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Iterative solutions of singular boundary value problems of third-order differential equation. (English) Zbl 1225.34031

Summary: We consider the following third-order boundary value problem
\[ \begin{gathered} u'''(t)+f(t,u(t))=0,\quad t\in (0,1),\\ u(0)=u'(0)=0,\qquad u'(1)=\alpha u'(\eta),\end{gathered}\tag{1} \]
where \(f\in C((0,1)\times (-\infty,+\infty),(-\infty,+\infty))\), \(0<\eta<1\).
We give the unique solution of the boundary value problem (1) under the conditions that \(\alpha\eta\neq 1\) and \(f(t,x)\) is mixed nonmonotone in \(x\) by using the cone theory and the Banach contraction mapping principle.

MSC:

34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34A45 Theoretical approximation of solutions to ordinary differential equations
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References:

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