Best Proximity Point for Generalized and -Geraphty Contractions

Abstract

This paper introduces a new class of mappings called -Geraphty-contractions and provides sufficient conditions for the existence and uniqueness of a best proximity point for such mappings. It also presents the best proximity point result for generalized contractions as well. Our results extend and generalize some theorems in the literature.

1. Introduction and Preliminaries

The center of interest of fixed point theory is the solving of the equation where is a mapping defined on a subset of a metric space, a normed linear space, or a topological vector space. Ever since its appearance, the well-known Banach contraction principle has been extensively studied, and the literature contains numerous interesting extensions and generalizations of the aforementioned result, in particular, Geraphty’s generalization of the Banach contraction principle.

In 1973, Geraphty ([1]) introduced the class of functions satisfying the following condition:

And proved the following result.

Theorem 1. Letbe a complete metric space andbe an operator satisfying the the following inequality for some

Then, has a unique fixed point.

Since contains the class of constant functions the previous theorem extends that of Banach.

Another interesting extension of the Banach contraction principle is due to Kirk et al. ([2]). The authors introduced the class of cyclic mappings, i.e., such that and And, under a suitable condition on , proved a fixed point theorem which extends that of Banach. Interestingly, a more important problem than the extension of the Banach principle arose.

Whereas a cyclic mapping does not necessarily have a fixed point, it is desirable to determine an element which is somehow closest to More precisely, an element for which the error assumes the least possible value where such a point is called a best proximity point of the cyclic mapping Since 2003, research on best proximity points of cyclic mapping became an important topic in nonlinear analysis and has been studied by many authors [2–8].

In 2012 Caballero and introduced the following contraction.

Definition 2 (see [9]). Letbe nonempty subsets of a metric spaceA nonself mappingis said to be a Geraphty-contraction if there exists asuch that

Just like cyclic mappings, nonself mappings may not have fixed points and a best proximity point for a nonself mapping is a point such that

Later on, the current authors ([10]) introduced the notion of tricyclic mappings and the best proximity point thereof. Let , and be nonempty subsets of a metric space A mapping is said to be tricyclic provided that and . A best proximity point of is a point such that where the mapping is defined by and

Some results about the best proximity points of tricyclic mappings can be found in [10–15].

In the next section of this paper, taking inspiration from our recent works, we introduce the new class of mappings, which stands somewhere between the two classes of cyclic and tricyclic mappings. We define Geraphty-contractions and establish a best proximity point theorem for such mappings. As a special case, we obtain the best proximity point and fixed point theorem for cyclic mapping.

To describe our results, we need some definitions and notations. Given a triad of nonempty subsets of a metric space then, the proximal pair of is given by:

A pair is said to be in if We subsequently use the following notations:

Note that is included in but the inverse does not always hold, and it is obvious that Let us illustrate the cases and with simple examples. (i)Consider , and Clearly and For all we have

Example 3. Letbeendowed with its Euclidean distance.

Hence, and so (ii)Let and Then and

Let

And then, (iii)In the special case where we get and

Let and be defined as

Clearly, , and

Definition 4. Let, andbe nonempty subsets of a metric spaceA mappingis said to be aifandA best proximity point foris a pointprovided that

Definition 5. Letbe nonempty subsets of a metric spaceAmappingis said to be a-Geraphty-contraction if there exists asuch that:

Notice that since the is strictly smaller than one, we have

In the special case where we obtain which means

Thus, every -Geraphty-contraction is continuous on

Example 6. Consider a mappingsuch thatandSupposeis Geraphty-contraction for some nondecreasingWe havefor all Then Which means, is a -Geraphty-contraction.

Definition 7 (see [16]). Letandbe nonempty subsets of a metric spaceThe pairis said to have theif and only ifwhere and

The concept of of pairs can naturally be extended to triads.

Definition 8. Let, andbe nonempty subsets of a metric spaceThe triadis said to have theif and only ifwhere , , and

Example 9. Let, , andsuch that(1)Suppose We necessarily haveIt is plain to see that if the pairs , and have the then, the triad does as well. (2)If thenWhich implies , and

Thus, the triad has the if the pair does.

2. -Geraphty-Contractions

Next, we state the main result of this section.

Theorem 10. Letbe a triad of nonempty closed subsets of a complete metric spacesuch thatis nonempty andLetbe a-Geraphty-contraction satisfyingandAssume the triadhas the-property. Then, there exists a uniquesuch that

Proof. Let since and There then exists such that By the same argument, we obtain some for which Keeping on this process, we can get a sequence in satisfying Since has the -property, we have Taking into consideration that is a -Geraphty-contraction, we have In the case where for some we have Hence, Consequently And the desired result then follows. Therefore, we suppose the contrary case, that is, for every We note that is a decreasing sequence of positive real numbers, hence, there exists such that Suppose we have Which implies that And since we get and this contradicts our assumption. Thus,
In the sequel, we show that is a Cauchy sequence. Notice that for any and since satisfies the -property, Suppose is not a Cauchy sequence, hence, We have Which means Since and . Then, by passage to the limit in the last inequality, we obtain Therefore, But, thus, which is contradictory with our assumption. Consequently is a Cauchy sequence.
Since and is a close subset of the complete metric space Since is continuous and Which implies, On the other hand, is a constant sequence with the value Thus That means is a best proximity point of . As for the uniqueness, suppose that and are two distinct best proximity points of That is Taking into account that has the -property, we get Which is contradictory with the fact that is a -Geraphty-contraction. Indeed, we have And the proof is completed.

Example 11. Considerwith its usual metric. Let, andbe defined by

It is easy to see that and Hence, is nonempty and

Let and let be the mapping defined as

Let

Since the constant functions where belong to is a -Geraphty-contraction. Also, has the -property. Indeed, if

Then , , and Thus, All things considered, has a unique best proximity point, clearly

Corollary 12. Letbe a pair of nonempty closed subsets of a complete metric spacesuch thatis nonempty.Letbe a-Geraphty-contraction satisfyingandAssume the triadhas the-property. Then there exists a uniquethat is both a best proximity point for the cyclic mappingand a fixed point for the self-mapping

Proof. The triad satisfies the condition of the previous theorem. Hence, there exists a unique such that , which implies and that is the wanted result.

3. -Min–Max Condition and Generalized Contractions

In this section, using the -min–max condition, we obtain a best proximity point result for nonself generalized contractions. First, we recall and fix some notions and notations that will subsequently be used. Given mappings , and where , and are nonempty subsets of a metric space .

Definition 13 (see [6]). Letbe nonempty subsets of a metric spaceThe mappingis called a generalized contraction if, given real numbersandwith, there exists a real numbersuch thatfor all in .

Obviously, every generalized contraction is a contractive mapping.

Definition 14 (see [17]). Letbe nonempty subsets of a metric spaceIt is stated that the pairsatisfies the min-max condition if, for alland, we havewhere min and max are defined as

Definition 15. Letbe nonempty subsets of a metric spaceThe triadis said to satisfy-min–max condition if for all, , and, where

Now, we are at liberty to state the main result of this section.

Theorem 16. Suppose, andare nonempty closed subsets of a metric spaceand the mappings, andverify the following conditions:

For a fixed element in , let

Then, the sequence must converge to a best proximity point of and the sequence must converge to a point such that

Proof. Let a sequence of real numbers be defined as follows: From the fact that is a generalized contraction and from the nonexpansiveness of and , it follows that is a bounded below, decreasing sequence of nonnegative real numbers and hence converges to some nonnegative real number, say We show that must be nil. If not, then we choose a positive integer such that Therefore, Similarly, we get Keeping on that process, we obtain When , we deduce that and this is a contradiction. Hence, it can be deduced that as . Next, we shall prove that is a Cauchy sequence. Let be given. Since , it is possible to choose a positive integer such that It is sufficient to prove that whenever is an element of the closed ball then so is If satisfies then And, if satisfies then We conclude that for all and that means is a Cauchy sequence. Taking the completeness of the space under consideration, and from the continuity of , and , we get for some and Then, we deduce that , and We also have And since the triad satisfies the -min–max condition, we obtain Then, we necessarily have And that finishes the proof.

As a special case of our result, we get the following best proximity point theorem, which was proved in [18].

Corollary 17 (see [18]). Let and be nonempty, closed subsets of a complete metric space. Let and satisfy the following conditions: Further, for a fixed element in , let Then, the sequence must converge to a best proximity point of and the sequence must converge to a best proximity point of such that

Further, if has two distinct best proximity points, then does not vanish and hence the sets and should be disjoint.

Proof. The triad , where is the identity mapping, satisfies the -min–max condition. Indeed, let such that and We have and And since satisfies the min-max condition, Thus, the triad fulfills all the conditions of the previous theorem. For , define Then, must converge to a best proximity point of , the sequence must converge to a point and sequence must converge to such that It suffices to notice that Now, if has two distinct best proximity points then Which means and are disjoint.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.