On Sufficient Conditions for the Existence of Past-Present-Future Dependent Fixed Point in the Razumikhin Class and Application

Abstract

We introduce the new type of nonself mapping and study sufficient conditions for the existence of past-present-future (for short PPF) dependent fixed point for such mapping in the Razumikhin class. Also, we apply our result to prove the PPF dependent coincidence point theorems. Finally, we use PPF dependence techniques to obtain solution for a nonlinear integral problem with delay.

1. Introduction

It is well known that many problems in many branches of mathematics, such as optimization problems, equilibrium problems, and variational problems, can be transformed to fixed point problem of the form for self-mapping defined on framework of metric space or vector space . Therefore, the applications of fixed point theory are very important in diverse disciplines of mathematics. The famous Banach’s contraction mapping principle is one of the cornerstones in the development of fixed point theory. From inspiration of this work, several researchers heavily studied this field. For example, see works of Kannan [1], Chatterjea [2], Berinde [3], Ćirić [4], Geraghty [5], Meir and Keeler [6], Suzuki [7], Mizogushi and Takahashi [8], Dass and Gupta [9], Jaggi [10], Lou [11], and so forth.

On the other hand, Bernfeld et al. [12] introduced the concept of Past-Present-Future (for short PPF) dependent fixed point or the fixed point with PPF dependence which is one type of fixed points for mappings that have different domains and ranges. They also established the existence of PPF dependent fixed point theorems in the Razumikhin class for Banach type contraction mappings. These results are useful for proving the solutions of nonlinear functional differential and integral equations which may depend upon the past history, present data, and future consideration. The generalizations of this result have been investigated heavily by many mathematicians (see [13–18] and references therein).

In this paper, we will introduce the new type of nonself mapping called Ciric-rational type contraction mapping. Also, we will study the sufficient conditions for the existence of PPF dependent fixed point theorems for such mapping in Razumikhin class. Furthermore, we apply the main result to the existence of PPF dependence coincidence point theorems. In the last section, an application to an integral problem with delay is also given.

2. Preliminaries

In this section, we recall some concepts and definitions that will be required in the sequel. Throughout this paper, let denote a Banach space with the norm , denote a closed interval in , and denote the set of all continuous -valued functions on equips with the supremum norm defined by

A point is said to be a PPF dependent fixed point or a fixed point with PPF dependence of a nonself mapping if for some .

For a fixed element , the Razumikhin or minimal class of functions in is defined by It is easy to see that constant functions are member of .

The class is algebraically closed with respect to difference if whenever . Similarly, is topologically closed if it is closed with respect to the topology on generated by the norm .

Definition 1 (see Bernfeld et al. [12]). The mapping is said to be Banach type contraction if there exists a real number such that for all .
The following PPF dependent fixed point theorem is proved by Bernfeld et al. [12].

Theorem 2 (see Bernfeld et al. [12]). Let be a Banach type contraction. If is topologically closed and algebraically closed with respect to difference, then has a unique PPF dependent fixed point in .

3. PPF Dependent Fixed Point Theorems

In this section, we introduce the concept of the Ciric-rational type contraction mappings. Also, we study sufficient condition for the existence of PPF dependent fixed point for such mapping.

Definition 3. The mapping is called Ciric-rational type contraction if there exist real numbers with and such that for all .

Remark 4. (i) All Banach type, Kannan type, and Chatterjea type mappings are Ciric-rational type contraction mapping.
(ii) If , then Ciric-rational type contraction mapping reduces to Ciric-type contraction.
(iii) If , then is a generalization and improvement of rational type contraction mapping.
Here, we prove PPF dependent fixed point theorems for Ciric-rational type contraction mappings.

Theorem 5. Let be a Ciric-rational type contraction mapping. If is topologically closed and algebraically closed with respect to difference, then has a unique PPF dependent fixed point in .
Moreover, for a fixed , if a sequence of iterates of in is defined by for all , then converges to a PPF dependent fixed point of in .

Proof. Let be an arbitrary function in . Since , there exists such that . Choose such that Since and by hypothesis, we get . This implies that there exists such that . Thus, we can choose such that By continuing this process, we can construct the sequence in such that for all . Since is algebraically closed with respect to difference, we have for all .
Next, we will show that is a Cauchy sequence in .
For each , we have For fixed , if , then we get This implies that On the other hand, if , then we get This implies that Now, we let From (12) and (14), we get for all . Repeated application of the above relation yields for all .
For with , we obtain that Since , we have . This shows that the sequence is a Cauchy sequence in . By the completeness of , we get converges to a limit point . Therefore, ; that is, Further, since is topologically closed, we have and thus
Now we prove that is a PPF dependent fixed point of . From the assumption of Ciric-rational type contraction of , we get for all . Taking the limit as in the above inequality, by (19) and (20), we have This implies that and then Therefore, is a PPF dependent fixed point of in .
Finally, we prove the uniqueness of PPF dependent fixed point of in . Let and be two PPF dependent fixed points of in . Therefore, Since , we have and hence . Therefore, has a unique PPF dependent fixed point in . This completes the proof.

Remark 6. If the Razumikhin class is not topologically closed, then the limit of the sequence in Theorem 5 may be outside of . Therefore, a PPF dependent fixed point of may not be unique.

By applying Theorem 5, we obtain the following result.

Corollary 7. Let be a nonself mapping and there exists a real number such that for all .
If there exists such that is topologically closed and algebraically closed with respect to difference, then has a unique PPF dependent fixed point in .
Moreover, for a fixed , if a sequence of iterates of in is defined by for all , then converges to a PPF dependent fixed point of in .

If we set in Theorem 5, we get the PPF dependent fixed point result for Ciric-type contraction mapping.

Corollary 8. Let be a nonself mapping and there exist real number and such that for all .
If is topologically closed and algebraically closed with respect to difference, then has a unique PPF dependent fixed point in .
Moreover, for a fixed , if a sequence of iterates of in is defined by for all , then converges to a PPF dependent fixed point of in .

If we set in Theorem 5, we get the PPF dependent fixed point result for generalized ratio type contraction mapping.

Corollary 9. Let be a nonself mapping and there exist real numbers with and such that for all .
If is topologically closed and algebraically closed with respect to difference, then has a unique PPF dependent fixed point in .
Moreover, for a fixed , if a sequence of iterates of in is defined by for all , then converges to a PPF dependent fixed point of in .

4. PPF Dependent Coincidence Point Theorems

Definition 10. Let and be two nonself mappings. A point is said to be a PPF dependent coincidence point or a coincidence point with PPF dependence of and if for some .

Next, we introduce the condition of the Ciric-rational type contraction for a pair of two nonself mappings.

Definition 11. Let and be two nonself mappings. The ordered pair is said to satisfy the condition of Ciric-rational type contraction if there exist real numbers with and such that for all .

Remark 12. It is easy to see that satisfies the condition of Ciric-rational type contraction and is identity mapping   is a Ciric-rational type contraction mapping.

Now, we apply our result to the previous section to the PPF dependent coincidence point theorem.

Theorem 13. Let and be two nonself mappings. Suppose that the following conditions hold:      satisfies the condition of Ciric-rational type contraction;    .
If is topologically closed and algebraically closed with respect to difference, then and have a PPF dependent coincidence point.

Proof. For self-mapping , it is well know that there exists such that and is one-to-one. Since we can define a nonself mapping by for all . Since is one-to-one mapping, we have is well-defined.
By the condition of Ciric-rational type contraction of and the construction of , we get for all . This implies that is a Ciric-rational type contraction mapping.
Using Theorem 5 with a mapping , we can find a unique PPF dependent fixed point of . Let a unique PPF dependent fixed point of be ; that is, . Since , we can find such that . Now, we have Therefore, is a PPF dependent coincidence point of and . This completes the proof.

By applying Theorem 13, we obtain the following corollaries.

Corollary 14. Let and be two nonself mappings. Suppose that the following conditions hold:    there exists a real number such that  for all ;   there exists such that .
If is topologically closed and algebraically closed with respect to difference, then and have a PPF dependent coincidence point in .

Corollary 15. Let and be two nonself mappings. Suppose that the following conditions hold:     there exist real numbers and such that  for all ;    .If is topologically closed and algebraically closed with respect to difference, then and have a PPF dependent coincidence point in .

Corollary 16. Let and be two nonself mappings. Suppose that the following conditions hold:     there exist real numbers with and such that  for all ;    .
If is topologically closed and algebraically closed with respect to difference, then and have a PPF dependent coincidence point in .

5. Application to a Nonlinear Integral Equation

In this section, we apply our result to study the existence and uniqueness of solution of a nonlinear integral equation.

Given a closed interval such that , let denote the space of continuous real-valued functions defined on . We equip the space with supremum normed defined by It well known that is a Banach space with this normed.

For fixed , for each , define a function by where the argument represents the delay in the argument of solutions.

Given , we will consider the following nonlinear integral problem: for all , where , , and .

Theorem 17. Problem (42) has only one solution defined on if the following conditions hold:       there exist nonnegative real number such that, for all and , one has

Proof. Define the following set: Also, define the normed in by We obtain that . Next, we show that is complete. Consider a Cauchy sequence in . It is easy to see that is a Cauchy sequence in for all . This implies that is a Cauchy sequence in for each . So converges to for each . Since is a sequence of uniformly continuous functions for a fixed , is also continuous in . Thus going backwards we get that converges to in . Therefore, is complete.
Next, we define the function by For , we have This implies that is a Ciric-rational type contraction.
Moreover, the Razumikhin is which is topologically closed and algebraically closed with respect to difference. Now all hypotheses of Theorem 5 are automatically satisfied with . Therefore, there exists PPF dependence fixed point of ; that is, . This implies that Hence, the integral equation (42) has a solution. This completes the proof.

Conflict of Interests

The authors declare that they have no competing interests.

Acknowledgment

The first author gratefully acknowledges the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) during this research.