Coupled Fixed Point Theorems for (φ,ψ)-Contractive Mixed Monotone Mappings in Partially Ordered Metric Spaces and Applications

Abstract

The object of this paper is to establish the existence and uniqueness of coupled fixed points under a (φ, ψ)-contractive condition for mixed monotone operators in the setup of partially ordered metric spaces. Presented work generalizes the recent results of Berinde (2011, 2012) and weakens the contractive conditions involved in the well-known results of Bhaskar and Lakshmikantham (2006), and Luong and Thuan (2011). The effectiveness of our work is validated with the help of a suitable example. As an application, we give a result of existence and uniqueness for the solutions of a class of nonlinear integral equations.

Coupled fixed point theorems in complete metric spaces endowed with a directed graph and application

“…The proof of the existence and uniqueness of the coupled common fixed point for our main results can be obtained by using a similar assertion as in Theorem 9 in [18]. …”

“…F .x n ; y n / D gx nC1 and F .y n ; x n / D gy nC1 ; F .u n ; v n / D gu nC1 and F .v n ; u n / D gv nC1 ; F .t n ; s n / D gt nC1 and F .s n ; t n / D gs nC1 ; (18) for all n 2 N. From the properties of coincidence points, x D x n , y D y n and u D u n , v D v n , namely, F .x; y/ D gx n , F .y; x/ D gy n and F .u; v/ D gu n , F .v; u/ D gv n for all n 2 N. As .gx; gt/, .gy; gs/ 2 E .G/, we get .gx; gt 0 /, .gy; gs 0 / 2 E .G/. Since F and g are G edge preserving, we obtain .F .x; y/ ; F .t 0 ; s 0 // D .gx; gt 1 / and .F .y; x/ ; F .s 0 ; t 0 // D .gy; gs 1 / 2 E .G/.…”

“…x n ; y n // ; F .gx n ; gy n // D 0 and lim n!1 d .g .F .y n ; x n // ; F .gy n ; gx n // D 0 whenever fx n g and fy n g are sequences in X such that lim n!1 F .x n ; y n / D lim n!1 gx n D x and lim n!1 F .y n ; x n / D lim n!1 gy n D y with x; y 2 X: 18]). Letˆdenote the class of functions ' W OE0; 1/ !…”

Some Results on N-Tupled Coincidence and Fixed Points of Graphs on Metric Spaces and an Application to Integral Equations

“…Definition (see [24]). Let Φ be the set of all functions : [0, ∞) → [0, ∞) such that, for any ∈ Φ, all following conditions are satisfied:…”