The Scientific World Journal

Volume 2014 (2014), Article ID 673647, 10 pages

http://dx.doi.org/10.1155/2014/673647

## PPF Dependent Fixed Point Results for Triangular -Admissible Mappings

^{1}Faculty of Mathematics Engineering, Kraljice Marije 16, 11 000 Belgrade, Serbia^{2}Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia^{3}Young Researchers and Elite Club, Rasht Branch, Islamic Azad University, Rasht, Iran^{4}Università degli Studi di Palermo, Dipartimento di Matematica e Informatica, Via Archirafi 34, 90123 Palermo, Italy

Received 30 August 2013; Accepted 27 October 2013; Published 2 February 2014

Academic Editors: K. Sadarangani and P. Wang

Copyright © 2014 Ljubomir Ćirić et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We introduce the concept of triangular -admissible mappings (pair of mappings) with respect to nonself-mappings and establish the existence of PPF dependent fixed (coincidence) point theorems for contraction mappings involving triangular -admissible mappings (pair of mappings) with respect to nonself-mappings in Razumikhin class. Several interesting consequences of our theorems are also given.

#### 1. Introduction and Preliminaries

The applications of fixed point theory are very important and useful in diverse disciplines of mathematics. In fact, fixed point theory can be applied for solving equilibrium problems, variational inequalities, and optimization problems. In particular, a very powerful tool is the Banach fixed point theorem, which was generalized and extended in various directions: modifying Banach’s contractive condition, changing the space, or extending single-valued mapping to multivalued mapping (see [1–8] and references therein). In 1997, Bernfeld et al. [9] introduced the concept of fixed point for mappings that have different domains and ranges, which is called PPF dependent fixed point or the fixed point with PPF dependence. Furthermore, they gave the notion of Banach type contraction for nonself-mapping and also proved the existence of PPF dependent fixed point theorems in the Razumikhin class for Banach type contraction mappings (also see [10]). The PPF dependent fixed point theorems 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. On the other hand, Samet et al. [11] first introduced the concept of -admissible self-mappings and proved the existence of fixed point results using contractive conditions involving -admissible mappings in complete metric spaces. They also gave some examples and applications of the obtained results to ordinary differential equations. In this paper, we will introduce the concept of triangular -admissible mappings (pair of mappings) with respect to nonself-mappings and establish the existence of PPF dependent fixed point theorems for contraction mappings involving triangular -admissible mappings (pair of mappings) with respect to nonself-mappings in Razumikhin class.

Throughout this paper, we assume that is a Banach space, denotes a closed interval in , and denotes the sets of all continuous -valued functions on equipped with the supremum norm defined by

For a fixed element , the Razumikhin or minimal class of functions in is defined by Clearly, every constant function from to belongs to .

*Definition 1. *Let be the Razumikhin class; then(i)the class is algebraically closed with respect to difference, if when ;(ii)the class is topologically closed if it is closed with respect to the topology on generated by the norm .

*Definition 2 (see [9]). *A mapping is said to be a PPF dependent fixed point or a fixed point with PPF dependence of mapping if for some .

*Definition 3 (see [10]). *Let and . A point is said to be a PPF dependent coincidence point or a coincidence point with PPF dependence of and if for some .

*Definition 4 (see [9]). *The mapping is called a Banach type contraction if there exists such that
for all .

In 2012, Samet et al. [11] introduced the concepts of --contractive and -admissible mappings and established various fixed point theorems for such mappings in complete metric spaces. Afterwards, Karapinar and Samet [12] generalized these notions to obtain fixed point results. More recently, Salimi et al. [13] modified the notions of --contractive and -admissible mappings and established fixed point theorems which are proper generalizations of the recent results in [11, 12].

Samet et al. [11] defined the notion of -admissible mappings as follows.

*Definition 5. *Let be a self-mapping on and let be a function. We say that is an -admissible mapping if

In [11] the authors consider the family of nondecreasing functions such that for each , where is the th iterate of and give the following theorem.

Theorem 6. *Let be a complete metric space and let be an -admissible mapping. Assume that
**
for all , where . Also, suppose that the following assertions hold:*(i)*there exists such that ,*(ii)*either is continuous or for any sequence in with for all and as , one has for all .**Then has a fixed point.*

Salimi et al. [13] modified and generalized the notions of --contractive mappings and -admissible mappings by the following ways.

*Definition 7 (see [13]). *Let be a self-mapping on and two functions. We say that is an -admissible mapping with respect to if
Note that if we take , then this definition reduces to Definition 5. Also, if we take , then we say that is an -subadmissible mapping.

The following result was proved by Salimi et al. [13].

Theorem 8 (see [13]). *Let be a complete metric space and let be an -admissible mapping. Assume that
**
where and
**
Also, suppose that the following assertions hold:*(i)*there exists such that ,*(ii)*either is continuous or for any sequence in with for all and as , one has for all .**Then has a fixed point.*

Recently Karapinar et al. [14] introduced the notion of triangular -admissible mapping as follows.

*Definition 9 (see [14]). *Let and . We say that is a triangular -admissible mapping if

For more details and applications of this line of research, we refer the reader to some related papers [15–21].

Now, motivated by Salimi et al. [13] and Karapinar et al. [14] (see also [15–21]), we introduce the following notion.

*Definition 10. *Let and , . We say that is a triangular -admissible mapping with respect to if, for ,

Note that if we take for all , then we say that is a triangular -admissible mapping. Also, if we take for all , then we say that is a triangular -subadmissible mapping.

*Example 11. *Let be a real Banach space with usual norm and let . Define by for all and by
. Then is a triangular -admissible mapping with respect to . Indeed, if , then and so . That is, which implies . Also, if
then and and so . That is, .

The following lemma is necessary later on.

Lemma 12. *Let be a triangular -admissible mapping with respect to . Define the sequence by the following way:
**
for all , where is such that . Then
*

*Proof. *Since is a triangular -admissible mapping with respect to ,
and so
By continuing this process we get,
Since
then by we get . By continuing this process, we get

*2. Main Results*

*One of our main theorems is a result of Geraghty type [22] obtained by a modification of the approach in [13]. Let denote the class of all functions satisfying the following condition:
*

*Theorem 13. Let , be three mappings satisfying the following assertions:(i)there exists such that is topologically closed and algebraically closed with respect to difference;(ii) is a triangular -admissible mapping with respect to ;(iii)there exists such that
for all ;(iv)if is a sequence in such that as and for all , then for all ;(v)there exists such that .Then, has a dependent fixed point .*

*Proof. *Let such that . Since , there exists such that . Choose such that
By continuing this process, by induction, we can build a sequence in such that,
Hence, from Lemma 12, we have
Since is algebraically closed with respect to difference, it follows that
Then, by (iii), we get
and so
for all . This implies that the sequence is decreasing in . Then, there exists such that . Assume . Now, by taking limit as in (24), we get
which implies . That is,
and since , which is a contradiction. Hence, . That is,

Now, we prove that the sequence is Cauchy in . Assume the contrary; then there exist and two sequences and with such that
From
letting , we get
By triangle inequality, we have

On the other hand, by (iii) and (21), we have

Therefore, we get
which implies
Taking limit as in the above inequality and applying (28) and (31), we get
which implies and since , we deduce
which is a contradiction. Consequently
and hence is a Cauchy sequence in . By the completeness of we get that converges to a point ; that is, as . Since is topologically closed, we deduce . From (iv) we have for all . Then, from (iii) we get
for all . Taking limit as in the above inequality, we get
that is,
which implies that is a dependent fixed point of in .

*If in Theorem 13 we take for all , then we deduce the following corollary.*

*Corollary 14. Let and be two mappings satisfying the following assertions:(i)there exists such that is topologically closed and algebraically closed with respect to difference;(ii) is a triangular -admissible mapping;(iii)there exists such that
for all ;(iv)if is a sequence in such that as and for all , then for all ;(v)there exists such that .Then, has a dependent fixed point .*

*If in Theorem 13 we take for all , then we deduce the following corollary.*

*Corollary 15. Let and be two mappings satisfying the following assertions:(i)there exists such that is topologically closed and algebraically closed with respect to difference;(ii) is a triangular -subadmissible mapping;(iii)there exists such that
for all ;(iv)if is a sequence in such that as and for all , then for all ;(v)there exists such that .Then, has a dependent fixed point .*

*Definition 16. *Let and , , . We say that is a triangular -admissible pair with respect to if, for ,
Note that if we take , then, we say that is a triangular -admissible pair. Also, if we take , then we say that is a triangular -subadmissible pair.

*The following theorem gives a result of existence of dependent coincidence points.*

*Theorem 17. Let , , and be four mappings satisfying the following assertions:(i)there exists such that is algebraically closed with respect to difference;(ii) is a triangular -admissible pair with respect to ;(iii)there exists such that
for all ;(iv)if is a sequence in such that as and for all , then for all ;(v)there exists such that ;(vi) is complete in ;(vii).Then, there exists such that is a dependent fixed point of and hence is a dependent coincidence point of and .*

*Proof. *Let such that . By condition (vii), there exists such that
By continuing this process, by induction, we can build a sequence in such that
Hence, from Lemma 12, we have
Since is algebraically closed with respect to difference, it follows that
Then, by (iii), we get
and so
for all . This implies that the sequence is decreasing in . Then, there exists such that . Assume . Now by taking limit as in (50) we get
which implies . That is,
and since , which is a contradiction. Hence, . That is,

Now, we prove that the sequence is Cauchy in . Assume the contrary; then there exist and two sequences and with such that
From
letting , we get
By triangle inequality, we have

On the other hand, by (iii) and (47), we have

Therefore, we get
which implies
Taking limit as in the above inequality and applying (54) and (57), we get
which implies and since , we deduce
which is a contradiction. Consequently
and hence is a Cauchy sequence in . By the completeness of , there exists such that as . From (iv), we have for all . Then from (iii) we get
for all . Taking limit as in the above inequality, we get
That is,
which implies that is a dependent fixed point of in and hence is a dependent coincidence point of and .

*If in Theorem 17 we take for all , then we deduce the following corollary.*

*Corollary 18. Let , , and be three mappings satisfying the following assertions:(i)there exists such that is algebraically closed with respect to difference;(ii) is a triangular -admissible pair;(iii)there exists such that
for all ;(iv)if is a sequence in such that as and for all , then for all ;(v)there exists such that ;(vi) is complete in ;(vii).Then, and have a dependent coincidence point .*

*If in Theorem 13 we take for all , then we deduce the following corollary.*

*Corollary 19. Let , , and be three mappings satisfying the following assertions:(i)there exists such that is algebraically closed with respect to difference;(ii) is a triangular -subadmissible pair;(iii)there exists such that
for all ;(iv)if is a sequence in such that as and for all , then ;(v)there exists such that ;(vi) is complete in ;(vii).Then, and have a dependent coincidence point .*

*2.1. Consequences of Corollary 14*

*Theorem 20. Let and be two mappings satisfying the following assertions:(i)there exists such that is topologically closed and algebraically closed with respect to difference;(ii) is a triangular -admissible mapping;(iii)there exists such that
for all ;(iv)if is a sequence in such that as and for all , then for all ;(v)there exists such that .Then, has a dependent fixed point .*

*Proof. *Let ; then by (iii) we have
That is, all conditions of Corollary 14 hold and has a dependent fixed point .

*Theorem 21. Let and be two mappings satisfying the following assertions:(i)there exists such that is topologically closed and algebraically closed with respect to difference;(ii) is a triangular -admissible mapping;(iii)there exists such that
for all , where ;(iv)if is a sequence in such that as and for all , then for all ;(v)there exists such that .Then, has a dependent fixed point .*

*Proof. *Let ; then by (iii) we have
which implies . That is, all conditions of Corollary 14 hold and has a dependent fixed point .

*Theorem 22. Let and be two mappings satisfying the following assertions:(i)there exists such that is topologically closed and algebraically closed with respect to difference;(ii) is a triangular -admissible mapping;(iii)there exists such that
for all , where ;(iv)if is a sequence in such that as and for all , then for all ;(v)there exists such that .Then, has a dependent fixed point .*

*Proof. *Let ; then by (iii) we have
which implies . That is, all conditions of Corollary 14 hold and has a dependent fixed point .

*2.2. Consequences of Corollary 15*

*Theorem 23. Let and be two mappings satisfying the following assertions:(i)there exists such that is topologically closed and algebraically closed with respect to difference;(ii) is a triangular -subadmissible mapping;(iii)there exists such that
for all ;(iv)if is a sequence in such that as and for all , then for all ;(v)there exists such that .Then, has a dependent fixed point .*

*Theorem 24. Let and be two mappings satisfying the following assertions:(i)there exists such that is topologically closed and algebraically closed with respect to difference;(ii) is a triangular -subadmissible mapping;(iii)there exists such that
for all , where and ;(iv)if is a sequence in such that as and for all , then for all ;(v)there exists such that .Then, has a dependent fixed point .*

*Theorem 25. Let and be two mappings satisfying the following assertions:(i)there exists such that is topologically closed and algebraically closed with respect to difference;(ii) is a triangular -subadmissible mapping;(iii)there exists such that
for all , where and ;(iv)if is a sequence in such that as and for all , then for all ;(v)there exists such that .Then, has a dependent fixed point .*

*2.3. Consequences of Corollary 18*

*Theorem 26. Let , , and be three mappings satisfying the following assertions:(i)there exists such that is algebraically closed with respect to difference;(ii) is a triangular -admissible pair;(iii)there exists such that
for all ;(iv)if is a sequence in such that as and for all , then for all ;(v)there exists such that ;(vi) is complete in ;(vii).Then, and have a dependent coincidence point .*

*Theorem 27. Let , , and be three mappings satisfying the following assertions:(i)there exists such that is algebraically closed with respect to difference;(ii) is a triangular -admissible pair;(iii)there exists such that
for all where ;(iv)if is a sequence in such that as and for all , then for all ;(v)there exists such that ;(vi) is complete in ;(vii).Then, and have a dependent coincidence point .*

*Theorem 28. Let , , and be three mappings satisfying the following assertions:(i)there exists such that is topologically closed and algebraically closed with respect to difference;(ii) is a triangular -admissible pair;(iii)there exists such that
for all , where ;(iv)if is a sequence in such that as and for all , then for all ;(v)there exists such that ;(vi) is complete in ;(vii)*