On the C-class functions of fixed point and best proximity point results for generalised cyclic coupled mappings

Existence of fixed point for C-class functions was first proved by Ansari in 2014. Then, many authors gave interesting results using C-class functions. In this paper, we prove the existence of strong coupled proximity point for generalised cyclic-coupled proximal maps. Our result generalises the results of Kadwin and Marudai.


Introduction and mathematical preliminaries
Initially, in 1922, Banach proved the existence and uniqueness of fixed point for contraction mapping. Later, among several interesting results given by various authors, Kannan (1969) introduced a kind of mapping which has its own significance, as it also admits fixed point on discontinuous maps. In spite of many authors proving the existence of fixed point on self-mappings, it has been proved by Kirk, Srinivasan and Veeramani (2003) that fixed points do exist on a special kind of map called cyclic maps.
Let A and B be two non-empty subsets of metric space (X, d)

. A mapping T:A ∪ B → A ∪ B is said to be cyclic if T(A) ⊂ B and T(B) ⊂ A.
Meanwhile, another class of mappings called coupled maps were introduced by Lakshmikantham and Ciric (2009) to find coupled fixed point which has wide range of applications to partial differential equations and boundary value problems.
Definition 1.1 An element (x, y) ∈ X × X in a non-empty set X is said to be a coupled fixed point for a mapping F:X × X → X if F(x, y) = x and F(y, x) = y.
These kind of maps were later generalised by Kumam, Pragadeeswarar, Marudai and Sitthithakerngkiet (2014) finding out coupled best proximity points for coupled proximal maps with respect to A and B as non-empty closed subsets of metric space (X, d) with A ∩ B = �. Very recently, (Choudhury & Maity, 2014) extended concept of cyclic maps by introducing cyclic coupled Kannantype contraction as follows.
Definition 1.2 A mapping T:X × X → X is said to be cyclic with respect to A and B if T(A, B) ⊂ B and T(B, A) ⊂ A. Definition 1.3 Let A and B be two non-empty subsets of a metric space (X, d). A mapping F:X × X → X is called cyclic coupled Kannan-type mapping if F is cyclic with respect to A and B satisfying, for some k ∈ 0, 1 2 , the inequality where x, v ∈ A and y, u ∈ B.
Definition 1.4 Let X be a non-empty set. An element (x, x) ∈ X × X is said to be strong coupled fixed The following theorem was proved by Choudhury and Maity (2014 Immediately, Udo-utun (2014) extended the result of (Choudhury & Maity, 2014) using Ciric-type contractions.
The existence and convergence of best proximity points is an interesting topic on optimisation theory on which several interesting results were published (Abkar & Gabeleh, 2013;Aydi, Felhi, & Karapinar, 2016;Eldred & Veeramani, 2006;Gupta, Rajput, & Kaurav, 2014;Latif, Abbas, & Hussain, 2016;Mursaleen, Srivastava, & Sharma, 2016). Such results may sometimes assume a sequential property on metric spaces called UC-property. Definition 1.6 Let A and B be non-empty subsets of a metric space (X, d). Then (A, B) is said to satisfy the UC property if {x n } and {z n } are sequences in A and {y n } is a sequence in B such that lim n→∞ d(x n , y n ) = d(A, B) and lim n→∞ d(z n , y n ) = d(A, B), then lim n→∞ d(x n , z n ) = 0.
In 2014, the concept of C-class functions was introduced by Ansari (2014). Using this concept, we can generalise many fixed point theorems in the literature.
Note for some f we have that f (0, 0) = 0.
We denote C-class functions as .
Let A and B be two non-empty disjoint subsets of a metric space (X, d). A mapping F:X × X → X is called cyclic coupled proximal mappings of type I if F is cyclic with respect to A and B satisfying the inequality where x, v ∈ A and y, u ∈ B for some k ∈ (0, 1).

Definition 1.10
Let A and B be two non-empty disjoint subsets of a metric space (X, d). A mapping F:X × X → X is called cyclic coupled proximal mappings of type II if F is cyclic with respect to A and B satisfying the inequality where x, v ∈ A and y, u ∈ B for some k ∈ 0, 1 2 .
In this paper, we define new generalised cyclic coupled mappings using C-class functions and prove the existence of strong coupled proximity points.

Best proximity points for cyclic coupled mappings
In this part, we introduce cyclic coupled proximal maps and prove the existence of proximity points for those maps under suitable conditions. Definition 2.1 Let A and B be two non-empty disjoint subsets of a metric space (X, d). A mapping F:X × X → X is called cyclic coupled proximal mappings of type I f . if F is cyclic with respect to A and B satisfying the inequality Remark 2.2 With choice F(s, t) = ks, 0<k<1, in Definition 2.1 we obtain Definition 1.9. Definition 2.3 Let A and B be two non-empty disjoint subsets of a metric space (X, d). A mapping F:X × X → X is called cyclic coupled proximal mappings of type II f if F is cyclic with respect to A and B satisfying the inequality where x, v ∈ A and y, u ∈ B for some ∈ Φ, (or ∈ Φ 1 ), f ∈ ℂ.
Remark 2.4 With choice F(s, t) = ks, 0<k<1, in Definition 2.3 we obtain Definition 1.10. Proof Let x 0 ∈ A, y 0 ∈ B be any two arbitrary elements of X. Let {x n } and {y n } be two sequences defined as F(x n , y n ) = y n+1 and F(y n , x n ) = x n+1 . Then, for n = 1, and Similarly, for n = 2, we get In general, we have and For, d(y n−1 , x n )) ≤ d(x n , y n+1 ). Then, Equation (2.1) reduces to For, d(x n , y n+1 ) ≤ d(y n−1 , x n )). Then, Equation (2.1) reduces to Thus, from (2.3) and (2.4), we conclude that

Similarly, Equation (2.2) reduces as and hence
Hence, using (2.5), {d(x 2n , y 2n+1 )} and {d(y 2n−1 , x 2n )} are decreasing sequences and converge to some r ≥ 0 as n→ ∞. Therefore, the Equation ( Therefore, using UC-property, we get lim n→∞ d(x n , x m ) = 0 (i.e. for given > 0, ∃n 0 ∈ ℕ such that ∀ m > n > n 0 , d(x n , x m ) < ). Hence, {x n } is a Cauchy sequence and converges to some point x ∈ A. Similarly, it can be proved that {y n } is a Cauchy sequence and converges to some point y ∈ B.
Since lim m→∞ (x m , y m ) = d(A, B) and d is uniformly continuous, d(x, y) = d (A, B).
Similarly, we can prove that d(y, F(y, x)) = d(A, B) which concludes that (x, y) is the strong coupled proximal point of F. Corollary 2.7 Let (X, d) be a complete metric space and A, B be two non-empty closed subsets of X such that A ∩ B = �. Let F:X × X → X be cyclic coupled proximal mapping of type I. Then F has strong coupled proximal point if (A, B) satisfies UC property.
Proof Letting, f (s, t) = ks, 0 < k < 1, we have Proof Let x 0 ∈ A, y 0 ∈ B be any two arbitrary elements of X. Define F(x n , y n ) = y n+1 and F(y n , x n ) = x n+1 .

.
From above, we have Hence, using (2.12) {d(x 2n , y 2n+1 )} and {d(y 2n−1 , x 2n )} are decreasing sequences and converge to some r ≥ 0 as n→ ∞. Therefore, the Equation ( Using the above arguments, we conclude that Now, using UC-property, we get Claim: {x n } is a Cauchy sequence. Letting m < n and using (2.9), we get Hence, as lim n→∞ d(x n , x m ) = 0.
Therefore, {x n } is a Cauchy sequence and hence converges to some point x ∈ A.

Now, we observe that
Now letting n → ∞, the above inequality reduces to So,