Triple positive solutions of conjugate boundary value problems
The author considers the following continuous boundary value problem: \[ \begin{gathered} (-1)^{n-p}y^{(n)}=a(t)h(y), \quad n\geq 2, \;t\in (0,1),\tag{1}\\ y^{(i)}(0)=0, \quad 0\leq i \leq p-1,\qquad y^{(i)}(1)=0, \quad 0\leq i \leq n-p-1, \end{gathered} \] and the discrete one: \[ \begin{gathered} (-1)^{n-p}\Delta^n y= F(k,y,\Delta y, \dots, \Delta^{n-1}y) , \quad n\geq 2, \;0\leq k\leq m, \tag{2}\\ \Delta^i y(0)=0, \quad 0\leq i \leq p-1, \qquad \Delta^i y(m+p+1)=0, \quad 0\leq i \leq n-p-1,\end{gathered} \] where \(1\leq p\leq n-1\) is fixed, \(h\) is a positive and nondecreasing function, \(a(t)\) is positive. Criteria are given for the existence of at least three positive solutions to boundary value problems (1) and (2). The criteria provide estimates on the norms of these positive solutions. As an application, the existence of radial solutions to certain partial difference equations is proved. The main tools of this study are fixed point theorems for operators on a cone.
scientific article; zbMATH DE number 1371503