@ Appl. Gen. Topol. 17, no. 2(2016), 173-183 doi:10.4995/agt.2016.5180 c© AGT, UPV, 2016 Global optimization using α-ordered proximal contractions in metric spaces with partial orders Somayya Komal a and Poom Kumam a,b,∗ a Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT),126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok 10140, Thai- land (somayya.komal@mail.kmutt.ac.th, poom.kumam@mail.kmutt.ac.th) b Theoretical and Computational Science (TaCS) Center, Science Laboratory Building, Faculty of Science King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok 10140, Thailand (poom.kumam@mail.kmutt.ac.th) Abstract The purpose of this article is to establish the global optimization with partial orders for the pair of non-self mappings, by introducing a new type of contractions like α-ordered contraction and α-ordered proxi- mal contraction in the frame work of complete metric spaces. Also to calculate some fixed point theorems with the help of these generalized contractions. In addition, to establish an example which shows the va- lidity of our main result. These results extend and unify many existing results in the literature. 2010 MSC: 58C30; 47H10. Keywords: common best proximity point; global optimal approximate so- lution; proximally increasing mappings; α-ordered contractions; α-ordered proximal contraction; α-ordered proximal cyclic con- traction. 1. Introduction It is obvious that best proximity point serves as an optimal approximate solution to the equation Zx = x, where Z is a non-self mapping from any two non-empty subsets of a metric space, a normed linear space or any other ∗Corresponding author: poom.kumam@mail.kmutt.ac.th and poom.kum@kmutt.ac.th Received 25 March 2016 – Accepted 19 July 2016 http://dx.doi.org/10.4995/agt.2016.5180 S. Komal and P. Kumam topological space. Also it is very interesting point that best proximity point theorems actually generalize the fixed point theorems in natural fashion by taking self mapping instead of non-self mapping in best proximity point theo- rem then we can get fixed point. Since d(x,Zx) ≥ d(A,B), for any x ∈ A, we obtain the global minimum of the mapping x 7→ d(x,Zx) as a best proximity point. For more details on this approach, we refer the reader to [2], [3], [4], [5], [6], [10], [7], [13], [11], [12], [14], [16], [1] and [15]. The basic purpose of this article is to establish some generalized notions and to derive new theorem of global optimization with partial orders in metric spaces. We have defined in this work an α-ordered contraction to find common best proximity points. The motivation of this paper is [9], we generalized that contraction of [9]. Also presented an example to verify the results. 2. Preliminaries In this section let us take that A and B are non-void subsets of a metric space (X,d). we recall some definitions and notations in this section which will be used throughout this work. Definition 2.1 ([8]). Let X be a metric space, A and B two nonempty subsets of X. Define d(A,B) = inf{d(a,b) : a ∈ A,b ∈ B}, A0 = {a ∈ A : there exists some b ∈ B such that d(a,b) = d(A,B)}, B0 = {b ∈ B : there exists some a ∈ A such that d(a,b) = d(A,B)}. Definition 2.2 ([8]). Given non-self mappings S : A → B and T : A → B, an element x∗ is called common best proximity point of the mappings if this condition satisfied: d(x∗,Sx∗) = d(x∗,Tx∗) = d(A,B). We noticed here that common best proximity point is that element at which both functions S and T attain their global minimum, since d(x,Sx) ≥ d(A,B) and d(x,Tx) ≥ d(A,B) for all x. Definition 2.3 ([9]). A mapping S : A → B is said to be an ordered contrac- tion if there exists a non-negative real number γ < 1 such that x1 � x2 ⇒ d(Sx1,Sx2) ≤ γd(x1,x2), for all x1,x2 ∈ A. Definition 2.4 ([9]). A mapping S : A → B is said to be an ordered proximal contraction if there exists γ < 1 such that x1 � x2, d(u1,Sx1) = d(A,B) and d(u2,Sx2) = d(A,B), implies that d(u1,u2) ≤ γd(x1,x2), for all u1,u2,x1,x2 ∈ A. c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 2 174 Global optimization using α-ordered proximal contractions in metric spaces with partial orders Definition 2.5 ([9]). Given non-self mappings S : A → B and T : B → A, the pair (S,T) is said to form an ordered proximal cyclic contraction if there exists a non-negative real number k < 1 such that x � y, d(u,Sx) = d(A,B) and d(v,Ty) = d(A,B), implies that d(u,v) ≤ kd(x,y) + (1 −k)d(A,B), for all u,x ∈ A and v,y ∈ B. Definition 2.6 ([9]). Given non-self mappings S : A → B and T : B → A, the pair (S,T) is said to be proximally increasing if x � y, d(u,Sx) = d(A,B) and d(v,Ty) = d(A,B), implies that u ≤ v, for all u,x ∈ A and v,y ∈ B. Definition 2.7 ([9]). Given non-self mapping S : A → B is said to be proxi- mally increasing if it satisfies the condition: x � y, d(u,Sx) = d(A,B) and d(v,Sy) = d(A,B), implies that u ≤ v, for all u,v,x,y ∈ A. Definition 2.8 ([9]). Given non-self mapping S : A → B is said to be increas- ing if it satisfies the condition: x � y, implies that Sx ≤ Sy, for all x,y ∈ A. Similarly, iteratively Snx ≤ Sny, for n ∈ N. 3. Main Results Now, we are in position to define some notions and to prove some results. Definition 3.1. A mapping S : A → B is said to be an α-ordered contraction if there exists β ∈F and α : X ×X → R+ be a function such that x1 � x2 ⇒ α(x1,x2)d(Sx1,Sx2) ≤ β(d(x1,x2))d(x1,x2), for all x1,x2 ∈ A. We denote by F the class of all functions β : [0,∞) → [0, 1) satisfying β(tn) → 1, implies tn → 0 as n →∞. c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 2 175 S. Komal and P. Kumam Definition 3.2. A mapping S : A → B is said to be an α-ordered proximal contraction if there exists β ∈F and α : X ×X → R+ such that x1 � x2, d(u1,Sx1) = d(A,B) and d(u2,Sx2) = d(A,B), implies that α(x1,x2)d(u1,u2) ≤ βd(x1,x2), for all u1,u2,x1,x2 ∈ A. Definition 3.3. Given non-self mappings S : A → B and T : B → A, the pair (S,T) is said to form an α-ordered proximal cyclic contraction if there exists a non-negative real number k < 1 such that x � y, d(u,Sx) = d(A,B) and d(v,Ty) = d(A,B), implies that α(x,y)d(u,v) ≤ kd(x,y) + (1 − k)d(A,B), for all u,x ∈ A and v,y ∈ B. Theorem 3.4. Let X be a non-empty set such that (X,�) is a partially ordered set and (X,d) is a complete metric space, α : X ×X → R+ be a function and let A,B be nonempty closed subsets of (X,d) such that A0 and B0 are non- void. Let S : A → B, T : B → A and g : A∪B → A∪B satisfy the following conditions: (1) S and T are α-ordered proximal contractions, proximally increasing; (2) g is surjective isometry, its inverse is an increasing mapping; (3) The pair (S,T) is proximally increasing, α-ordered proximal cyclic con- traction; (4) S(A0) ⊆ B0, T(B0) ⊆ A0; (5) A0 ⊆ g(A0) and B0 ⊆ g(B0); (6) S and T are α-proximal admissible maps; (7) α(x0,x1) ≥ 1 for x0,x1 ∈ X; (8) There exist elements x0 and x1 in A0 and y0,y1 ∈ B0 such that d(gx1,Sx0) = d(A,B), and d(gy1,Ty0) = d(A,B). x0 � x1, y0 � y1, x0 � y0. (9) If {xn} is an increasing sequence of elements in A converging to x, then xn � x, for all n. Also, if {yn} is an increasing sequence of elements in B converging to y, then yn � y for all n. c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 2 176 Global optimization using α-ordered proximal contractions in metric spaces with partial orders Then there exists a point x ∈ A and a point y ∈ B such that d(gx,Sx) = d(gy,Ty) = d(x,y) = d(A,B). Moreover, the sequence {xn} in A0, defined by d(gxn+1,Sxn) = d(A,B) (n ≥ 1), converges to the element x, and the sequence {yn} in B0, defined by d(gyn+1,Tyn) = d(A,B) (n ≥ 1), converges to the element y. Proof. Since α(x0,x1) ≥ 1 for x0,x1 ∈ X, and for x1 ∈ A0, S(A0) ⊆ B0 there exists x2 ∈ A0 such that d(x2,Sx1) = d(A,B), for x2 ∈ A0, S(A0) ⊆ B0 there exists x3 ∈ A0 such that d(x3,Sx2) = d(A,B). Since S is α-proximal admissible mapping, then from d(x2,Sx1) = d(A,B) d(x3,Sx2) = d(A,B), implies that α(x2,x3) ≥ 1. Proceeding in the same manner, we have α(xn,xn+1) ≥ 1, for n ∈ N. The hypothesis (8) implies the existence of elements x0 and x1 in A0 such that d(gx1,Sx0) = d(A,B) and x0 � x1. In view of the fact that S(A0) ⊆ B0, also it is given that A0 ⊆ g(A0), there exists an element x2 ∈ A0 such that d(gx2,Sx1) = d(A,B). Since S is proximally increasing, gx1 � gx2. As the inverse of mapping g is increasing, so x1 � x2. Again, since S(A0) ⊆ B0 and A0 ⊆ g(A0), there exists an element x3 ∈ A0 such that d(gx3,Sx2) = d(A,B). Continuing in a similar fashion, one can find an element xn in A0 such that d(gxn,Sxn−1) = d(A,B) and xn−1 � xn. In light of the fact that g is an isometry and that S is α-ordered proximal contraction, we obtain d(xn,xn+1) = d(gxn,gxn+1) ≤ α(xn−1,xn)d(gxn,gxn+1) ≤ β(d(xn−1,xn))d(xn−1,xn). c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 2 177 S. Komal and P. Kumam This shows that {d(xn+1,xn)} is a decreasing sequence and bounded below. Hence there exists r ≥ 0 such that limn→∞d(xn+1,xn) = r. Suppose that r > 0. Observed that d(xn+1,xn) d(xn,xn−1) ≤ β(d(xn−1,xn)). Taking limit as n →∞, we get lim n→∞ β(d(xn,xn−1)) = 1. Since β ∈ F, so that r = 0, which is a contradiction to our supposition and hence (3.1) lim n→∞ d(xn,xn−1) = 0. Now, we claim that {xn} is a Cauchy sequence. Suppose that {xn} is not Cauchy sequence. Then there exists � > 0 and subsequences {xmk},{xnk} of {xn} such that for any positive integers nk > mk ≥ k rk := d(xmk,xnk ) ≥ �, d(xmk,xnk−1) < �, for any k ∈{1, 2, 3, ...}. For each n ≥ 1, let αn := d(xn+1,xn). Then, we have � ≤ rk = d(xmk,xnk ) ≤ d(xmk,xnk−1) + d(xnk−1,xnk ) < � + γnk−1.(3.2) Taking limit as k →∞, we get � ≤ lim k→∞ rk < � + lim k→∞ γnk−1. (3.3) It follows that � ≤ lim k→∞ rk < � + 0 (3.4) lim k→∞ d(xmk,xnk ) = �. Notice also that � ≤ rk = d(xmk,xnk ) ≤ d(xmk,xmk+1) + d(xnk+1,xnk ) + d(xmk+1,xnk+1) = γmk + γnk + d(xmk+1,xnk+1) = γmk + γnk + d(gxmk+1,gxnk+1) ≤ γmk + γnk + α(xmk,xnk )d(gxmk+1,gxnk+1) ≤ γmk + γnk + β(d(xmk,xnk ))d(xmk,xnk ), c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 2 178 Global optimization using α-ordered proximal contractions in metric spaces with partial orders implies that d(xmk,xnk ) −γmk −γnk d(xmk,xnk ) ≤ β(d(xmk,xnk )). Taking limit as k →∞, we obtain lim k→∞ β(d(xmk,xnk )) = 1, since β ∈F, so lim k→∞ d(xmk,xnk ) = 0. Hence � = 0, which is a contradiction. So {xn} is a Cauchy sequence and converges to some element x ∈ A. So, we have xn � x for any n. Similarly, in view of the fact that T(B0) ⊆ A0 and B0 ⊆ g(B0), it is ascertained that there is a sequence {yn} of elements in B0 such that (gyn+1,Tyn) = d(A,B). Since T is proximally increasing and the inverse of g is an increasing mapping, yn � yn+1. Since g is an isometry and T is an α-ordered proximal contraction, it follows that d(yn,yn+1) = d(gyn,gyn+1) ≤ α(yn−1,yn)d(gyn,gyn+1) ≤ β(d(yn−1,yn))d(yn−1,yn). Similarly, there exists a Cauchy sequence {yn} such that it converges to some element y ∈ B. Therefore, it follows that yn � y for all n. Further, since the pair (S,T) is proximally increasing and the inverse of g is an increasing mapping, we have xn � yn, for all n. Since the pair (S,T) forms an α-ordered proximal cyclic contraction and g is an isometry, it follows that d(xn+1,yn+1) = d(gxn+1,gyn+1) ≤ α(xn,yn)d(gxn+1,gyn+1) ≤ kd(xn,yn) + (1 −k)d(A,B). Letting n →∞, it follows that d(x,y) = kd(x,y) + (1 −k)d(A,B) (3.5) ⇒ d(x,y) = d(A,B). Thus x ∈ A0 and y ∈ B0. Since S(A0) ⊆ B0 and T(B0) ⊆ A0, there exists u ∈ A and v ∈ B such that (3.6) d(u,Sx) = d(A,B) d(v,Ty) = d(A,B). } Since S is α-ordered proximal contraction, we get from d(u,Sx) = d(A,B) and d(gxn+1,Sxn) = d(A,B) as (3.7) d(u,gxn+1) ≤ α(xn,x)d(u,gxn+1) ≤ β(d(x,xn))d(x,xn). c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 2 179 S. Komal and P. Kumam Letting n →∞ in the above inequality, we have d(u,gx) = 0 and so u = gx. It follows that {gxn} converges to u. Further, as g is an isometry, the sequence {gxn} converges to gx as well. Thus, we write as d(gx,Sx) = d(u,Sx) = d(A,B). In the same manner, we have v = gy and so it can be prove that d(gy,Ty) = d(v,Ty) = d(A,B). � Example 3.5. Consider X = R2 be an Euclidean metric space with partially ordered set X. Let us define the sets A = {1}× [0,∞) and B = {2}× [0,∞). Take A0 = A and B0 = B. Obviously, d(A,B) = 1. Let g : A∪B → A∪B be an identity mapping, the mapping g is surjective isometry, its inverse is an increasing mapping, A0 ⊆ g(A0) and B0 ⊆ g(B0). Let us define S : A → B and T : B → A as: S(1,x) = (2, x x + 1 ), and T(2,x) = (1, x x + 1 ). where (1,x) ∈ A, (2,x) ∈ B and x ∈ [0,∞). Let α : R2 ×R2 → [0,∞) defined as: α(x,y) = { 1 if x=1 or x=2 and y ∈ [0,∞), 0 elsewhere. Clearly, S and T are proximally increasing and α-ordered proximal contractions with these assumptions such that S(A0) ⊆ B0 and T(B0) ⊆ A0. The pair (S,T) is proximally increasing, α-ordered proximal cyclic contraction. Thus, all other assumptions of the Theorem (3.1) are also satisfied. Finally, very easily one can say that the element (1, 0) in A and the element (2, 0) in B satisfy the conclusion of the preceding result. If g is the identity mapping in the Theorem 3.4, then we obtain the following: Corollary 3.6. Let X be a non-empty set such that (X,�) is a partially ordered set and (X,d) is a complete metric space, α : X ×X → R+ be a function and let A,B be nonempty closed subsets of (X,d) such that A0 and B0 are non-void. Let S : A → B, T : B → A satisfy the following conditions: (1) S and T are α-ordered proximal contractions, proximally increasing; (2) The pair (S,T) is proximally increasing, α-ordered proximal cyclic con- traction; (3) S(A0) ⊆ B0, T(B0) ⊆ A0; (4) S and T are α-proximal admissible maps; (5) α(x0,x1) ≥ 1 for x0,x1 ∈ X; c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 2 180 Global optimization using α-ordered proximal contractions in metric spaces with partial orders (6) There exist elements x0 and x1 in A0 and y0,y1 ∈ B0 such that d(gx1,Sx0) = d(A,B), and d(gy1,Ty0) = d(A,B). x0 � x1, y0 � y1, x0 � y0. (7) If {xn} is an increasing sequence of elements in A converging to x, then xn � x, for all n. Also, if {yn} is an increasing sequence of elements in B converging to y, then yn � y for all n. Then there exists a point x ∈ A and a point y ∈ B such that d(x,Sx) = d(y,Ty) = d(x,y) = d(A,B). Moreover, the sequence {xn} in A0, defined by d(xn+1,Sxn) = d(A,B) (n ≥ 1), converges to the element x, and the sequence {yn} in B0, defined by d(yn+1,Tyn) = d(A,B) (n ≥ 1), converges to the element y. If α(x0,x1) = 1 and β(t) = k, where k ∈ [0, 1) in the Corollary (3.1), then we obtain the following corollary of [9]. Corollary 3.7. Let X be a non-empty set such that (X,�) is a partially ordered set and (X,d) is a complete metric space, let A,B be nonempty closed subsets of (X,d) such that A0 and B0 are non-void. Let S : A → B, T : B → A satisfy the following conditions: (1) S and T are ordered proximal contractions, proximally increasing; (2) The pair (S,T) is proximally increasing, ordered proximal cyclic con- traction; (3) S(A0) ⊆ B0, T(B0) ⊆ A0; (4) There exist elements x0 and x1 in A0 and y0,y1 ∈ B0 such that d(gx1,Sx0) = d(A,B), and d(gy1,Ty0) = d(A,B). x0 � x1, y0 � y1, x0 � y0. (5) If {xn} is an increasing sequence of elements in A converging to x, then xn � x, for all n. Also, if {yn} is an increasing sequence of elements in B converging to y, then yn � y for all n. Then there exists a point x ∈ A and a point y ∈ B such that d(x,Sx) = d(y,Ty) = d(x,y) = d(A,B). Moreover, the sequence {xn} in A0, defined by d(xn+1,Sxn) = d(A,B) (n ≥ 1), c© AGT, UPV, 2016 Appl. Gen. Topol. 17, no. 2 181 S. Komal and P. Kumam converges to the element x, and the sequence {yn} in B0, defined by d(yn+1,Tyn) = d(A,B) (n ≥ 1), converges to the element y. By taking α(x0,x1) = 1 and β(t) = k, where k ∈ [0, 1) in the Theorem (3.1), we get the main result of [9] as: Corollary 3.8. Let X be a non-empty set such that (X,�) is a partially ordered set and (X,d) is a complete metric space and let A,B be nonempty closed subsets of (X,d) such that A0 and B0 are non-void. Let S : A → B, T : B → A and g : A∪B → A∪B satisfy the following conditions: (1) S and T are ordered proximal contractions, proximally increasing; (2) g is surjective isometry, its inverse is an increasing mapping; (3) The pair (S,T) is proximally increasing, ordered proximal cyclic con- traction; (4) S(A0) ⊆ B0, T(B0) ⊆ A0; (5) A0 ⊆ g(A0) and B0 ⊆ g(B0); (6) There exist elements x0 and x1 in A0 and y0,y1 ∈ B0 such that d(gx1,Sx0) = d(A,B), and d(gy1,Ty0) = d(A,B). x0 � x1, y0 � y1, x0 � y0. (7) If {xn} is an increasing sequence of elements in A converging to x, then xn � x, for all n. Also, if {yn} is an increasing sequence of elements in B converging to y, then yn � y for all n. Then there exists a point x ∈ A and a point y ∈ B such that d(gx,Sx) = d(gy,Ty) = d(x,y) = d(A,B). Moreover, the sequence {xn} in A0, defined by d(gxn+1,Sxn) = d(A,B) (n ≥ 1), converges to the element x, and the sequence {yn} in B0, defined by d(gyn+1,Tyn) = d(A,B) (n ≥ 1), converges to the element y. If we take A = B = X, and α(x0,x1) = 1 in our main result (3.3), we get the following fixed point corollary, which is also the result of [9]. Corollary 3.9. Let X be a non-empty set such that (X,�) is a partially ordered set and (X,d) is a complete metric space. Let S : X → X satisfy the following conditions: (1) S is increasing, ordered contraction; (2) There exist elements x0 in A such that x0 � Sx0; (3) If {xn} is an increasing sequence of elements in A converging to x, then xn � x, for all n. 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