Some New Results on Coincidence Points for Multivalued Suzuki-Type Mappings in Fairly Complete Spaces

In this paper, we introduce Suzuki-type (α, β, γg)−generalized and modified proximal contractive mappings. We establish some coincidence and best proximity point results in fairly complete spaces. Also, we provide coincidence and best proximity point results in partially ordered complete metric spaces for Suzuki-type (α, β, γg)−generalized and modified proximal contractive mappings. Furthermore, some examples are presented in each section to elaborate and explain the usability of the obtained results. As an application, we obtain fixed-point results in metric spaces and in partially ordered metric spaces. The results obtained in this article further extend, modify and generalize the various results in the literature.


Introduction and Preliminaries
S. Banach [1] stated and proved the Banach contraction principle. This principle has wide applications due to its simple and constructive nature of proof. The constructive proof leads to developing algorithms and can be easily applied in computer and data sciences (see in [2]) as well. The application of this principle is not limited to these areas, it is extensively used in dynamically programming ( [3]) and biosciences as well. Due to wide range of applications, researchers around the globe are attracted towards this principle to generalize, modify and extend this pioneer result (for detail, see [4][5][6][7][8][9][10][11][12]). These modifications are consisting upon three pillars (1) generalizing the contractive conditions, (2) generalizing the underlying space and (3) modifying the single valued mapping with multivalued mapping. In all the three modifications, the Banach contraction principle gets modification with three different aspects.
The "fixed point" q of a self-mapping M is actually a solution of an operator equation Mq = q (i.e., d(q, Mq) = 0). Among these three aspects of generalization of "Banach contraction principle", it would be quite interesting to discuss, if the operator equation Mq = q has no solution. In this case, when d(q, Mq) = 0 then it is evident to minimize the distance between q and Mq which leads to the following optimization problem: min q∈Y d(q, Mq). for all q, r ∈ Y where α ∈ [0, 1) and β ∈ [0, ∞) then the mapping M has a "fixed point".
T. Suzuki ([14]) introduced "Suzuki contraction", which generalized the "Banach contraction" and he proved the following "fixed-point theorem". Theorem 2. ( [14]) Let (Y, d) be a complete metric space and mapping M : Y → Y satisfies for all q, r ∈ Y then mapping M has a unique fixed point in Y.
In 2014, M. Gabeleh ([15]) revised and generalized the contractions presented in Theorem 1 and in Theorem 2 to prove the single valued and multivalued "best proximity point results" . Recently S. Basha ([16]) introduced the concept of "fairly and proximally complete spaces" and proved "best proximity point results" in these spaces.
In this paper, we will modify Suzuki-type "best proximity point results" of M. Gabeleh ([15]) and prove Suzuki-type "coincidence best proximity point results" in the setting of "fairly complete space" and "partially ordered fairly complete space".
We will use the following notations in the entire article and assume that Q and R are nonempty subsets of a metric space (Y, d), further d(Q, R) = inf{d(q, r) : q ∈ Q and r ∈ R} (distance between two sets Q and R), Q 0 = {q ∈ Q such that d(q, r) = d(Q, R), for some r ∈ R}, R 0 = {r ∈ R such that d(q, r) = d(Q, R), for some q ∈ Q}, also d * (q, r) = d(q, r) − d(Q, R), for some q ∈ Q and r ∈ R.
We recall the following notions of cyclically Cauchy sequence and fairly Cauchy sequence.

Definition 2.
( [16]) Consider two sequences {q n } in Q and {r n } in R. The sequence {(q n , r n )} in (Q, R) is said to be: • A cyclically Cauchy sequence if there exists a natural number N such that d(q m , r n ) < d(Q, R) + , for every > 0 and for all m, n ≥ N ∈ N • A fairly Cauchy sequence if the following conditions are satisfied (1) {(q n , r n )} is a cyclically Cauchy sequence, (2) {q n } and {r n } are Cauchy sequences, for all n ≥ N ∈ N.
Next, we recall a special type of completeness for a pair of nonempty subsets (Q, R) of (Y, d).
for all q 1 , q 2 , r 1 and r 2 in Q.
for all q, r ∈ Y.
The concept of α−admissible mapping was generalized and extended in many directions. Jleli et al. ( [19]) introduced α−proximal admissible mapping as follows. Definition 6. Let Q and R be the nonempty subsets of metric space (Y, d).
Please note that if Q = R = Y then every α−proximal admissible mapping is an α−admissible mapping.
for all q, r ∈ Q.
Let C B (Y) be a closed and bounded subset of the metric space (Y, d). Then the Pompeiu-Hausdroff metric ( [21]) on C B (Y), is defined as and D * (q, r) = D(q, r) − d(Q, R), for all q ∈ Q and r ∈ R.
The constants γ and β satisfies the condition C, if γ ∈ (0, 1] and 0 ≤ β < γ such that 0 < β + γ ≤ 1. In the first result we will prove that the pair (g, M) which satisfies the Suzuki-type (α, β, γ g )−generalized proximal contraction has a coincidence best proximity point in the frame work of fairly complete spaces. Theorem 3. Let Q and R be nonempty closed subsets of a complete metric space (Y, d) such that the pair (Q, R) is a fairly complete space and satisfy the P −property. Consider a pair (g, M) satisfying Suzuki-type (α, β, γ g )−generalized proximal contractive condition with M(Q 0 ) ⊆ R 0 , Q 0 ⊆ g(Q 0 ) where mapping g satisfy the α R −property and mapping M is an α−proximal admissible. Furthermore, suppose that there exist some q 0 , q 1 ∈ Q 0 such that D(gq 1 , Mq 0 ) = d(Q, R) and α(q 0 , q 1 ) ≥ 1.
Then the pair (g, M) possesses a coincidence best proximity point.
Since the pair (Q, R) satisfy the P −property, using the P −property the above inequality can be written as which shows that H(Mq 1 , Mq 2 ) ≤ γH(Mq 0 , Mq 1 ). Continuing on the same lines for q 2 we can verify the following and so the above inequality can be written as Since α(q 2 , q 3 ) ≥ 1 and the pair (g, M) satisfies Suzuki-type (α, β, γ g )−generalized proximal contraction which implies that which can be written as The pair (Q, R) satisfy the P −property, using inequality (3) and the P −property in above inequality, we obtain H(Mq 2 , which leads {Mq n } to be a Cauchy sequence in R and (Q, R) is a pair of nonempty closed subsets of a complete metric space (Y, d) and so {Mq n } converges to some point q ∈ R 0 . In the same way, the sequence {gq n } is convergent to some point gp ∈ Q 0 . So, we have By Equations (4) and (6) we have using inequality (4), the inequality (7) becomes for all n ≥ N 1 ∈ N. Since 1 + β + γ > 1 and 1 1+β+γ < 1 we can write Since α(q n , q) ≥ 1 and the pair (g, M) satisfies Suzuki-type (α, β, γ g )−generalized proximal contraction which implies that above inequality can be written as We can write using inequalities (4) and (9), inequality (10) becomes after simplification above inequality can be written as From triangular inequality we have Using inequality (11), inequality (12) becomes after simplification we have Since α(q n , p) ≥ 1 and furthermore the pair (g, M) satisfy Suzuki-type (α, β, γ g )−generalized proximal contraction which implies that Therefore, p is a coincidence best proximity point of the pair (g, M).
The subsequent example corroborates the result proved in Theorem 3.

Example 1.
Let Y = R 2 be a metric space with Euclidean metric d.
Define inequality (14) holds. If we choose q = (−1, 0) and r = (−1, 1) then the inequality (13) does not holds. This shows that the pair (g, M) satisfy Suzuki-type (α, β, γ g )−generalized proximal contractive condition; further remaining conditions of Theorem 3 holds, therefore the pair (g, M) has two coincidence best proximity points (−1, 1) and (−1, −1). Please note that in this example the contractive condition of Theorem 3.1 of M. Gabeleh ([15]) does not hold. Indeed, q = (−1, 1) and r = (−1, 0) we have M. Gabeleh in ( [15]) proved the best proximity point results but did not discussed the uniqueness of the best proximity point results. In this paper, we will need an additional condition C (2) to prove the uniqueness of coincidence best proximity point results for Suzuki-type generalized proximal contractions. Theorem 4. Let Q and R be nonempty closed subsets of a complete metric space (Y, d) such that the pair (Q, R) is a fairly complete space and satisfies the P −property. Consider a pair (g, M) satisfying Suzuki-type (α, β, γ g )−generalized proximal contractive condition with M(Q 0 ) ⊆ R 0 , Q 0 ⊆ g(Q 0 ) where g is a one-to-one mapping and satisfies the α R −property. Mapping M is an α−proximal admissible further suppose that there exist some q 0 , q 1 ∈ Q 0 such that If the constants β and γ satisfy the condition C (2) then the pair (g, M) possesses a unique coincidence best proximity point.

Proof.
Following arguments similar to those in the proof of Theorem 3, we get the existence of the coincidence best proximity point of the pair of mappings (g, M). Now, we must prove the uniqueness of coincidence best proximity point of the pair of mappings (g, M). On contrary suppose that q 1 , q 2 ∈ Q are two coincidence best proximity points of the pair of mappings (g, M) with q 1 = q 2 that is the pair (Q, R) possesses the P −property and g : Q → Q is a one-to-one mapping, we can write Since 1 + β + γ ≥ 1 and 1 1+β+γ ≤ 1 we have As α(q 2 , q 1 ) ≥ 1 and the pair (g, M) satisfies Suzuki-type (α, β, γ g )−generalized proximal contraction which implies that from above inequality it can be written as H(Mq 2 , Mq 1 ) ≤ γd(gq 2 , gq 1 ) + βD * (gq 1 , Mq 2 ), and by using Equation (15) After simple calculation we have 1 ≤ γ + β which is a contradiction. Hence, q 1 = q 2 and the pair (g, M) possesses a unique coincidence best proximity point.
Let us visualize Theorem 4 with the example which follows. After calculation we can see that d(Q, R) = 3 and the pair (Q, R) satisfies the P −property. Define mappings g : Q → Q, M : Q → C B (R) as: The coincidence best proximity point results discussed below can be obtained directly from Theorem 3.

Corollary 1.
Let Q and R be nonempty closed subsets of a complete metric space (Y, d) such that the pair (Q, R) is a fairly complete space and satisfies the P −property. Consider g : Q → Q and M : Q → R satisfy the following, if with M(Q 0 ) ⊆ R 0 , Q 0 ⊆ g(Q 0 ) where mapping g satisfies the α R −property and M is an α−proximal admissible mapping. Furthermore, suppose that there exists some q 0 , q 1 ∈ Q 0 such that d(gq 1 , Mq 0 ) = d(Q, R) and α(q 0 , q 1 ) ≥ 1, where α : Q × Q → [0, ∞). Then the pair (g, M) has a coincidence best proximity point.

Corollary 2.
Let Q and R be nonempty closed subsets of a complete metric space (Y, d) such that the pair (Q, R) is a fairly complete space and satisfy the P −property. Consider g : Q → Q be a one-to-one mapping and M : Q → R satisfy with M(Q 0 ) ⊆ R 0 , Q 0 ⊆ g(Q 0 ) where mapping g satisfies the α R −property and mapping M is an α−proximal admissible. Furthermore, suppose that there exist some q 0 , q 1 ∈ Q 0 such that where α : Q × Q → [0, ∞). Then the pair (g, M) has unique coincidence best proximity point if the constants β, γ satisfies the condition C (2).
The subsequent result is a best proximity point theorem for the Suzuki-type (α, β, γ)−generalized proximal contraction in the framework of fairly complete space.
Theorem 5. Let Q and R be nonempty closed subsets of a complete metric space (Y, d) such that the pair (Q, R) is a fairly complete space and satisfy the P −property. Consider the mapping M satisfy the Suzuki-type (α, β, γ)−generalized proximal contractive condition with M(Q 0 ) ⊆ R 0 and M is an α−proximal admissible mapping. Furthermore, suppose that there exist some q 0 , q 1 ∈ Q 0 such that D(q 1 , Mq 0 ) = d(Q, R) and α(q 0 , q 1 ) ≥ 1.
Then the mapping M has a best proximity point.
Proof. If we take g = I Q in Theorem 3 then Suzuki-type (α, β, γ g )−generalized proximal mapping becomes Suzuki-type (α, β, γ)−generalized proximal mapping, remaining aspects of Theorem 5 are same as in the proof of Theorem 3. Hence we have a best proximity point of mapping M. Corollary 3. Let Q and R be nonempty closed subsets of a complete metric space (Y, d) such that the pair (Q, R) is a fairly complete space and satisfies the P −property. Consider the mapping M satisfies the Suzuki-type (α, β, γ)−generalized proximal contractive condition with M(Q 0 ) ⊆ R 0 and M is an α−proximal admissible mapping. Suppose that there exist some q 0 , q 1 ∈ Q 0 such that D(q 1 , Mq 0 ) = d(Q, R) and α(q 0 , q 1 ) ≥ 1.
Furthermore, if the constants β, γ satisfy the condition C (2) then unique best proximity point of mapping M exists.
Now we must show that the subsequent condition of Suzuki-type (α, β, γ)−generalized proximal contraction holds for all q, r ∈ Q 0 .   The following results are the nice consequences of Theorem 5.

Corollary 4.
Let Q and R be nonempty closed subsets of a complete metric space (Y, d) such that the pair (Q, R) is a fairly complete space and satisfy the P −property. Consider an α−proximal admissible mapping M : Q → R satisfy the following contractive condition with M(Q 0 ) ⊆ R 0 . Furthermore, suppose that there exist some q 0 , q 1 ∈ Q 0 such that d(gq 1 , Mq 0 ) = d(Q, R) and α(q 0 , q 1 ) ≥ 1, where α : Q × Q → [0, ∞). Then the mapping M has a best proximity point.

Corollary 5.
If we add the condition (2) to the statement of Corollary 4 we obtain that the mapping M possesses a unique best proximity point.

Suzuki-Type (α, β, γ g )−Modified Proximal Contractive Mapping
We begin this section with the subsequent definitions.
The following result is a coincidence best proximity point theorem for Suzuki-type (α, β, γ g )−modified proximal contraction in the setting of a fairly complete space. Theorem 6. Let Q and R be nonempty closed subsets of a complete metric space (Y, d) such that the pair (Q, R) is fairly complete space. Consider the pair of mappings (g, M) satisfy Suzuki-type (α, β, γ g )−modified proximal contraction with M(Q 0 ) ⊆ R 0 , Q 0 ⊆ g(Q 0 ). Set R has the property of uniform M−approximation in set Q and mapping g satisfy the α R −property. Furthermore, assume the existence of some q 0 , q 1 ∈ Q 0 such that D(gq 1 , Mq 0 ) = d(Q, R) and α(q 0 , q 1 ) ≥ 1. Then the pair (g, M) possesses a coincidence best proximity point.
Since α(q 2 , q 3 ) ≥ 1 and the pair (g, M) satisfy Suzuki-type (α, β, γ g )−modified proximal contraction which implies that from above inequality we have using inequality (20), above inequality becomes Thus, for a sequence {Mq n } in R 0 , we have which implies that {Mq n } is a Cauchy sequence and (Q, R) is a pair of nonempty closed subsets of a complete metric space (Y, d), {Mq n } converges to some point q ∈ R. Therefore, we have for any δ > 0 Since the set R has the property of uniform M−approximation in set Q which implies that d(gq n+1 , gq n+2 ) < , hence {gq n } is a Cauchy sequence and converges to gp ∈ Q and we have d(gp, q) = lim n→∞ D(gq n , Mq n−1 ) = d(Q, R).
The following example is given to support the usability of Theorem 6.

Example 4.
Let Y = R with metric d be defined as d(q, r) = |q − r|. Also suppose that Q = {2, 4, 6, 9, 12} and R = {3, 5, 7, 8, 14} are the nonempty subsets of Y. We have d(Q, R) = 1, Q 0 = {2, 4, 6, 9} and R 0 = {3, 5, 7, 8}, further the pair (Q, R) does not satisfy the P −property. Now consider mappings M : Q → C B (R), g : Q → Q be defined as: clearly M(Q 0 ) ⊆ R 0 and Q 0 ⊆ g(Q 0 ). If we choose q = 4 and r = 2 then after simple calculation we can show that the following inequality does not hold and for all the remaining cases above contraction holds for β = 0.3, γ = 0.4. Now it must be shown that the subsequent condition of Suzuki-type (α, β, γ g )−modified proximal contractive condition holds. The next coincidence best proximity point result follows from Theorem 6 directly.

Corollary 6.
Let Q and R be nonempty closed subsets of a complete metric space (Y, d) such that the pair (Q, R) is a fairly complete space. Consider mappings g : Q → Q and M : Q → R satisfy the following contractive condition with M(Q 0 ) ⊆ R 0 , Q 0 ⊆ g(Q 0 ). Set R has the property of uniform M−approximation in set Q and mapping g satisfies the α R −property. Furthermore, suppose that there exist some q 0 , q 1 ∈ Q 0 such that d(gq 1 , Mq 0 ) = d(Q, R) and α(q 0 , q 1 ) ≥ 1 where α : Q × Q → [0, ∞). Then the pair (g, M) has a coincidence best proximity point.
Next example is given to corroborates the usability of Corollary 6. The subsequent result is a best proximity point theorem for the Suzuki-type (α, β, γ)−modified proximal contraction in the framework of fairly complete space. Theorem 7. Let Q and R be nonempty closed subsets of a complete metric space (Y, d) such that the pair is a fairly complete space. Consider a mapping M is an α−proximal admissible and satisfy Suzuki-type (α, β, γ)−modified proximal contraction with M(Q 0 ) ⊆ R 0 . Further set R has the property of uniform M−approximation in set Q and suppose that there exist some q 0 , q 1 ∈ Q 0 such that D(q 1 , Mq 0 ) = d(Q, R) with α(q 0 , q 1 ) ≥ 1. Then mapping M possesses a best proximity point.
Proof. If we take g = I Q in Theorem 6, the remaining aspects follow from the same lines.
The next best proximity point result directly follows from Theorem 7. Corollary 7. Let Q and R be nonempty closed subsets of a complete metric space (Y, d) such that the pair (Q, R) is a fairly complete space. Consider a mapping M : Q → R satisfies the following contractive condition with M(Q 0 ) ⊆ R 0 and set R has the property of uniform M−approximation in set Q. Furthermore, suppose that there exist some q 0 , q 1 ∈ Q 0 such that d(q 1 , Mq 0 ) = d(Q, R) and α(q 0 , q 1 ) ≥ 1 where α : Q × Q → [0, ∞). Then mapping M has a best proximity point.

Some Results Related to Partially Ordered Metric Space
Here, we are concerned with coincidence best proximity point results for generalized and modified Suzuki-type contractions in partially ordered metric space.
From now and onward ∆ defines: ∆ = {q, r ∈ Q such that q r or r q}.
Definition 10. ( [22]) Suppose Y be a nonempty set, a triplet (Y, d, ) is called a partially ordered metric space if it satisfies the following conditions: is partial order on Y.
Definition 12. A pair (g, M) where g : Q → Q and M : Q → C B (R) is an:
Then the pair (g, M) possesses a unique coincidence best proximity point.
Let us consider {q n } as a sequence then α(q n , q n+1 ) ≥ 1 with q n → q as n → ∞ then it follows that (q n , q n+1 ) ∈ ∆ with q n → q as n → ∞. Hence remaining conditions of Theorem 3 fulfilled so that pair (g, M) possesses a coincidence best proximity point.
Theorem 9. Let Q and R are the same sets as in Theorem 8. Suppose that the pair (g, M) where g : Q → Q and M : Q → C B (R) satisfies an ordered Suzuki-type (β, γ g )−modified proximal contractive condition with all assumptions of Theorem 8. Then the pair (g, M) possesses a unique coincidence best proximity point.

Application to Fixed-Point Theory
Here, we will discuss some results about the fixed-point theory for generalized and modified Suzuki-type contraction.
If Q = R = Y then the following contractive conditions can be define.

Definition 13.
A mapping M : Y → C B (Y) is called a: 1.
From Theorems 5 and 7 we can find following new fixed-point results.
Theorem 10. Suppose that if there exists q 0 with α(q 0 , Mq 0 ) ≥ 1 then the mapping M : Y → C B (Y) which satisfy Suzuki-type (α, β, γ)−generalized contractive condition on a complete metric space (Y, d) has a unique fixed point.
However, here we have Q = R = Y, so we have D(q, Mq) = 0 and there exists a fixed point q of Suzuki-type (α, β, γ)−generalized contraction of mapping M.
Theorem 11. Suppose that if there exists q 0 with α(q 0 , Mq 0 ) ≥ 1 then the mapping M : Y → C B (Y) which satisfy Suzuki-type (α, β, γ)−modified contractive condition on a complete metric space (Y, d) has unique fixed point.
Proof. By following the prove of Theorem 8, we can say that for self mapping every ordered Suzuki-type (α, β, γ)−generalized contractive condition implies ordered Suzuki-type (β, γ)−generalized contractive condition. The remaining aspects of Theorem 8 fulfilled on the same lines and mapping M possesses a unique fixed point.
Theorem 13. If a mapping M : Y → C B (Y) satisfies an ordered Suzuki-type (β, γ)−modified contractive condition with q 0 ∈ Y such that (q 0 , Mq 0 ) ∈ ∆ on complete partially ordered metric space (Y, d, ) then mapping M possesses a unique fixed point.

Conclusions
In this article, we defined Suzuki-type (α, β, γ g )−generalized and modified proximal contractive mappings. Furthermore, some coincidence and best proximity point results are obtained in fairly complete spaces, which generalized the result discussed by M. Gabeleh in ( [15]). As an application, we obtained some fixed point and coincidence point results in partially ordered metric spaces for modified and generalized Suzuki-type contractions. Some illustrative examples are also provided to visualize and support to the results obtained herein.

Conflicts of Interest:
The authors declare no conflict of interest.