Two Fixed-point Theorems for Mappings Satisfying a General Contractive Condition of Integral Type
نویسنده
چکیده
We establish two fixed-point theorems for mappings satisfying a general contrac-tive inequality of integral type. These results substantially extend the theorem of Branciari (2002). In a recent paper [1], Branciari established the following theorem. Theorem 1. Let (X, d) be a complete metric space, c ∈ [0, 1), f : X → X a mapping such that, for each x, y ∈ X, d(f x,f y) 0 ϕ(t)dt ≤ c d(x,y) 0 ϕ(t)dt, (1) where ϕ : R + → R + is a Lebesgue-integrable mapping which is summable, non-negative, and such that, for each > 0, 0 ϕ(t)dt > 0. Then f has a unique fixed point z ∈ X such that, for each x ∈ X, lim n f n x = z. In [1], it was mentioned that (1) could be extended to more general contrac-tive conditions. It is the purpose of this paper to make such an extension to two of the most general contractive conditions. Define m(x, y) = max d(x,y),d(x,f x),d(y,f y), d(x, f y) + d(y, f x) 2. (2) Our first result is the following theorem. Theorem 2. Let (X, d) be a complete metric space, k ∈ [0, 1), f : X → X a mapping such that, for each x, y ∈ X, d(f x,f y) 0 ϕ(t)dt ≤ k m(x,y) 0 ϕ(t)dt, (3)
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تاریخ انتشار 2002