Based on classical Lie Group method, we construct a class of explicit solutions of two-dimensional ideal incompressible magnetohydrodynamics (MHD) equation by its infinitesimal generator. Via these explicit solutions we study the uniqueness and stability of initial-boundary problem on MHD.

A resolvent for a non-self-adjoint differential operator with a block-triangular operator potential, increasing at infinity, is constructed. Sufficient conditions under which the spectrum is real and discrete are obtained.

The paper deals with global existence of weak solutions to a one-dimensional mathematical model describing magnetoelastic interactions. The model is described by a fractional Landau-Lifshitz-Gilbert equation for the magnetization field coupled to an evolution equation for the displacement. We prove global existence by using Faedo-Galerkin/penalty method. Some commutator estimates are used to prove the convergence of nonlinear terms.

We prove the existence of a curve (with respect to the scalar delay) of periodic positive solutions for a smooth model of Cooke-Kaplan’s integral equation by using the implicit function theorem under suitable conditions. We also show a situation in which any bounded solution with a sufficiently small delay is isolated, clearing an asymptotic stability result of Cooke and Kaplan.

We study the existence of solutions for nonlinear boundary value problems ${\left(\phi \left(u^{\mathrm{\prime }}\right)\right)}^{\mathrm{\prime }}=f\left(t,u,u^{\mathrm{\prime }}\right),\hspace{0.17em}\hspace{0.17em}l\left(u,u^{\mathrm{\prime }}\right)=\mathrm{0}$, where $l(u,u^{\mathrm{\prime }})=\mathrm{0}$ denotes the boundary conditions on a compact interval $\left[\mathrm{0},T\right]$, $\phi$ is a homeomorphism such that $\phi (\mathrm{0})=\mathrm{0}$, and $f:\left[\mathrm{0},T\right]\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ is a continuous function. All the contemplated boundary value problems are reduced to finding a fixed point for one operator defined on a space of functions, and Schauder fixed point theorem or Leray-Schauder degree is used.