Fixed Points of Multivalued Nonself Almost Contractions

The study of fixed points of single-valued self-mappings or multivalued self-mappings satisfying certain contraction conditions has a great majority of results inmetric fixed point theory. All these results are mainly generalizations of Banach contraction principle. The Banach contraction principle guarantees the existence and uniqueness of fixed points of certain self-maps in complete metric spaces. This result has various applications to operator theory and variational analysis. So, it has been extended in many ways until now. One of these is related to multivalued mappings. Its starting point is due to Nadler Jr. [1]. The fixed point theory for multivalued nonself-mappings developed rapidly after the publication of Assad and Kirk’s paper [2] in which they proved a non-self-multivalued version of Banach’s contraction principle. Further results for multivalued non-self-mappings were proved in, for example, [3–7]. For other related results, see also [8–38]. On the other hand, Berinde [11–13] introduced a new class of self-mappings (usually called weak contractions or almost contractions) that satisfy a simple but general contraction condition that includes most of the conditions in Rhoades’ classification [39]. He obtained a fixed point theorem for such mappings which generalized the results of Kannan [40], Chatterjea [41], and Zamfirescu [42]. As shown in [43], the weakly contractive metric-type fixed point result in [12] is “almost” covered by the related altering metric one due to Khan et al. [21]. In [9], M. Berinde and V. Berinde extendedTheorem 8 to the case of multivalued weak contractions.