M.N. Nkashama

A generalized upper and lower solutions method and multiplicity results for nonlinear first-order ordinary differential equations

Abstract We introduce a generalized upper and lower solutions method for the solvability of first-order ordinary differential equations u′(t) = ƒ(t, u(t)), u(0) = u(1) in order to cover the case when the function ƒ satisfies Caratheodory conditions. This method is then applied to get multiplicity results when the nonlinearity ƒ interacts with the real eigenvalue of the linearized problem. Our proofs are based on differential inequalities and classical Leray-Schauder degree.

Nonlinear two-point boundary value problems at resonance without Landesman-Lazer condition

The purpose of this paper is to study the solvability of a semilinear two-point boundary value problem of resonance type in which the nonlinear perturbation is not (necessarily) required to satisfy Landesman-Lazer condition or the monotonicity assumption. The nonlinearity may be unbounded.

Unbounded perturbations of forced second order ordinary differential equations at resonance

Abstract We prove the existence of at least one solution for the differential equation x″(t) + m2x(t) + g(t, x(t)) = e(t) with periodicity conditions x(0) − x(2Π) = x′(0) − x′(2Π) = 0, where m⩾0 is an integer, e is integrable and g satisfies Caratheodory conditions. Our results are obtained for the case when there is resonance at the eigenvalue m2 of the linear second order differential equation λ″(t) + λx(t) = 0, λϵ R with x(0) − x(2Π) = x′(0) − x′(2Π) = 0. The function g may be unbounded and “touching” of the eigenvalue (m + 1)2 (resp. (m − 1)2 if m>0) on a subset of positive measure is allowed. Our approach also works when periodicity conditions are replaced by Dirichlet or Neumann boundary conditions. The proofs are based on Topological degree, Mawhins continuation theorem and Leray-Schauder techniques.

Existence of multiple solutions for some nonlinear boundary value problems

Abstract By imposing one-sided conditions on the nonlinearity, where neither regularity nor uniformity is required, we prove the existence of either a nonnegative or a nonpositive solution for first and second order ordinary differential equations with periodic, Neumann, or Dirichlet boundary conditions. Positone and nonpositone problems are considered. Some nonexistence results are also obtained. When using generalized Ambrosetti-Prodi type conditions we get the existence of nonnegative and nonpositive solutions for first and second order periodic or Neumann boundary value problems. Our method of proof makes use of topological degree arguments in Cones.

Asymptotic constancy for pseudo monotone dynamical systems on function spaces

Abstract A pseudo monotone dynamical system is a dynamical system which preserves the order relation between initial points and equilibrium points. The purpose of this paper is to present some convergence, oscillation, and order stability criteria for pseudo monotone dynamical systems on function spaces for which each constant function is an equilibrium point. Some applications to neutral functional differential equations and semilinear parabolic partial differential equations with Neumann boundary condition are given.

On periodic solutions of nonlinear second order vector differential equations

This paper considers existence of periodic solutions for vector Lienard differential equations In our main result we write where Q(t, x) is a symmetric matrix and h(t, x) is sublinear. The key assumption relates the asymptotic behaviour as x →+ ∞ of the eigenvalues of Q(t, x) to the spectrum of the linear operator −d 2 /dt 2 Several choices for Q(t, x) are considered which lead to known theorems and extend others. In the case of the Duffing equation the assumptions are weakened. Our approach is based on Leray-Schauders degree theory and a priori estimates.

Periodic solutions of forced Liénard equations with jumping nonlinearities under nonuniform conditions

This paper is devoted to the existence of periodic solutions for the scalar forced Lienard differential equation The key assumptions relate the asymptotic behaviour as x →± ∞of g ( t ; x )/ x to the “critical values” of the positively 1-homogeneous problem No condition on f , except continuity, is assumed. Our approach is based on Leray–Schauder degree techniques and a priori estimates.

Periodic-solutions of the Boundary-value Problem for the Nonlinear Heat-equation

We prove the existence of generalized periodic solutions of the boundary value problem for the nonlinear heat equation. The proof is based on classical Leray-Schauders techniques and coincidence degree.

Periodic solutions of Liénard systems at resonance

SummaryWe use classical Leray-Schauder techniques in order to derive the existence of periodic solutions for Liénard differential systems.

Periodically perturbed nonconservative systems of Liénard type

We give sufficient conditions for the solvability of forced, strongly coupled nonlinear vector Lienard equations. These conditions guarantee the existence of periodic solutions for any forcing term. They include sublinear as well as superlinear nonlinearities. They do not require the symmetry of the restoring term. The method of proof makes use of Leray-Schauder degree.