In the present paper, considering a new concept of multivalued almost F-contraction, we give a general class of multivalued weakly Picard operators on complete metric spaces. Also, we give some illustrative examples showing that our results are proper generalizations of some previous theorems.

In the present work, two fixed point theorems for self maps on ordered cone metric spaces are proved motivated by [7, L. G. Huang and X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 332 (2007) 1468-1476] and [15, A. C. M. Ran and M. C. B. Reuring, A fixed point theorem in partially ordered sets and some application to matrix equations, Proc.

In his recent paper, Wardowski [16] introduced the concept of Fcontraction, which is a proper generalization of ordinary contraction on a complete metric space. Then, some generalizations of F -contractions including multivalued case are obtained in [2,4,7,13]. In this paper, by considering both F -contractions and fixed point result on ordered metric spaces, we introduce a new concept of ordered F -contraction on ordered metric space. Then, we give a fixed point theorem for such mapping. To support our result, we give an example showing that our main theorem is applicable, but both results of Ran and Reurings [12] and Wardowski [16] are not.

The concept of partial metric p on a nonempty set X was introduced by Matthews [8]. One of the most interesting properties of a partial metric is that p(x, x) may not be zero for x ∈ X. Also, each partial metric p on a nonempty set X generates a T0 topology on X. By omitting the small self-distance axiom of partial metric, Heckmann [7] defined the weak partial metric space. In the present paper, we give some fixed point results on weak partial metric spaces.

The concept of partial metric p on a nonempty set X was introduced by Matthews [13] and it was slightly modified by Heckmann [11] as weak partial metric. In [12], the authors studied fixed point result of new extension of Banach's contraction principle to partial metric space and give some generalized versions of the fixed point theorem of Matthews. In the present paper, we extend and generalize the previous results to weak partial metric spaces.