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.

In this paper, we discuss the existence of solutions for a boundary value problem of fractional differential inclusions with nonlocal Riemann-Liouville integral boundary conditions. Our results include the cases when the multivalued map involved in the problem is (i) convex valued, (ii) lower semicontinuous with nonempty closed and decomposable values and (iii) nonconvex valued. In case (i) we apply a nonlinear alternative of Leray-Schauder type, in the second case we combine the nonlinear alternative...