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

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
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