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We study Ulam-Hyers stability and the well-posedness of the fixed point problem for new type of generalized contraction mapping, so called α-λ-contraction mapping. The results in this paper generalize and unify several results in the literature such as the Banach contraction principle. [ABSTRACT FROM AUTHOR]

We introduce the class of mixed equilibriumproblemswith theweakly relaxed α-monotone bi-function in Banach spaces.Using the KKM technique, we obtain the existence of solutions for mixed equilibrium problem with weakly relaxed α-monotone bi-function in Banach spaces.The results presented in this paper extend and improve the corresponding results in the existing literature. [ABSTRACT FROM AUTHOR]

Recently, Basha (2011) established the best proximity point theorems for proximal contractions of the first and second kinds which are extension of Banach's contraction principle in the case of non-self-mappings. The aim of this paper is to extend and generalize the notions of proximal contractions of the first and second kinds which aremore general than the notion of self-contractions, establish the existence of an optimal approximate solution theorems for these non-self-mappings, and also give examples to validate ourmain results. [ABSTRACT FROM AUTHOR]