A Generalization of Ćirić Quasicontractions

Abstract

and Applied Analysis 3 for every x, y ∈ X and every Cauchy sequence of the form {TSnx} for x ∈ X converges in X. Then i d TS, TS ≤ qδ O Tx, n , for all i, j ∈ {1, 2, . . . , n}, for all x ∈ X and n ∈ N, ii δ O Tx,∞ ≤ 1/ 1 − q d Tx, TSx , for all x ∈ X, iii S has a unique fixed point b ∈ X, iv lim TSx Tb. Proof. We will mainly follow the arguments in the proof of the Ćirić’s theorem. Let T, S be defined as in Theorem 2.1. We start with the proof of i . Let x ∈ X be arbitrary element. Let n ≥ 1 be integer and i, j ∈ {1, 2, . . . , n}. Then TSi−1x, TS, TSj−1x, TSx ∈ O Tx, n . 2.2 Note that we use S0 as the identity self-mapping. Due to 2.1 , we have d ( TSx, TSx ) d ( TS ( Si−1x ) , TS ( Sj−1x )) ≤ qmax { d ( TSi−1x, TSj−1x ) ;d ( TSi−1x, TSx ) , d ( TSj−1x, TSx ) , d ( TSi−1x, TSx ) , d ( TSx, TSj−1x )}

Cite this paper

@inproceedings{Karapnar2014AGO, title={A Generalization of Ćirić Quasicontractions}, author={Erdal Karapınar and Kieu Phuong Chi and Tran Duc Thanh}, year={2014} }