Abstract

The aim of this paper to discuss generalized iterative process of F-contraction and establish new fixed point theorems in complete metric spaces. As an application of our results, we prove existence and uniqueness of functional equations and system of differential equations. Our results provide extension as well as substantial generalizations and improvements of several well known results in the existing comparable literature.

1. Introduction

In 2012, Wardowski ²⁹ introduce a new type of contractions called F-contraction and proved new fixed point theorems concerning F-contraction. Afterwards Se-celean ²⁸, proved fixed point theorems consisting of F-contractions by Iterated function systems. He generalized the Banach contraction principle in a different way than as it was done by different investigators.

Cosentino et al. ¹¹ established some fixed point results of Hardy-Rogers-type for self-mappings on complete metric spaces or complete ordered metric spaces. Lately, Acar et al. ¹ introduced the concept of generalized multivalued F-contraction mappings further Altun et al. ² extended multivalued mappings with δ-Distance and established fixed point results in complete metric space. Sgroi et al. ²⁴ established fixed point theorems for multivalued F-contractions and obtained the solution of certain functional and integral equations, which was a proper generalization of some multivalued fixed point theorems including Nadler’s. Thereafter, many papers have published on F-contractive mappings in various spaces. For more detail see ^{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20} and references therein.

Definition 1 ¹⁶ Let be a metric space. A mapping is said to be an F contraction if there exists such that

(1.1)

where is a mapping satisfying the following conditions:

(F1) F is strictly increasing, i.e. for all such that x < y,

(F2) For each sequence of positive numbers, if and only if

(F3) There exists such that

We denote by the set of all functions satisfying the conditions (F1)-(F3).

Example 2 ²⁹ Let be given by the formula It is clear that F satisfied (F1)-(F2)-(F3) for any Each mapping satisfying (1:1) is an F-contraction such that

It is clear that for such that the inequality also holds, i.e. T is a Banach contraction.

Example 3 ²⁹ If then F satis.es (F1)-(F3) and the condition (1.1) is of the form

Remark 4 From (F1) and (1.1) it is easy to conclude that every F-contraction is necessarily continuous.

Wardowski ²⁹ stated a modified version of the Banach contraction principle as follows.

Theorem 5 ²⁹ Let be a complete metric space and let be an F contraction. Then has a unique fixed point and for every the sequence converges to

2. Preliminaries

We recollect some essential notations, required definitions, and primary results coherent with the literature.

Let be a metric space. We denote by the class of all nonempty closed subsets of For a nonempty set we denote by the class of all nonempty subsets of For define a set

The Hausdorff metric H on induced by metric d is given as:

Let and A point in is called a fixed point of if The set of all fixed points of is denoted by Furthermore, a point in is called a coincidence point of and if The set of all such points is denoted by If for some point in we have then a point is called a common fixed point of and We denote set of all common fixed points of and by

Definition 6 Let and Let be an arbitrary but fixed element in A sequence in is called iterative sequence of starting with if Then is called a generalized iterative process of f and T starting at Note that reduces to dynamic process of starting at if (an identity map on ) ¹⁶. The generalized iterative process will simply be written as

Example 7 ¹⁶ Consider a Banach space X = C(I) with a norm where Let be such that for any is a family of the functions where i.e.,

and let Then the sequence is a dynamic process of the operator T starting at

Definition 8 Let be a self map on a metric space A multivalued mapping is called generalized multivalued Ciric type F-contraction with respect to a iterative process if there exist and such that

where

and

Example 9 Let and be defined as for all we obtain a sequence these can be many f iterative sequence of T starting at

Let be an arbitrary point in Then

Then clearly We obtain a generalized iterative process

is called a generalized dynamic process of f and T starting at So you can construct many f iterative sequences of T starting at x0 for different values.

3. Main Result

Throughout this section, we assume that the mapping F is right continuous. In the following we will consider only the dynamic processes satisfying the following condition:

If dynamic processes does not satisfy property then there exists such that and which implies that that is, the set of coincidence point of hybrid pair is nonempty. Under suitable conditions on hybrid pair one obtaines the existence of common fixed point of

Our main result is the following.

Theorem 10 Let and a generalized multivalued Ciric type F- contraction with respect to generalized iterative process Then provided that is complete and F is continuous or T is closed multivalued mapping. Moreover if one of the following conditions holds:

(a) for some f is - weakly commuting at x, (b) is a singleton subset of

Proof. Let be a generalized iterative process of the mapping and starting at We observe that if there exists such that then the existence of a fixed point is obvious. Hence we can assume that for all Since is a generalized multivalued Ciric type F- contraction with respect to a generalized iterative process, it follows that

(2.1)

implies

Implies that

(2.2)

Since F is strictly increasing, therefore

If

for some n, then,

gives a contradiction. So we have

Consequently

(2.3)

for all By definition 8, there exists and such that for all Thus, we obtain

On taking limit as we have By (F1), we get By (F3) there exists an such that

(2.4)

Hence it follows that

On taking limit as tends to we obtain that is, This implies that is convergent and hence the sequence is a Cauchy sequence in so there is such that Suppose that is in such that Next we prove that Indeed, assume the contrary, then because is closed: Since is strictly increasing, we deduce that

for all Therefore

Since from definition 8, we have

(2.5)

for all

Next suppose that F is continuous. Since

we deduce that

so, by continuity of F,

which provides a contradiction. We conclude that and thus Now let (a) holds, that is for is -weakly commuting at So we get By the given hypothesis and hence Consequently (b) Since (say) and this implies that Thus

Example 11 Let and be defined as and d be the usual metric on X. Define and by and for all Then for all we obtain

Thus all conditions of above Theorem 10 is satisfied and 0 is a fixed point of T.

Example 12 Let be the usual metric space. Define and by and for all and and for all Note that so is complete. It is easy to check that for all with (equivalently with ), one has

So we can apply Theorem 10.

Corollary 13 Let be a complete metric space, be an arbitrary point in and a multivalued Ciric type F-contraction with respect to dynamic process either F is continuous or T is closed multivalued mapping. Then there exists a fixed point of T.

(1) Applications

Decision space and a state space are two basic components of dynamic programming problem. State space is a set of states including initial states, action states and transitional states. So a state space is set of parameters representing different states. A decision space is the set of possible actions that can be taken to solve the problem. These general settings allow us to formulate many problems in mathematical optimization and computer programming. In particular the problem of dynamic programming related to multistage process reduces to the problem of solving functional equations

(2.6)

(2.7)

where and are Banach spaces, and and

for more details on dynamic programming we refer to ^{6, 7, 8, 9, 23}. Suppose that and are the state and decision spaces respectively. We aim to give the existence and uniqueness of common and bounded solution of functional equations given in (2.6) and (2.7). Let denotes the set of all bounded real valued functions on For an arbitrary define Then is a Banach space endowed with the metric defined as

(2.8)

Suppose that the following conditions hold:

(C1): and are bounded.

(C2): For and define

(2.9)

(2.10)

Moreover assume that and such that for every and implies

(2.11)

where

(C3): For any there exists such that for

(C4): There exists such that

Theorem 14 Assume that the conditions (C1)-(C4) are satisfied. If is a closed convex subspace of , then the functional equations (2.6) and (2.7) have a unique, common and bounded solution.

Proof. Note that is a complete metric space. By (C1), J, K are self-maps of The condition (C3) implies that It follows from (C4) that J and K commute at their coincidence points. Let be an arbitrary positive number and Choose and such that

(2.12)

where Further from (2.9) and (2.10), we have

(2.13)

(2.14)

Then (2.12) and (2.14) together with (2.11) imply

(2.15)

Then (2.12) and (2.13) together with (2.11) imply

(2.16)

From (2.15) and (2.16), we have

(2.17)

The inequality (2.17) implies

(2.18)

(2.19)

Therefore by Theorem 10, the pair (K, J) has a common fixed point , that is, is unique, bounded and common solution of (2.6) and (2.7).

(1) Application of system of integral equations:

Now we discuss an application of fixed point theorem we proved in the previous section in solving the system of Volterra type integral equations. Such system is given by the following equations:

(2.20)

(2.21)

for where We find the solution of the system (2.20) and (2.21). Let be the space of all continuous functions defined on For define supremum norm as: where is taken as a function. Let be endowed with the metric

(2.22)

for all With these setting becomes Banach space.

Now we prove the following theorem to ensure the existence of solution of system of integral equations. For more details on such applications we refer the reader to ^{3, 21}.

Theorem 15 Assume the following conditions are satisfied:

(i) and are continuous;

(ii) Define

Suppose there exist and such that

for all and where

(iii) there exists such that implies Then the system of integral equations given in (2:20) and (2:21) has a solution.

Proof. By assumption (iii)

This implies

That is

which further implies

So all the conditions of Theorem 10 are satisfied. Hence the system of integral equations given in (2.20) and (2.21) has a unique common solution.

4. Conclusion

This paper presents fixed point theorems for generalized iterative process under the improved notion of dynamic process. The presented theorem provide extension as well as substantial generalizations and improvements of several well known results. The present version of these results make significant and useful contribution in the existing literature.

Conflict of Interests

The authors declare that they have no competing interests.

References