Existence of three solutions for a first-order problem with nonlinear non-local boundary conditions

TL;DR

This paper establishes conditions under which a nonlinear first-order differential equation with a nonlocal boundary condition has at least three positive solutions, using fixed point theory.

Contribution

It provides new existence criteria for multiple positive solutions in nonlinear boundary value problems with nonlocal conditions, employing the Leggett-Williams fixed point theorem.

Findings

Existence of at least three positive solutions for large λ

Conditions involving nonlinear nonlocal boundary terms

Application of fixed point theorem to boundary value problems

Abstract

Conditions for the existence of at least three positive solutions to the nonlinear first-order problem with a nonlinear nonlocal boundary condition given by && y'(t) - p(t)y(t) = \sum_{i=1}^m f_i\big(t,y(t)\big), \quad t\in[0,1], && \lambda y(0) = y(1) + \sum_{j=1}^n \Phi_j(\tau_j,y(\tau_j)), \quad \tau_j\in[0,1], are discussed, for sufficiently large $\lambda>1$. The Leggett-Williams fixed point theorem is utilized.