We give conditions which guarantee the existence of positive solutions for a variety of arbitrary order boundary value problems for which all boundary conditions involve functionals, using the well-known Krasnosel'skiĭ fixed point theorem. The conditions presented here deal with a variety of problems, which correspond to various functionals, in a uniform way. The applicability of the results obtained is demonstrated by a numerical application.

We study the existence of positive solutions of the nonlinear fourth order problem $u^{\left(4\right)}\left(x\right)=\lambda a\left(x\right)f\left(u\left(x\right)\right)$, u(0) = u’(0) = u”(1) = u”’(1) = 0, where a: [0,1] → ℝ may change sign, f(0) < 0, and λ < 0 is sufficiently small. Our approach is based on the Leray-Schauder fixed point theorem.

Let α,β,γ,δ ≥ 0 and ϱ:= γβ + αγ + αδ > 0. Let ψ(t) = β + αt, ϕ(t) = γ + δ - γt, t ∈ [0,1]. We study the existence of positive solutions for the m-point boundary value problem ⎧u” + h(t)f(u) = 0, 0 < t < 1, ⎨$\alpha u\left(0\right)-\beta u^{\text{'}}\left(0\right)={\sum }_{i=1}^{m-2}{a}_{i}u\left({\xi }_i\right)$ ⎩$\gamma u\left(1\right)+\delta u^{\text{'}}\left(1\right)={\sum }_{i=1}^{m-2}{b}_{i}u\left({\xi }_i\right)$, where ${\xi }_i\in (0,1)$, $a_i,b_i\in (0,\infty )$ (for i ∈ 1,…,m-2) are given constants satisfying $\varrho -{\sum }_{i=1}^{m-2}{a}_i\varphi \left({\xi }_i\right)>0$, $\varrho -{\sum }_{i=1}^{m-2}{b}_i\psi \left({\xi }_i\right)>0$ and $\Delta :=\left|\begin{array}{cc}-{\sum }_{i=1}^{m-2}{a}_i\psi \left({\xi }_i\right)& \varrho -{\sum }_{i=1}^{m-2}{a}_i\varphi \left({\xi }_i\right)\\ \varrho -{\sum }_{i=1}^{m-2}{b}_i\psi \left({\xi }_i\right)& -{\sum }_{i=1}^{m-2}{b}_i\varphi \left({\xi }_i\right)\end{array}\right|<0$. We show the existence of positive solutions if f is either superlinear or sublinear by a simple application of a fixed point theorem in cones. Our result extends a result established by Erbe and Wang for two-point...

In this paper we deal with the four-point singular boundary value problem $$\left\{\begin{array}{cc}{\left({\phi }_p\left(u^{\text{'}}\left(t\right)\right)\right)}^{\text{'}}+q\left(t\right)f(t,u\left(t\right),u^{\text{'}}\left(t\right))=0,\hfill & t\in (0,1),\hfill \\ u^{\text{'}}\left(0\right)-\alpha u\left(\xi \right)=0,u^{\text{'}}\left(1\right)+\beta u\left(\eta \right)=0,\hfill \end{array}\right.$$ where ${\phi }_p\left(s\right)={\left|s\right|}^{p-2}s$, $p>1$, $0<\xi <\eta <1$, $\alpha ,\beta >0$, $q\in C[0,1]$, $q\left(t\right)>0$, $t\in (0,1)$, and $f\in C\left(\right[0,1]\times (0,+\infty )\times ℝ,(0,+\infty \left)\right)$ may be singular at $u=0$. By using the well-known theory of the Leray-Schauder degree, sufficient conditions are given for the existence of positive solutions.