The boundary value problem is considered provided that f : [0, +∞) → [0, +∞) is Lipschitzian and is continuous and Lipschitzian in xand x′. We assume that f is bounded by two linear functions kx and lx, where k > l > 0, and h is bounded. We find the conditions on (λ, µ) which guarantee the existence of a solution to the problem. These conditions are of geometrical nature.

Bioenergy is one of the widespread renewable energy sources. Energy from biomass thermal conversion can reduce greenhouse emissions and have a positive effect on climate change. Biomass conversion is generally carried out in reactors of cylindrical shape. From a modelling point of view many factors have to be taken into account in order to optimize thermal efficiency of the conversion process. One of the important methods for the analysis of complex fluid flows is hydrodynamic stability theory. Base flow solution in classical hydrodynamic stability problems is usually found as a simple analytical solution of the equations of motion. Biomass conversion problems lead to nonlinear boundary value problems, which can be either solved numerically or analyzed using the bifurcation theory. In the present paper we analyze a mathematical model of heat transfer in the presence of nonlinear heat sources. This model includes the study of positive solutions to a nonlinear boundary value problem with certain boundary conditions. The equations in a problem contain several parameters, which essentially affect the behaviour and the number of solutions. Bifurcation analysis of the problem, conducted with respect to the parameters, allows obtaining somewhat precise results on the number of positive solutions. Generally, two, one and zero positive solutions are possible, depending on the values of the parameters. The obtained solutions represent base flow for the hydrodynamic stability problem, which can be solved with the objective to identify the factors affecting the conversion process.

Eigenvalue problems of the form x” = −λf(x+ ) + μg(x− ), x‘(a) = 0, x' (b) = 0 are considered. We are looking for (λ,μ) such that the problem (i), (ii) has a nontrivial solution. This problem generalizes the famous Fučík problem for piece‐wise linear equations. In our considerations functions f and g may be nonlinear. Consequently spectra may differ essentially from those for the Fučík equation.

Eigenvalue problems of the form x" = -A/{x' *") + fi.g{x~), x(0) = 0, rE(l) = 0 are considered, where x"^ and x~ are respectively the positive and the negative parts of x. We are looking for (A,/x) such that the problem has a nontrivial solution. This problem generalizes the famous Fu(5ik problem for piece-wise linear equations. In order to show that nonlinear PuCik spectra may differ essentially from the classical ones, we consider piece-wise linear functions f(x) and ^(x). We show that the first branches of the Fucik spectrum may contain bounded components.

We consider the Emden‐Fowler equation x” = ‐q(t)|x|2εx, ε > 0, in the interval [a,b]. The coefficient q(t) is a positive valued continuous function. The Nehari characteristic number An associated with the Emden‐Fowler equation coincides with a minimal value of the functional [] over all solutions of the boundary value problem x” = ‐q(t)|x|2εx, x(a) = x(b) = 0, x(t) has exactly (n ‐ 1) zeros in (a, b). The respective solution is called the Nehari solution. We construct an example which shows that the Nehari extremal problem may have more than one solution.