Coincidence Points for Mappings under Generalized Contraction

Transcription

1 Int. Journal of Math. Analysis, Vol. 6, 2012, no. 59, Coincidence Points for Mappings under Generalized Contraction Ajay Gairola and Mahesh C. Joshi Department of Mathematics, D.S.B. Campus Kumaun University, Nainital, India Abstract In this paper we establish some results on the existence of coincidence and fixed points for multi-valued and single valued mappings extending the result of Feng and Liu [2] and Liu et.al [5]. It is also proved with counter example that our results generalize and extend some well known results. Mathematics Subject Classification: 47H50, 47H54 Keywords: common fixed point, Coincidence point, multi-valued mappings 1. Introduction and Preliminaries Generalizing Banach Contraction Principle, Nadler [3] introduced the concept of multivalued contraction mapping. Let (X, d) be a metric space. Following Nadler [3] and Liu et.al [5] we follow following notations throughout this paper. CB(X) (resp.cl(x)) denote the family of all closed and bounded (resp. closed) subsets of X. C(X) represents set of all compact subsets of X. The Hausdorff distance for two subsets A, B of X is defined as,h(a, B) = max ({sup d(a, B) : a A}, {sup d(a, b) : b B}) where d(a, B) =inf{d(a, b) : b B}. It is well known that CB(X) (resp. CL(X)) isa metric space with Hausdorff distance function. Let T : X CL(X). Using the concept of Hausdorff distance, Nadler [3] defined multivalued contraction as H(Tx,Ty) d(x, y) x, y X and

2 2922 A. Gairola and M. C. Joshi α<1. Nadler proved that for a multivalued contraction in a complete metric space there exists a fixed point. Recently Feng and Liu [2] and Liu et.al [5] generalized the Nadler s result. Feng and Liu [2] gave an example to establish that if the mapping T does not satisfy the above contractive condition even then it has a fixed point. Feng and Liu [2] generalized the above contractive condition by considering the point y T (x) for any x X in place of x, y X and proved the following result. Theorem 1 [4]. Let (X, d) be a complete metric space, and let T be a multivalued mapping from X to CL(X). If there exist constant b, c (0, 1), c<b, such that for any x X there is y T (x) satisfying bd(x, y) f(x), f(y) cd(x, y). Then T has a fixed point in X provided the function f(x) = d(x, T(x)), x X is lower semi continuous. Generalizing above result and the result of Ciric [1], Liu et.al [5] relax the contractive condition by taking α(f(x)) and β(f(x)) in place of constant b and c, where α : B (0, 1],β : B [0, 1) (1) and B = { [0, supf(x)] if supf(x) < + [0, + ) if supf(x) < + }. In this paper we extend the result of [2] and [5] for the existence of coincidence points. 2. Main Result Let (X, d) be a metric space, T : X CL(X) and f : X X. An orbit of the multivalued map T at a point x 0 in X is a sequence {x n : x n Tx n 1,n= 1, 2, 3...}. The space X is T orbitally complete if every Cauchy sequence of the from {x ni : x ni Tx ni 1} converges in X. If for a point x 0 in X, there exists a sequence {x n } X such that fx n+1 Tx n, n = 0, 1, 2,... then O f (x 0 )={fx n : n =1, 2...} is an orbit of (T, f) at x 0. A space X is (T, f) - orbitally complete if every Cauchy sequence of the form {fx ni : fx ni Tx ni 1} converges in X. A function φ(x) =d(fx,tx) is called (T, f) -orbitally lower semi continuous if for any point z X an orbit {f(x n )} of (T, f) with Limfx n = fz implying that φ(z) Lim n φ(x n ).

3 Coincidence points for mappings 2923 Theorem 2.1 Let (X, d) be a metric space. T : X CL(X) and f : X X such that T (X) f(x) and f(x) is (T, f) -orbitally complete. If for any x X there exists y X such that f(y) T (x) and α(φ(x))d(fx,fy) φ(x) and φ(y) β(φ(x))d(fx,fy), where α and β are defined as (1) satisfying, supβ(r) Lim r 0 +α(r) > 0, Lim r t + < 1 t [0, supφ(x)) (2) α(r) and the function φ is (T, f) -orbitally lower semi continuous at z. Then there exist a coincidence point z of f and T. Proof. Letγ(t) = β(t), t [0, supφ(x)) (3) α(t) Let x 0 X, since T (x) f(x) we choose x 1 X such that fx 1 Tx 0, and α(φ(x 0 ))d(fx 0,fx 1 ) φ(x 0 )=d(fx 0,Tx 0 ), φ(x 1 )=d(fx 1,Tx 1 ) β(φ(x 0 ))d(fx 0,fx 1 ), imples φ(x 1 ) β(φ(x 0 )) φ(x 0). Using (3) we get, φ(x α(φ(x 0 )) 1) β(φ(x 0 )) φ(x 0) = α(φ(x 0 )) γ(φ(x 0 ))(φ(x 0 ). Continuing the process we get an orbit {fx n } n 0 of T satisfying, α(φ(x n ))d(fx n,fx n+1 ) φ(x n )=d(fx n,tx n ), and Using (3) we get φ(x n+1 )=d(fx n+1,tx n+1 ) β(φ(x n ))d(f n,f n+1 ), n 0 (4) φ(x n ) φ(x n+1 ) β(φ(x n )) α(φ(x n )) = γ(φ(x n))(φ(x n )). (5) since 0 γ(t) < 1 and by (5) it is clear that {φx n } n 0 is a nonnegative and decreasing sequence. Hence φ(x n ) is convergent. Let Lim n φ(x n )=a (6) where a 0, suppose a>0, taking Limit n in (5) and by (2)(3) and (6) a = Lim n supφ(x n+1 ) Lim n [γ(φ(x n ))] Lim n supγ(φ(x n ))Lim n supφ(x n ) = asupγ(φ(x n )) <a, which is contradiction hence a = 0, i.e. To prove that {fx n },n 0 is a Cauchy sequence. Lim n φ(x n ) = 0 (7) Let b = Lim n [supγ(φ(x n ))], c= Lim n infφ(x n ) (8)

4 2924 A. Gairola and M. C. Joshi Then from (2)(3) and (8), 0 b<1,c>0. Let p (o, c),q (b, 1) then from (8), γ(φ(x n )) <q, α(φ(x n ) > p, n 0, which together with (4) and (5) gives φ(x n+1 ) qφ(x n ),d(fx n,fx n+1 ) φ(xn). p Continuing similar calculation, we get, φ(x n+1 ) q n+1 n 0 fx n0, d(fx n fx n+1 ) q n n 0, which gives φ(x n) p n 1 d(fx n fx m ) k=1 d(fx k,fx k+1 ) φ(x n 0 ) p n 1 k=n q k n 0 φ(x n 0 ) p(1 q) qn+1 n 0 (9) Since q<1 therefore (9) implies that {fx n } is a Cauchy sequence. And since f(x) is (T, f) -orbitally complete, z X such that Lim n f(x n )=fz. Now we will prove that z is coincidence point of f and T. Since φ is (T, f) -orbitally lower semi continuous therefore 0 d(fz,tz)=φ(z) Lim n φ(x n )=0 (by (7)) implies φ(z) =0ord(fz, Tz) = 0 i.e. f(z) T (z) orf and T have a coincidence point. In Theorem 2.1, taking constants α and β in place of αφ(x) and βφ(x) respectively we get following result as a corollary. The following corollary is also serves as a generalization of Singh and Kulsrestha[4]. Corollary 2.1 Let (X, d) be a metric space. T : X CL(X) and f : X X are mappings such that T (X) f(x) and f(x) is (T, f) -orbitally complete. If for any x X there exists y X such that f(y) T (x) satisfying, βd(fx,fy) φ(x) and φ(y) αd(fx,fy), where α, β (0, 1) and α<βand the function φ is is lower semi continuous. Then T and f has a coincidence point in X. Proof. Let x 0 X, since T (x) f(x) we choose x 1 X so that fx 1 Tx 0. By the given contraction condition, β(fx 0,fx 1 ) φ(x 0 ) = d(f 0,Tx 0 ) and φ(x 1 )=d(fx 1,Tx 1 ) αd(fx 0,fx 1 ). In similar way we choose x n+1 X such that fx n+1 T (x n ) and d(fx n+1,tx n+1 ) αd(fx n,f n+1 ), βd(d(fx n,fx n+1 ) φ(x n )=d(fx n,tx n ) which implies d(fx n+1,tx n+1 ) α β d(fx n,t n )ord(fx n+1,fx n+2 ) α β d(fx n,fx n+1 ). Hence d(fx n,tx n ) ( α β )n d(fx 0,Tx 0 )or d(fx n,fx n+1 ) ( α β )n d(fx 0,fx 1 ). (10)

5 Coincidence points for mappings 2925 Using (10) for m, n N,m > n, d(fx m,fx n ) d(fx m,fx m 1 )+d(fx m 1,fx m 2 ) d(fx n+1,fx n ) ( α β )m 1 d(fx 0,fx 1 )+( α β )m 2 d(fx 0,fx 1 )+( α β )m 3 d(fx 0,fx 1 )+...+( α β )n d(fx 0,fx 1 ), implies d(fx m,fx n ) ( α β )n 1 ( α β ) (11) As n,( α β )n 0, hence {fx n } is a cauchy sequence. Since f(x) is (T, f) -orbitally complete there exists z X such that {fx n } converges to fz. Now using the condition of (T, f) -orbitally lower semi continuity of φ one can easily prove that z is coincidence point of f and T. In theorem 2.1 taking C(X) in place of CL(X) we get we get following result as corollary. Corollary 2.2 Let (X, d) be a metric space. T : X C(X) and f : X X such that T and f satisfy all conditions as in theorem 2.1 then there exist a coincidence point z of f and T. Proof. Proof is same as of theorem 2.1. In corollary 2.1 taking C(X) in place of CL(X) we get the following result as a corrolary. Corollary 2.3 Let (X, d) be a metric space. T : X C(X) and f : X X are mapping such that all conditions as in corollary 2.1 are satisfied then T and f has a coincidence point in X. Proof. Proof is same as of corollary 2.1. Example. Let X = { 1, 1,... 1 } {0, 1},d(x, y) = x y, for all x, y X, n then X is complete metric space. Define mappings T : X CL(X) as T (x) = { [ 1, 1] 2 2n+2 x = 1,n= o, 1, n [0, 1] 2 x =0 and f(x) = x 2,x X. Obviously, T and f does not satisfy hybrid contraction condition [4]. H(T ( 1 ),T(0)) = 1 1 = 1 0 = d( 1, 0) on the other 2 2n 2 2 2n 2 2n 2 2n hand, we have φ(x) = d(fx, Tx), T (x) = { 1 2 2n+2, x = 1 2 n,n= o, 1, , x =0, 1 }. }.

6 2926 A. Gairola and M. C. Joshi it shows that φ is continuous. Further, there exists y X for any x X such that 1 d(fx,fy) φ(x) and 2 d(fy,ty) 1 d(fx,fy). Then from corollary 2.1 there exists a coincidence point of f and T. References [1]. L.B. Ciric, multivalued nonlinear contraction mappings, Nonlinear Analysis: Theory method and application, vol.57, no. (7-8)(2009) ( ). [2]. Y.Feng, and S. Liu, Fixed Points theorems for multi-valued contractive mappings and multi-valued caristi type mappings, J. Math. Anal. Appl. 317 (2006) [3]. S.B. Jr Nadler, Multi-valued Contraction mappings, Pacific J. Math. 30(1969) [4]. S.L.Singh,, C. Kulsresrtha, Coincidence Theorems, Indian J. Phy. Natur. Sci. 3B(1983) [5]. Z.Liu, Wei Sun, Shin Min Kang, Jeong Sheok Ume, On Fixed Point theorem for multivalued Contraction, Fixed Point theory and applications, Received: May, 2012