A fixed point theorem for a pair of maps satisfying a general contractive condition of integral type

We give a general condition which enables one to easily establish fixed point theorems for a pair of maps satisfying a contractive inequality of integral type. Branciari [1] obtained a fixed point result for a single mapping satisfying an analogue of Banach’s contraction principle for an integral-type inequality. The second author [3] proved two fixed point theorems involving more general contractive conditions. In this paper, we establish a general principle, which makes it possible to prove many fixed point theorems for a pair of maps of integral type. Define Φ = {ϕ: ϕ:R+→R} such that ϕ is nonnegative, Lebesgue integrable, and sat-isfies 0 ϕ(t)dt> 0 for each > 0. (1) Let ψ:R+→R+ satisfy that (i) ψ is nonnegative and nondecreasing on R+, (ii) ψ(t) < t for each t> 0, (iii) n=1ψn(t) < ∞ for each fixed t> 0. Define Ψ = {ψ: ψ satisfies (i)–(iii)}. Lemma 1. Let S and T be self-maps of a metric space (X,d). Suppose that there exists a sequence {xn} ⊂ X with x0 ∈ X, x2n+1: = Sx2n, x2n+2: = Tx2n+1, such that {xn} is complete and there exists a k ∈ [0,1) such that ∫ d(Sx,Ty) 0 ϕ(t)dt ≤ ψ ( ∫ d(x,y) 0 ϕ(t)d