Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition -- a non-monotone case
The authors are concerned with the existence and uniqueness of positive solutions to the boundary value problem \[ -w''+k^2w=\int_0^\pi f_\alpha(x,y)b(w(y))\,dy,\quad 0<x<\pi, \] subject to the Dirichlet boundary conditions \(w(0)=w(\pi)=0\), where the nonlinearity \(b\) represents the birth function and satisfies some biological sign, monotonicity and boundedness assumptions; the kernel \(f_\alpha\) is the sum of a sine series. The problem corresponds to searching steady state solutions for a reaction-diffusion equation. The authors consider separately the monotone and the non-monotone cases; then using the super-sub solutions, they prove existence and uniqueness of positive solutions. Two examples of applications illustrate the obtained results.
nonlocal second-order differential equations
scientific article; zbMATH DE number 6159400