The existence of positive solutions for a nonlocal boundary-value problem with vector-valued response is investigated. We develop duality and variational principles for this problem. Our variational approach enables us to approximate solutions and give a measure of a duality gap between the primal and dual functional for minimizing sequences.

We study the existence of one-signed periodic solutions of the equations $$\begin{array}{ccc}& x^{\text{'}\text{'}}\left(t\right)-a^2\left(t\right)x\left(t\right)+\mu f(t,x\left(t\right),x^{\text{'}}\left(t\right))=0,\hfill & \hfill x^{\text{'}\text{'}}\left(t\right)+a^2\left(t\right)x\left(t\right)=\mu f(t,x\left(t\right),x^{\text{'}}\left(t\right)),\end{array}$$ where $\mu >0$, $a:(-\infty ,+\infty )\to (0,\infty )$ is continuous and 1-periodic, $f$ is a continuous and 1-periodic in the first variable and may take values of different signs. The Krasnosielski fixed point theorem on cone is used.

In the paper it is proved that the generalized linear boundary value problem generates a Fredholm operator. Its index depends on the number of boundary conditions. The existence results of Landesman-Lazer type are given as an application to nonlinear problems by using dual generalized boundary value problems.

We are concerned with the solvability of the fourth-order four-point boundary value problem ⎧ $u^{\left(4\right)}\left(t\right)=f(t,u\left(t\right),u^{\text{'}\text{'}}\left(t\right))$, t ∈ [0,1], ⎨ u(0) = u(1) = 0, ⎩ au”(ζ₁) - bu”’(ζ₁) = 0, cu”(ζ₂) + du”’(ζ₂) = 0, where 0 ≤ ζ₁ < ζ₂ ≤ 1, f ∈ C([0,1] × [0,∞) × (-∞,0],[0,∞)). By using Guo-Krasnosel’skiĭ’s fixed point theorem on cones, some criteria are established to ensure the existence, nonexistence and multiplicity of positive solutions for this problem.

Let $f:[0,1]\times {ℝ}^2\to ℝ$ be a function satisfying Caratheodory’s conditions and let $e\left(t\right)\in L^1[0,1]$. Let ${\xi }_i,{\tau }_j\in (0,1)$, $c_i,a_j\in ℝ$, all of the $c_i$’s, (respectively, $a_j$’s) having the same sign, $i=1,2,...,m-2$, $j=1,2,...,n-2$, $0<{\xi }_1<{\xi }_2<...<{\xi }_{m-2}<1$, $0<{\tau }_1<{\tau }_2<...<{\tau }_{n-2}<1$ be given. This paper is concerned with the problem of existence of a solution for the multi-point boundary value problems $$\begin{array}{c}\hfill x^{\text{'}\text{'}}\left(t\right)=f(t,x\left(t\right),x^{\text{'}}\left(t\right))+e\left(t\right),t\in (0,1)E\\ \hfill x\left(0\right)=\sum _{i=1}^{m-2}{c}_i{x}^{\text{'}}\left({\xi }_i\right),x\left(1\right)=\sum _{j=1}^{n-2}{a}_{j}x\left({\tau }_j\right)B{C}_{mn}\end{array}$$ and $$\begin{array}{c}\hfill x^{\text{'}\text{'}}\left(t\right)=f(t,x\left(t\right),x^{\text{'}}\left(t\right))+e\left(t\right),t\in (0,1)E\\ \hfill x\left(0\right)=\sum _{i=1}^{m-2}{c}_i{x}^{\text{'}}\left({\xi }_i\right),x^{\text{'}}\left(1\right)=\sum _{j=1}^{n-2}{a}_j{x}^{\text{'}}\left({\tau }_j\right),B{C}_{mn}^{\text{'}}\end{array}$$ Conditions for the existence of a solution for the above boundary value problems are given using Leray-Schauder Continuation theorem.