Motivated by some non-local boundary-value problems (BVPs) that arise in heat-flow problems, we establish new results for the existence of non-zero solutions of integral equations of the form$$ u(t)=\gamma(t)\alpha[u]+\int_{G}k(t,s)f(s,u(s))\,\mathrm{d}s, $$where $G$ is a compact set in $\mathbb{R}^{n}$. Here $\alpha[u]$ is a positive functional and $f$ is positive, while $k$ and $\gamma$ may change sign, so positive solutions need not exist. We prove the existence of multiple non-zero solutions of the BVPs under suitable conditions. We show that solutions of the BVPs lose positivity as a parameter decreases. For a certain parameter range not all solutions can be positive, but for one of the boundary conditions we consider we show that there are positive solutions for certain types of nonlinearity. We also prove a uniqueness result.

We establish new existence results for multiple positive solutions of fourth-order nonlinear equations which model deflections of an elastic beam. We consider the widely studied boundary conditions corresponding to clamped and hinged ends and many non-local boundary conditions, with a unified approach. Our method is to show that each boundary-value problem can be written as the same type of perturbed integral equation, in the space , because we obtain sharp results for the existence of one positive solution; for multiple solutions we seek optimal values of some of the constants that occur in the theory, which allows us to impose weaker assumptions on the nonlinear term than in previous works. Our non-local boundary conditions contain multi-point problems as special cases and, for the first time in fourth-order problems, we allow coefficients of both signs.