In this paper, we introduce the concept of -admissible non-self mappings and prove the existence and convergence of the past-present-future (briefly, PPF) dependent fixed point theorems for such mappings in the Razumikhin class. We use these results to prove the PPF dependent fixed point of Bernfeld et al. (Appl. Anal. 6:271-280, 1977) and also apply our results to PPF dependent coincidence point theorems.
MSC: 47H09, 47H10.
Keywords:PPF fixed points; Razumikhin classes; rational type contraction
The applications of fixed point theory are very important and useful in diverse disciplines of mathematics. The theory can be applied to solve many problem in real world, for example: equilibrium problems, variational inequalities and optimization problems. A very powerful tool in fixed point theory is the Banach fixed point theorem or Banach’s contraction principle for a single-valued mapping. It is no surprise that there is a great number of generalizations of this principle. Several mathematicians have gone in several directions modifying Banach’s contractive condition, changing the space or extending a single-valued mapping to a multivalued mapping (see [1-10]).
One of the most interesting results is the extension of Banach’s contraction principle in case of non-self mappings. In 1997, Bernfeld et al. introduced the concept of fixed point for mappings that have different domains and ranges, the so called past-present-future (briefly, PPF) dependent fixed point or the fixed point with PPF dependence. Furthermore, they gave the notion of Banach-type contraction for a non-self mapping and also proved 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. Several PPF dependence fixed point theorems have been proved by many researchers (see [12-15]).
On the other hand, Samet et al. were first to introduce the concept of α-admissible self-mappings and they proved the existence of fixed point results using contractive conditions involving an α-admissible mapping in complete metric spaces. They also gave some examples and applications to ordinary differential equations of the obtained results. Subsequently, there are a number of results proved for contraction mappings via the concept of α-admissible mapping in metric spaces and other spaces (see [17-19] and references therein).
To the best of our knowledge, there has been no discussion so far concerning the PPF dependent fixed point theorems via α-admissible mappings. In this paper, we introduce the concept of -admissible non-self mappings and establish the existence and convergence of PPF dependent fixed point theorems for contraction mappings involving -admissible non-self mappings in the Razumikhin class. Furthermore, we apply our results to the existence of PPF dependent fixed point theorems in  and also apply to PPF dependent coincidence point theorems.
Throughout this paper, E denotes a Banach space with the norm , I denotes a closed interval in ℝ, and denotes the set of all continuous E-valued functions on I equipped with the supremum norm defined by
It is easy to see that the constant function is one of the mapping in . The class is said to be algebraically closed with respect to difference if whenever . Also, we say that the class is topologically closed if it is closed with respect to the topology on generated by the norm .
Definition 2.1 (Bernfeld et al.)
Definition 2.2 (Bernfeld et al.)
Definition 2.3 (Samet et al.)
Then T is α-admissible.
Then T is α-admissible.
Remark 2.6 In the setting of Examples 2.4 and 2.5, every nondecreasing self-mapping T is ß-admissible.
Then T is α-admissible.
Next, we prove the following result for a PPF dependent fixed point.
By repeating the above relation, we get
By repeating this process and by induction, we get
This implies that
We also obtain that
In this section, we show that many existing results in the literature can be deduced from and applied easily to our theorems.
4.1 Banach contraction theorem
By applying Theorems 3.3, 3.4 and 3.5, we obtain the following results.
Proof Let be the mapping defined by for all . Then T is an -admissible mapping. It is easy to show that all the hypotheses of Theorems 3.3, 3.4 and 3.5 are satisfied. Consequently, T has a unique PPF dependent fixed point in . □
4.2 PPF dependent coincidence point theorems
In this section, we discuss some relation between PPF dependent fixed point results and PPF dependent coincidence point results. First, we give the concept of PPF dependent coincidence point.
Now, we indicate that Theorem 3.3 can be utilized to derive a PPF dependent coincidence point theorem.
From (4.3) and condition (c), we have
This implies that ω is a PPF dependent coincidence point of T and S. This completes the proof. □
Similarly, we can apply Theorems 3.4 and 3.5 to the Theorems 4.6 and 4.7. Then, in order to avoid repetition, the proof is omitted.
The authors declare that they have no competing interests.
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
This work was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (under NRU-CSEC project No. NRU56000508).
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