## Existence and Uniqueness of Fixed Point in Various Abstract Spaces and Related Applications

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Ljiljana Gajić, Mila Stojaković, Biljana Carić, "On Angrisani and Clavelli Synthetic Approaches to Problems of Fixed Points in Convex Metric Space", *Abstract and Applied Analysis*, vol. 2014, Article ID 406759, 5 pages, 2014. https://doi.org/10.1155/2014/406759

# On Angrisani and Clavelli Synthetic Approaches to Problems of Fixed Points in Convex Metric Space

**Academic Editor:**Poom Kumam

#### Abstract

The purpose of this paper is to prove some fixed point results for mapping without continuity condition on Takahashi convex metric space as an application of synthetic approaches to fixed point problems of Angrisani and Clavelli. Our results are generalizations in Banach space of fixed point results proved by Kirk and Saliga, 2000; Ahmed and Zeyada, 2010.

#### 1. Introduction and Preliminaries

It is well-known that continuity is an ideal property, while in some applications the mapping under consideration may not be continuous, yet at the same time it may be “not very discontinuous.”

In [1] Angrisani and Clavelli introduced regular-global-inf functions. Such functions satisfy a condition weaker than continuity, yet in many circumstances it is precisely the condition needed to assure either the uniqueness or compactness of the set of solutions in fixed point problems.

*Definition 1. *Function , defined on topological space , is regular-global-inf (r.g.i.) in if implies that there exist an such that and a neighbourhood such that for each . If this condition holds for each , then is said to be an r.g.i. on .

An equivalent condition to be r.g.i. on metric space for is proved by Kirk and Saliga.

Proposition 2 (see [2]). *Let be a metric space and . Then is an r.g.i. on if and only if, for any sequence , the conditions
**
imply .*

One of the basic results in [1] is the following one. (Here we use to denote the usual Kuratowski measure of noncompactness on metric space and for .)

Theorem 3 (see [1]). *Let be an r.g.i. defined on a complete metric space . If , then the set of global minimum points of is nonempty and compact.*

*Remark 4. *The last theorem assures that if is a mapping of compact metric space into itself with , and if , is an r.g.i. on , then the fixed point set of is nonempty and compact even when is discontinuous.

*Example 5. *Let be a complete metric space and a mapping such that, for some and all ,
(iri quasi-contraction). Then is discontinuous and , , is r.g.i. (see [1]).

Let be a bounded subset of metric space . The* Kuratowski measure of noncompactness * means the of numbers such that can be covered by a finite number of sets with a diameter less than or equal to . With we are going to denote the* Hausdorff measure of noncompactness*, where is the infimum of numbers such that can be covered by a finite number of balls of radii smaller than .

It is easy to prove that for and bounded subsets (1) is totally bounded;(2);(3);(4).

Moreover, these two measures of noncompactness are equivalent in the sense that so if and only if (for any sequence of bounded subsets of ). The last property indicates that fixed point results are independent of choice of measure of noncompactness.

In Banach spaces this function has some additional properties connected with the linear structure. One of these is ( is a convex hull of —the intersection of all convex sets in containing ).

This property has a great importance in fixed point theory. In locally convex spaces this is always true, but when topological vector space is not locally convex it need not be true (see [3]).

In the absence of linear structure the concept of convexity can be introduced in an abstract form. In metric spaces at first it was done by Menger in 1928. In 1970 Takahashi [4] introduced a new concept of convexity in metric space.

*Definition 6 (see [4]). *Let be a metric space and a closed unit interval. A mapping is said to be* convex structure* on if for all ,
together with a convex structure is called a* (Takahashi) convex metric space * or abbreviated TCS.

Any convex subset of a normed space is a convex metric space with .

*Definition 7 (see [4]). *Let be a TCS. A nonempty subset of is said to be convex if and only if whenever and .

Proposition 8 (see [4]). *Let be a TCS. If and , then *(a)* and ;*(b)*;*(c)* and ;*(d)*balls (either open or closed) in are convex;*(e)*intersections of convex subsets of are convex.*

For fixed let .

*Definition 9. *A TCS has* property * if for every ,

Obviously in a normed space the last inequality is always satisfied.

*Example 10 (see [4]). *Let be a linear metric space with the following properties: (1), for all ;(2), for all and .

For is a TCS with property .

*Remark 11. *Property implies that convex structure is continuous at least in first two variables which gives that the closure of convex set is convex.

*Definition 12. *A TCS has* property * if for any finite subset conv is a compact set.

*Example 13 (see [4]). *Let be a compact convex subset of Banach space and let be the set of all nonexpansive mappings on into itself. Define a metric on by and by , for and . Then is a compact TCS, so is with property . The property is also satisfied.

Talman in [5] introduced a new notion of convex structure for metric space based on Takahashi notion—the so called* strong convex structure* (SCS for short). In SCS condition is always satisfied so it seems to be “natural.”

Any TCS satisfying and has the next important property.

Proposition 14 (see [5]). *Let be a TCS with properties and . Then for any bounded subset *

Some, among the many studies concerning the fixed point theory in convex metric spaces, can be found in [6–13].

#### 2. Main Results

Measures of noncompactness which arise in the study of fixed point theory usually involve the study of either condensing mappings or -set contractions. Continuity is always implicit in the definitions of these classes of mappings. Kirk and Saliga [2] show that in many instances it suffices to replace the continuity assumption with the weaker r.g.i. condition. We are going to follow this idea in frame of TCS.

Theorem 15. *Let be a complete TCS with properties and , a closed convex bounded subset of , and a mapping satisfying the following:*(i)* for any nonempty closed convex -invariant subset of , where ;*(ii)* for all for which ;*(iii)* is r.g.i. on .** Then the fixed point set of is nonempty and compact.*

*Proof. *Choose a point . Let denote the family of all closed convex subsets of for which and . Since , . Let
Convex structure has property so is a convex set as a closure of convex set. We are going to prove that .

Since is a closed convex set containing and . This implies that so and hence . The last two statements clearly force .

Properties (1)–(4) of measure and Proposition 14 imply that
so in view of (ii) must be compact.

Now, Proposition 2 ensures that has a fixed point on so is nonempty. Condition (ii) implies that is totally bounded. Since is r.g.i. has to be closed. Finally, we conclude that is compact.

The assumption is strong, especially in the absence of conditions which at the same time imply continuity. So we are going to give some sufficient conditions which are easier to check and more suitable for application.

Let us recall some well-known definitions. A mapping is called* nonexpansive* if , for all , and* directionally nonexpansive* if for each and . If there exists such that this inequality holds for , then we say that is* uniformly locally directionally nonexpansive*.

Proposition 16. *Let be a complete TCS with property , a closed convex bounded subset of , and a uniformly locally directionally nonexpansive. Let . For the fixed , sequences and are defined as follows:
**
Then for each *

*Proof. *We prove (10) by induction on . For inequality (10) is trivial. Assume that (10) holds for given and all .

In order to prove that (10) holds for , we proceed as follows: replacing with in (10) yields
Also
since and is uniformly locally directionally nonexpansive. Combining (12) and (13)
By Proposition 8 (c),
so
On the other hand,
for any , meaning that is a decreasing sequence.

Now, using inequality , we have that
Thus (10) holds for , completing the proof of inequality.

Further, the sequence is decreasing, so there exists . Let us suppose that . Select positive integer , and , satisfying . Now choose positive integer such that
Using (10), we obtain

By the last contradiction we conclude that and what we had to prove.

*Remark 17. *This statement is a generalization of Lemma 9.4 from [14].

Combining the last result with Theorem 15 we have the following consequence.

Corollary 18. *Let be a bounded closed convex subset of complete TCS with properties and and let satisfy the following: *(i)* is uniformly locally directionally nonexpansive on ;*(ii)*, for all for which ;*(iii)* is r.g.i. on .** Then the fixed point set of is nonempty and compact.*

Moreover, using Proposition 16 we also get generalizations of some other Kirk and Saliga [2] fixed point results.

Corollary 19. *Let be a bounded closed convex subset of a complete TCS with properties and and let satisfy the following: *(i)* is uniformly locally directionally nonexpansive on ;*(ii)*, where is any function for which .** Then has a unique fixed point if and only if is an r.g.i. on .*

*Proof. *Proposition 16 gives and as in [2] one can prove that . By Theorem 1.2 [1], has a unique fixed point if and only if is r.g.i. on .

Theorem 20. *Let be a bounded closed convex subset of a complete TCS with properties and and suppose satisfies the following: *(i)* is directionally nonexpansive on ;*(ii)*, for some and all ;*(iii)* is an r.g.i. on .** Then the fixed point set of is nonempty and compact. Moreover, if satisfies , then .*

*Proof. *By Proposition 16, . Since (i) implies that
the conclusion follows immediately from Theorem 2.3 [2].

We established that for every sequence defined by , , where and . Therefore meaning that converges to the set , but the convergence to the specific point from is not provided. Putting some additional assumption, we could arrange that the sequence converges to a fixed point of the mapping .

Next, we recall the concept of* weakly quasi-nonexpansive* mappings with respect to sequence introduced by Ahmed and Zeyada in [15].

*Definition 21 (see [15]). *Let be a metric space and let be a sequence in . Assume that is a mapping with satisfying . Thus, for a given there exists such that for all . Mapping is called weakly quasi-nonexpansive with respect to if for each there exists such that, for all with .

The next result is improvement of Theorem 20 and also a generalisation of Theorem 2.24 from [15].

Theorem 22. *Let be a bounded closed convex subset of a complete TCS with properties and and let satisfy the following: *(i)* is directionally nonexpansive on ;*(ii)* for some and all ;*(iii)* is r.g.i. on ;*(iv)* satisfies and is weakly quasi-nonexpansive with respect to .** Then converges to a point in .*

*Proof. *Our assertion is a consequence of Theorem 20 and Theorem 2.5(b) from [15].

Using Proposition 16, the next corollary holds.

Corollary 23. *Let be a bounded closed convex subset of a complete TCS with properties and and let satisfy the following: *(i)* is directionally nonexpansive on ;*(ii)* for some and all ;*(iii)* is r.g.i. on ;*(iv)* is weakly quasi-nonexpansive with respect to sequence .** Then converges to a point in .*

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The authors are very grateful to the anonymous referees for their careful reading of the paper and suggestions which have contributed to the improvement of the paper. This paper is partially supported by Ministarstvo nauke i ivotne sredine Republike Srbije.

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#### Copyright

Copyright © 2014 Ljiljana Gajić 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.