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Positive solutions for singular three-point boundary-value problems. (English) Zbl 1179.34019

The paper is devoted to the following singular three-point boundary value problem \[ \begin{cases} -y''(t)=a(t)f(t,y(t),y'(t)),\quad t\in(0,1)\\ y'(0)=0,\quad y(1)=\alpha y(\eta), \end{cases} \] where \(0<\alpha,\eta<1\) and \(a\in C((0,1),(0,+\infty)).\) The nonlinearity \(f=f(t,x,z)\) may have singularities at \(x=0\) and \(z=0.\) Some growth conditions are imposed on \(f.\) One of the main hypotheses states that \(f\) should be dominated by separate-variable functions, say \[ f(t,x,z)\leq\Phi(t)h(x)g(| z|). \] The conditions \[ \int_0^1a(s)\Phi(s)ds=+\infty\,\text{ and }\,\int_0^z\frac{dr}{g(r)}<+\infty\;\forall\,z\geq0 \] are shown to be necessary. The proofs of the existence theorems use the index fixed point theory, an integral formulation and several technical lemmas. The positive solutions lie in special cones of a weighted Banach space. Some examples illustrate the obtained results.

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
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