Noncompact‐type Krasnoselskii fixed‐point theorems and their applications

Abstract

Communicated by W. SprößigIn this paper, we first establish some user-friendly versions of fixed-point theorems for the sum of two operators in the setting that the involved operators are not necessarily compact and continuous. These fixed-point results generalize, encompass, and complement a number of previously known generalizations of the Krasnoselskii fixed-point theorem. Next, with these obtained fixed-point results, we study the existence of solutions for a class of transport equations, the existence of global solutions for a class of Darboux problems on the first quadrant, the existence and/or uniqueness of periodic solutions for a class of difference equations, and the existence and/or uniqueness of solutions for some kind of perturbed Volterra-type integral equations.

Classical solutions for a class of nonlinear wave equations

“…The related results are [1,2,[4][5][6][7] and the references therein. The fixed point theory for sums of operators has recently been developed (see [3,6,7,13,15,17,18] and the references therein). In [7], the authors have developed a new fixed point index for the sum of an expansive mapping and a 𝑘-set contraction defined in cones of Banach spaces.…”

Normal forms of parabolic logarithmic transseries

“…In the proof of the Main Theorem we use Proposition 3.3 below, which gives a fixed point theorem motivated by the Krasnoselskii Fixed Point Theorem (see e.g. [XG16]), and which follows directly by the Banach Fixed Point Theorem. Proposition 3.3 is proven (very elementary) in [PRRS21, Proposition 3.2].…”

Normal forms of hyperbolic logarithmic transseries

“…Hence, one of our goals here is to dispense, as far as possible, with this former impractical transfinite process. We design a new algorithm based on a version of the Banach fixed point theorem, inspired by Krasnoselskii's fixed point theorem [XG16]. To this end, we endow our spaces of transseries with a distance which gives them a structure of complete metric spaces.…”

“…The following proposition, which is an easy consequence of the Banach fixed point theorem, is inspired by a version of the fixed point theorem due to Krasnoselskii (see e.g. [XG16]). Proposition 3.2 (A fixed point theorem).…”