Abstract

The authors establish a very nice generalization of Banach’s fixed point theorem for selfmappings defined on complete metric spaces by employing the following contractive condition of integral type: Let (X,d) be a complete metric space and φ:[0,+∞)→[0,+∞) a Lebesgue-integrable mapping. A mapping T:X→X is said to be ψ∫φ-weakly contractive if ∀x,y∈X, ψ(∫d(Tx,Ty)0φ(t)dt)≤ψ(∫d(x,y)0φ(t)dt)−ϕ(∫d(x,y)0φ(t)dt), where ψ:[0,+∞)→[0,+∞) is a continuous and nondecreasing function and ϕ:[0,+∞)→[0,+∞) is a lower semicontinuous and nondecreasing function such that ψ(t)=0=ϕ(t)⇔t=0. Their result indeed generalizes several known results in the literature of which the recent result of [P. N. Dutta and B. S. Choudhury, Fixed Point Theory Appl. 2008, Article ID 406368, 8 p. (2008; Zbl 1177.54024)] is a particular case. The reference section of the present paper enlists a number of articles on the study of Banach’s fixed point theorem.