On a generalization of Krasnoselskii's theorem

Abstract

In this paper we prove a generalization of the well known theorem of Krasnoselskii on the superposition operator in which the domain of Nemytskii's operator is a product space. We also give an application of this result.

Stability of Nonlinear Dirichlet BVPs Governed by Fractional Laplacian

“…Applying the Krasnoselskii theorem (cf. [ 37 , 38 ]) the continuity of the operator L s × L p ∋( u , w ) ↦ ϕ (·, u (·), w (·)) ∈ L 1 follows and which together with condition (A2) implies I k ( u k ) → 0 as k → ∞ . Thus we have got a contradiction with the inequality | I k ( u k )| > ε 0 .…”

Optimal control of systems governed by fractional Laplacian in the minimax framework

“…The lower estimate by ε leads to the contradiction with the upper bound as all the above terms tend to zero. To observe this it is enough to apply Krasnoselskii Theorem [19,Theorem 2] on the continuity of the superposition of the operators:…”

Application of Mountain Pass Theorem to superlinear equations with fractional Laplacian controlled by distributed parameters and boundary data

“…By (C6) and (C7) the cost functional is well-defined and continuous with respect to (z, u) ∈ R × U variables. Let (z k , u k ) , k ∈ N be a minimizing sequence for problem (29) Entire class U λ is equicontinuous and uniformly bounded, so certainly {u k } is such as well. By Arzéla-Ascoli's Theorem, there exists a subsequence {u k } such that u k → u 0 uniformly on Ω and u 0 ∈ U λ .…”