Abstract
In this paper, we introduce the concept of admissible nonself mappings and prove the existence and convergence of the pastpresentfuture (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. 6:271280, 1977) and also apply our results to PPF dependent coincidence point theorems.
MSC: 47H09, 47H10.
Keywords:
PPF fixed points; Razumikhin classes; rational type contraction1 Introduction
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 singlevalued 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 singlevalued mapping to a multivalued mapping (see [110]).
One of the most interesting results is the extension of Banach’s contraction principle in case of nonself mappings. In 1997, Bernfeld et al.[11] introduced the concept of fixed point for mappings that have different domains and ranges, the so called pastpresentfuture (briefly, PPF) dependent fixed point or the fixed point with PPF dependence. Furthermore, they gave the notion of Banachtype contraction for a nonself mapping and also proved the existence of PPF dependent fixed point theorems in the Razumikhin class for Banachtype 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 [1215]).
On the other hand, Samet et al.[16] were first to introduce the concept of αadmissible selfmappings and they proved the existence of fixed point results using contractive conditions involving an αadmissible 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 αadmissible mapping in metric spaces and other spaces (see [1719] and references therein).
To the best of our knowledge, there has been no discussion so far concerning the PPF dependent fixed point theorems via αadmissible mappings. In this paper, we introduce the concept of admissible nonself mappings and establish the existence and convergence of PPF dependent fixed point theorems for contraction mappings involving admissible nonself mappings in the Razumikhin class. Furthermore, we apply our results to the existence of PPF dependent fixed point theorems in [11] and also apply to PPF dependent coincidence point theorems.
2 Preliminaries
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 Evalued functions on I equipped with the supremum norm defined by
For a fixed element , the Razumikhin or minimal class of functions in is 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.[11])
A point is said to be a PPF dependent fixed point or a fixed point with PPF dependence of the nonself mapping if for some .
Definition 2.2 (Bernfeld et al.[11])
The mapping is called a Banachtype contraction if there exists a real number such that
Definition 2.3 (Samet et al.[16])
Let X be a nonempty set, and . We say that T is an αadmissible mapping if it satisfies the following condition:
Example 2.4 Let . Define and by and
Then T is αadmissible.
Example 2.5 Let . Define and by for all and
Then T is αadmissible.
Remark 2.6 In the setting of Examples 2.4 and 2.5, every nondecreasing selfmapping T is ßadmissible.
Example 2.7 Let . Define and by
and
Then T is αadmissible.
3 PPF dependent fixed point theorems for admissible mappings
First of all, we introduce the concept of admissible nonself mappings.
Definition 3.1 Let and , . We say that T is an admissible mapping if for ,
Example 3.2 Let be real Banach spaces with usual norms and . Define and by and
Next, we prove the following result for a PPF dependent fixed point.
Theorem 3.3Let, be two mappings satisfying the following conditions:
(a) There existssuch thatis topologically closed and algebraically closed with respect to difference.
(d) Ifis a sequence insuch thatasandfor all, then.
If there existssuch that, thenThas a unique PPF dependent fixed pointinsuch that.
Moreover, for a fixedsuch that, if a sequenceof iterates ofTinis defined by
for all, thenconverges to a PPF dependent fixed point ofTin.
Proof Let be a point in such that . 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, by induction, we can construct the sequence in such that
It follows from the fact that is algebraically closed with respect to difference that
Since T is admissible and , we deduce that
By continuing this process, we get for all .
Next, we show that is a Cauchy sequence in . For each , we have
By repeating the above relation, we get
This implies that the sequence is a Cauchy sequence in . By the completeness of , we get that converges to a limit point , that is, . Since is topologically closed, we have .
Now we prove that is a PPF dependent fixed point of T. By (d), we have . From assumption (c), we get
for all . Taking the limit as in the above inequality, we have
and so
This implies that is a PPF dependent fixed point of T in .
Finally, we prove the uniqueness of a PPF dependent fixed point of T in . Let and be two PPF dependent fixed points of T in such that and . Now we obtain that
Since , we get and then . Therefore, T has a unique PPF dependent fixed point in . This completes the proof. □
Theorem 3.4Let, be two mappings satisfying the following conditions:
(a) There existssuch thatis topologically closed and algebraically closed with respect to difference.
(d) Ifis a sequence insuch thatasandfor all, then.
If there existssuch that, thenThas a unique PPF dependent fixed pointinsuch that.
Moreover, for a fixedsuch that, if a sequenceof iterates ofTinis defined by
for all, thenconverges to a PPF dependent fixed point ofTin.
Proof Let be a point in such that . Since , there exists such that . Now, we choose such that
From the fact that , we obtain that there exists such that . Thus, we can choose such that
By continuing this process, we can construct the sequence in such that
By algebraic closedness with respect to difference of , we get
Since T is admissible and , we have
By repeating this process and by induction, we get
Next, we show that is a Cauchy sequence in . For each , we have
This implies that
for all . Repeated application of the above relation yields
This implies that the sequence is a Cauchy sequence in . Since is topologically closed and is complete, we get converges to a limit point , that is, .
Now we show that is a PPF dependent fixed point of T. By (3.3) and assumption (d), we get . From assumption (c), we get
for all . Taking the limit as in the above inequality, we have
This implies that and so is a PPF dependent fixed point of T in .
Finally, we prove the uniqueness of a PPF dependent fixed point of T in . Let and be two PPF dependent fixed points of T in such that and . By assumption (c), we have
and so . Since , we have and hence . Therefore, T has a unique PPF dependent fixed point in . This completes the proof. □
Theorem 3.5Let, be two mappings satisfying the following conditions:
(a) There existssuch thatis topologically closed and algebraically closed with respect to difference.
(d) Ifis a sequence insuch thatasandfor all, then.
If there existssuch that, thenThas a unique PPF dependent fixed pointinsuch that.
Moreover, for a fixedsuch that, if a sequenceof iterates ofTinis defined by
for all, thenconverges to a PPF dependent fixed point ofTin .
Proof For fixed in such that . Here we construct the sequence in .
Since , there exists such that . Choose such that
Since , we can find such that . By the same argument, we can choose such that
By induction, we produce the sequence in such that
We also obtain that
for all since is algebraically closed with respect to difference.
Since T is admissible and , we have
By continuing this process, we get for all .
Next, we show that is a Cauchy sequence in . For each , we have
for all . By repeating this inequality, we have
This implies that the sequence is a Cauchy sequence in .
Since is topologically closed, by the completeness of , we get converges to a limit point , that is, .
Now we prove that is a PPF dependent fixed point of T. Since for all and , by using condition (d), we have . From condition (c), we get
Since the exponential function is a real continuous function, we can take the limit as in the above inequality, and so
This implies that and hence is a PPF dependent fixed point of T in .
Finally, we prove the uniqueness of PPF dependent fixed point of T in . Let and be two PPF dependent fixed points of T in such that and . Now we obtain that
and then . Since , we get and then . Therefore, T has a unique PPF dependent fixed point in . This completes the proof. □
Remark 3.6 If the Razumikhin class is not topologically closed, then the limit of the sequence in Theorems 3.3, 3.4 and 3.5 may be outside of , which may not be unique.
4 Consequences
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.
Theorem 4.1Let, and there exists a real numbersuch that
If there existssuch thatis topologically closed and algebraically closed with respect to difference, thenThas a unique PPF dependent fixed point in.
Moreover, for a fixed, if a sequenceof iterates ofTinis defined by
for all, thenconverges to a PPF dependent fixed point ofTin.
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.
Definition 4.2 Let and . A point is said to be a PPF dependent coincidence point or a coincidence point with PPF dependence of S and T if for some .
Definition 4.3 Let and , , . We say that is an admissible pair if for ,
Remark 4.4 It easy to see that if is an admissible pair and S is an identity mapping, then T is also an admissible mapping.
Now, we indicate that Theorem 3.3 can be utilized to derive a PPF dependent coincidence point theorem.
Theorem 4.5Let, , be three mappings satisfying the following conditions:
(a) There existssuch thatis topologically closed and algebraically closed with respect to difference.
(d) Ifis a sequence insuch thatasandfor all, then.
If there existssuch that, thenSandThave a PPF dependent coincidence pointωinsuch that.
Proof Consider the mapping . We obtain that there exists such that and is onetoone. Since , we can define a mapping by
for all . Since is onetoone, then is well defined.
From (4.3) and condition (c), we have
for all . This shows that satisfies condition (c) of Theorem 3.3.
Now, we use Theorem 3.3 with a mapping , then there exists a unique PPF dependent fixed point of , that is, and . Since , we can find such that . Therefore, we get
and
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.
Theorem 4.6Let, , be three mappings satisfying the following conditions:
(a) There existssuch thatis topologically closed and algebraically closed with respect to difference.
(d) Ifis a sequence insuch thatasandfor all, then.
If there existssuch that, thenSandThave a PPF dependent coincidence pointωinsuch that.
Theorem 4.7Let, , be three mappings satisfying the following conditions:
(a) There existssuch thatis topologically closed and algebraically closed with respect to difference.
(d) Ifis a sequence insuch thatasandfor all, then.
If there existssuch that, thenSandThave a PPF dependent coincidence pointωinsuch that.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Acknowledgements
This work was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (under NRUCSEC project No. NRU56000508).
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