Fixed points and coincidence points for multimaps with not necessarily bounded images
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In metric spaces, single-valued self-maps and multimaps with closed images are considered and fixed point and coincidence point theorems for such maps have been obtained without using the (extended) Hausdorff metric, thereby generalizing many results in the literature including those on the famous conjecture of Reich on multimaps. 1. Introduction Many authors have been using the Hausdorff metric to obtain fixed point and coincidence point theorems for multimaps on a metric space. In most cases, the metric nature of the Hausdorff metric is not used and the existence part of theorems can be proved without using the concept of Hausdorff metric under much less stringent conditions on maps. The aim of this paper is to illustrate this and to obtain fixed point and coincidence point theorems for multimaps with not necessarily bounded images. Incidentally we obtain improvements over the results of Chang [3], Daffer et al. [6], Jachymski [9], Mizoguchi and Takahashi [12], and We¸grzyk [17] on the famous conjecture of Reich on multimaps (Conjecture 3.12). 2. Notation Throughout this paper, unless otherwise stated, (X,d) is a metric space; C(X) is the collection of all nonempty, closed subsets of X; B(X) is the collection of all nonempty, bounded subsets of X; CB(X) is the collection of all nonempty, bounded, closed subsets of X; S, T are self-maps on X; I is the identity map on X; F, G are mappings from X into C(X); for a nonempty subset A of X and x ∈ X, d(x,A) = inf {d(x, y) : y ∈ A}; for nonempty subsets A, B of X,
H(A,B) = max sup d(x,B),sup d(y,A) ; x∈A
Copyright © 2004 Hindawi Publishing Corporation Fixed Point Theory and Applications 2004:3 (2004) 221–242 2000 Mathematics Subject Classification: 47H10, 54H25 URL: http://dx.doi.org/10.1155/S1687182004308090
y ∈B
(2.1)
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Fixed points and coincidence points
f , g, and ρ are functions on X defined as f (x) = d(Sx,Fx), g(x) = d(Tx,Gx), and ρ(x) = d(x,Fx) for all x in X; for a nonempty subset A of X, αA = inf { f (x) : x ∈ A}, βA = inf {g(x) : x ∈ A}, γA = inf {ρ(x) : x ∈ A}, and δ(A) = sup{d(x, y) : x, y ∈ A}; for x, y in X and a nonnegative constant k, A(x, y) = max{d(Sx,T y),d(Sx,Fx),d(T y,Gy)}, Bk (x, y) = max{A(x, y),k[d(Sx,Gy) + d(T y,Fx)]}, A0 (x, y) = max{d(Sx,Sy),d(Sx,Fx),d(Sy,F y)}, C0 (x, y) = max{A0 (x, y),(1/2)[d(Sx,F y) + d(Sy,Fx)]}, A1 (x, y) = max{d(x, y),d(x,Fx),d(y,F y)}, C1 (x, y) = max{A1 (x, y),(1/2)[d(x,F y) + d(y,Fx)]}, m(x, y) = max{d(x, y),d(x,Fx),d(y,Gy),(1/2)[d(x,Gy) + d(y,Fx)]}; N is the set of all positive integers; R+ is the set of all nonnegative real numbers; ϕ : R+ → R+ ; for a real-valued function θ on a subset E of the real line, θ˜ and θˆ are the functions ˜ = limsup ˆ ˜ on E defined as θ(t) r →t+ θ(r) and θ(t) = max{θ(t), θ(t)} for all t in E; for a self1 map h on an arbitrary set E, h = h, and for a positive integer n, hn+1 is the composition n of h and hn ; for s ∈ (0, ∞], Γs = {ϕ : ϕ is increasing on [0,s) and ∞ n=1 ϕ (t) < +∞ ∀t in ∗ [0,s)}; Γ = {ϕ : ϕ ∈ Γs for some s ∈ (0, ∞]}; Γ = {ϕ ∈ Γ : ϕ(t) < t ∀t ∈ (0, ∞)}, Γ = {ϕ
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