Positive solutions of a derivative dependent second-order problem subject to Stieltjes integral boundary conditions. (English) Zbl 1449.34084

Ming, Zhongyang; Zhang, Guowei; Li, Hongyu

Electron. J. Qual. Theory Differ. Equ. 2019, Paper No. 98, 15 p. (2019).

Summary: In this paper, we investigate the derivative dependent second-order problem subject to Stieltjes integral boundary conditions \[-u''(t)=f(t,u(t),u'(t)),\quad t\in[0,1],\] \[au(0)-bu'(0)=\alpha[u],\, cu(1)+du'(1)=\beta[u],\] where \(f\): \([0,1]\times \mathbb{R}^+\times \mathbb{R}\rightarrow \mathbb{R}^+\) is continuous, \(\alpha[u]\) and \(\beta[u]\) are linear functionals involving Stieltjes integrals. Some conditions on the nonlinearity \(f\) and the spectral radius of the linear operator are presented that guarantee the existence of positive solutions to the problem by the theory of fixed point index. The conditions allow that \(f(t,x_1,x_2)\) has superlinear or sublinear growth in \(x_1,x_2\). Two examples are provided to illustrate the theorems under multi-point and integral boundary conditions with sign-changing coefficients.

MSC:

34B18	Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B10	Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15	Nonlinear boundary value problems for ordinary differential equations
47N20	Applications of operator theory to differential and integral equations

Keywords:

positive solution; fixed point index; cone; spectral radius