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The linear stability of a convective flow in a vertical fluid layer caused by nonlinear heat sources in the presence of cross-flow through the walls of the channel is investigated in this paper. This study is relevant to the analysis of factors that affect the effectiveness of biomass thermal conversion. The nonlinear problem for the base flow temperature is investigated in detail using the Krasnosel’skiĭ–Guo cone expansion/contraction theorem. It is shown that a different number of solutions can exist depending on the values of the parameters. Estimates for the norm of the solutions are obtained. The linear stability problem is solved numerically by a collocation method based on Chebyshev polynomials. It is shown that the increase in the cross-flow intensity stabilizes the flow, but there is also a small region of the radial Reynolds numbers where the flow is destabilized.

We consider a second order scalar conservative differential equation whose potential function is a Morse function with a finite number of critical points and is unbounded at infinity. We give an upper bound for the number of nonglobal nontrivial period annuli of the equation and prove that the upper bound obtained is sharp. We use tree theory in our considerations.

Linear stability analysis of a combined convective flow in an annulus is performed in the paper. The base flow is generated by two factors: (a) different constant wall temperatures and (b) heat release as a result of a chemical reaction that takes place in the fluid. The nonlinear boundary value problem for the distribution of the base flow temperature is analyzed using bifurcation analysis. The linear stability problem is solved numerically using a collocation method. Two separate cases are considered: Case 1 (non-zero different constant wall temperatures) and Case 2 (zero wall temperatures). Numerical calculations show that the development of instability is different for Cases 1 and 2. Multiple minima on the marginal stability curves are found for Case 1 as the Prandtl number increases. Concurrence between local minima leads to the selection of the global minimum in such a way that a finite jump in the value of the wave number is observed for some values of the Prandtl number. All marginal stability curves for Case 2 have one minimum in the range of the Prandtl numbers considered. The corresponding critical values of the Grashof number decrease monotonically as the Prandtl number grows.

We consider a third order system ${\boldsymbol x}'''={\boldsymbol f}({\boldsymbol x})$ with the two-point boundary conditions ${\boldsymbol x}(0)={\boldsymbol 0}$, ${\boldsymbol x}'(0)={\boldsymbol 0}$, ${\boldsymbol x}(1)={\boldsymbol 0}$, where ${\boldsymbol f}({\boldsymbol 0})={\boldsymbol 0}$ and the vector field ${\boldsymbol f}\in C^1(\mathbb{R}^n,\mathbb{R}^n)$ is asymptotically linear with the derivative at infinity ${\boldsymbol f}'(\infty)$. We introduce an asymptotically linear vector field ${\boldsymbol \phi}$ such that its singular points (zeros) are in a one-to-one correspondence with the solutions of the boundary value problem. Using the vector field rotation theory, we prove that under the non-resonance conditions for the linearized problems at zero and infinity the indices of ${\boldsymbol \phi}$ at zero and infinity can be expressed in the terms of the eigenvalues of the matrices ${\boldsymbol f}'({\boldsymbol 0})$ and ${\boldsymbol f}'(\infty)$, respectively. This proof constitutes an essential part of our article. If these indices are different, then standard arguments of the vector field rotation theory ensure the existence of at least one nontrivial solution to the boundary value problem. At the end of the article we consider the consequences for the scalar case.

We consider systems of the form $$\displaylines{ x_1''+ g_1(x_1) = h_1(x_1,x_2,\ldots,x_n),\cr x_2''+ g_2(x_2) = h_2(x_1,x_2,\ldots,x_n),\cr \cdots \cr x_n''+ g_n(x_n) = h_n(x_1,x_2,\ldots,x_n) }$$ along with the boundary conditions $$ x_1(0)=x_2(0)=\dots=x_n(0)=0=x_1(1)=x_2(1)=\dots=x_n(1)\,. $$ We assume that right sides are bounded continuous functions, and satisfy $h_i(0,0,\ldots,0)=0$. Also we assume that $g_i(x_i)$ are asymptotically asymmetric functions. By using vector field rotation theory, we provide the existence of solutions.

We consider the second order system x′′=f(x) with the Dirichlet boundary conditions x(0)=0=x(1), where the vector field f∈C1(Rn,Rn) is asymptotically linear and f(0)=0. We provide the existence and multiplicity results using the vector field rotation theory.

Eigenvalue problems of the form $x'' = -\lambda f(x^+) + \mu g(x^-),$ $\quad (i),$ $x(0) = 0, \; x(1) = 0,$ $\quad (ii)$ are considered, where $x^+$ and $x^-$ are the positive and negative parts of $x$ respectively. We are looking for $(\lambda, \mu)$ such that the problem $(i), (ii)$ has a nontrivial solution. This problem generalizes the famous Fu\v{c}\'{i}k problem for piece-wise linear equations. In our considerations functions $f$ and $g$ may be nonlinear functions of super-, sub- and quasi-linear growth in various combinations. The spectra obtained under the normalization condition $|x'(0)|=1$ are sometimes similar to usual Fu\v{c}\'{i}k spectrum for the Dirichlet problem and sometimes they are quite different. This depends on monotonicity properties of the functions $\xi t_1 (\xi)$ and $\eta \tau_1 (\eta),$ where $t_1 (\xi)$ and $\tau_1 (\eta)$ are the first zero functions of the Cauchy problems $x''= -f(x),$ $\: x(0)=0,$ $\: x'(0)=\xi> 0,$ $y''= g(y),$ $\: y(0)=0,$ $\: y'(0)=-\eta,$ $(\eta > 0)$ respectively.

We state the conditions of geometrical nature which guarantee the existence of a solution to the boundary value problem x ′′ + 2δx ′ + λf (x + ) − µg(x −) = h(t, x, x ′ ), x(0) = 0 = x(1) with a damping term 2δx ′ and nonnegative parameters λ, µ, provided that f (x +) − g(x −) is a sector-bounded nonlinearity.

Properties of asymmetric oscillator described by the equation (i), where and , are studied. A set of such that the problem (i), (ii), and (iii) have a nontrivial solution, is called α-spectrum. We give full description of α-spectra in terms of solution sets and solution surfaces. The exact number of nontrivial solutions of the two-parameter Dirichlet boundary value problem (i), and (ii) is given.

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.

Boundary value problems of the form are considered, where In our considerations functions f and g are generally nonlinear. We give a description of a solution set of the problem (i), (ii). It consist of all triples () such that (λ,μ,x(t)) nontrivially ′solves the problem(i),(ii) and |x (z)| = α at zero points z of the function x(t) (iii). We show that this solution set is a union of solution surfaces which are centro-affine equivalent. Each solution surface is associated with nontrivial solutions with definite nodal type. Properties of solution surfaces are studied. It is shown, in particular, that solution surface associated with solutions with exactly i zeroes in the interval (a,b) is centro-affne equivalent to a solution surface of the Dirichlet problem (i), x(a) = 0 = x(b), (iii) corresponding to solutions with odd number of zeros 2j − 1 (i ≠ 2j)in the interval (a,b).

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. First published online: 14 Oct 2010

Eigenvalue problems of the form x″ = -λf(x+) + μg(x-), x(0) = 0, x(1) = 0 are considered, where x+ and x− are respectively the positive and the negative parts of x. We are looking for (λ, μ) such that the problem has a nontrivial solution. This problem generalizes the famous Fučik problem for piece‐wise linear equations. In order to show that nonlinear Fučik spectra may differ essentially from the classical ones, we consider piece‐wise linear functions f(x) and g(x). We show that the first branches of the Fučik spectrum may contain bounded components. First Published Online: 14 Oct 2010

Eigenvalue problems of the form x 00 = −λf(x) + µg(x), (i), x(0) = 0, x(1) = 0 (ii) are considered. We are looking for (λ, µ) such that the problem (i), (ii) has a nontrivial solution. This problem generalizes the famous Fuchik problem for piece-wise linear equations. In our considerations functions f and g may be super-, sub- and quasi-linear in various combinations. The spectra obtained under the normalization condition (otherwise problems may have continuous spectra) structurally are similar to usual Fuchik spectrum for the Dirichlet problem. We provide explicit formulas for Fuchik spectra for super and super, super and sub, sub and super, sub and sub cases, where superlinear and sublinear parts of equations are of the form |x| 2α x and |x| 1 2β+1 respectively (α > 0, β > 0.)

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. First Published Online: 14 Oct 2010

We present the explicit non‐recursive formulas for the Taylor series expansion coefficients for the functions Sn (t) defined as solutions of the Emden ‐ Fowler type equations x” = –nx 2n−1 with the initial conditions x(0) = 0, x‘(0) = 1, where n = 1,2,… Pateikiamos vadinamos Emdeno‐Faulerio lygties x” = –nx 2n−1 pradinio uždavinio x(0) = 0, x‘(0) = 1,(n = 1,2,…) sprendiniu Sn (t) Teiloro eilute koeficientu išreikštines formules. Jos yra nerekursyvine. First Published Online: 14 Oct 2010