Existence of fixed points for the sum of two operators

Abstract

Key words Fixed points, sum of operators, dissipative operators, integral equations, Dirichlet problems MSC (2000) 47H10, 47H17, 47B44The purpose of this paper is to study the existence of fixed points for the sum of two nonlinear operators in the framework of real Banach spaces. Later on, we give some examples of applications of this type of results.

Well-posedness of a nonlinear evolution equation arising in growing cell population

“…is bijective and moreover (I−S) −1 is continuous (for instance see [15]), then given 1 (h) ∈ L 1 (l 1 ,l 2 ) there exists a unique function ∈ L 1 (l 1 ,l 2 ) such that (I−S)( ) = 1 (h).…”

Fixed Point Theory for 1-Set Contractions: a Survey

“…Further applications to nonlinear Volterra and Hammerstein equations can be found, e.g., in [2,45,62,63]. …”

Existence of fixed points and measures of weak noncompactness

“…It was proved in [12] that each nonexpansive mapping T : K → K , where K is a bounded closed and convex subset of a Banach space X which satisfies that I − T : K → X is φ-expansive, has a unique fixed point. Moreover, it is seen there that such mappings are a wider family than the strict contraction ones.…”

“…In the above two theorems the weak continuity of the mapping is required. Motivated by a nonlinear equation arising in transport theory, Latrach, Aziz Taoudi, Zeghal, in Theorem 2.1 of [11], established generalizations of the Schauder, Darbo and Krasnoselskii fixed point theorems for the weak topology (see also [12]). Such results also use the concept of the De Blasi measure of weak noncompactness and, in contrast with the previous two results, do not assume the weak continuity of the mappings.…”