Quasicontraction nonself-mappings on convex metric spaces and common fixed point theorems
- PDF / 144,547 Bytes
- 12 Pages / 468 x 680 pts Page_size
- 33 Downloads / 179 Views
e consider quasicontraction nonself-mappings on Takahashi convex metric spaces and common fixed point theorems for a pair of maps. Results generalizing and unifying fixed ´ c are established. point theorems of Ivanov, Jungck, Das and Naik, and Ciri´ 1. Introduction and preliminaries Let X be a complete metric space. A map T : X → X such that for some constant λ ∈ (0,1) and for every x, y ∈ X
d(Tx,T y) ≤ λ · max d(x, y),d(x,Tx),d(y,T y),d(x,T y),d(y,Tx)
(1.1)
´ c [1] introduced and studied quasiconis called quasicontraction. Let us remark that Ciri´ ´ c’s result traction as one of the most general contractive type map. The well known Ciri´ (see, e.g., [1, 6, 11]) is that quasicontraction T possesses a unique fixed point. ´ c’s result. For the convenience of the reader we recall the following recent Ciri´ Theorem 1.1 [2, Theorem 2.1]. Let X be a Banach space, C a nonempty closed subset of X, and ∂C the boundary of C. Let T : C → X be a nonself mapping such that for some constant λ ∈ (0,1) and for every x, y ∈ C
d(Tx,T y) ≤ λ · max d(x, y),d(x,Tx),d(y,T y),d(x,T y),d(y,Tx) .
(1.2)
Suppose that T(∂C) ⊂ C.
(1.3)
Then T has a unique fixed point in C. ´ c [3], let us remark that problem to extend the known fixed point theorem Following Ciri´ for self mappings T : C → C, defined by (1.1), to corresponding nonself mappings T : C → X, C = X, was open more than 20 years. In 1970, Takahashi [15] introduced the definition of convexity in metric space and generalized same important fixed point theorems previously proved for Banach spaces. In Copyright © 2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005:3 (2005) 365–375 DOI: 10.1155/FPTA.2005.365
366
Quasicontraction nonself-mappings
this paper we consider quasicontraction nonself-mappings on Takahashi convex metric spaces and common fixed point theorems for a pair of maps. Results generalizing and unifying fixed point theorems of Ivanov [7], Jungck [8], Das and Naik [3], Ciri´c [2], Gaji´c [5] and Rakoˇcevi´c [12] are established. Let us recall that (see Jungck [9]) the self maps f and g on a metric space (X,d) are said to be a compatible pair if
lim d g f xn , f gxn = 0
(1.4)
n→∞
whenever {xn } is a sequence in X such that lim gxn = lim f xn = x
n→∞
(1.5)
n→∞
for some x in X. Following Sessa [14] we will say that f ,g : X → X are weakly commuting if d( f gx,g f x) ≤ d( f x,gx) for every x ∈ X.
(1.6)
Clearly weak commutativity of f and g is a generalization of the conventional commutativity of f and g, and the concept of compatibility of two mappings includes weakly commuting mappings as a proper subclass. We recall the following definition of a convex metric space (see [15]). Definition 1.2. Let X be a metric space and I = [0,1] the closed unit interval. A Takahashi convex structure on X is a function W : X × X × I → X which has the property that for every x, y ∈ X and λ ∈ I
d z,W(x, y,λ) ≤ λd(z,x) + (1 − λ)d(z, y)
(1.7)
for every z ∈ X. If (X,d) is equipped with a Takahashi convex structure, then X is
Data Loading...
