Remark on a Couple Coincidence Point in Cone Normed Spaces

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1 International Journal of Mathematical Analysis Vol. 8, 2014, no. 50, HIKARI Ltd, Remark on a Couple Coincidence Point in Cone Normed Spaces Snježana Maksimović Faculty of Electrical Engineering, University of Banja Luka, Patre 5, Banja Luka, Bosnia and Herzegovina Zoran D. Mitrović Faculty of Electrical Engineering, University of Banja Luka, Patre 5, Banja Luka, Bosnia and Herzegovina Copyright c 2014 Snježana Maksimović and Zoran D. Mitrović. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, using the generalization of a Theorem of Ky Fan s, we will derive the results on coupled coincidence points and coupled fixed points in cone normed spaces. Mathematics Subject Classification: 47H10, 54H25 Keywords: Fixed point, Coincidence point, Best approximation, KKM map 1 Introduction The definition of normed space was first given (independently) by Banach, Hahn and Wiener. The theory developed rapidly, as seen from the treatise by Banach published only ten years later. In [5], Rzepecki introduced not only a generalized metric but also a generalized norm. As a generalization of normed spaces, cone normed spaces (in short CNSs) play a very important

2 2462 Snježana Maksimović and Zoran D. Mitrović role in fixed point theory, computer science and some other research areas as well as in function analysis. The main properties of a CNSs seem to be first studied by Sonmez [7], Karapinar [2], and Abdeljawad [11]. In [3], Karapinar and Turkoglu proved Theorem 2.10 for a CNSs using Ky Fan s theorem for topological vector spaces. As a consequence of Theorem 2.3, [6], we have that every cone normed space (in short CNS) satisfies Ky Fan s theorem. Using Ky Fan s theorem we will derive some results on coupled coincidence points and coupled fixed points in CNSs. 2 Preliminaries Let E to be a Hausdorff topological vector space with its zero vector θ. A proper nonempty and closed subset P of E is called a (convex) cone if αp + βp P for α, β 0 and P ( P ) = {θ}. We shall always assume that the cone P has a nonempty interior int P (such cones are called solid). Each cone P induces a partial order on E by x y if and only if y x P, and x y will stand for x y and x y, while x y will stand for y x int P. The cone P is called normal if there is a constant number K 1 such that for all x, y E 0 x y implies x K y. The least positive number satisfying the above inequality is called the normal constant of P. Let X and Y be non-empty sets and we denote by 2 X a family of all non-empty subsets of X. A multivalued map or map F from X into Y is a map F : X 2 Y. Definition 2.1. [1] Let X and Y be two topological Hausdorff spaces and F : X 2 Y be a map with nonempty values. Then, F is said to be: (i) upper semicontinuous (u.s.c.), if for each closed set B Y, F (B) = {x X : F (x) B } is closed in X. (ii) lower semicontinuous (l.s.c.), if for each open set B Y, F (B) = {x X : F (x) B } is open in X. (iii) continuous, if it is both u.s.c. and l.s.c. Now, we will give the definition of a CNS, which is a generalization of a norm space. Definition 2.2. [5]-[7],[11] Let X be a real vector space and let (E, P ) be a real ordered Banach space. Suppose that the map p : X E satisfies the following. (CN1) x p = θ x = θ X ;

3 Remark on a couple coincidence point in cone normed spaces 2463 (CN2) αx p = α x p for any scalar α and any x X; (CN3) x + y p x p + y p for all x, y X. Then p is called a cone norm on X, and we call (X, p ) a CNS. If X is a CNS, then by A p = inf{ a p : a A} is defined a cone norm of non-empty set A X (see Definition 3 in [8]). A cone P is called minihedral cone if sup{x, y} exists for all x, y E and and strongly minihedral if every subset of E which is bounded from above has a supremum. Without the assumption of strong minihedralness for the cone, A p has no sense. We will always assume that cone P is solid and strongly minihederal. Definition 2.3. [3] Let X be a CNS, x X and {x n } n 1 sequence in X. Then one has the following. (a) {x n } converges to x whenever for every c E with 0 c there is a natural number N, such that x n x p c for all n > N. (b) {x n } is a Cauchy sequence whenever for every c E with 0 c there is a natural number N, such that x n x m p c for all n, m > N. (c) (X, p ) is a complete cone normed space if every Cauchy sequence is convergent. Complete cone-normed spaces will be called cone Banach spaces. Throughout this paper, let X be partially ordered set and let p be a cone norme on X such that X is a complete CNS over the normal cone with the normal constant K. We will define p : X X E with (x 1, y 1 ) (x 2, y 2 ) p = x 1 y 1 p + x 2 y 2 p. Definition 2.4. [5] Let X be a CNS. A function f : X X is said to be sequentially continuous if x n x p 0 implies that f(x n ) f(x) p 0. Analogously, F : X X X is said to be sequentially continuous if (x n, y n ) (x, y) p 0 implies that F (x n, y n ) F (x, y)) p 0. Lemma 2.5. [10] Let X be a CNS. Then f : X X is continuous if and only if is sequentially continuous. Note that Definition 2.4 and Lemma 2.5 hold when F is a multivalued map. Definition 2.6. [3] Let K be nonempty subset of a CNS X. A map H : K 2 X is called a KKM map if, for every finite subset {x 1,..., x n } of K, where co denotes convex hull. co{x 1,..., x n } n H(x i ),

4 2464 Snježana Maksimović and Zoran D. Mitrović Definition 2.7. Let X be CNS and K and U the nonempty convex subsets of X. A map F : K 2 X is said to be convex with respect to U if F (λx + (1 λ)y u p λ F (x) u p + (1 λ) F (y) u p for all u U, x, y K and λ [0, 1]. One can simply prove the following lemma. Lemma 2.8. Let K be a convex subset of a CNS X. If the map F : K 2 X is convex with respect U, then F ( λ i x i ) u p λ i F (x i ) u p (1) for all x i K, u U and λ i [0, 1], i = 1,..., n, n N such that λ i = 1. Definition 2.9. [3] Let (X, p ) be CNS and K and U the nonempty convex subsets of X. A map f : K X is said to be almost quasiconvex with respect to U if f(λx + (1 λ)y) u p c f ([λx + (1 λ)y], u) where c f ([λx + (1 λ)y], u) { f(x) u p, f(y) u p }, for all u U, x, y K and λ [0, 1]. Theorem [6] Let X be a CNS, K be a nonempty subset of X and H : K 2 X be a KKM map with closed values. If H(x) is compact for at least one x K, then x K H(x). 3 Main Results Theorem 3.1. Let X be a CNS over cone P, K a nonempty convex compact subset of X, F : K K X and G : K 2 X are continuous maps. If there exists a convex map H : K 2 X with respect to F (K K) such that G(x) F (x, y) p H(x) F (x, y) p, for all x, y K, (2) then there exists (x 0, y 0 ) K K such that G(x 0 ) F (x 0, y 0 ) p + G(y 0 ) F (y 0, x 0 ) p = inf (x,y) K K { H(x) F (x 0, y 0 ) p + H(y) F (y 0, x 0 ) p }. Moreover, if G : K 2 K and H : K 2 K is onto map, then there exists (x 0, y 0 ) K K such that G(x 0 ) F (x 0, y 0 ) p + G(y 0 ) F (y 0, x 0 ) p = inf (x,y) K K { x F (x 0, y 0 ) p + x F (y 0, x 0 ) p }.

5 Remark on a couple coincidence point in cone normed spaces 2465 Proof. Define a map S : K K 2 K K by S(z, t) = {(x, y) K K : G(x) F (x, y) p + G(y) F (y, x) p H(z) F (x, y) p + H(t) F (y, x) p } for each (z, t) K K. From (2) we have that (z, t) S(z, t), hence S(z, t) is nonempty for all (z, t) K K. The mappings F and G are continuous and we have that S(z, t) is closed for each (z, t) K K (see Theorem 2.3, [6]). Since K K is compact set, then S(z, t) is compact for each (z, t) K K. Let us show that S is KKM mapping. Supose that for any (z i, t i ) K K, i {1,..., n}, there exists (z 0, t 0 ) co{(z 1, t 1 ),..., (z n, t n )} such that (z 0, t 0 ) / n S(z i, t i ). (3) Then we know that there exist λ i 0, i {1,..., n} such that (z 0, t 0 ) = n λ i(z i, t i ) and n λ i = 1. Regarding that H is a convex with respect to F (K K) yields H(z 0 ) F (z 0, t 0 ) p n λ i H(z i ) F (z 0, t 0 ) p H(t 0 ) F (t 0, z 0 ) p n λ i H(t i ) F (t 0, z 0 ) p. In other hand, from (3) we have for all i {1,..., n}. Then G(z 0 ) F (z 0, t 0 ) p + G(t 0 ) F (t 0, z 0 ) p H(z i ) F (z 0, t 0 ) p + H(t i ) F (t 0, z 0 ) p G(z 0 ) F (z 0, t 0 ) p + G(t 0 ) F (t 0, z 0 ) p λ i H(z i ) F (z 0, t 0 ) p + λ i H(t i ) F (t 0, z 0 ) p H(z 0 ) F (z 0, t 0 ) p + H(t 0 ) F (t 0, z 0 ) p. This is a contradiction with (2). From Theorem 2.10 there exists (x 0, y 0 ) K K such that (x 0, y 0 ) S(x, y) for all (x, y) K K and this completes the proof. As corollary of Theorem 3.1 we obtain the following result for a coupled coincidence points.

6 2466 Snježana Maksimović and Zoran D. Mitrović Corollary 3.2. Let X be a CNS over cone P, K a nonempty convex compact subset of X, F : K K K and G : K 2 K are continuous maps. If there exists a convex onto map H : K 2 K, respect to F (K K) such that G(x) F (x, y) p H(x) F (x, y) p for all x, y K, then there exists (x 0, y 0 ) K K such that F (x 0, y 0 ) G(x 0 ) and F (y 0, x 0 ) G(y 0 ). Corollary 3.3. Let X be a CNS over a cone P, K nonempty convex compact subset of X, F : K K X continuous map. Then F has a coupled fixed point, i.e. there exists (x 0, y 0 ) K K such that F (x 0, y 0 ) = x 0 and F (y 0, x 0 ) = y 0. Proof. Let G(x) = {x} and apply Theorem 3.1. Theorem 3.4. Let X be a CNS over a cone P, K a nonempty convex compact subset of X, f : K X and g : K K continuous maps. If there exists an almost quasiconvex onto map h : K K with respect to f(k) such that g(x) f(x) p h(x) f(x) p for all x K, (4) then there exists x 0 K such that Proof. Define a map S : K 2 K by g(x 0 ) f(x 0 ) p = inf x K x f(x 0) p. S(y) = {x K : g(x) f(x) p h(y) f(x) p } for each z K. From (4) we have that z S(z), hence S(z) is nonempty for all z K. The maps f and g are continuous and we have that S(z) is closed for each z K (see Theorem 2.3, [6]). Since K is compact set, then S(z) is compact for each z K. Let us show that S is KKM map. Suppose that for any z i K, i {1,..., n}, there exists z 0 co{z 1,..., z n } such that z 0 / n S(z i ). (5) Then we know that there exist λ i 0, i {1,..., n} such that z 0 = n λ iz i and n λ i = 1. Regarding that h is an almost convex with respect to f(k) yields h(z 0 ) f(z 0 ) p sup h(z i ) f(z 0 ) p. i

7 Remark on a couple coincidence point in cone normed spaces 2467 In other hand, from (5) we have g(z 0 ) f(z 0 ) p h(z i ) f(z 0 ) p for all i {1,..., n}. Then g(z 0 ) f(z 0 ) p sup h(z i ) f(z 0 ) p h(z 0 ) f(z 0 ) p. i This is a contradiction with (4). From Theorem 2.10 (we will take H(x) = {h(x)}) there exists x 0 K such that x 0 S(x) for all x K and this completes the proof. Corollary 3.5. Let X be a CNS over a cone P, K a nonempty convex compact subset of X, f : K K and g : K K continuous maps. If there exists an almost quasiconvex onto map h : K K with respect to f(k) such that g(x) f(x) p h(x) f(x) p for all x K, then there exists x 0 K such that g(x 0 ) = f(x 0 ). Note that Theorem 3.4 proved in [4] for a normed spaces (see [4] Theorem 3.3). References [1] A. Amini, M. Fakhar, J. Zafarani, KKM mapping in metric spaces, Nonlinear Anal. (60) (2005) [2] E. Karapinar, Fixed point theorems in cone Banach spaces, Fixed Point Theory and Appl. (2009) Article ID , 1-9 doi: /2009/ [3] E. Karapinar, D. Turkoglu, Best approximations theorem for a couple in cone Banach spaces, Fixed Point Theory, Appl., (2010), Article ID , 9 pages doi: /2010/ [4] Z. D. Mitrović, I.D. Arandjelović, Existence of generalised best approximations, J. Nonlinear Convex Anal., (4) (2014), [5] B. Rzepecki, On fixed point theorems of Maia type, Publications de l Institut Mathématique 28 (42) (1980) [6] S. Simić, A note on Stone s, Baire s, Ky Fan s and Dugandji s theorem in tvs-cone metric space, Appl. Math. Lett. (24) (2011)

8 2468 Snježana Maksimović and Zoran D. Mitrović [7] A. Sonmez, Dissertation, Istanbul University, June, [8] A. Sonmez, On paracompactness in cone metric spaces, Appl. Math. Lett. 23 (2010) [9] A. Sonmez, H. Cakalli, Cone normed spaces and weighted mean, Math. Comput. Modelling (52) (2010) [10] D. Turkoglu, M. Abuloha, Cone metric spaces and fixed point theorems in diametrically contractive mappings, Acta Mathematica Sinica. 26 (3) (2010), [11] D. Turkoglu, M. Abuloha, T. Abdeljawad, Some theorems and examples of cone Banach spaces, Journal of Computational Analysis and Applications 12 (4) (2010) Received: September 25, 2014; Published: October 29, 2014