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Green’s function and existence of solutions for a functional focal differential equation. (English) Zbl 1112.34041

The paper deals with the boundary value problem for the higher-order functional-differential equation
\[ x^{(n)}(t)=f(t,x(t+\theta)), \qquad t_1\leq t\leq t_3,\qquad -\tau\leq\theta\leq 0,\tag{1} \]
\[ x^{(i)}(t_1)=0,\qquad 0\leq i\leq n-4,\qquad n\geq 4,\tag{2} \]
\[ \alpha x^{(n-3)}(t)-\beta x^{(n-2)}(t)=\sigma(t),\qquad t_1-\tau\leq t\leq t_1,\tag{3} \]
\[ x^{(n-2)}(t_2)=0,\qquad x^{(n-1)}(t_3)=0,\tag{4} \]
where \(t_1<t_2<t_3\), \(\alpha,\beta>0\), \(t_3-t_1\geq \tau\geq 0\), \(\theta\in[-\tau,0]\) is a constant, \(\sigma:[t_1-\tau,t_1]\to \mathbb{R}\) is continuous with \(\sigma(t_1)=0\), and \(f:\mathbb{R}^2\to \mathbb{R}\) is continuous and nonnegative for \(x\geq 0\).
The first part of the paper is devoted to determining Green’s function for a homogeneous third-order three-point mixed boundary value problem and finding conditions to ensure its positivity. Then, this is applied to establish conditions guaranteeing the existence of at least one positive solution to (1)–(4) via Krasnosel’skij’s fixed-point theorem. Finally, the result on the existence of at least two positive solutions to (1)–(4) is proved by using the Avery-Henderson fixed-point theorem.
Reviewer: Robert Hakl (Brno)

MSC:

34K10 Boundary value problems for functional-differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B27 Green’s functions for ordinary differential equations
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