Multiplicity Results for Operator Systems via Fixed Point Index

Abstract

In this paper, we introduce and discuss the concept of a mutual control problem. Our analysis relies on a vector fixed-point approach based on the fixed-point theorems of Perov, Schauder, Leray-Schauder, and Avramescu. Additionally, for a related semi-observability problem, we employ a novel technique utilizing Bielecki equivalent norms.

Multiplicity Solutions for Integral Boundary Value Problem of Fractional Differential Systems

“…is result is based on an extension of the Krasnosel'skiǐ's fixed-point theorem due to Radu Precup and Jorge Rodriguez-Lopez in [46]. We rewrite the original fractional differential systems as equivalent fractional integral systems.…”

“…□ e main proof of this research uses the following theorem in [46]. Theorem 1 (see [46]). Let (X, ‖ • ‖) be a Banat space, K 1 , K 2 ⊂ X two cones, and…”

“…However, in recent years, many scholars have begun to use the fixed-point index to study the existence and multiplicity of operator equation and operator systems [44][45][46][47][48][49][50][51]. For example, in [46], the authors use the fixed-point index in cones to study the existence, localization, and multiplicity of positive solutions to operator systems of the following form:…”

Existence of positive solutions of a Hammerstein integral equation using the layered compression-expansion fixed point theorem

“…In this paper, we show that under the same assumptions on the kernel G, the Layered Expansion-Compression Fixed Point Theorem can be applied to show the existence of positive and positive symmetric solutions of the Hammerstein integral equation. More examples of recent work on Hammerstein integral equations, can be found in [14,15,19,21,26].…”

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

“…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].…”