The fixed point theory of set-valued contractions which was initiated by Nadler [S.B. Nadler Jr., Multivalued contraction mappings, Pacific J. Math. 30 (1969) 475-488] was developed in different directions by many authors, in particular, by [S. Reich, Fixed points of contractive functions, Boll. Unione Mat. Ital. 5 (1972) 26-42; N. Mizoguchi, W. Takahashi, Fixed point theorems for multivalued mappings on complete metric spaces, J. Math. Anal. Appl. 141 (1989) 177-188; Y. Feng, S. Liu, Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings, J. Math. Anal. Appl. 317 (2006) 103-112]. In the present paper, the concept of contraction for set-valued maps in metric spaces is introduced and the conditions guaranteeing the existence of a fixed point for such a contraction are established. One of our results essentially generalizes the Nadler and Feng-Liu theorems and is different from the Mizoguchi-Takahashi result. The second result is different from the Reich and Mizoguchi-Takahashi results. The method used in the proofs of our results is inspired by Mizoguchi-Takahashi and Feng-Liu's ideas. Comparisons and examples are given.

The article is a continuation of the investigations concerning F-contractions which have been recently introduced in [Wardowski in Fixed Point Theory Appl. 2012:94,2012. The authors extend the concept of F-contractive mappings to the case of nonlinear F-contractions and prove a fixed point theorem via the dynamic processes. The paper includes a non-trivial example which shows the motivation for such investigations. The work is summarized by the application of the introduced nonlinear F-contractions to functional equations. MSC: 47H09; 47H10; 46N10; 54E50

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Investigations of problems of set-valued asymptotic fixed point theory of set-valued dynamic systems are initiated. The concepts of the set-valued asymptotic contractions of set-valued dynamic systems T : M → 2 M in metric space M are introduced and conditions guaranteeing the existence and uniqueness of endpoints v ∈ M of set-valued dynamic systems of set-valued asymptotic contractions T : M → 2 M and convergence to v of all generalized sequences of iterations of T are established. Examples and remarks show a fundamental difference between our results and the well-known ones.

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