Coexistence fixed point theorems in product Banach spaces and applications

Abstract

New coexistence fixed point theorems in product Banach spaces are established via the classical theory of fixed point index for compact maps defined on cones. Each component of a coexistence fixed point is nonzero. These coexistence fixed point theorems are applied to obtain results on coexistence solutions of systems of Hammerstein integral equations, second‐order ordinary differential equations with general separated boundary conditions, and one‐dimensional competition model of Ricker types.

A fixed point index approach to Krasnosel'skii-Precup fixed point theorem in cones and applications

“…Note that Krasnosel'skiȋ-Precup fixed point theorem has been already employed by several authors in order to study the existence, localization and multiplicity of positive solutions for different types of systems of boundary value problems, see for instance [7,23,[27][28][29]. Our intention is to emphasize the applicability of the new fixed point theorems established here and so we present a multiplicity result for a system of Hammerstein type equations, which complements previous results in the literature, see [2,8,14,16,25] and the references therein. Moreover, concerning radial solutions of (p 1 , p 2 )-Laplacian systems, our sufficient conditions provide not only the existence of positive solutions, but also a novel localization of them, cf.…”

“…This fact motivated Precup to establish the vector version of Krasnosel'skiȋ fixed point theorem [21,22] (see Theorem 2.3 below), which provides a component-wise localization of the fixed points. Thus, it gives sufficient conditions for the existence of a coexistence fixed point as coined by Lan [16], that is, a fixed point with all the components different from zero. As far as we know, there are only a few papers in the literature which deal with theoretical results concerning coexistence fixed points of compact maps, see [3,16,21,22,25].…”

“…Thus, it gives sufficient conditions for the existence of a coexistence fixed point as coined by Lan [16], that is, a fixed point with all the components different from zero. As far as we know, there are only a few papers in the literature which deal with theoretical results concerning coexistence fixed points of compact maps, see [3,16,21,22,25].…”

Nonzero nonnegative solutions of population models of Ricker types and theory of limit inferiors and superiors

“…Let u (1) , u (1) 1 ∈ D 1 m with u (1) 1 ≤ u (1) and u (2) , u (2) 1 ∈ D 2 m with u (2) ≤ u (2) 1 , where…”

“…By (10), we have for u (1) ∈ D 1 m and u (2) ∈ D 2 m with |u (2) | > max{|u (2) 0 |, r}, 𝑓 ∞ (r) ≤ 𝑓 (u (1) , u (2) ).…”