We prove some fixed point results for -weakly contractive maps inG-metric spaces, we show that these maps satisfy propertyP. The results presented in this paper generalize several well-known comparable results in the literature.

LetXbe a Banach space and letLΦ(I,X)denote the space of OrliczX-valued integrable functions on the unit intervalIequipped with the Luxemburg norm. In this paper, we present a distance formula dist(f1,f2,LΦ(I,G))Φ, whereGis a closed subspace ofX, andf1,f2∈LΦ(I,X). Moreover, some related results concerning best simultaneous approximation inLΦ(I,X)are presented.

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Abstract. In this paper we have established some coupled coincidence and coupled common fixed point theorems on (ψ, φ)-weakly contractive condition for mapping having the g-mixed monotone property in partially ordered generalized metric spaces which generalize some recent fixed point theorems given in the literature. IntroductionThe Banach contraction principle have played a major role in solving problems in many branches of pure and applied mathematics as Boyd and Wong Aydi et. al.[12] has proved some coupled coincidence and coupled common fixed point theorems for a mixed g-monotone mapping satisfying nonlinear contractions in partially ordered generalized metric spaces. Saadati et. al. [13] has established some fixed point results in generalized ordered metric space. Recently, Choudhury and Maity [14], and Cho et. al. [15] initiated the study of coupled fixed point in generalized ordered metric spaces.The purpose of this work is to establish the coupled coincidence and coupled common fixed point theorems for nonlinear contraction condition related to a pair of altering distance functions for mappings with mixed monotone property in partially ordered G-metric spaces.Consistent with Mustafa and Sims [8], the following definitions and results will be needed in the sequel.

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