Ansari et al., Cogent Mathematics (2016), 3: 1235354 http://dx.doi.org/10.1080/23311835.2016.1235354
PURE MATHEMATICS | RESEARCH ARTICLE
On the C-class functions of fixed point and best proximity point results for generalised cyclic coupled mappings Received: 03 July 2016 Accepted: 02 September 2016 First Published: 16 September 2016 *Corresponding author: Poom Kumam, KMUTT-Fixed Point Theory and Applications Research Group (KMUTTFPTA), Theoretical and Computational Science Center (TaCS), Science Laboratory Building, Faculty of Science, King Mongkuts University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand; KMUTTFixed Point Research Laboratory, Faculty of Science, Department of Mathematics, King Mongkuts University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand E-mail:
[email protected] Reviewing editor: Hari M. Srivastava, University of Victoria, Canada Additional information is available at the end of the article
Arslan Hojat Ansari1, Geno Kadwin Jacob2, Muthiah Marudai3 and Poom Kumam4,5*
Abstract: 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. Subjects: Foundations & Theorems; Mathematics & Statistics; Science Keywords: cyclic coupled contraction; best proximity point; multivalued mapping; fixed point; C-class function 2010 Mathematics subject classifications: Primary 47H10; Secondary 54H25 1. 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.
ABOUT THE AUTHORS
PUBLIC INTEREST STATEMENT
Arslan Hojat Ansari is a research scholar and PhD student at the Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran. His area of research includes fixed point theory and special functions. Geno Kadwin Jacob is a research scholar at the Department of Mathematics, Bharathidasan University, India. His area of research includes metric fixed point theory and linear complementarity problem. Dr Muthiah Marudai is a professor and the chair at the Department of Mathematics, Bharathidasan University, India. His area of research includes metric fixed point theory and fuzzy analysis. He had published many research articles in international journals. Dr Poom Kumam is the head of Theoretical and Computational Science (TaCS) Center and KMUTTFixed Point Theory and Applications Research Group. His area of research is fixed point theory with applications. He had published more than 350 research articles in international journals around the world.
Many mathematical problems can be formulated as a fixed point equation of the form T(x) = x, where T is a self-mapping in some suitable framework. However, if T is non-self mapping, the abovementioned equation does not necessarily have a solution. In such case, it is worthy to determine an approximate solution x such that the error d(x, T(x)) is minimum.
© 2016 The Author(s). This open access article is distributed under a Creative Commons Attribution (CC-BY) 4.0 license.
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Ansari et al., Cogent Mathematics (2016), 3: 1235354 http://dx.doi.org/10.1080/23311835.2016.1235354
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, 12 , the inequality d(F(x, y), F(u, v)) ≤ k[d(x, F(x, y)) + d(u, F(u, v))].
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 point if F(x, x) = x. The following theorem was proved by Choudhury and Maity (2014). Theorem 1.5 Let A and B be two non-empty closed subsets of a complete metric space (X, d) with A ∩ B ≠ � and F:X × X → X be a cyclic coupled Kannan-type mapping with respect to A and B with A ∩ B ≠ �. Then F has a strong coupled fixed point on A ∩ B. 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, 2016; 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 {xn } and {zn } are sequences in A and {yn } is a sequence in B such that limn→∞ d(xn , yn ) = d(A, B) and limn→∞ d(zn , yn ) = d(A, B), then limn→∞ d(xn , zn ) = 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.
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Ansari et al., Cogent Mathematics (2016), 3: 1235354 http://dx.doi.org/10.1080/23311835.2016.1235354
Definition 1.7 Ansari (2014) A mapping f :[0, ∞)2 → ℝ is called C-class function if it is continuous and satisfies the following axioms: f (s, t) ≤ s; (1)
(2) f (s, t) = s implies that either s = 0 or t = 0; for all s, t ∈ [0, ∞). Note for some f we have that f (0, 0) = 0. We denote C-class functions as . Example 1.8 Ansari (2014), Liu, Ansari, Chandok, and Park (2016) The following functions f :[0, ∞)2 → ℝ are elements of , for all s, t ∈ [0, ∞): (1) f (s, t) = s − t, f (s, t) = s ⇒ t = 0; (2) f (s, t) = ms, 0 e, f (s, 1) = s⇒ s = 0; r
(6) f (s, t) = (s + l)(1∕(1+t) ) − l, l > 1, r ∈ (0, ∞), f (s, t) = s ⇒t = 0; (7) f (s, t) = s logt+a a, a > 1, F(s, t) = s ⇒s = 0 or t = 0; ( )( ) t , f (s, t) = s ⇒ t = 0; (8) f (s, t) = s − 1+s 2+s 1+t
(9) f (s, t) = s𝛽(s), 𝛽:[0, ∞) → [0, 1), and is continuous, f (s, t) = s ⇒ s = 0;
(10) f (s, t) = s −
t , f (s, t) k+t
= s ⇒ t = 0;
(11) f (s, t) = s − 𝜑(s), f (s, t) = s ⇒ s = 0, here 𝜑:[0, ∞) → [0, ∞) is a continuous function such that 𝜑(t) = 0 ⇔ t = 0; (12) f (s, t) = sh(s, t), f (s, t) = s ⇒ s = 0, here h:[0, ∞) × [0, ∞) → [0, ∞) is a continuous function such that h(t, s) < 1 for all t, s > 0; ( ) ( )( ) t , f (s, t) = s ⇒ t = 0; (13) f (s, t) = s − 2+t t, f (s, t) = s ⇒ t = 0. (8) f (s, t) = s − 1+s 1+t 2+s 1+t √ n n (14) f (s, t) = ln(1 + s ), f (s, t) = s ⇒ s = 0; (15) f (s, t) = 𝜙(s), f (s, t) = s ⇒ s = 0, here 𝜙:[0, ∞) → [0, ∞) is a upper semi-continuous function such that 𝜙(0) = 0, and 𝜙(t) < t for t > 0, f (s, t) = (16)
s ; (1+s)r
r ∈ (0, ∞), f (s, t) = s ⇒ s = 0 ;
(17) f (s, t) = 𝜗(s); 𝜗:ℝ+ × ℝ+ → ℝ is a generalised Mizoguchi-Takahashi-type function, f (s, t) = s⇒ s = 0; f (s, t) = (18)
s Γ(1∕2)
∞
∫0
e−x √ x+t
dx, where Γ is the Euler gamma function.
Let Φ denote the set of all functions 𝜑:[0, +∞) → [0, +∞) that satisfy the following conditions:
𝜑 is lower semi-continuous on [0, +∞); (1) 𝜑(0) = 0; (2) 𝜑(s) > 0 for each s > 0. (3) Let Φ1 denote the set of all functions 𝜑:[0, +∞) → [0, +∞) that satisfy the following conditions:
𝜑 is lower semi-continuous on [0, +∞); (1) 𝜑(0) ≥ 0; (2) 𝜑(s) > 0 for each s > 0. (3)
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Ansari et al., Cogent Mathematics (2016), 3: 1235354 http://dx.doi.org/10.1080/23311835.2016.1235354
Let Ψ denote all the functions 𝜓:[0, ∞) → [0, ∞) which satisfy
𝜓 (t) = 0 if and only if t = 0; (i) 𝜓 is continuous; (ii) 𝜓 (s) ≤ s, ∀s > 0. (iii) Definition 1.9 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 d(F(x, y), F(u, v)) ≤ k max[ d(x, F(x, y)), d(u, F(u, v))] + (1 − k) d(A, B)
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 d(F(x, y), F(u, v)) ≤ k[ d(x, F(x, y)) + d(u, F(u, v))] + (1 − 2k) d(A, B) ( ) where x, v ∈ A and y, u ∈ B for some k ∈ 0, 12 .
In this paper, we define new generalised cyclic coupled mappings using C-class functions and prove the existence of strong coupled proximity points.
2. 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 If 𝜑. if F is cyclic with respect to A and B satisfying the inequality ( d(F(x, y), F(u, v)) ≤ f max{ d(x, F(x, y)), d(u, F(u, v))} − d(A, B), ( )) 𝜑 max{ d(x, F(x, y)), d(u, F(u, v))} − d(A, B) + d(A, B),
where x, v ∈ A and y, u ∈ B for some 𝜑 ∈ Φ, (or 𝜑 ∈ Φ1 ), f ∈ ℂ. Remark 2.2 With choice F(s, t) = ks, 0
