We consider the boundary-value problem

where H: [0,+∞) → ℝ and f : [0, 1] × ℝ → ℝ are continuous and λ > 0 is a parameter. We show that if H satisfies a boundedness condition on a specified compact set, then this, together with an assumption that H is either affine or superlinear at +∞, implies existence of at least one positive solution to the problem, even in the case where we impose no growth conditions on f. Finally, since it can hold that f(t, y) < 0 for all (t, y) ∈ [0, 1]×ℝ, the semipositone problem is included as a special case of the existence result.

We prove new results on the existence, non-existence, localization and multiplicity of non-trivial solutions for perturbed Hammerstein integral equations. Our approach is topological and relies on the classical fixed-point index. Some of the criteria involve a comparison with the spectral radius of some related linear operators. We apply our results to some boundary-value problems with local and non-local boundary conditions of Neumann type. We illustrate in some examples the methodologies used.

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 deal with a class of $p$-Laplacian Dirichlet boundary-value problems where the combined effects of ‘sublinear’ and‘superlinear’ growths allow us to establish the existence of at least two positive solutions.

AMS 2000 Mathematics subject classification: Primary 35J25; 35J60. Secondary 35J65; 35J70

Working on a suitable cone of continuous functions, we give new results for integral equations of the form$\lambda u(t)=\int_{G}k(t,s)f(s,u(s))\,\mathrm{d} s:=Tu(t)$, where $G$ is a compact set in $\mathbb{R}^{n}$ and $k$ is apossibly discontinuous function that is allowed to change sign. We apply our results to prove existence of eigenvaluesof some non-local boundary-value problems.

AMS 2000 Mathematics subject classification: Primary 34B10. Secondary 34B18; 47H10; 47H30