A multivalued version of Krasnoselskii’s theorem in generalized Banach spaces

Abstract

The purpose of this paper is to extend Krasnoselskii's fixed point theorem to the case of generalized Banach spaces for multivalued operators. As application, we will give an existence result for a system of Fredholm-Volterra type differential inclusions.

Some Krasnosel’skii type random fixed point theorems

“…Schauder's fixed point theorem was extended to generalized Banach spaces by Viorel [35]. Very recently, in vector-valued Banach spaces, Krasnosel'skii type fixed point theorems for single and multivalued operators were studied by Petre and Petruşel [36], and Petre [37]. In [38], the authors obtained a random version of the Perov and Schauder fixed point theorems.…”

Fixed point theorems in vector metric spaces for multivalued operators

“…Notice that in the setting of a vector metric space can appear some difficulties, especially when we want to give an example or an application of our result to a differential inclusion. For example, in the case when the vector metric space is R n , we need to introduce an absolute value property for a matrix which converges to zero to can give an application to a differential inclusions system, see I.-R. Petre [12] (Theorem 2.2), in the case when the vector metric space is a Riesz space E, for a working application we need to assume that E is order complete which guarantees that a certain infimum exists in E, see I.-R. Petre [11] (Theorem 2.28), and also for an application in generalized b-metric spaces, see I.-R. Petre, M. Bota [13] (Theorem 3.13).…”

Existence results and Ulam–Hyers stability to impulsive coupled system fractional differential equations

“…The classical Banach contraction principle is a very useful tool in nonlinear analysis with many applications to operational equations, fractal theory, optimization theory and other topics. The classical Banach contraction principle was extended for contractive maps on spaces endowed with vector-valued metrics by Perov in 1964 [40] and Perov and Kibenko [41]. In 1966 Perov formulated a fixed point theorem which extends the well-known contraction mapping principle for the case when the metric d takes values in R m , that is, in the case when we have a generalized metric space.…”