Existence of positive solutions of third-order boundary value problems with integral boundary conditions in Banach spaces. (English) Zbl 1380.34045
Summary: This paper deals with positive solutions of a third-order differential equation in ordered Banach spaces,
\[
(\varphi(-x''(t)))'=f(t, x(t)), \quad t\in J,
\]
subject to the following integral boundary conditions:
\[
x(0)=\theta, \quad x''(0)=\theta, \quad x(1)=\int^1_0 g(t)x(t) dt,
\]
where \(\theta\) is the zero element of \(E\), \(g \in L[0, 1]\) is nonnegative, \(\varphi : \mathbb R \to \mathbb R\) is an increasing
and positive homomorphism, and \(\varphi (0) = \theta_ 1\). The arguments are based upon the fixed-point principle in cone for strict set contraction operators. Meanwhile, as an application, we also give an example to illustrate our results.
MSC:
| 34B18 | Positive solutions to nonlinear boundary value problems for ordinary differential equations |
| 34G20 | Nonlinear differential equations in abstract spaces |
| 34A12 | Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations |
Keywords:
positive solutions; boundary-value problem; fixed-point principle; cone; measure of noncompactnessReferences:
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