Non-local second-order boundary value problems with derivative-dependent nonlinearity

Abstract

We prove the existence of multiple positive solutions of nonlinear second-order nonlocal boundary value problems with nonlinear term having derivative dependence. We allow the nonlinearity to grow quadratically with respect to derivatives. We obtain a priori bounds for norms of derivatives by using a recently obtained Gronwall-type inequality. Three examples illustrate some of the results. This article is part of the theme issue ‘Topological degree and fixed point theories in differential and difference equations’.

Existence of multiple positive solutions for a third-order boundary value problem with nonlocal conditions

“…Note that α 2 is negative. All constants satisfy inequalities (11), (12). Let us find Thus, by Theorem 2, boundary value problem (13) has at least one positive solution x(t) such that 1 x 8.…”

“…In [9], the authors obtain asymptotic formulas for eigenvalues and eigenfunctions of the one-dimensional Sturm-Liouville equation with one classical-type Dirichlet boundary condition and integral-type nonlocal boundary condition. In [11], the author proves the existence of multiple positive solutions of nonlinear second-order nonlocal boundary value problems with a nonlinear term having derivative dependence. In [6], the authors solve the two-dimensional nonlinear elliptic equation with the boundary integral condition depending on two parameters.…”

Boundary value problems for nonlinear second‐order functional differential equations with piecewise constant arguments

“…Some other models for thermostats have been studied in previous studies 28,29 (see also the study in Webb 30 ). For some models with this type of applications from the perspective of fractional calculus, we refer to previous works, 31,32 and also Rezapour et al 33 as example of variable order fractional thermostat models.…”