Fixed and Common Fixed Point Theorems in Metric Spaces

Transcription

1 Int. Journal of Math. Analysis, Vol. 6, 2012, no. 4, Fixed and Common Fixed Point Theorems in Metric Spaces Saleh A. Al-Mezel Department of Mathematics University of Tabuk P.O. Box 741, Tabuk 7149, Saudi Arabia Abstract In this paper, without using the extended Hausdorff metric, we prove fixed point and common fixed point results for multivalued maps under certain conditions. Consequently, Our results either improve or generalize a number of known results including the corresponding results due to Caristi [2], Mizoguchi and Takahashi [12] Latif and Beg [9], Suzuki and Takahashi [16]. Mathematics Subject Classification: 47H09, 54H25 Keywords: metric space, w-distance, contraction map, fixed point, Caristi s fixed point theorem, multivalued map 1 Introduction The well-known Banach contraction principle has been extended and generalized in many different directions. One of its most important extensions is known as Caristi s fixed point theorem. It is well known that Caristi s fixed point theorem, and the other two classical results, namely, Ekland variational principle and Takahashi existence theorem are equivalent [15]. Many authors generalized and improved these theorems in metric spaces; see [1, 6, 8, 14, 18, 19] and references therein. In [5], Kada et al. introduced a notion of w-distance on a metric space and using this notion, they improved the Caristi s fixed point theorem, Ekland variational principle and Takahashi existence theorem. Using the w-distance distance, Suzuki and Takahashi [16] have introduced notions of single-valued and multivalued weakly contractive(in short, w-contractive) maps and proved

2 174 S. A. Al-Mezel fixed point results for such maps. Consequently, they generalized the Banach Contraction principle and Nadler s fixed point result [13]. Some others fixed point results concerning w-distance can be found in [7, 10, 11, 14, 17, 18] In this paper, first we prove a multivalued version of the Caristi s fixed point theorem with respect to w-distance, which is equivalent to the generalized Caristi s fixed point theorem [5, Theorem 2]. Applying this result we prove a theorem on the existence of fixed points for multivalued w-contractive maps. Also, we obtain a common fixed point result for multivalued maps in the setting of metric spaces. Our results either improve or generalize the corresponding results due to Caristi [2], Mizoguchi and Takahashi [12] Latif and Beg [9], Suzuki and Takahashi [16], Husain and Latif [4] and many others. 2 Preliminaries Let X be a metric space with metric d. We use 2 X to denote the collection of all nonempty subsets of X and Cl(X) for the collection of all nonempty closed subsets of X. Recall that a real-valued function ϕ defined on X is said to be lower semicontinuous if for any sequence {x n } X with x n x X imply that ϕ(x) lim inf n ϕ(x n ). A point x X is called a fixed point of T : X 2 X if x T (x) and the set of fixed points of T is denoted by Fix(T ). A point x X is called a common fixed point of f : X X and T if f(x) =x T (x). In 1976, Caristi [2] obtained the following fixed point theorem on complete metric spaces, known as Caristi s fixed point theorem. Theorem 2.1 Let X be a complete metric space. Let ψ : X [0, ) be a lower semicontinuous function and let f : X X be a single valued map such that for any x X Then f has a fixed point. d( x, f(x)) ψ(x) ψ(f(x)). (1) A function ω : X X [0, ) is called a w-distance [5] on X if it satisfies the following for any x, y, z X: (w 1 ) ω(x, z) ω(x, y)+ω(y, z); (w 2 ) a map ω(x,.) :X [0, ) is lower semicontinuous; (w 3 ) for any ɛ>0, there exists δ>0 such that ω(z, x) δ and ω(z, y) δ

3 Fixed and common fixed point theorems 175 imply d(x, y) ɛ. Note that, in general for x, y X, ω(x, y) ω(y, x). Clearly, the metric d is a w-distance on X. Let (Y,. ) be a normed space. Then the functions ω 1,ω 2 : Y Y [0, ) defined by ω 1 (x, y) = y and ω 2 (x, y) = x + y for all x, y Y are w-distances [5]. Many other examples of w-distance are given in [5, 16]. Using the concept of w-distance distance, Suzuki and Takahashi [16] have introduced the following notion of multivalued w-contractive maps. A multivalued map T : X Cl(X) is called w-contractive [16] if there exist a w-distance ω on X and a constant h (0, 1) such that for any x, y X and u T (x) there is v T (y) with ω(u, v) hω(x, y). In particular, if we take ω = d, then w-contractive map is a contractive type map [?, 4]. In the sequel, otherwise specified, we shall assume that ψ : X (, ] is proper, lower semicontinuous and bounded below function and ω is a w- distance on X. Forx X, define ω(x, T (x)) = inf{ω(x, y) :y T (x)}. Using w-distance, Kada et al. theorem as follows: [5] have generalized Caristi s fixed point Theorem 2.2 ([5]) Let X be a complete metric space. Let f be a singlevalued self map on X. Assume that for each x X, ψ(f(x)) + ω(x, f(x)) ψ(x). Then, there exists x o X such that f(x o )=x o and ω(x o,x o )=0. Using w-distance, Suzuki and Takahashi [16] have proved the following fixed result which is an improved version of the Multivalued contraction principle due to Nadler [13]. Theorem 2.3 ([16]) Let X be a complete metric space. Then each multivalued w-contractive map T : X Cl(X) has a fixed point x o X such that ω(x o,x o )=0.

4 176 S. A. Al-Mezel 3 The Results Using the concept of w-distance, we prove a multivalued version of the Caristi s fixed point theorem. Theorem 3.1 Let X be a complete metric space. Let T : X 2 X be a multivalued map such that for each x X there exists y T (x) satisfying ψ(y)+ω(x, y) ψ(x), Then, T has a fixed point x 0 X such that ω(x 0,x 0 )=0. Proof. Define a function f : X X by f(x) =y T (x) X. Note that for each x X, ω(x, f(x)) ψ(x) ψ(f(x)). Since the map ψ is proper, there exists u X with ψ(u) < + and we get ω(u, u) =0. Now, let M = {x X : ψ(x) ψ(u) ω(u, x)}. Then, M is nonempty because u M and using the facts that the functions ψ and ω(u,.) are lower semicontinuous, it is easy to show that the set M is closed in X. Thus M is a complete metric space. Now, we show that f(m) M. Note that for each x M, we have and thus ψ(f(x)) + ω(x, f(x)) ψ(x) ψ(u) ω(u, x) ψ(f(x)) ψ(u) {ω(u, x)+ω(x, f(x))} ψ(u) ω(u, f(x)). It follows that f(x) M and hence f is a self map on M. Applying Theorem 2.2, there exists x 0 M such that f(x 0 )=x 0 T (x 0 ) and ω(x 0,x 0 )=0. For a multivalued map T : X Cl(X) and for ɛ>0, define a set T ɛ (x) ={y T (x) :ω(x, y) (1 + ɛ)ω(x, T (x))}, x X. Now, if the map T is w-contractive with contractive constant h, then for any ɛ< 1 h 1, the set T ɛ(x) for every x X. Further, using the fact that for each x X the function ω(x,.) is lower semicontinuous, it is easy to show that the set T ɛ (x) is closed. Thus, in fact we get a multivalued map T ɛ : X Cl(X) Now, applying Theorem 3.1, we obtain the following fixed point result for multivalued w-contractive maps.

5 Fixed and common fixed point theorems 177 Theorem 3.2 Let X be a complete metric space. Let T : X Cl(X) be a w-contractive map with contraction constant h. Assume that for any nonempty closed subset M of X there exists a positive number ɛ<1/h 1 with T ɛ (x) M φ for all x M and the map x ω(x, T (x)) is lower semicontinuous. Then T has a fixed point x 0 M such that ω(x 0,x 0 )=0. Proof. Define a map J : M Cl(M) by J(x) = T ɛ (x) M, x M. Since for each x M, the set T ɛ (x) M is nonempty and closed subset of M, thus J does carry M into Cl(M). Let y J(x) then y T (x) and ω(x, y) (1 + ɛ)ω(x, T (x)). Since T is w-contractive, there exists z T (y) such that Thus ω(y, T(y)) ω(y, z) hω(x, y). ω(x, T (x)) ω(y, T(y)) ω(x, T (x)) hω(x, y) 1 ( )ω(x, y) hω(x, y) 1+ɛ = 1 ( h)ω(x, y). 1+ɛ Defining ϕ : M Rby 1 ϕ(x) =( 1+ɛ h) 1 ω(x, T (x)), x M. Then, clearly ω(x, y) ϕ(x) ϕ(y). Now, applying Theorem 3.1 we get a point x 0 M such that such that x 0 J(x 0 ) and ω(x 0,x 0 )=0. Since J(x 0 )=T ɛ (x 0 ) M, and hence we get x 0 T (x 0 ) M. Theorem 3.3 Let X be a complete metric space. Let f be a single valued self-map of X with f(x) =M complete. Let T : X 2 X be a multivalued map such that T (X) M and for each x X and any y T (x) ω(x, f(y)) ψ(x) ψ(f(y)), where ψ is a proper, bounded below and lower semicontinuous function of M into (, + ]. Then, there exits a point x 0 M such that x 0 ft(x 0 ).

6 178 S. A. Al-Mezel Proof. For each y M, define J(y) =ft(y) = x T (y) {f(x)}. Clearly, J carries M into 2 M. Now, for each u J(y) there exists some v T (y) with u = f(v) and ω(y, f(v)) ψ(y) ψ(f(v)), that is; ω(y, u) ψ(y) ψ(u). Since ψ is proper, there exists z M with ψ(z) < +. Let Y = {y M : ψ(y) ψ(z) ω(z, y)}. Clearly, Y is nonempty subset of M. By the lower semicontinuity of ψ and ω(z,.), Y is closed. Thus Y is a complete metric space. Now we show that Y is invariant under the map J. Now, let u J(y), y Y. By definition of J, there exists v T (y) such that u = f(v), also ψ(u)+ω(y, u) ψ(y) ψ(z) ω(z, y) and hence ψ(u) ψ(z) ω(z, u), proving that u Y and hence J(y) Y for all y Y. Now, Theorem 3.1 guarantees that there exits x 0 M such that x 0 J(x 0 )=ft(x 0 ). Finally, we obtain a common fixed point result. Theorem 3.4 Suppose that X, M, f and T satisfy the assumptions of Theorem 3.3 and Moreover the following conditions hold: (a) f and T commute weakly. (b) x Fix(f) implies x ft(x). Then T and f have a common fixed point in M. Proof. By the proof of Theorem 3.3, there exits x 0 M such that x 0 ft(x 0 ). Using condition (a) and (b), we obtain x 0 = f(x 0 ) ft(x 0 ) Tf(x 0 )=T (x 0 ). Thus, x 0 must be a common fixed point of f and T. Remark 3.5 (1) Obviously, Theorem 3.1 implies Theorem 2.2. Hence Theorem 2.2. and Theorem 3.1 are equivalent.

7 Fixed and common fixed point theorems 179 (2) Theorem 3.1 is an improved version of the fixed point result due to Mizoguchi and Takahashi [12, Theorem 1]. (3) Theorem 3.2 improves the fixed point result due to Mizoguchi and Takahashi [12, Theorem 2]. Further, note that Theorem 2.3 is a direct consequence of the Theorem 3.2. (4) Theorem 3.4 contains [9, Theorem 2.5] as a special case. Acknowledgement. The author gratefully acknowledges the financial support from the Deanship of Scientific Research (DSR) at University of Tabuk through the grant number 46/21/1432. References [1] J. S. Bae, Fixed point theorems for weakly contractive multivalued maps, J. Math. Analy. Appl., 284 (2003), [2] J. Caristi, Fixed point theorem for mapping satisfying inwardness conditions, Trans. Am. Math. Soc 215 (1976) [3] I. Ekland, Nonconvex minimization problems, Bull. Math. Soc. 1 (1979) [4] T. Husain and A. Latif, Fixed points of multivalued nonexpansive maps, Internat. J. Math. & Math. Sci., Vol., 14 (1991), [5] O. Kada, T. Susuki and W. Takahashi, Nonconvex minimization theorems and fixed point theorems in complete metric spaces, Math. Japon., 44 (1996), [6] M. A. Khamsi, Remarks on Caristi s fixed point theorem, Nonlinear Analysis, Vol. 71 (2009), [7] T.H. Kim, K. Kim and J.S. Ume, Fixed point theorems on complete metric spaces, Panamer. Math. J., 7 (1997), [8] A. Latif, Generalized Caristi s fixed point theorems, Fixed Point Theory and Applications, Vol. 2009, Article ID , 7 pages. [9] A. Latif and I. Beg, Geometric Fixed Points For Single And Multivalued Mappings, Demonstratio Mathematica, Vol. 30, No. 4 (1997), [10] A. Latif and Afrah A.N. Abdou, Multivalued generalized nonlinear contractive maps and fixed points, Nonlinear Analysis, 74 (2011)),

8 180 S. A. Al-Mezel [11] A. Latif and W. A. Albar, Fixed point results in complete metric spaces, Demonstratio Mathematica, XLI (2008), [12] N. Mizoguchi and W. Takahashi, Fixed points theorems for multivalued mappings on complete metric spaces, J. Math. Anal. Appl., 141 (1989), [13] S. B. Nadler, Multivalued contraction mappings, Pacific J. Math., 30 (1969), [14] S. Park, On generalizations of the Ekland-type variational principles, Nonlinear Anal., 39 (2000), [15] R.R. Phelps, Convex functions, Monotone operators and Differentiability, Lecture notes in Math. Vol. 1364, Springer-Verlag, Berlin, [16] T. Suzuki and W. Takahashi, Fixed point Theorems and characterizations of metric completeness, Topol. Methods Nonlinear Anal., 8 (1996), [17] T. Suzuki, Several fixed point theorems in complete metric spaces, Yokohama Math. J., 44 (1997), [18] J. S. Ume, B. S. Lee and S. J. Cho, Some results on fixed point theorems for multivalued mappings in complete metric spaces, IJMMS, 30 (2002), [19] Z. Wu, Equivalent formulations of Ekeland s variational principle, Nonlinear Anal. 55 (2003) Received: July, 2011