Research Article | Open Access
Best Proximity Point Results for Some Contractive Mappings in Uniform Spaces
We introduce the concept of -distance (an analogue of -metric), -proximal contraction, and -proximal cyclic contraction for non-self-mappings in Hausdorff uniform spaces. We investigate the existence and uniqueness of best proximity points for these modified contractive mappings. The results obtained extended and generalised some fixed and best proximity points results in literature. Examples are given to validate the main results.
The importance of fixed point theory emerges from the fact that it furnishes a unified approach and constitutes an important tool in solving equations which are not necessarily linear. A large number of problems can be formulated as nonlinear equations of the form , where is a self-mapping in some framework; see [1–4] and other references therein. Nevertheless, an equation of the type does not necessarily possess a solution if happens to be a non-self-mapping. In this case, one seeks an appropriate solution that is optimal in the sense that is minimum. That is, we resolve a problem of finding an element such that is in best proximity to in some sense.
Best proximity point theorem analyzes the condition under which the optimisation problem, namely, , has a solution. The point is called the best proximity point of , if , where . Note that the best proximity point reduces to a fixed point if is a self-mapping.
A best proximity point problem is a problem of achieving the minimum distance between two sets through a function defined on one of the sets to the other.
The very popular best approximation theorem is due to Fan . If is a nonempty compact subset of a Hausdorff locally convex topological vector space and is a continuous mapping, then there exists an element such that . Fan’s results are not without shortcomings; the best approximation theorem only ensures the existence of approximate solutions, without necessarily yielding an optimal solution. But the best proximity point theorem provides sufficient conditions that ensure the existence of approximate solutions which are also optimal.
Afterwards many authors such as Eldred and Veeramani  have derived extensions of Fan’s Theorem and the best approximation theorems in many directions. Significant best proximity point results are in [7–11] and other references therein.
In fixed point theory, other spaces of study other than metric spaces have been used by different authors. Pseudometric spaces interestingly generalise metric spaces. One of the spaces in literature that generalises the metric and pseudometric spaces is the uniform space.
Weil  was the first to characterise uniform spaces in terms of a family of pseudometrics and Bourbaki  provided the definition of a uniform structure in terms of entourages. Aamri and El Moutawakil  gave some results on common fixed point for some contractive and expansive maps in uniform spaces and provided the definition of -distance and -distance.
Also, Olatinwo  established some common fixed point theorems for self-mappings in uniform spaces by using the - and -distances. Dhagat et al.  proved some common fixed point theorems for pairs of weakly and semicompatible mappings using -distances in uniform spaces. Hussain et al.  apply the concept of cyclic () contractions to establish certain fixed and common point theorems on a Hausdorff uniform space.
In another development, Geraghty  introduced the generalised contraction self-map using comparison functions.
Another useful result is by Karapinar and Erhan  who gave the definition of a -contractive map for non-self-mappings and Karapinar  who established some results on best proximity points of -Geraghty contractive non-self-mappings.
Also, Basha  gave some necessary and sufficient conditions to claim the existence of unique best proximity points for proximal contractions in metric spaces. Mongkolkeha et al.  introduced proximal cyclic contractions in metric spaces which are more general than the class of proximal contractions given by Basha .
Motivated by the results above, we develop the concept of -proximal contraction and -proximal cyclic contractions in uniform spaces and obtain the existence and uniqueness of best proximity points of these non-self-contractive mappings using -distance function.
The following definitions are fundamental to our work.
Definition 1 (see ). A uniform space is a nonempty set equipped with a uniform structure which is a family of subsets of Cartesian product which satisfy the following conditions:(i)If , then contains the diagonal .(ii)If , then is also in .(iii)If , then .(iv)If , and which contains , then .(v)If , then there exists such that whenever and are in , then is in . is called the uniform structure or uniformity of and its elements are called entourages, neighborhoods, surroundings, or vicinities.
Definition 2 (see ). Let be a uniform space. A function is said to be an(a)-distance if, for any , there exists such that if and for some , then ;(b)-distance if is an -distance and .
Another extension of a metric space is the -metric space.
Definition 3 (see ). Let be a nonempty set and be a given real number. A map is said to be a -metric if and only if, for all , the following conditions are satisfied:(i) with and if and only if .(ii).(iii)The pair is called a -metric space. If , it becomes a metric space.
Examples in literature to show that -metric is a generalisation of a metric space are in [16, 21].
Now, we introduce the concept of -distance.
Definition 4. Let be a uniform space. A function is said to be a -distance if (i) is an -distance,(ii).Note that the function reduces to an -distance if the constant is taken as 1.
Example 5. Let be a uniform space and let be a metric on . It is clear that is a uniform space where is the set of all subsets of satisfying for some . Moreover, if , then is an -distance on .
Also, the following definition is required.
Definition 6 (see ). Let be a uniform space and an -distance on (a)If , and and are said to be -close. A sequence is a Cauchy sequence for if, for any , there exists such that and are -close for . The sequence is a -Cauchy sequence if for every there exists such that for all (b) is -complete if for any -Cauchy sequence , there exists such that .(c) is -continuous if implies .(d) is said to be -bounded if .To guarantee the uniqueness of the limit of the Cauchy sequence for , the uniform space needs to be Hausdorff.
Definition 7 (see ). A uniform space is said to be Hausdorff if and only if the intersection of all the reduces to the diagonal of , . In other words, for all implies .
A uniform structure defines a unique topology on for which the neighborhoods of are the sets .
is continuous if with respect to implies with respect to .
Observe that all the above maps are self-mappings.
A large number of articles investigate non-self-contractive mappings on metric spaces. Some of these are given below.
Definition 8 (see ). Let be a metric space and and be nonempty subsets of . A mapping is said to be a -contraction if there exists such that
Definition 9 (see ). Let and be nonempty subsets of a metric space and let such that ,(i) is cyclic if and .(ii) is called a cyclic contraction if for some (iii) is called a cyclic -contraction if is a strictly increasing map Note that (4) becomes (3) with for all But the converse is not true in general (see ).
Definition 10 (see ). Let be a nonempty subset of a complete metric space . A mapping is said to be a proximal contraction if there exists a nonnegative real number such that for all , .Basha  proved the following theorem.
Theorem 11 (see ). Let be two nonempty subsets of a complete metric space . Suppose that is nonempty and closed. Let satisfy the following conditions: (a) is a proximal contraction,(b).Then there exists a point such that . Moreover, if is injective on , then the point such that is unique.
Definition 12 (see ). Let and . The pair is called a proximal cyclic contraction pair if there exists such that for all .
Given nonempty subsets and of a uniform space , we adopt the following notations and definitions used for metric spaces to the context of uniform spaces.
Definition 13. Let and be an isometry. The mapping is said to preserve the isometric distance with respect to if
Definition 14. An element is called a best proximity point of a mapping if it satisfies the condition that
Now, we give the definition of -proximal contraction and -proximal cyclic contraction for non-self-mapping in uniform spaces.
Definition 15. Let be a pair of nonempty subsets of an -complete Hausdorff uniform space such that is an -distance on . A mapping is said to be a -proximal contraction if there exists a nondecreasing continuous weak comparison function satisfying the following. For each and ,,, converges for any , such that , such that
Definition 16. Let be a pair of nonempty subsets of -complete Hausdorff uniform space such that is an -distance on . Suppose and are mappings. The pair is said to be a -proximal cyclic contraction if there exists a nondecreasing continuous weak comparison function satisfying above, such that for all and
It is easy to see that a self-mapping that is a -proximal contraction is a contraction. But a non-self -proximal contraction is not necessarily a contraction map. If and -distance is replaced with a metric , (9) reduces to (6). Similarly, (8) reduces to (5). Also, (9) and (8) reduce to (2) if , and if the -distance is replaced with a metric , in the sense that .
The following example shows that -distance function is different from the metric distance function . In fact, the -distance function reduces to the metric distance function when is a metric space.
Example 17. Let and be nonempty closed subsets of with the usual metric. Let be a mapping given by and and let . It is easy to see that
Clearly, is not a -proximal contraction; that is, .
has no best proximity point since there is no such that .
Now, taking And consider defined as .
Clearly, for all is a -proximal contraction and −1 is the unique best proximity point of .
The following Lemma, which is true for self-mappings (see Lemma ) can be proved for non-self-mappings.
Lemma 18 (see ). Let be a Hausdorff uniform space and be an -distance on . Let be arbitrary sequences in and be sequences in converging to 0. Then, for , the following holds:(a)If and , then . In particular, and , and then .(b)If and , then .(c)If and , then, converges to .(d)If , then is a -Cauchy sequence in .
3. Main Results
We give the first theorem.
Theorem 19. Let be a pair of nonempty subset of an -complete Hausdorff uniform space such that is an -distance on and is . Suppose a map is such that is a -proximal contraction. Then there exists a unique point such that
Proof. Let , since and There exists such that . Also, since , there exists such that . Furthermore, we obtain the sequences and subsets of such that We show that is a complete -Cauchy sequence whose limit is the unique best proximity point of . Since is a -proximal contraction, from (10) and (11) we have Thus by induction, for any .
Since is an -distance, we have . Now for , Let Then Suppose , and since is a weak comparison function, by Definition 15, it follows that So there exists a such that . Then by (15), Repeating the same argument, we obtain
Therefore, the sequence is a -Cauchy in the -complete space . Hence there exists such that since is closed. We prove that is the best proximity point of ; that is,
Also, since and , there exists an element such that Using (19) and (11) and since is a -proximal contraction, As since . Therefore, and thus . So from (19), To guarantee the uniqueness of , we show that is Hausdorff. Suppose there exists such that By the -proximal contraction , which implies . Similarly, . But by the second property of -distance, Hence, We conclude that .
Corollary 20. Let be a complete metric space. Suppose satisfies ; then has a unique fixed point.
Proof. Set , , and in Theorem 19, to obtain the result.
Corollary 21 (see ). Let and be two nonempty subsets of a complete metric space . Suppose satisfies . Then has a unique best proximity point.
Proof. Set and in Theorem 19, to obtain the corollary.
Corollary 22 (see ). Let and be two nonempty subsets of a complete metric space . Suppose is nonempty and closed and satisfies the following conditions:(a) is a proximal contraction,(b).Then there exists a unique point such that . Moreover, , and there exists a sequence such that for every and .
Proof. Set and in Theorem 19.
Now, we establish some results of best proximity point for -proximal cyclic contractions in uniform spaces.
Theorem 23. Let be a pair of nonempty closed subset of a -bounded and -complete Hausdorff uniform space such that and is an -distance on . Let , , and satisfy the following conditions:(i)the pair () is a -proximal cyclic contraction,(ii), ,(iii) and ,(iv) is isometry.Then there exist unique points and such that Further, if is any fixed element in and is any fixed element in , the sequences and , defined by converge to the best proximity points and , respectively.
Proof. Let be fixed element in . Since and , it follows that there exists an element such that Again, since and , there exists an element such that Following the steps in the proof of Theorem 19, we can find such that By induction, one can determine an element such thatAlso, since is an isometry and by the -proximity cyclic contraction using (30) and (31), it follows that, for each ,Since is an -distance, we have . Now for , Let , and then Next we show that is -Cauchy in the -complete space ; that is, for any .
Recall that if there exists such that , we are done, and is the required best proximity point of . Thus we assume that .
Suppose . Now using Definition 15, we have so there exists a such that
Then by (35), Repeating the same argument, we obtain So the sequence is -Cauchy in the -complete space .
Hence, converges to some element . Similarly, since and , there exists a sequence such that it converges to some element and from (31), Since the pair () is a -proximal cyclic contraction and is isometry, using (31) and (41), we have By (33), on taking limit as , we have We show that Assume , from (43), , a contradiction. Hence, Thus, and . Since and , there exist and such that Now, we show that and .
Since is a -proximal cyclic contraction, using (44) and (31) we have Letting in (46), , and since is an -distance, Again letting , we get and so, . Therefore we have Similarly, we can obtain and so, Thus, from (44), (48), and (49), we getNext we prove the uniqueness of and . Suppose that there exist and with and such that Since is an isometry, and is a -proximal cyclic contraction, using (48) and (51), we have , a contradiction. Hence, . Similarly, we show that . But since is a -distance, we have Therefore, Now we have and By Lemma 18(a), we conclude that Similarly,
Corollary 24 (see ). Let be a -complete Hausdorff uniform space and an -distance on Suppose is a cyclic -contraction such that , for all , where is a weak comparison function. Then has a unique fixed point.
Proof. The proof follows from Theorem 23 if , , and -distance is reduced to -distance function.
Corollary 25 (see ). Let be a complete metric space and be a Geraghty contraction satisfying for each , where Then has a unique fixed point.
Proof. The proof follows from Theorem 23 if and is a metric distance.
Example 26. Let such that . Clearly, . Suppose , and .
Let , and be defined byNow, for we obtain , Now, (9) generalises (6) in the sense that
(1) for all .
Hence, (6) fails. () is not a proximal cyclic contraction. We see that has no unique best proximity point since there is no such that .
But taking ,
(2) becomes . is a -proximal contraction. Clearly, and is the unique best proximity point of the pair , while is the unique best proximity point of the pair . Hence, (9) is different from (6).
We give the following examples to show that the -proximal cyclic contraction is different from the Geraghty contraction.
Example 27. Consider the usual metric and and let and Obviously, , and , . Let , be defined as taking and . Also, consider . And and are defined as follows:We show that is not a Geraghty contraction , a contradiction. is not a Geraghty contraction.
We see that has no best proximity points since there is no such that .
But is a -proximal cyclic contraction. Clearly taking , implies and .
is a -proximal cyclic contraction and is the unique best proximity point of while is the unique best proximity point of .
Conflicts of Interest
The authors declare that they have no conflicts of interest.
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
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