FIXED POINT THEOREM ON MULTI-VALUED MAPPINGS

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1 International Journal of Analysis and Applications ISSN Volume 1, Number 2 (2013), FIXED POINT THEOREM ON MULTI-VALUED MAPPINGS J.MARIA JOSEPH 1,, E. RAMGANESH 2 Abstract. In this paper, we prove a common fixed point theorem for two multivalued self-mappings in complete metric spaces. 1. Introduction and preliminaries: The study of fixed points for set valued contractions and nonexpansive maps using the Hausdorff metric was initiated by Markin. Later, an interesting and rich fixed point theory for such maps has been developed. The theory of set valued maps has applications in control theory, convex optimization, differential inclusions and economics. Following the Banach contraction principle Nadler introduced the concept of set valued contractions and established that a set valued contraction possesses a fixed point in a complete metric space. Subsequently many authors generalized Nadlers fixed point theorem in different ways[[1],[2]]. Definition 1.1. Let X and Y be nonempty sets. T is said to be a multi-valued mapping from X to Y if T is a function from X to the power set of Y. We denote a multi-valued map by T : X 2 Y. Definition 1.2. A point x 0 X is said to be a fixed point of the multi-valued mapping T if x 0 T x 0. Example 1.3. Every single valued mapping can be viewed as a multi-valued mapping. Let f : X Y be a single valued mapping. Define T : X 2 Y by T x = {f(x)}. Note that T is multi-valued mapping iff for each x X, T x Y. Unless otherwise stated we always assume T x is non-empty for each x X. Definition 1.4. Let (X, d) be a metric space. A map T : X X is called contraction if there exists 0 λ < 1 such that d(t x, T y) λd(x, y), for all x, y X. Definition 1.5. Let (X, d) be a metric space. We define the Hausdorff metric on CB(X) induced by d. That is { } H(A, B) = max sup d(x, B), sup d(y, A) x A y B for all A, B CB(X), where CB(X) denotes the family of all nonempty closed and bounded subsets of X and d(x, B) = inf{d(x, b) : b B}, for all x X Mathematics Subject Classification. 54H25,47H10. Key words and phrases. Fixed point, Multivalued map, Hausdorff metric. c 2013 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 123

2 124 JOSEPH AND RAMGANESH Definition 1.6. Let (X, d) be a metric space. A map T : X CB(X) is said to be multi valued contraction if there exists 0 λ < 1 such that H(T x, T y) λd(x, y), for all x, y X. Lemma 1.7. [3] If A, B CB(X) and a A, then for each ε > 0, there exists b B such that d(a, b) H(A, B) ε. 2. Main Results Theorem 2.1. Let(X, d) be complete metric space and let S, T : X CB(X) be multivalued maps satisfying H(T x, Sy) ad(x, T y) b(d(x, Sy) d(t y, T x)), where 0 < a 2b < 1, a, b 0, for all x, y X. Then F (T ) = F (S) and T x = Sx = F (T ), for all x F (T ). Proof. Fix any x X. Define x 0 = x and let x 1 T x 0. By lemma(1.7), we may choose x 2 Sx 0 such that d(x 1, x 2 ) H(T x 0, Sx 0 ) (a b). Now d(x 1, x 2 ) H(T x 0, Sx 0 ) (a b) ad(x 0, x 1 ) b(d(x 0, x 1 ) d(x 1, x 2 )) (a b) a b d(x 0, x 1 ) a b. By lemma(1.7), there exists x 3 T x 2 such that Now d(x 3, x 2 ) H(T x 2, Sx 0 ) d(x 3, x 2 ) H(T x 2, Sx 0 ) (a b)2 (a b)2. ad(x 2, x 1 ) b(d(x 2, x 2 ) d(x 1, x 3 )) a b d(x 2, x 1 ) ( a b )2 ( a b )2 d(x 0, x 1 ) 2( a b )2 (a b)2 Continuing this process, we obtain by induction a sequence {x n } such that x 2n Sx 2n 2, x 2n1 T x 2n, such that d(x 2n1, x 2n2 ) H(T x 2n, Sx 2n ) () 2n, d(x 2n, x 2n1 ) H(Sx 2n 2, T x 2n ) () 2n 1,

3 FIXED POINT THEOREM ON MULTI-VALUED MAPPINGS 125 Now, Also, d(x 2n, x 2n1 ) H(Sx 2n 2, T x 2n ) () 2n 1 ad(x 2n, T x 2n 2 ) b(d(x 2n, Sx 2n 2 ) d(t x 2n 2, T x 2n ) () 2n 1 ad(x 2n, x 2n 1 ) b(d(x 2n 1, x 2n ) d(x 2n, x 2n1 ) () 2n 1 (a b) () d(x 2n 1, x 2n ) () 2n d(x 2n1, x 2n2 ) H(T x 2n, Sx 2n ) () 2n ad(x 2n, T x 2n ) b(d(x 2n, Sx 2n ) d(t x 2n, T x 2n ) Therefore, () 2n ad(x 2n, x 2n1 ) b(d(x 2n, x 2n1 ) d(x 2n1, x 2n2 ) d(x n, x n1 ) for all n N and let k = (ab) (1 b) () 2n (a b) () d(x 2n, x 2n1 ) () 2n1 (a b) () d(x (a b)n n 1, x n ) () n d(x n, x n1 ) kd(x n 1, x n ) k n k(kd(x n 2, x n 1 ) k n 1 ) k n = k 2 (d(x n 2, x n 1 )) kk n 1 k n... k n d(x 0, x 1 ) nk n. Since k < 1, k n and nk n have same radius of convergence, {x n } is a Cauchy sequence. Since (X, d) is complete, there exists z X such that x n z. d(t z, z) d(z, x 2n2 ) d(x 2n2, T z) d(z, x 2n2 ) H(T z, Sx 2n ) d(z, x 2n2 ) [ad(z, T x 2n ) b(d(z, Sx 2n ) d(t x 2n, T z))] d(z, x 2n2 ) [ad(z, x 2n1 ) b(d(z, x 2n2 ) d(x 2n1, T z))] ad(z, z) b[d(z, z) d(z, T z)] as n. Therefore d(t z, z)() 0. Hence d(t z, z) = 0

4 126 JOSEPH AND RAMGANESH H(T z, Sz) ad(z, T z) b(d(z, Sz) d(t z, T z))] = ad(z, T z) bd(z, Sz) ad(z, T z) bd(z, T z) bd(t z, Sz) (a b)d(z, T z) bh(t z, Sz) H(T z, Sz) ( a b )d(z, T z) Hence, H(T z, Sz) = 0, z T z = Sz and therefore z F (T ), z F (S), To complete the proof, it is enough to show following four cases: (i) F (T ) T z and Sx = T x for all x F (T ). (ii) T z F (T ) (iii) T x = T z for all x F (T ) (iv) F (S) T z For any x F (T ), d(x, T z) H(T x, Sz) ( a b )d(x, T z) This shows that d(x, T z) = 0 and x T z.further H(Sx, T x) ( a b )d(x, T x) = 0 and x Sx = T x.for any x T z d(x, T x) H(Sz, T x) ( a b )d(t z, x) = 0 This shows that x T x. Now, we see thatt z = F (T ) F (S)and Sx = T x for all x F (T ). For any x F (T ), Hence, T x = Sz = T z. x F (S), H(T x, Sz) ( a b )d(x, T z) = ( a b )d(x, F (T )) = 0 It remains to show that F (S) T z = F (T ). For any d(x, T z) H(T x, Sz) ( a b )d(t x, z) ( a b )H(T x, Sz) ( a b )2 d(x, T z) Hence, d(x, T z) = 0. Then x T z and F (S) T z. In what follows, let denote multimap. Corollary 2.1. Let T : X X be a multivalued map with nonempty compact values and r [0, 1) such that H(T x, T 2 y) rd(x, T y),

5 FIXED POINT THEOREM ON MULTI-VALUED MAPPINGS 127 for all x, y X. Then, F (T ) and T x = F (T ) for all x F (T ). Remark 2.2. Let S be a self mapping (multi valued or single valued) defined on X, we denote F (S) the collection of all fixed points of S. If one of S and T in Theorem 2.1 is single valued, then the set F (T ) = F (S) is singleton and the maps S and T have a unique common fixed point in X. 3. Acknowledgments The authors would like to thank the editor of the paper and the referees for their precise remarks to improve the presentation of the paper. References [1] Lai-Jiu Lin and Sung-Yu Wang,Common Fixed Point Theorems for a Finite Family of Discontinuous and Noncommutative Maps,Fixed Point Theory and Applications,vol.2011,Article ID ,19 pages. [2] J. Maria Joseph, M.Marudai, Common Fixed Point Theorem for Set-Valued Maps and a Stationary Point Theorem, Int. Journal of Math. Analysis, 6, (2012), no. 33, [3] S.B. Nadler, Multi-valued contraction mappings, Pacific Journal of Mathematics, 30 (1969), Department of Mathematics, St.Joseph s college,tiruchirappalli,tamil Nadu,India 2 Department of Educational Technology, Bharathidasan University,Tiruchirappalli,Tamil Nadu,India Corresponding author