Coincidence Point Results for Multivalued Suzuki Type Mappings Using θ-Contraction in b-Metric Spaces

In this paper, we introduce the concept of coincidence best proximity point for multivalued Suzuki-type α -admissible mapping using θ -contraction in b-metric space. Some examples are presented here to understand the use of the main results and to support the results proved herein. The obtained results extend and generalize various existing results in literature.


Introduction and Preliminaries
In 1922, Stefan Banach [1] proved his famous result "Banach contraction principle", which states that "let (X, d) be a complete metric space and T : X → X be a contraction, then T has a unique fixed point". The constructive proof of theorem helps the researchers working in Computer Sciences to develop algorithm based upon the proof of theorem, and it able them to solve complex networking problem by relating it with "fixed point problem". This is one of its application in Computer Sciences. Later, researchers found its applications in several branches of sciences, specially, Economics, Data Science, Physics, Medical Science, Game Theory, etc. Due to several application of "fixed point theory", researchers was motivated to further generalize it in different directions, by generalizing the contractive conditions, underlying space and concept of completeness. Among the several generalizations of "Banach fixed point theorem", weak contractive conditions were introduced for finding unique "fixed point". Often these weak conditions are related with metric spaces and some time are related with contractive conditions. In case of self-mappings, the solution u * of the operator equation Tu = u is the "fixed point" of mapping T (such that d(u, Tu) = 0, if mapping T is nonself, then "fixed point" of T will not exist. In this case, if T is nonself-mapping, then we cannot find any such u * that satisfy the "fixed point" problem u = Tu (or d(u, Tu) = 0), then it is evident to minimize the d(u, Tu); any such u * that minimize the given optimization problem: is known as the "approximate fixed point" of T.
Further, for nonself mappings T : U → V, where sets U and V are nonempty subsets of metric space (X, d), also U ∩ V = ∅. In this case, u ∈ U, then Tu ∈ V, where U ∩ V = ∅, in this scenario, is the minimization/optimization problem (1) that reduces to best proximity point problem, and any point u * that satisfies d(u, Tu) = d(U, V) is called "best proximity point" of T. Note that if condition U ∩ V = ∅ is removed then d(U, V) = 0, in this case, every best proximity point can be reduced to "fixed point" of T.
Finding the "best proximity points" for two mappings is another kind of generalization of "best proximity point"; any u * ∈ X that satisfies d(g(u * ), Tu * ) = d(U, V); here, U and V are nonempty subsets of (X, d) and T : U → V and let g : U → U be any mapping. Point u * is called "coincidence best proximity point" of mappings g and T. If g = I U (identity over U) then every "coincidence best proximity point" will reduced to "best proximity point" of mapping T.
Extreme values are the largest and smallest values a function attains in specific interval. These extreme values of functions peaked our interest by observing how it knew the highest/lowest values of a stock or the fastest/slowest a body is moving. All these kinds of problems are related (to lower the risk and increase the benefit/profit) with optimization problem. The best proximity points are actually approximate fixed points with least error; we model the given optimization problem with a functional equation or operator, then we optimize the given model using best approximation technique. Now, these functions observe some very specific properties that would be hard to find in real-world problems, so as to relate these functions with specific constraints.
In 1989 and 1993, Bakhtin [2] and Czerwik [3], respectively, introduced the concept of b-metric space. As an application, Equation (2) is used in several iterative schemes, and the homotopy perturbation method (see , for details, in [4,5]. After the revolution in mathematics due to L. Zadeh ([6]), by presenting the concept of fuzzy sets, Kramosil and Veeramani [7][8][9] introduced the revolutionary idea of fuzzy metric spaces. Several authors around the globe studied fixed point theory in a new and different environment of fuzzy metric space. It gets more exposure due to the vast applications of fuzzy metric spaces in controlling the noise in data, smoothing the data, and decision-making, but the authors did not pay attention to study the best proximity point theory in fuzzy metric spaces. In 2012, N. Saleem et al. investigated best proximity and coincidence point results in fuzzy metric spaces [10][11][12][13][14][15].
Among the several generalization of fixed and best proximity point theory, one is to generalize the contractive conditions and generalize the underlying spaces. Also, researcher try to study the best proximity point results for multivalued mapping (this was not an easy task). Several authors obtained best proximity points for multivalued mapping, for details, see [13]).
In generalization of contractive conditions, the existence and convergence of best proximity points were discussed by various author (for details, see [16][17][18][19]).
We will use the following notions in our main results.
. Let X be a nonempty set and the mapping d b : where s is any real number such that s ≥ 1, then (X, d) is known as b-metric space.

Definition 2 ([2]
). Let X be a b-metric space and u ∈ X, then

•
A sequence {u n } is convergent and converges to u in X if, for every ε > 0, there exists n 0 ∈ N such that d b (u n , u) < ε, for all n > n 0 , is represented as lim n→∞ u n = u or u n → u as n → ∞.
In 2012, Samet et al. [30] introduced the concept of α-ψ-contraction and α-admissible mapping and proved various fixed point theorems. Further, Samet introduced the concept of α-admissible mapping, defined as follows.
(Θ 1 ) θ is continuous and increasing function; (Θ 2 ) lim n→∞ α n = 0 if and only if lim n→∞ θ(α n ) = 1; here, α n is a sequence from the domain of θ, A function θ ∈ θ if it satisfies the properties Θ 1 − Θ 3 and a function θ ∈ θ * if θ satisfies all the conditions of θ and additional property Θ * . Now, we are going to define some classes of comparison functions which carry some particular properties as follows.
A function ψ ∈ Ψ 1 is called comparison function, which is continuous at u = 0, and for any p ≥ 1, p th -iteration of a comparison function ψ is also a comparison function, further for any positive u ψ(u) < u. (b) Ψ 2 is class of functions, consisting upon the nondecreasing functions ψ, and ∑ ∞ n=1 ψ n (u) is finite, for all u > 0. Clearly, Ψ 2 ⊆ Ψ 1 . (c) Ψ 3 is class of functions, consisting upon increasing functions, and there exists n 0 ∈ N, a ∈ (0, 1) and a series of non-negative numbers is convergent ∑ ∞ n=1 u n , such that for any u ≥ 0, ψ n+1 (u) ≤ aψ n (u) + u n for all n ≥ n 0 .
The function ψ ∈ Ψ 3 is known as a c-comparison function. (d) Ψ 4 is class of function, consisting upon monotone increasing functions and there exists an n 0 ∈ N, a ∈ (0, 1), s ∈ [1, ∞) and a convergent series of non-negative numbers ∑ ∞ n=1 u n such that for any u ≥ 0, s n+1 ψ n+1 (u) ≤ as n ψ n (u) + u n for all n ≥ n 0 .
The function ψ ∈ Ψ 4 is known as a b-comparison function.

Lemma 1 ([34]
). If ψ is a b-comparison function with s ≥ 1, then the series ∑ ∞ n=0 s n ψ n (u) is convergent for u > 0 and the function r b (u) = ∞ ∑ n=0 s n ψ n (u) : R + → R + is increasing and continuous at u = 0.
Note that through out this article, we assume that d b (b-metric) is continuous.

Main Results
Now, we will introduce the Suzuki-type α-ψ g -modified proximal contraction and Suzuki-type α-ψ-modified proximal contraction as follows.

1.
A pair of mappings (g, T) where g : U → U and T : U → CB(V) is called Suzuki-type α-ψ g -modified proximal contraction, if T is α-proximal admissible, and

2.
A mapping T : Note that from now an onward, we will use for all u, v ∈ U, and CB(V) denotes the closed and bounded subsets of V. Our first result related with "coincidence best proximity point" for a pair of mappings (g, T), which satisfy Suzuki-type α-ψ g -modified proximal contraction is as follows. Theorem 1. Let U and V be nonempty and closed subsets of a complete b-metric space (X, d b ). Consider a pair of continuous mappings (g, T) that satisfy Suzuki-type α-ψ g -modified proximal contractive condition with , where g is an isometry mapping satisfying α R -property. Also, the pair of subsets (U, V) satisfies the weak P-property. Further suppose that there exist some u 0 , u 1 ∈ U 0 , such that then, mappings (g, T) has a unique coincidence best proximity point.
As T is α-proximal admissible, we have α(gu 1 , gu 2 ) ≥ 1; also, g satisfies α R -property, and therefore α(gu 1 , gu 2 ) ≥ 1 implies α(u 1 , u 2 ) ≥ 1. Further, As α(u 0 , u 1 ) ≥ 1 and the pair of mappings (g, T) are Suzuki-type α-ψ g -modified proximal contractions, we have where As the pair of sets (U, V) satisfies the weak P-property and the mapping g is an isometry mapping, we have which shows that u 0 is the coincidence best proximity point of pair (g, T) and the proof is complete.
which holds true if u 2 = u 1 , then proof is finished, and we will obtain u 1 as a "coincidence best proximity point" of the mappings g and T, so from (3), we have If u 2 = u 1 , then from (7), which is a contradiction, therefore Thus, there exist some q > 1 such that where t 0 = d(u 0 , u 1 ). Now, consider two distinct elements, from (9), we can write ψ(d b (u 1 , u 2 )) < ψ(qψ(t 0 )) as ψ ∈ Ψ 4 . If we set q 1 = ψ(qψ(t 0 )) ψ(d b (u 1 ,u 2 )) , then q 1 > 1. If u 3 = u 2 then from (10), u 2 will be the coincidence best proximity point of mappings g and T, then the proof of theorem is finished. Now, consider u 3 = u 2 , then we have After simplification, we have As α(u 1 , u 2 ) ≥ 1 and mapping T is Suzuki-type α-ψ g -modified proximal contraction, then we have where As the pair of sets (U, V) satisfies the weak P-property and mapping g is isometry, so we have which holds true if u 2 = u 3 ; in this case, u 2 becomes coincidence best proximity point for pair of mappings (g, T) and the proof is finished. If u 2 = u 3 , then inequality (14) implies As ψ ∈ Ψ 4 , then, from inequality (16), we have Continuing in this way, we can obtain a sequence {u n } in U 0 such that Then, 1 2s 2 D * b (gu n−1 , Tu n−1 ) < D b (u n−1, u n ).
As α(u n−1 , u n ) ≥ 1 and mapping T is Suzuki-type α-ψ g -modified proximal contractive condition, we can write where Therefore, we have As the pair of sets (U, V) satisfies the weak P-property and g is isometry mapping, we have If u n 0 = u n 0 +1 for some n 0 ∈ N, then, from (17), we have which shows that u n 0 is the coincidence best proximity point of pair (g, T). Suppose u n = u n+1 , (20) can be written as which is a contradiction, therefore max{d b (u n−1 , u n ), d b (u n , u n+1 )} = d b (u n−1 , u n ); then, from inequality (20), we have and where t 0 = d(u 0 , u 1 ). Now, we have to prove that {u n } is a Cauchy sequence in U. Note that That is, Assume S n = ∑ n i=0 s i ψ i (qψ(t 0 )). Then, the above inequality can be written as It follows from Lemma (1) that ∑ ∞ i=0 s i ψ i (t) converges for any t ≥ 0. Thus, lim n→∞ S n−2 = S, for some S ∈ [0, ∞). If s = 1, then from inequality (24), we have If s > 1, then from inequality (24), we have Therefore, lim m,n→∞ d b (u n , u m ) = 0 and {u n } is a Cauchy sequence in U 0 . As U 0 is a closed subset of complete b− metric space (X, d b ), then there exist z ∈ U 0 ⊆ X, such that d b (u n , z) → 0, as n → ∞.
As g, T are continuous mappings, we can deduce that H(Tu n , Tz) → 0, as n → ∞. Therefore, which shows that z is the coincidence best proximity point of pair (g, T).
For the uniqueness of coincidence best proximity point of T, suppose to the contrary that u, v ∈ U 0 are two coincidence best proximity points of pair (g, T) with u = v, so we have As the pair (U, V) satisfies the weak P-property and mapping g is isometry, then we have Here, After simple calculations, which is a contradiction, and therefore the coincidence best proximity point is unique.
In our next result, we proved the existence and uniqueness of best proximity point for Suzuki-type α-ψ-modified proximal contraction T in complete b-metric space.
Theorem 2. Let U and V be nonempty closed subsets of a complete b-metric space X. Consider a continuous mapping, T, that satisfies the Suzuki-type α-ψ-modified proximal contractive condition, and T(U 0 ) ⊆ V 0 . Also, the pair of subsets (U, V) satisfy the weak P-property. Further, suppose that there exist some u 0 , then mapping T has a unique best proximity point.
Proof. By taking mapping g = I U (identity mapping over U is isometry mapping), the remaining proof is in line with Theorem (1).
The following example is presented to elaborate the result presented in Theorem (2).  For all u 1 , u 2 ∈ U 0 ⊆ U and v 1 , v 2 ∈ V 0 ⊆ V; further, pair (U, V) satisfies weak P-property, as (X, d b ) is b-metric with s = 2. Now, consider a mapping, T : U → CB(V), defined as clearly T(U 0 ) ⊆ V 0 . Now, we have to show that mapping T satisfy the Suzuki-type α-ψ-modified proximal contraction. The following part of Suzuki-type α-ψ-modified proximal contraction holds for all u, v ∈ U 0 , Now, we must show that the second part of Suzuki-type α-ψ-modified proximal contraction holds for all u, v ∈ U 0 α(u, v)H(Tu, Tv) ≤ ψ (M(u, v)). then, after simple calculation, inequality (25) holds true for all u = v ∈ U 0 . By considering s ≥ 2, α(u, v) = 1 for all u, v ∈ U, and ψ(t) = 999 1000 t ∈ Ψ 4 , then inequality (26) holds true for all u, v ∈ U, which shows that T satisfy the Suzuki-type α-ψ-modified proximal contractive condition; further, all conditions of Theorem (1) hold true, therefore T has best proximity points in U.

Corollary 1.
Let U, V be two nonempty and closed subsets of a complete b-metric space X. Suppose T : U → V be a continuous Suzuki-type α-ψ-modified proximal contraction with T(U 0 ) ⊆ V 0 and pair (U, V) satisfies the weak P-property. Further, suppose that if there exist some u 0 , u 1 ∈ U 0 , such that then mapping T has a unique best proximity point.  d(u, v)).
for all u, v ∈ U. Further, if there exist some u 0 , u 1 ∈ U 0 , such that D b (u 1 , Tu 0 ) = d b (U, V) and α(u 0 , u 1 ) ≥ 1, then mapping T has unique best proximity point.

Proof. After simple calculations, we have
and the rest proof of this corollary is on the same lines as Theorem (1).

Remark 2.
It is clear that all the above results hold for complete metric space by taking s = 1.

Suzuki Type α-θ-Modified Proximal Contractive Mapping
This section is dedicated to stating and proving the coincidence best proximity point result for Suzuki-type α-θ g -modified proximal contraction.

Definition 10.
A pair of mappings (g, T), where g : U → U and T : U → CB(V), is said to satisfy the following.  In our next result, we will state and prove a coincidence best proximity point theorem for Suzuki-type α-θ g -modified proximal contraction in complete b-metric space. Theorem 3. Suppose U and V are nonempty closed subsets of a complete b-metric space (X, d b ) with U 0 = ∅. Suppose a pair of continuous mappings (g, T) of Suzuki-type α-θ g -modified proximal contraction, where T : U → CB(V) and g : U → U. Moreover, g is isometry mapping satisfying α R -property; further, T(U 0 ) ⊆ V 0 , U 0 ⊆ g(U 0 ) and (U, V) satisfy the weak P-property, and suppose that there exist u 0 , u 1 ∈ U 0 , such that

Suzuki-type α-θ g -modified proximal contraction, if T is α-proximal admissible
Then, pair (g, T) has a unique coincidence best proximity point.
Proof. Let u n be the n th term of the sequence {u n } generated by following the same line of proof as in Theorem (1), we can construct a sequence {u n } in U 0 , satisfying the following, As pair (g, T) is Suzuki-type α-θ g -modified proximal contraction, then we have θ(H(Tu n−1 , Tu n )) ≤ α(u n−1 , u n )θ(H(Tu n , Tu n−1 )) ≤ r s (M g (u n , u n−1 )) k .
As α(u n−1 , u n ) ≥ 1, using (19) from Theorem (1), we have Choose a real number r 1 such that 0 ≤ r < t < r 1 < 1, with 1 √ r 1 > 1; also, u n−1 and u n are the given points in U 0 . As pair (U, V) satisfies the weak P-property, θ is increasing, and If max{d b (u n−1 , u n ), d b (u n , u n+1 )} = d b (u n , u n+1 ), then from above inequalities, we have holds true if u n = u n+1 , then u n is a coincidence best proximity point of pair (g, T) and proof is finished; if u n = u n+1 , then it is a contradiction, as √ r 1 < 1 and s > 1. Therefore, we have Set r = √ r 1 s as r < 1 and rs = √ r 1 < 1, it follows from Lemma (2) that {u n } is a Cauchy sequence in U 0 , where U 0 is closed subset of complete b-metric space (X, d b ). Thus, there exists an element z ∈ U 0 ⊆ U, such that u n → z, as n → ∞. As g and T are continuous mappings, Tu n → Tz as n → ∞, which implies that as required. Uniqueness: On the contrary, suppose that pair of mappings (g, T) has more that one coincidence best proximity points, suppose u and v are two distinct coincidence best proximity points of mappings (g, T), so we have As the pair (U, V) satisfy the weak P-property and g is an isometry mapping, we have Here, implies that, a contradiction, therefore the coincidence best proximity point of (g, T) is unique.
Let T : U → CB(V) be a continuous Suzuki-type α-θ-modified proximal contraction. Moreover, T(U 0 ) ⊆ V 0 and (U, V) satisfy the weak P-property, further suppose that there exist u 0 , u 1 ∈ U 0 such that Then, mapping T has a unique best proximity point.
Proof. If we take g = I U (mapping g as Identity on U), the remaining proof follows the same lines.
Clearly, T(U 0 ) ⊆ V 0 and pair (U, V) satisfy the weak P-property. Now, we will show that mapping T satisfy the Suzuki-type α-θ-modified proximal contractive condition: implies that cases (33) and (31) hold. Therefore, u = 2 is the best proximity point of T in U.

Corollary 3.
Let U, V be nonempty closed subsets of a complete b-metric space X. Let mapping T : U → V be a continuous Suzuki-type α-θ-modified proximal contraction with T(U 0 ) ⊆ V 0 , also pair (U, V) satisfies the weak P-property, further suppose that there exist some u 0 , u 1 ∈ U 0 , such that then the mapping T has a unique best proximity point.

Corollary 4.
Let U, V be nonempty closed subsets of a complete b-metric space X and pair (U, V) satisfy weak P-property. Suppose T : U → CB(V) be a continuous, satisfying which implies that, for all u, v ∈ U, r, k ∈ (0, 1) and s ≥ 1. Further, suppose that if there exist some u 0 , u 1 ∈ U 0 such that then mapping T has a unique best proximity point.
Proof. After simple calculations, as discussed in proof of Theorem (3), we have remaining proof of this Corollary is on the same lines as Theorem (3).

Remark 3.
All the above results holds for complete metric space with s = 1, as every b-metric space is a metric space for s = 1.

Results in Partially Ordered B-Metric Space
In this section, we will discuss coincidence best proximity point theorem for modified Suzuki-type contraction in partially ordered b-metric space. Henceforth, we will consider the following notion, for all u 1 , u 2 ∈ U. Definition 13. A pair of mappings (g, T), where g : U → U and T : U → CB(V) is ordered Suzuki-type ψ g -modified proximal contraction, if for u, v ∈ U, Theorem 5. Let U and V be nonempty and closed subsets of a complete partially ordered b-metric space (X, d V , ). Suppose a pair of continuous mappings (g, T) is an ordered Suzuki-type ψ g -modified proximal contraction with T(U 0 ) ⊆ V 0 and U 0 ⊆ g(U 0 ), where g is an isometry mapping satisfying α R -property; also, T is proximal order preserving and pair (U, V) satisfies the weak P-property. Further, suppose that there exist some u 0 , u 1 ∈ U 0 , such that D b (u 1 , Tu 0 ) = d(U, V) and (u 0 , u 1 ) ∈ ∆, then(g, T) has a unique coincidence best proximity point.
As T is α-proximal admissible mapping, as defined below, As T is proximally ordered preserving (u 1 , u 2 ) ∈ ∆, that is, α(u 1 , u 2 ) ≥ 1. As T is proximally ordered preserving, we have Note that if (u, v) ∈ ∆, then α(u, v) = 1; otherwise, α(u, v) = 0. As mapping T is ordered Suzuki-type α-ψ g -modified proximal mapping, we have Let us consider {u n } as a sequence, then α(u n , u n+1 ) ≥ 1 for all n ∈ N ∪ {0} with u n → u as n → ∞, then we can say that (u n , u n+1 ) ∈ ∆, for all n ∈ N ∪ {0}, with u n → u as n → ∞. Therefore, all conditions of Theorem (1) hold and the coincidence best proximity point of mappings (g, T) exist.
Similarly, we can prove the following theorem. Theorem 6. Suppose X, U, U 0 , and V are as in Theorem (5), let pair (g, T) be an ordered Suzuki-type α-θ g -modified proximal contractive mappings, where g : U → U and T : U → CB(V) with all assumptions of Theorem (5). Then unique coincidence best proximity point of mappings (g, T) exist.

Application to Fixed Point Theory
In this section, we will provide some results related fixed point theory for modified Suzuki contraction. Our result extends [21] and also generalize the main theorem of Suzuki [39].
Here, if we consider U = V = X, then we have the following definitions.
Theorem 7. Let (X, d b ) be a complete b-metric space and consider a continuous mapping T : X → CB(X) be a Suzuki-type α-ψ-modified contraction; further, if there exist u 0 with α(u 0 , Tu 0 ) ≥ 1, then mapping T has a unique fixed point.

Proof.
We take U = V = X in Theorem (2), as for self-mapping every proximal Suzuki-type α-ψ-modified contraction becomes Suzuki-type α-ψ-modified contraction, and from (1), for self mapping, every proximal α-admissible mapping becomes α-admissible mapping, all conditions of Theorem (2) are satisfied; therefore, according to Theorem (2), we can find u as a best proximity point of mapping T, which implies that , from above, we can say in case of self-mapping every Suzuki-type α-ψ-modified contraction mapping T has a unique fixed point.
Theorem 8. Suppose X be a complete b-metric space and T : X → CB(X) is a Suzuki-type α-θ-modified contraction that satisfies all the conditions of Theorem (7). Then, T has a unique fixed point.

Proof.
We take U = V = X in Theorem (4), as for self-mapping every proximal Suzuki-type α-θ-modified contraction becomes Suzuki type α-θ-modified contraction, and from (1), for self mappings, every proximal α-admissible mapping becomes α-admissible mapping, all conditions of Theorem (4) are satisfied; therefore, according to Theorem (4), we can find u a best proximity point of mapping T, which implies Tu); therefore, for self-mapping, every Suzuki-type α-θ-modified contraction mapping T has a unique fixed point.
Theorem 9. Let (X, d b , ) is a complete partially ordered b-metric space, consider an increasing continuous mapping T : X → CB(X) be an ordered Suzuki-type ψ-modified contraction with u 0 ∈ X, such that (u 0 , Tu 0 ) ∈ ∆, then T has a unique fixed point.
Proof. Following the same lines of proof of Theorem (5), and taking in account for self-mapping such that (u 0 , Tu 0 ) ∈ ∆, we have α(u 0 , Tu 0 ) = 1, then every ordered Suzuki-type α-ψ-modified contraction becomes ordered Suzuki-type ψ-modified contraction and the remaining conditions of Theorem (5) holds. Then, T has a unique fixed point.
Finally, we have a fixed point theorem for Suzuki-type ordered θ-modified contraction in complete partial ordered b-metric space: Theorem 10. Let (X, d b , ) is a complete partially ordered b-metric space and T : X → CB(X) is Suzuki-type ordered θ-modified contraction satisfying the condition of Theorem (9), then T has a unique fixed point.

Conclusions
In this article, a multivalued Suzuki-type α-ψ g -modified proximal contraction and Suzuki-type α-ψ-modified proximal contraction are introduced; further, some coincidence best proximity point and best proximity point results are proved, which generalized the main results in [40] in the sense of b-metric space. Some of the best proximity point results are also proved for multivalued Suzuki-type α-ψ-modified proximal contraction and Suzuki-type α-θ-modified proximal contraction. Further, some coincidence best proximity point theorem for multivalued modified Suzuki-type contraction in partially ordered b-metric space are proved. An application of the main results related to fixed point theorems for modified Suzuki contraction are presented. The obtained results extend from those in [21] and also generalized the main theorem of T. Suzuki ([39]). Some examples are presented to explain and support the obtained results.