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A fixed-point theorem for integral type inequality depending on another function. (English) Zbl 1216.54015

Let (X,d) be a complete metric space, 0α<1, and S,T be selfmaps of X such that T is injective, continuous, and subsequentially convergent. Suppose that there exists xX such that 0 d(Tsy,TS 2 y) ϕ(t)dtα 0 d(T,TSy) ϕ(t)dt for each y in the orbit of x, where ϕ:=[0,+)[0,+) is a Lebesgue integrable mapping which is summable, nonnegative, and such that 0 ε ϕ(t)dt>0 for each ε>0. Then the authors show that

(i) lim n TS n x=Tq,

(ii) 0 d(Tq,TS n x) ϕ(t)dtα n 0 d(Tq,Tx) ϕ(t)dt, and

(iii) q is a fixed point of S if and only if G(x):=d(TSx,Tx) is S T -orbitally lower semicontinuous at q.

54H25Fixed-point and coincidence theorems in topological spaces