Fixed Point Theory and Applications for Function SpacesView this Special Issue
3D Dynamic Programming Approach to Functional Equations with Applications
This article demonstrates the graphical existence of a single fixed point while imposing the contractive condition of Chatterjea type -contraction on -metric space (briefly as -MS). We present two examples that verify the validity of the results given in the paper. The paper further explains the subsistence of the fixed point even if the contractive condition is valid only for a closed ball inside the space rather than imposing it on the whole -MS. Moreover, the application of the mentioned results in finding a single solution of functional equations is described that is widely used in computer programming and optimization.
1. Introduction and Preliminaries
After Banach presented his renowned Banach Contraction Principle, his idea was generalized by various authors into many interesting generalizations (see [1–8]). Wardowski  extended his idea to a more generalized form which he named as -contraction. An additional strictly increasing function with certain other restrictions was used to modify the Banach theorem. He investigated the fixed point of the contraction and explained the generality of his theorem with the help of concrete examples. This idea was furthered by Klim and Wardowski  into set-valued maps using a dynamic process instead of the ordinary Picard sequence. Later, Nazam et al.  extended Wardowski’s theorem into the form of Kannan’s theorem and hence proved the theorem for noncontinuous maps. He also described that a fixed point for such maps can be iterated even if the contractive inequality holds true only for a subset closed ball of the MS. The notion of -contraction was extended by other authors as well (see [12–16]).
This article relaxes the map by eliminating one of its restrictions, (F3) and hence iterates a fixed point for it. The investigation is carried out for single as well as set-valued maps. Our work is new which extends the preexisting theorems of Wardowski and their consequent results. This paper demonstrates the main idea of this research with the help of graphs which unifies work from the previous research carried out on the topic.
Some basic definitions are given below which will be needed in a sequel.
Definition 1 (see ). Assume is a set of functions satisfying the following conditions:
(F1) is a nondecreasing function, i.e.,
(F2) for any sequence , we have
Definition 2 (see ). Assume a nonempty setand is a map. Assume that there is some such that
(d3) for every for each , and for every with , we have
Then is called an-metric on , while is named as -MS.
Example 3 (see ). Let and defined by for all . Then, is an -MS.
Example 4 (see ). Let and is defined as for all . Then, is an -metric on
Definition 5 (see ). Let . If (i) for some . Then is -convergent to (ii) then the sequence is -Cauchy(iii)For each implies is -convergent. Then, the space is known as -complete
Definition 6 (see ). Let be an -MS. A subset of is said to be -open if, for every , there is some such that , where We say that a subset of is -closed if is -open.
Definition 7 (see ). Let be a nonempty subset of and be an -metric, then, the following statements are equivalent: (i)-closed(ii)For any sequence , we have
Theorem 8 (see ). Assume an -complete -MS, and be a given map. Let there is some such that Then, for at most one. Moreover, for any , the sequence defined by is -convergent to .
Theorem 9 (see ). Assume that is a complete MS with metric , and consider a function such that for all , where are nonnegative numbers satisfying . Then, has a unique fixed point.
Lemma 10 (see ). Let is a Banach space and is a metric defined by Then, is an -MS.
2. Common Fixed Points Results of Reich Type -Contractions
This section of the paper investigates the fixed point of single-valued -contractions for two maps and single map in -MS.
Theorem 11. Assume, is an -complete -MS and are self-mappings. Assume that for nonnegative functionsand with , there is somesuch that with , for all . Then, for some in .
Proof. Choose an arbitrary point and iterate a sequence by
Using (9) and (10), we can write
Using (F1), we have
Hence, for , we have
Using (15), we can write
Since , for any , ∃ some such that
Further, let satisfies (d3) andis fixed. By (F2), there is some such that
By (17) and (18), we write
Using the above equation and (d3), we have
Hence, () is-Cauchy in . Since is -complete, ∃ such that is -convergent to i.e.,
Now, assume that. Then,
By (F1) and letting , we have
which is a contradiction, as is nonnegative. Hence, i.e.,.
Following the same steps, we get . Hence, .
Uniqueness: assume another common fixedof the maps and
Using (F1), we get ( which is a contradiction. Hence,
Example 12. Assume that , and are defined by and On the other hand, we define by One can verify that fulfill conditions (F1) and (F2) and that is an -metric. Assume , then whenever . One can verify that . For , the inequality (9) is true. Furthermore, is a unique point such that .
Choosing in the previous result, a result of Chatterjea type -contraction is obtained.
Corollary 13. Assume, is an -complete -MS and are self-maps. Assume that for , there is some such that with , for all . Then, for a unique in .
Substituting with in Corollary 13, we obtain the following result.
Corollary 14. Assume that is an -complete -MS, and is a self-map. Assume that for , there is some such that with , for all . Then, has at most one fixed point in .
3. Investigation of Fixed Points of-Contractions on -Closed Balls
This section of the paper investigates a fixed point of single-valued -contractions for two maps and single map imposed only on a -closed subset of -MS.
Definition 15. Assume an -MS which is -complete,, and are self-maps, let , be nonnegative functions with Then, is named as Reich type -contraction on if there is some satisfying
Theorem 16. Assume, an -MS which is -complete and is a Reich type -contraction on . Assume and the conditions given below are fulfilled: (a) is -closed(b), for and (c)∃ such as , where
Then, for some in .
Proof. Choose and arbitrary pointand iterate a sequence by
Using mathematical induction, we show that is in for all . By hypothesis
Therefore, . Assume for some . Now, if , then by (33), we can write
From (F1), we have
On the other hand, if Continuity this way, for , we deduce from inequality (37) and (38) that
From (39) and (40), we write
Now, using (41), we have
From (b) and (c), we obtain
Hence by (F1), we deduce that
Therefore, for all . Now for we have by (33)
Using (a) and repeating the steps done in heorem 11, we get to the conclusion that is -Cauchy to a point in Proceeding in a similar way as in heorem 11, we obtain that
Substituting by in the previous theorem, the following result is obtained.
Corollary 17. Assume, , is an -complete -MS and is a self-map. Let such that Assume that and the below conditions are fulfilled: (a) is -closed(b), for and (c)(d), with Then there is a unique in such that
Example 18. Let and. Define by
See Figure 1. Define
One can verify that fulfill conditions (F1) and (F2) and that is and -metric.
On the other hand, define by
Fix , then . Clearly, is -closed hence is satisfied. Now, since , therefore, and , which implies that and
Therefore, condition (b) is obeyed. Moreover, assume , then is satisfied. i.e., and In a similar way, for each , ∃ some and satisfying condition (c). Now checking for condition (d), we have two cases:
Case 19. If , then
Therefore, for all condition (d) is also satisfied.
Case 20. If , e.g., and , then
Hence, condition (b) holds only for and not on . Moreover, is the fixed point of . Given below Figure 1 shows two maps and
As we see in Figure 1, is an increasing function. Therefore, it will not change the inequality, i.e., the right side of the inequality will still be dominant. Note that -axis represents the values of the function , and it can be observed that for every value of and , and hence satisfy the inequality of the above example.
Corollary 21. Assume ,such that and is an -complete -MS. Let are self-maps and Assume that for and the below conditions are fulfilled: (a) is -closed(b), for all (c), for and (d)∃ such as , where
Then for a unique in .
4. Application to Functional Equations
This section discusses the application of our results in finding a common solution of functional equations that are used in dynamic programming.
The study of dynamic programming splits into two parts. A state space is a set of parameters of various states, i.e., initial states, transitional states, and action states. On the other hand, a decision space is a series of actions taking place for finding the possible solution to the indicated problem. The problem of dynamic program is transformed into functional equations: where and are Banach spaces such that and and
Assume and are state space and decision spaces, respectively. Assumedenotes a set of all-bounded real-valued maps on . Letand say. Then, is a Banach space and is the metric defined as
Suppose the following conditions are satisfied:
(): , and are bounded.
(): For and , define by
Observe that the functions , and are bounded hence and are well-defined.
(): For , , and , we have where for such that, where .
Based on the above hypothesis, we present the below theorem.