Common Fixed Point for Three Pairs of Self-Maps Satisfying Common (E.A) Property in Generalized Metric Spaces

Abstract

and Applied Analysis 3 Moreover, if the pairs (f, R), (g, S), and (h, T) are weakly compatible, then f, g, h, R, S, and T have a unique common fixed point in X. Proof. First, we suppose that the subspace RX is closed inX, fX ⊆ SX, gX ⊆ TX, and two pairs of (f, R) and (g, S) satisfy common (E.A) property.Then by Definition 11 we know that, there exist two sequences {x n } and {y n } inX such that lim n→∞ fx n = lim n→∞ Rx n = lim n→∞ gy n = lim n→∞ Sy n = t (4) for some t ∈ X. Since gX ⊆ TX, there exists a sequence {z n } in X such that gy n = Tz n . Hence lim n→∞ Tz n = t. Next, we will show lim n→∞ hz n = t. In fact, if lim n→∞ hz n ̸ = t, then from condition (3), we can get G (fx n , gy n , hz n ) ≤ φ (max {G (Rx n , Sy n , Tz n ) , G (fx n , Rx n , Rx n ) , G (gy n , Sy n , Sy n ) , G (hz n , Tz n , Tz n ) , 1 3 [G (fx n , Sy n , Tz n ) + G (Rx n , gy n , Tz n ) +G (Rx n , Sy n , hz n )] , 1 3 [G (fx n , gy n , Tz n ) + G (fx n , Sy n , hz n ) +G (Rx n , gy n , hz n )] }) . (5) On letting n → ∞ and based on the property of φ, we can obtain G(t, t, lim n→∞ hz n ) ≤ φ (G ( lim n→∞ hz n , t, t)) < G ( lim n→∞ hz n , t, t) . (6) It is contradiction; so lim n→∞ hz n = t. Since RX is a closed subspace of X and lim n→∞ Rx n = t, there exists a p in X such that t = Rp. We claim that fp = t. Suppose not; then by using (3) we obtain G (fp, gy n , hz n ) ≤ φ (max {G (Rp, Sy n , Tz n ) , G (fp, Rp, Rp) , G (gy n , Sy n , Sy n ) , G (hz n , Tz n , Tz n ) ,

Cite this paper

@inproceedings{Gu2014CommonFP, title={Common Fixed Point for Three Pairs of Self-Maps Satisfying Common (E.A) Property in Generalized Metric Spaces}, author={Feng Gu and Yun Yin and Yisheng Song}, year={2014} }