In this paper, we introduce a new class of set-valued mappings which is called MD-type mappings. This class of mappings is a set-valued case of a class of the D-type mappings. The class of D-type mappings is a generalization of nonexpansive mappings that recently introduced by Kaewkhao and Sokhuma. The class of MD-type mappings includes upper semi-continuous Suzuki type mappings, upper semi-continuous L-type mappings and upper semi-continuous quasi-nonexpansive mappings. We study relationships between MD-type mappings and some other set-valued generalization of nonexpansive mappings. In the sequel we show that if E is a nonempty, WEAKly compact and convex subset of a Banach space (X, ‖ . ‖ ) for which every D-type self-mapping on E has a FIXED POINT, then every MD-type self-mapping on E has a FIXED POINT. This result gives a partial affirmative answer to an open problem of Rich. Moreover, we establish some FIXED theorems for MD-type mappings which generalize some well-known results in the literature.

In this paper, we prove some common FIXED POINT theorems for WEAKly compatible mappings in metric spaces satisfying generalized (Y , j) -contractive conditions under the common limit range PROPERTY. We present a FIXED POINT theorem for four finite families of self-mappings which can be utilized to derive common FIXED POINT theorems involving any number of finite mappings. Our results improve and extend the corresponding results of Radenovic et al. (Bull. Iranian Math. Soc.38 (3): 625-645, 2012). We also furnish some illustrative examples to support our main results.

In this paper we define WEAK F-contractions on a metric space into itself by extending F-contractions introduced by D. Wardowski (2012) and provide some FIXED POINT results in complete metric spaces and in partially ordered complete generalized metric spaces. Some relation- ships between WEAK j-contractions and φ-contractions are highlighted. We also give some applications on fractal theory improving the classical Hutchinson-Barnsley's theory of iterated function systems. Some illustrative examples are provided.

This paper extends the notion of WEAK convergence in metric spaces to the case of S-metric spaces. Moreover, some results on the WEAK convergence of FIXED POINT iterations of Banach's, Kannan's, Chatterjee's, Reich's, Hardy and Roger's types of contractions on S-metric spaces are obtained. In addition, an example is presented to demonstrate our primary result.

In this paper, we describe some topological properties of dualistic partial metric spaces and establish some  xed POINT theorems for WEAK contraction mappings of rational type de ned on dualistic partial metric spaces. These results are generalizations of some existing results in the literature. Moreover, we present examples to illustrate our result.