A unified approach to compression-expansion fixed point theorems for operators systems and applications

Abstract

In this paper, we present some fixed point theorems for operator systems in the line of Krasnosel'skii's theorem in cones. The cone-compression and cone-expansion type conditions are imposed in a component-wise manner. Unlike related results in the literature, the operators are allowed to be defined in the Cartesian product of conical regions delimited by nonconvex sets. Our approach, based on the fixed point index, ensures the existence of a coexistence fixed point--that is, one with nontrivial components. As a first application, we establish several localization results for systems of integral equations between strictly star-shaped sets defined by functionals. These results cannot be derived solely from previous studies dealing with operators in annular regions. A second application concerns nonlinear systems involving the Φ-Laplacian.

Figures (3)

• Figure 1: Illustration of the retraction $\rho^r_h$ on $\overline{\Omega}$, a strictly star convex set over a cone $K$ in $\mathbb{R}^2$.
• Figure 2: An illustrative idea of the relative position of the sets $(\overline{K}_j)_{r_j}$, $\overline{\Omega}_{r_j}^{\varphi_j}$ and $(\overline{K}_j)_{\frac{r_j}{c_j}}$.
• Figure 3: Graphical representation of mutual non-inclusion between $\Omega_{r_j}^{\varphi_j}$ and $(K_j)_{r_j+\varepsilon}$ (see Figure \ref{['fig2']}).