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On the solvability of a parameter-dependent cantilever-type BVP

Gennaro Infante

Abstract

We discuss the solvability of a parameter dependent cantilever-type boundary value problem. We provide an existence and localization result for the positive solutions via a Birkhoff-Kellogg type theorem. We also obtain, under additional growth conditions, upper and lower bounds for the involved parameters. An example is presented in order to illustrate the theoretical results.

On the solvability of a parameter-dependent cantilever-type BVP

Abstract

We discuss the solvability of a parameter dependent cantilever-type boundary value problem. We provide an existence and localization result for the positive solutions via a Birkhoff-Kellogg type theorem. We also obtain, under additional growth conditions, upper and lower bounds for the involved parameters. An example is presented in order to illustrate the theoretical results.
Paper Structure (2 sections, 4 theorems, 38 equations, 1 figure)

This paper contains 2 sections, 4 theorems, 38 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Existence and localization of the eigenvalues

Key Result

Theorem 2.2

Let $(X,\| \, \|)$ be a real Banach space, $U\subset X$ be an open bounded set with $0\in U$, $\mathcal{K}\subset X$ be a cone, $T:\mathcal{K}\cap \overline{U}\to \mathcal{K}$ be compact and suppose that Then there exist $\lambda_{0}\in (0,+\infty)$ and $x_{0}\in \mathcal{K}\cap \partial U$ such that $x_{0}=\lambda_{0} Tx_{0}$.

Figures (1)

  • Figure 1: Localization of $(u_{\rho}, \lambda_{\rho})$

Theorems & Definitions (8)

  • Definition 2.1
  • Theorem 2.2: Theorem 2.3.6, guolak
  • Theorem 2.3
  • proof
  • Corollary 2.4
  • Theorem 2.5
  • proof
  • Example 2.6