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Article Citation - WoS: 17Citation - Scopus: 27Fixed Point Theorems in New Generalized Metric Spaces(Springer Basel Ag, 2016) Karapinar, Erdal; Karapınar, Erdal; O'Regan, Donal; Roldan Lopez de Hierro, Antonio Francisco; Shahzad, Naseer; Karapınar, Erdal; Mathematics; MathematicsThe aim of our paper is to present new fixed point theorems under very general contractive conditions in generalized metric spaces which were recently introduced by Jleli and Samet in [Fixed Point Theory Appl. 2015 (2015), doi:10.1186/s13663-015-0312-7]. Although these spaces are not endowed with a triangle inequality, these spaces extend some well known abstract metric spaces (for example, b-metric spaces, Hitzler-Seda metric spaces, modular spaces with the Fatou property, etc.). We handle several types of contractive conditions. The main theorems we present involve a reflexive and transitive binary relation that is not necessarily a partial order. We give a counterexample to a recent fixed point result of Jleli and Samet. Our results extend and unify recent results in the context of partially ordered abstract metric spaces.Article Citation - WoS: 173Fixed Point Theory for Cyclic Weak Φ-Contraction(Pergamon-elsevier Science Ltd, 2011) Karapinar, Erdal; Karapınar, Erdal; Karapınar, Erdal; Mathematics; MathematicsIn this manuscript, the notion of cyclic weak phi-contraction is considered. It is shown that a self-mapping T on a complete metric space X has a fixed point if it satisfied cyclic weak phi-contraction. (C) 2010 Elsevier Ltd. All rights reserved.Article Citation - WoS: 146Citation - Scopus: 157On the Solutions of Fractional Differential Equations Via Geraghty Type Hybrid Contractions(Ministry Communications & High Technologies Republic Azerbaijan, 2021) Adiguzel, Rezan Sevinik; Sevinik Adıgüzel, Rezan; Aksoy, Umit; Aksoy, Ümit; Karapinar, Erdal; Karapınar, Erdal; Erhan, Inci M.; Erhan, İnci; Sevinik Adıgüzel, Rezan; Aksoy, Ümit; Karapınar, Erdal; Erhan, İnci; Mathematics; Mathematics; MathematicsThe aim of this article is twofold. Firstly, to study fixed points of mappings on b metric spaces satisfying a general contractive condition called Geraghty type hybrid contraction. Secondly, to apply the theoretical results to the problem of existence and uniqueness of solutions of boundary value problems with integral boundary conditions associated with a certain type of nonlinear fractional differential equations. The conditions for the existence of fixed points for Geraghty type hybrid contractions are determined and several consequences of the main results are deduced. Some examples on boundary value problems for nonlinear fractional differential equations of order 3 < alpha <= 4 are provided, where the existence and uniqueness of solutions are shown by using Geraghty type contractions.Article Citation - WoS: 3Citation - Scopus: 6A Short Note on the Equivalence Between 'best Proximity' Points and 'fixed Point' Results(Springeropen, 2014) Jleli, Mohamed; Karapınar, Erdal; Karapinar, Erdal; Samet, Bessem; Karapınar, Erdal; Mathematics; MathematicsIn this short note, we notice that, unexpectedly, some existing fixed point results and recently announced best proximity point results are equivalent.
