COMMON FIXED POINT THEOREMS OF CARISTI TYPE MAPPINGS WITH w-distance. Received April 10, 2010; revised April 28, 2010

Transcription

1 Scientiae Mathematicae Japonicae Online, e-2010, COMMON FIXED POINT THEOREMS OF CARISTI TYPE MAPPINGS WITH w-distance Tomoaki Obama Daishi Kuroiwa Received April 10, 2010; revised April 28, 2010 Abstract. In this paper, we obtain common fixed point theorems of Caristi type mappings [2] by using the concept of w-distance, which is introduced by Kada, Suzuki Takahashi [4]. Our results generalize theorems of Bhakta Basu [1]. 1 Introduction Preliminaries In 1981, Bhakta Basu [1] proved a common fixed point theorem of Caristi type mappings in a complete metric space. A mapping T from a metric space X to X is said to be orbitally continuous if for every x, x 0 X, T ni+1 x converges to T x 0 whenever T n i x converges to x 0. Theorem 1. ([1]) Let (X, d) be a complete metric space let S, T be two orbitally continuous mappings of X into itself. Suppose that there are two functions ϕ, ψ of X into [0, ) satisfying the following condition: For any two points x, y X, d(sx, T y) ϕ(x) ϕ(sx) + ψ(y) ψ(t y). On the other h, in 1996, Kada, Suzuki Takahashi [4] introduced the concept of w-distance on a metric space. Definition 1. ([4]) Let (X, d) be a metric space. Then a function ρ : X X [0, ) is called a w-distance on X if the following are satisfied: (1) ρ(x, z) ρ(x, y) + ρ(y, z) for any x, y, z X; (2) for any x X, ρ(x, ) : X [0, ) is lower semicontinuous; (3) for any ε > 0, there exists δ > 0 such that for any x, y, z X, ρ(z, x) δ ρ(z, y) δ imply d(x, y) ε. They generalized Caristi s fixed point theorem [2], Ekel s variational principle [3] Takahashi s nonconvex minimization theorem [6] by using w-distance ([4], [5]). In this paper, using the concept of w-distance, we obtain common fixed point theorems on complete metric spaces. Our results generalize theorems of Bhakta Basu [1]. The following lemma is very important in the proofs of our results. Lemma 1. ([4]) Let (X, d) be a metric space let ρ be a w-distance on X. Let {x n } {y n } be sequences in X, let {α n } {β n } be sequences in [0, ) converging to 0, let x, y, z X. Then the following hold: (i) If ρ(x n, y) α n ρ(x n, z) β n for any n N, then y = z. ρ(x, y) = 0 ρ(x, z) = 0, then y = z; 2000 Mathematics Subject Classification. 47H10, 54E50. Key words phrases. Common fixed point theorem, Caristi type mapping, w-distance. In particular, if

2 354 TOMOAKI OBAMA AND DAISHI KUROIWA (ii) if ρ(x n, y n ) α n ρ(x n, z) β n for any n N, then {y n } converges to z; (iii) if ρ(x n, x m ) α n for any n, m N with m > n, then {x n } is a Cauchy sequence; (iv) if ρ(y, x n ) α n for any n N, then {x n } is a Cauchy sequence. 2 Generalized common fixed point theorems of Bhakta Basu In this section, we generalize results of Bhakta Basu by using w-distance. Theorem 2. Let (X, d) be a complete metric space let ρ be a w-distance on X. Let S, T be two orbitally continuous mappings of X into itself. Suppose that there are two functions ϕ, ψ of X into [0, ) satisfying the following condition: For any two points x, y X, max{ρ(sx, T y), ρ(t y, Sx)} ϕ(x) ϕ(sx) + ψ(y) ψ(t y). Proof. Let x 0 y 0 be any two points of X. We consider the following sequences Then we have for all i {1, 2,..., n}. So, x n = S n x 0, y n = T n y 0 (n N). ρ(x i, y i ) = ρ(sx i 1, T y i 1 ) ϕ(x i 1 ) ϕ(sx i 1 ) + ψ(y i 1 ) ψ(t y i 1 ) = ϕ(x i 1 ) ϕ(x i ) + ψ(y i 1 ) ψ(y i ) ρ(x i, y i ) {ϕ(x i 1 ) ϕ(x i ) + ψ(y i 1 ) ψ(y i )} = ϕ(x 0 ) ϕ(x n ) + ψ(y 0 ) ψ(y n ) ϕ(x 0 ) + ψ(y 0 ). Again, ρ(y i, x i+1 ) = ρ(t y i 1, Sx i ) ϕ(x i ) ϕ(sx i ) + ψ(y i 1 ) ψ(t y i 1 ) = ϕ(x i ) ϕ(x i+1 ) + ψ(y i 1 ) ψ(y i ) for all i {1, 2,..., n}. So, ρ(y i, x i+1 ) {ϕ(x i ) ϕ(x i+1 ) + ψ(y i 1 ) ψ(y i )} = ϕ(x 1 ) ϕ(x n+1 ) + ψ(y 0 ) ψ(y n ) ϕ(x 1 ) + ψ(y 0 ). Since, ρ(x i, x i+1 ) ρ(x i, y i ) + ρ(y i, x i+1 ), we have ρ(x i, x i+1 ) {ρ(x i, y i ) + ρ(y i, x i+1 )} ϕ(x 0 ) + ϕ(x 1 ) + 2ψ(y 0 ).

3 COMMON FIXED POINT THEOREMS WITH w-distance 355 This gives that series with m > n. Then ρ(x i, x i+1 ) is convergent. Let n m be any two positive integers ρ(x n, x m ) m 1 i=n By Lemma 1(iii), {x n } is a Cauchy sequence. Similarly, we have for all i {1, 2,..., n}. So, ρ(x i+1, y i+1 ) = ρ(sx i, T y i ) ρ(x i, x i+1 ) 0 as n. ϕ(x i ) ϕ(sx i ) + ψ(y i ) ψ(t y i ) = ϕ(x i ) ϕ(x i+1 ) + ψ(y i ) ψ(y i+1 ) ρ(x i+1, y i+1 ) {ϕ(x i ) ϕ(x i+1 ) + ψ(y i ) ψ(y i+1 )} = ϕ(x 1 ) ϕ(x n+1 ) + ψ(y 1 ) ψ(y n+1 ) ϕ(x 1 ) + ψ(y 1 ). Since, ρ(y i, y i+1 ) ρ(y i, x i+1 ) + ρ(x i+1, y i+1 ), we have ρ(y i, y i+1 ) {ρ(y i, x i+1 ) + d(x i+1, y i+1 )} 2ϕ(x 1 ) + ψ(y 0 ) + ψ(y 1 ). This gives that series ρ(y i, y i+1 ) is convergent. sequence. Since X is complete, each of them is convergent, that is, x n x, y n ȳ In the same way, {y n } is a Cauchy as n for some x, ȳ X. Since S T are orbitally continuous, Sx n S x, T y n T ȳ, that is, x n+1 S x, y n+1 T ȳ as n. This gives that S x = x T ȳ = ȳ. Now, Hence, by Lemma 1(i), x = ȳ. ρ( x, x) ρ( x, ȳ) + ρ(ȳ, x) = ρ(s x, T ȳ) + ρ(t ȳ, S x) 2[ϕ( x) ϕ(s x) + ψ(ȳ) ψ(t ȳ)] = 0 ρ( x, ȳ) = ρ(s x, T ȳ) ϕ( x) ϕ(s x) + ψ(ȳ) ψ(t ȳ) = 0.

4 356 TOMOAKI OBAMA AND DAISHI KUROIWA Let z X satisfying S z = z. Then ρ( z, z) ρ( z, x) + ρ( x, z) = ρ(s z, T x) + ρ(t x, S z) 2[ϕ( z) ϕ(s z) + ψ( x) ψ(t x)] = 0 ρ( z, x) = ρ(s z, T x) ϕ( z) ϕ(s z) + ψ( x) ψ(t x) = 0. So, z = x. Thus x is the only fixed point of S. Similarly, we can show that x is the only fixed point of T. This proves the theorem. Corollary 1. Let (X, d) be a complete metric space let ρ be a w-distance on X. Let F = {S α α Λ} be a family of orbitally continuous mappings of X into itself. Suppose that for each mapping S F, there is a function ϕ S of X into [0, ) satisfying the following condition: For any two mappings S, T F for any x, y X, max{ρ(sx, T y), ρ(t y, Sx)} ϕ S (x) ϕ S (Sx) + ϕ T (y) ϕ T (T y). Then the family F have a unique common fixed point. Proof. Let S T be two mappings of F. Then S, T ϕ S, ϕ T satisfy the conditions of Theorem 2. Hence S T have a unique common fixed point x 0. Let S be any other mapping of F. Again by using Theorem 2, S S have a unique common fixed point x 1. As x 0 is the unique fixed point of S, x 0 = x 1. Hence x 0 is a unique common fixed point of S, T S. As S is an arbitrary mapping of F, it follows that x 0 is a unique common fixed point of the mappings of F. Next we consider common fixed point theorems for set-valued maps. A set-valued mapping S from a metric space X into 2 X is said to be upper semicontinuous if for every x X every open set V with Sx V, there exists a neighborhood U of x such that Sz V for all z U. See [7]. Lemma 2. Let (X, d) be a metric space let S be an upper semicontinuous mapping of X into 2 X. For any x X, Sx is nonempty closed. Let x 0 X {x n } is a sequence in X. Then, { xn+1 Sx n (n N) x n x 0 x 0 Sx 0. Proof. Assume that x n+1 Sx n x n x 0. Suppose that x 0 / Sx 0. Since X is a metric space Sx 0 is a closed set, there exists two open sets G 1 G 2 such that x 0 G 1, Sx 0 G 2 G 1 G 2 =. From upper semicontinuity of S, there exists a neighborhood U x0 of x 0 such that Sx G 2 for any x U x0. Since x n x 0, x n U x0 for large enough n, therefore x n+1 Sx n G 2, we have x 0 clg 2. This is a contradiction.

5 COMMON FIXED POINT THEOREMS WITH w-distance 357 Theorem 3. Let (X, d) be a complete metric space let ρ be a w-distance on X. Let S, T be two upper semicontinuous mappings of X into 2 X. For any x X, Sx T x are nonempty closed. Suppose that there are two functions ϕ, ψ of X into [0, ) satisfying the following condition: For any two points x, y X for any u Sx, v T y, max{ρ(u, v), ρ(v, u)} ϕ(x) ϕ(u) + ψ(y) ψ(v). Proof. Let x 0 y 0 be any two points of X {x n } {y n } be sequences satisfying Then we have x n Sx n 1, y n T y n 1 (n N). ρ(x i, y i ) ϕ(x i 1 ) ϕ(x i ) + ψ(y i 1 ) ψ(y i ), ρ(y i, x i+1 ) ϕ(x i ) ϕ(x i+1 ) + ψ(y i 1 ) ψ(y i ) ρ(x i+1, y i+1 ) ϕ(x i ) ϕ(x i+1 ) + ψ(y i ) ψ(y i+1 ) for all i N. In similar way to proof of Theorem 2, {x n } {y n } are Cauchy sequences. Since X is complete, each of them is convergent, that is, x n x, y n ȳ as n for some x, ȳ X. By Lemma 2, we have that x S x ȳ T ȳ. Now, ρ( x, x) ρ( x, ȳ) + ρ(ȳ, x) 2[ϕ( x) ϕ( x) + ψ(ȳ) ψ(ȳ)] = 0 ρ( x, ȳ) ϕ( x) ϕ( x) + ψ(ȳ) ψ(ȳ) = 0. Hence, by Lemma 1(i), x = ȳ. We obtain x S x T x. Let z X satisfying z S z. Then ρ( z, z) ρ( z, x) + ρ( x, z) 2[ϕ( z) ϕ( z) + ψ( x) ψ( x)] = 0 ρ( z, x) ϕ( z) ϕ( z) + ψ( x) ψ( x) = 0. So, z = x. Thus x is the only fixed point of S. Similarly, we can show that x is the only fixed point of T. This proves the theorem. Corollary 2. Let (X, d) be a complete metric space let ρ be a w-distance on X. Let F = {S α α Λ} be a family of upper semicontinuous mappings of X into 2 X. For any x X S F, Sx is nonempty closed. Suppose that there is a family {ϕ S S F} of functions of X into [0, ) satisfying the following condition: For any two mappings S, T F for any x, y X u Sx, v T y, max{ρ(u, v), ρ(v, u)} ϕ S (x) ϕ S (u) + ϕ T (y) ϕ T (v). Then the family F have a unique common fixed point.

6 358 TOMOAKI OBAMA AND DAISHI KUROIWA 3 More common fixed point theorems Let (X, d) be a metric space let ρ be w-distance on X. In this section, we consider the following condition instead of max{ρ(x, y), ρ(y, x)} + ρ(x, Sx) + ρ(y, T y) ϕ(x) ϕ(sx) + ψ(y) ψ(t y) max{ρ(sx, T y), ρ(t y, Sx)} ϕ(x) ϕ(sx) + ψ(y) ψ(t y) for orbitally continuous mappings S T of X into itself. The inequality max{ρ(sx, T y), ρ(t y, Sx)} max{ρ(x, y), ρ(y, x)} + ρ(x, Sx) + ρ(y, T y) does not hold in general. For example, let X = {a, b, c}, d(x, y) = 1 if x y, d(x, y) = 0 if x = y, ρ(a, b) = 3, ρ(x, y) = d(x, y) whenever (x, y) (b, a), Sa = Sb = a, Sc = b T a = T c = a, T b = c. If (x, y) = (c, a), max{ρ(sx, T y), ρ(t y, Sx)} > max{ρ(x, y), ρ(y, x)} + ρ(x, Sx) + ρ(y, T y). Theorem 4. Let (X, d) be a complete metric space let ρ be a w-distance on X. Let S, T be two orbitally continuous mappings of X into itself. Suppose that there are two functions ϕ, ψ of X into [0, ) satisfying the following condition: For any two points x, y X, max{ρ(x, y), ρ(y, x)} + ρ(x, Sx) + ρ(y, T y) ϕ(x) ϕ(sx) + ψ(y) ψ(t y). Proof. Let x 0 y 0 be any two points of X. We consider the following sequences Then we have x n = S n x 0, y n = T n y 0 (n N). ρ(x i 1, x i ) max{ρ(x i 1, y i 1 ), ρ(y i 1, x i 1 )} + ρ(x i 1, x i ) + ρ(y i 1, y i ) for all i {1, 2,..., n}. So, ϕ(x i 1 ) ϕ(x i ) + ψ(y i 1 ) ψ(y i ) ρ(x i 1, x i ) {ϕ(x i 1 ) ϕ(x i ) + ψ(y i 1 ) ψ(y i )} = ϕ(x 0 ) ϕ(x n ) + ψ(y 0 ) ψ(y n ) ϕ(x 0 ) + ψ(y 0 ). This gives that series ρ(x i 1, x i ) is convergent, {x n } is a Cauchy sequence in similar way to proof of Theorem 2. Also we have {y n } is a Cauchy sequence. Since X is complete, each of them is convergent, that is, x n x, y n ȳ as n for some x, ȳ X. Since S T are orbitally continuous, Sx n S x, T y n T ȳ, that is, x n+1 S x, y n+1 T ȳ

7 COMMON FIXED POINT THEOREMS WITH w-distance 359 as n. This gives that S x = x T ȳ = ȳ. Now, max{ρ( x, ȳ), ρ(ȳ, x)} max{ρ( x, ȳ), ρ(ȳ, x)} + ρ( x, S x) + ρ(ȳ, T ȳ) then ρ( x, ȳ) = ρ(ȳ, x) = 0. And also ϕ( x) ϕ(s x) + ψ(ȳ) ψ(t ȳ) = 0, ρ( x, x) ρ( x, ȳ) + ρ(ȳ, x) = 0, hence ρ( x, ȳ) = ρ( x, x) = 0. By Lemma 1(i), x = ȳ. Let z X satisfying S z = z. Then max{ρ( z, x), ρ( x, z)} max{ρ( z, x), ρ( x, z)} + ρ( z, S z) + ρ( x, T x) ϕ( z) ϕ(s z) + ψ( x) ψ(t x) = 0. In the same way, we have z = x. Thus x is the only fixed point of S. Similarly, we can show that x is the only fixed point of T. This proves the theorem. Corollary 3. Let (X, d) be a complete metric space let ρ be a w-distance on X. Let F = {S α α Λ} be a family of orbitally continuous mappings of X into itself. Suppose that there is a family {ϕ S S F} of functions of X into [0, ) satisfying the following condition: For any two mappings S, T F for any x, y X, max{ρ(x, y), ρ(y, x)} + ρ(x, Sx) + ρ(y, T y) ϕ S (x) ϕ S (Sx) + ϕ T (y) ϕ T (T y). Then the family F have a unique common fixed point. The proof is similar to Corollary 1, omitted. Theorem 5. Let (X, d) be a complete metric space let ρ be a w-distance on X. Let S, T be two upper semicontinuous mappings of X into 2 X. For any x X, Sx T x are nonempty closed. Suppose that there are two functions ϕ, ψ of X into [0, ) satisfying the following condition: For any two points x, y X for any u Sx, v T y, max{ρ(x, y), ρ(y, x)} + ρ(x, u) + ρ(y, v) ϕ(x) ϕ(u) + ψ(y) ψ(v). Proof. Let x 0 y 0 be any two points of X {x n } {y n } be sequences satisfying Then we have x n Sx n 1, y n T y n 1 (n N). ρ(x i 1, x i ) ϕ(x i 1 ) ϕ(x i ) + ψ(y i 1 ) ψ(y i ) ρ(y i 1, y i ) ϕ(x i 1 ) ϕ(x i ) + ψ(y i 1 ) ψ(y i ) for all i N. In similar way to proof of Theorem 4, {x n } {y n } are Cauchy sequences. Since X is complete, each of them is convergent, that is, x n x, y n ȳ as n for some x, ȳ X. By Lemma 2, we have that x S x ȳ T ȳ.

8 360 TOMOAKI OBAMA AND DAISHI KUROIWA Now, max{ρ( x, ȳ), ρ(ȳ, x)} max{ρ( x, ȳ), ρ(ȳ, x)} + ρ( x, x) + ρ(ȳ, ȳ) then ρ( x, ȳ) = ρ(ȳ, x) = 0. And also ϕ( x) ϕ( x) + ψ(ȳ) ψ(ȳ) = 0, ρ( x, x) ρ( x, ȳ) + ρ(ȳ, x) = 0, hence ρ( x, ȳ) = ρ( x, x) = 0. By Lemma 1(i), x = ȳ. We obtain x S x T x. Let z X satisfying z S z. Then max{ρ( z, x), ρ( x, z)} max{ρ( z, x), ρ( x, z)} + ρ( z, z) + ρ( x, x) ϕ( z) ϕ( z) + ψ( x) ψ( x) = 0. In the same way, we have z = x. Thus x is the only fixed point of S. Similarly, we can show that x is the only fixed point of T. This proves the theorem. Corollary 4. Let (X, d) be a complete metric space let ρ be a w-distance on X. Let F = {S α α Λ} be a family of upper semicontinuous mappings of X into 2 X. For any x X S F, Sx is nonempty closed. Suppose that there is a family {ϕ S S F} of functions of X into [0, ) satisfying the following condition: For any two mappings S, T F for any x, y X u Sx, v T y, max{ρ(x, y), ρ(y, x)} + ρ(x, u) + ρ(y, v) ϕ S (x) ϕ S (u) + ϕ T (y) ϕ T (v). Then the mappings of F have a unique common fixed point. Acknowledgements The authors are grateful to Prof. W. Takahashi for many comments suggestions improved the quality of the paper. References [1] P. C. Bhakta, T. Basu, Some fixed point theorems on metric spaces, J. Indian Math. Soc. 45 (1981), [2] J. Caristi, Fixed point theorems for mappings satisfying inwardness conditions, Trans. Amer. Math. Soc. 215 (1976), [3] I. Ekel, Nonconvex minimization problems, Bull. Amer. Math. Soc. 1 (1979), [4] O. Kada, T. Suzuki, W. Takahashi, Nonconvex minimization theorems fixed point theorems in complete metric spaces, Math. Japon. 44 (1996), [5] T. Suzuki, W. Takahashi, Fixed point theorems characterizations of metric completeness, Topol. Methods Nonlinear Anal. 8 (1996), [6] W. Takahashi, Existence theorems generalizing fixed point theorems for multivalued mappings, in Fixed Point Theory its Application (J.B.Baillon M.Thera, Eds.), Pitman Res. Notes in Math. Ser. #252, 1991, pp [7] W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, Interdisciplinary Graduate School of Science Engineering, Shimane University, 1060 Nishikawatsu, Matsue, Shimane , JAPAN Interdisciplinary Faculty of Science Engineering, Shimane University, 1060 Nishikawatsu, Matsue, Shimane , JAPAN address: kuroiwa@math.shimane-u.ac.jp