Three positive solutions for higher order \(m\)-point boundary value problems.
The authors consider the \(2n\)th-order m-point boundary value problem \[ y^{(2n)}(t)=f(t,y(t),y''(t),\dots ,y^{(2(n-1))}(t)), \] \(0\leq t\leq 1,\) \(y^{(2i)}(0)=0,\) \(y^{(2i)}(1)=\sum_{j=1}^{m-2}k_{ij}y^{(2i)}(\xi _{j}),\) \(0\leq i\leq n-1,\) where \(k_{ij}>0\), \(i=0,1,\dots,n-1;\) \(j=1,2,\dots,m-2\), \(0<\xi _1<\xi _2<\dots <\xi _{m-2}<1.\) The function \(f\) satisfying \((-1)^{n}f:[0,1] \times \mathbb R^{n}\to [0,\infty)\) is continuous. Under some growth conditions imposed on \(f\) (which are too long to be stated here), the authors use the Leggett-Williams fixed-point theorem to show the existence of at least three positive solutions. Two lemmas from the recent article [\textit{Y. Guo}, \textit{W. Shan} and \textit{W. Ge}, J. Comput. Appl. Math. 151, No. 2, 415--424 (2003; Zbl 1026.34016)] are used.
Green function, Leggett-Williams fixed-point theorem
higher-order \(m\)-point boundary value problem
scientific article; zbMATH DE number 2057707