In this paper, we look at a linear system of ordinary differential equations as derived from the two-dimensional Ginzburg–Landau equation. In two cases, it is known that this system admits bounded solutions coming from the invariance of the Ginzburg–Landau equation by translations and rotations. The specific contribution of our work is to prove that in the other cases, the system does not admit any bounded solutions. We show that this bounded solution problem is related to an eigenvalue problem.

A generalised Hermite spectral method for Fisher's equation in genetics with different asymptotic solution behaviour at infinities is proposed, involving a fully discrete scheme using a second order finite difference approximation in the time. The convergence and stability of the scheme are analysed, and some numerical results demonstrate its efficiency and substantiate our theoretical analysis.

Let l[y] be a formally self-adjoint differential expression of an even order on the interval [0, b〉(b ≤ ∞) and let L0 be the corresponding minimal operator. By using the concept of a decomposing boundary triplet, we consider the boundary problem formed by the equation l[y] − λy = f, f ∈ L2[0, b〉, and the Nevanlinna λ-dependent boundary conditions with constant values at the regular endpoint 0. For such a problem we introduce the concept of the m-function, which in the case of self-adjoint separated boundary conditions coincides with the classical characteristic (Titchmarsh–Weyl) function. Our method allows one to describe all minimal spectral functions of the boundary problem, i.e. all spectral functions of the minimally possible dimension. We also improve (in the case of intermediate deficiency indices n±(L0) and non-separated boundary conditions) the known estimate of the spectral multiplicity of the (exit space) self-adjoint extension à ⊃ L0. Results are obtained for expressions l[y] with operator-valued coefficients and arbitrary (equal or unequal) deficiency indices n±(L0).

The nonlinear eigenvalue problem

for 0 ≤ x < ∞, fixed p ∈ (1, ∞), and with y′(0)/y(0) specified is studied under various conditions on the coefficients s and q, leading to either oscillatory or non-oscillatory situations.

This paper is concerned with a boundary-value problem on the half-line for nonlinear two-dimensional delay differential systems with positive delays. A theorem is established, which provides sufficient conditions for the existence of positive solutions. The application of this theorem to the special case of second-order nonlinear delay differential equations is given. Also, the application of the theorem to two-dimensional Emden–Fowler-type delay differential systems with constant delays is presented. Moreover, some general examples demonstrating the applicability of the theorem are included.

The well-known Falkner–Skan equation is one of the most important equations in laminar boundary layer theory and is used to describe the steady two-dimensional flow of a slightly viscous incompressible fluid past wedge shaped bodies of angles related to $\text{ }\!\!\lambda\!\!\text{ }\!\!\pi\!\!\text{ /2}$, where $\text{ }\!\!\lambda\!\!\text{ }\,\in \,\mathbb{R}$ is a parameter involved in the equation. It is known that there exists ${{\text{ }\!\!\lambda\!\!\text{ }}^{*}}\,<\,0$ such that the equation with suitable boundary conditions has at least one positive solution for each $\text{ }\!\!\lambda\!\!\text{ }\,\ge \,{{\text{ }\!\!\lambda\!\!\text{ }}^{*}}$ and has no positive solutions for $\text{ }\!\!\lambda\!\!\text{ }\,<\,{{\text{ }\!\!\lambda\!\!\text{ }}^{*}}$ . The known numerical result shows ${{\text{ }\!\!\lambda\!\!\text{ }}^{*}}\,=\,-0.1988$ . In this paper, ${{\text{ }\!\!\lambda\!\!\text{ }}^{*}}\,\in \,[-0.4,\,-0.12]$ is proved analytically by establishing a singular integral equation which is equivalent to the Falkner–Skan equation. The equivalence result provides new techniques to study properties and existence of solutions of the Falkner–Skan equation.

Second order nonlinear delay differential equations with positive delays are considered, and sufficient conditions are given that guarantee the existence of positive increasing solutions on the half-line with first order derivatives tending to zero at infinity. The approach is elementary and is essentially based on an old idea which appeared in the author's paper Arch. Math. (Basel)36 (1981), 168–178. The application of the result obtained to second order Emden-Fowler type differential equations with constant delays and, especially, to second order linear differential equations with constant delays, is also presented. Moreover, some (general or specific) examples demonstrating the applicability of the main result are given.

A generalization of norm type cone-compression and -expansion results due to Krasnosel'skiì (see Dokl. Akad. Nauk SSSR NS 135 (1960) 527–530) is presented here for single-valued completely continuous maps defined on a Fréchet space. Applications to second-order differential equations on the half line are presented, and the existence of nontrivial solutions is established.