Multivalued self almost local contractions

We introduce a new class of contractive mappings: the almost local contractions, starting from the almost contractions presented by V. Berinde in [V. Berinde, Approximating fixed points of weak contractions using the Picard iteration Nonlinear Analysis Forum 9 (2004) No.1, 43-53], and also from the concept of local contraction presented by Filipe Martins da Rocha and Vailakis in [V. Filipe Martins-da-Rocha, Y. Vailakis, Existence and uniqueness of a fixed point for local contractions, Econometrica, vol.78, No.3 (May, 2010) 1127-1141]. First of all, we present the notion of multivalued self almost contractions with many examples. The main results of this paper are given by the extension to the case of multivalued self almost local contractions.