(4)for then
Definition 3 (see [1]). Let , and . Assume that such that
(D1) for all ,
(D2) , for all
(D3) for every , for every , , and with , we have
Then, is called an -metric space.
Example 4 (see [1]). Let be defined by with and . Then, () is an -metric space.
Theorem 5. [1].
Let be an -metric space and . Assume that these conditions are satisfied:
(i) is -complete(ii) such that
Then, such that Furthermore, for any , defined by is -convergent to .
Afterwards, many researchers [2–9] worked in this space.
In this article, we give the notions of twisted -admissible and twisted -rational contractions in the setting of -metric spaces and prove some new theorems.
3. Main Results
In 2012, Samet et al. [10] introduced the concepts of -admissibility mappings and --contraction in complete metric spaces.
Definition 6 (see [10]). Let and . Then, is said to be -admissible if ,
According to Samet et al. [10], represents the class of all nondecreasing functions such that for all , where is the th iterate of .
Lemma 7 (see [10]). If , then (i) as (ii) for all (iii) iff
Now, we give the concept of twisted -admissible in the -metric space as follows.
Definition 8. Let be an -metric space, and . Then, is said to be twisted -admissible if for
Now, we state our main result.
Theorem 9. Let be an -metric space and be twisted -admissible. Suppose that the following assertions are satisfied: (a) is -complete(b)there exists such that and (c) is continuous If any one of these assertions hold:(i) such that(ii) such thatwhere for all , then such that .
Proof. Let such that and Generate in by . If for some , then is a fixed point of . So we suppose that Then as is twisted -admissible, we get implies and By induction, we get and for all Suppose the inequality (7) holds. So with and , we have
where
If , then from (11), we obtain
a contradiction. Hence, ; therefore, (11) becomes
Consequently, we get
Assume inequality (8) holds and such that
which implies that
where
If then from (17), we obtain
a contradiction. Thus, ; therefore, (17) becomes
Consequently, we get
Assume inequality (9) holds and such that
which implies that
where
If then from (23), we obtain
a contradiction. Thus, ; therefore, (23) becomes
Consequently, we get
Let be such that (D3) is satisfied. Let be fixed. By (), such that
Let such that Hence, by (27), , and (), we have
for Using (D3) and (29), we get implies
which implies by () that This shows that is -Cauchy. As is -complete, such that is -convergent to . As is continuous, so we have Thus, such that .
In the next result, we omit the continuity of and use an adjunctive condition on .
Theorem 10. Let be an -metric space and be twisted -admissible. Suppose that the following assertions are satisfied: (a) is -complete(b) such that and (c)If is a sequence in such that and and as then and If any one of these assertions hold: (i) such that(ii) such thatwhere for all , then such that .
Proof. Let such that and Proceeding as in the proof of Theorem 9, we have such that is -convergent to , i.e.,
Suppose that and inequality (31) holds. By and (D3), we have
Similarly, if inequality (32) holds. So such that
which implies that
that is
Also if inequality (33) holds, then such that
that is
Thus, for all cases, by and (D3), we have
for If then
Letting and utilizing () and (35), we get
which implies that a contradiction.
If then
Letting and utilizing () and (35), we get
which implies that a contradiction. Thus, we have , i.e., .
For the uniqueness of the fixed point, we take the following property:
(P) and for all fixed points of
Theorem 11. If we add the property (P) in supposition of Theorem 10,then we get that the fixed point of the mapping is unique.
Proof. Let be such that and such that Then, by hypothesis (P), we have and Suppose (i) holds. Then, a contradiction. Hence, such that . Suppose (ii) holds. Then, there exists such that which implies that a contradiction. Hence, such that . Suppose (iii) holds. Then, there exists such that which implies that a contradiction. Hence, such that .
Corollary 12. Let be an -metric space and be -admissible. Assume that these assertions hold: (a) is -complete(b) such that (c) is continuous or if is a sequence in such that for all and as then If any one of these assertions hold:(i) such that(1)(ii) such that(2)where , then such that .
Proof. Taking for all in Theorem 10.
Corollary 13. Let be an -metric space and be -admissible mapping such that where . Suppose that the following assertions are satisfied: (a) is -complete(b) such that (c) is continuous or if is a sequence in such that for all and as then Then such that
Proof. If only (i) holds in Corollary 12.
Corollary 14. Let be an -metric space and be -admissible mapping such that . Assume that these assertions hold: (a) is -complete(b) such that and (c) is continuous or if is a sequence in such that and and as then and Then, such that
Proof. If only hypothesis (i) holds in Theorem 10 and .
Following is Boyd and Wong type in the setting of -metric space which is a consequence of Corollary 14.
Corollary 15. Let be an -metric space and be a self-mapping such that . Assume that these assertions hold: (a) is -complete(b) is continuous. Then, such that
Proof. Taking for all in Corollary 14.
Example 16. Let and -metric given by Take and Define by Now, we define by Evidently, is twisted - rational contraction of type (i) with and Actually, , we have
All the conditions of Theorem 9 are satisfied. Hence, which is unique
4. Applications in Dynamic Programming
In this section, we now establish the solution of functional equations arising from dynamic programming related to multistage process [11, 12] as an application of Theorem 10. Recall that a dynamic programming problem is a decision-making problem in variables in which the problem being subdivided into subproblems (stages), each being a decision-making problem in one variable only. The decision is the “goodness” of a selected alternative depending on satisfying the optimal policy of the problem. The state of the system at any stage is regarded as the information that links the stages together, such that the optimal decisions for the remaining stages can be made. The state allows us to consider each stage separately and guarantees that the solution is feasible for all the stages. This setting formulates the problems of mathematical optimization and computer programming which are converted into the problems of functional equations where and and and are Banach spaces, precisely, is a state space and is a decision space.
Let represent the set of all bounded real-valued functions defined on . For , consider
Clearly, (,) is a Banach space. We endow with the -metric (with and ) defined by
Evidently, () is a -complete -metric space. To show the existence of a solution of (64), we define by and . Clearly, is well-defined as and are bounded.
Theorem 17. Let is given by (67) and assume that these assertions are satisfied:
Assume that there exists such that
(i), where (ii), where , , , and (iii)if is a sequence in such that for all and as , then for all (iv)there exists such that
Then, functional equation (64) has a unique and bounded solution.
Proof. Note that is a -complete - metric space. Let be an arbitrary positive number and such that , then there exist such that Now, from (68) and (71), it follows easily that Hence, we get Similarly, from (69) and (70), we obtain Therefore, from (73) and (74), we have that implies, Since is arbitrary, then Define Consequently, we have that is, is a twisted -contractive mapping of type (i) with for all . Thus, by Theorem 10, has a fixed point. Thus, all the suppositions of Theorem 10 are satisfied. Therefore, there exists , such that , which is the bounded solution of the functional Equation (64).
5. Conclusion
In the present paper, we have defined the notions of twisted -admissible and twisted -rational contractions in the context of -metric spaces and established some generalized fixed point results. As an application of our main results, we investigate the problem of dynamic programming related to multistage process which reduces to the problem of solving the functional equation.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no competing interests.
Authors’ Contributions
All authors contributed equally and approved the final manuscript.