On a general class of multi-valued weakly Picard mappings

Abstract

The concept of weak contraction from the case of single-valued mappings is extended to multi-valued mappings and then corresponding convergence theorems for the Picard iteration associated to a multi-valued weak contraction are obtained. The main results in this paper extend, improve and unify a multitude of classical results in the fixed point theory of single and multi-valued contractive mappings and also improve recent results from the paper [P.Z. Daffer, H. Kaneko, Fixed points of generalized contractive multi-valued mappings,

Some theorems for a new type of multivalued contractivemaps on metric space

“…Later on, several studies were conducted on a variety of generalizations, extensions, and applications of this result of Nadler (see [1,3,6,7,13,14,16,18]). …”

On generalized weakly directional contractions and approximate fixed point property with applications

“…Since then a number of generalizations in various different directions of the Banach contraction principle have been investigated by several authors; see and references therein. A interesting direction of research is the extension of the Banach contraction principle to multivalued maps, known as Nadler's fixed point theorem [2], Mizoguchi-Takahashi's fixed point theorem [3], Berinde-Berinde's fixed point theorem [5] and references therein. Another interesting direction of research led to extend to the multivalued maps setting previous fixed point results valid for single-valued maps with so-called directional contraction properties (see [20][21][22][23][24]).…”

“…Recall that a multivalued map T : X → N (X) is called (1) a Nadler's type contraction (or a multivalued k-contraction [3]), if there exists a number 0 <k < 1 such that [5][6][7], if there exist two constants θ (0, 1) and L ≥ 0 such that…”

Some Suzuki Type Fixed Point Theorems for Generalized Contractive Multifunctions

“…Then Kikkawa and Suzuki [8] gave another generalization which generalized the work of Suzuki and the Nadler fixed point theorem. Very Recently Bose and Roychowdhury [3] presented a theorem concerning (θ, L) -multivalued weak contraction which generalized the work of Kikkawa and Suzuki, Nadler [10], Kamran [7], and Berinde and Berinde [1]. Also Mot and Petrusel [9] gave another generalization concerning special multivalued generalized contractions which extended the result of Kikkawa and Suzuki.…”