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In this paper, we introduce the new concept called partial generalized convex contractions and partial generalized convex contractions of order 2. Also, we establish some approximate fixed point theorems for such mappings in αcomplete metric spaces. Our results extend and unify the results of Miandaragh et al. [M. A. Miandaragh, M. Postolache, S. Rezapour, Approximate fixed points of generalized convex contractions, Fixed Point Theory and Applications 2013, 2013:255] and several well-known results in literature. We give some examples of a nonlinear contraction mapping, which is not applied to the existence of approximate fixed point and fixed point by using the results of Miandaragh et al. We also consider approximate fixed point results in metric space endowed with an arbitrary binary relation and approximate fixed point results in metric space endowed with graph.

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In this paper, we introduce the concept of partial α-ψ contractive mappings along with generalized metric distance. We also establish the existence of fixed point theorems for such mappings in generalized metric spaces. Our results extend and unify main results of Karapinar [E. Karapinar, Abstr. Appl. Anal., 2014 (2014), 7 pages] and several well-known results in literature. We give some examples to illustrate the usability of our results. Moreover, we prove the fixed point results in generalized metric space endowed with an arbitrary binary relation and the fixed point results in generalized metric space endowed with graph.

The aim of this work is to introduce some new notions of generalized convex contraction mappings and establish some approximate fixed point theorems for such mappings in the setting of complete metric spaces. Examples and application to approximate fixed point results for cyclic mappings are also given in order to illustrate the effectiveness of the obtained results. MSC: 47H09; 47H10