Existence of positive solutions for higher-order functional differential equations
The \(n\)th-order functional-differential equation \[ y^{(n)}(t)+ f(t, y(t+ \theta))= 0,\quad t\in [0,1],\;\theta\in [-\tau,a],\tag{1} \] together with the boundary conditions of Sturm-Liouville type \[ \begin{alignedat}{2} y^{(i)}(0)&= 0,&\quad &0\leq i\leq n-3,\\ \alpha y^{(n-2)}(t)- \beta y^{(n- 1)}(t)&= \eta(t),&\quad &t\in [-\tau, 0], \tag{2}\\ \gamma y^{(n-2)}(t)+ \delta y^{(n-1)}(t)&= \xi(t),&\quad &t\in [1,1+ a],\end{alignedat} \] is considered. Here, \(\tau\), \(a\), \(\alpha\), \(\beta\), \(\gamma\) and \(\delta\) are nonnegative constants, \(0\leq \tau+ a< 1\) and \(\gamma\beta+ \alpha\gamma+ \alpha\delta> 0\); \(\eta\in C^0[-\tau, 0]\), \(\xi\in C^0[1, 1+a]\), \(\eta(1)= 0\) and \(f: [0,1]\times C^+\to [0,\infty)\) is continuous, with \(C^+= \{\varphi: \varphi\in C^{n-2}[-\tau, a]\), \(\varphi(t)\geq 0\) for \(t\in [-\tau, a]\}\). The paper presents conditions guaranteeing the existence of one or two positive solutions to the problems posed by \textit{P. J. Y. Wong} and \textit{R. P. Agarwal} [Math. Probl. Eng. 2, No. 5, 401--434 (1996; Zbl 0921.34021)], \textit{R. P. Agarwal} and \textit{F.-H. Wong} [J. Comput. Appl. Math. 88, No. 1, 3--14 (1998; Zbl 0993.34017) and Nonlinear Stud. 5, No. 1, 15--24 (1998; Zbl 0928.34016)], \textit{R. P. Agarwal}, \textit{D. O'Regan} and \textit{V. Lakshmikantham} [Nonlinear Anal., Theory Methods Appl. 43, No. 1, 61--73 (2001; Zbl 0969.34021)] and \textit{I. Rachunková} and \textit{S. Stanek} [Georgian Math. J. 10, No. 1, 165--191 (2003; Zbl 1069.34029)]. Existence results are proved by means of Krasnoselskij's fixed-point theorem in a cone.
scientific article; zbMATH DE number 2105424