Stephen R. Bernfeld

Fixed point theorems of operators with ppf dependence in banach spaces

In this paper the theory of fixed point theorems of nonlinear whose domain and range are different Banach spaces are considered. Also the analogues of the Contraction Mapping Principle. Krasnoselskiis fixed point theorem and a result on the convergence of quasinonexpansive mappings are dealt with. As an application, the existence and uniqueness of solutions of differential equations with retarded argument in Banach spaces are discussed

Large-time asymptotic equivalence for a class of non-autonomous semilinear parabolic equations

Abstract In this article we prove new results concerning the long-time behavior of solutions to a class of non-autonomous semilinear parabolic Neumann boundary-value problems defined on open bounded connected subsets Ω of RN. The nature of the equations that we investigate leads us to consider two complementary situations, according to whether the time-dependent lower order terms in the equations possess recurrence properties. If the lower order terms are recurrent, we prove that every solution stabilizes around a spatially homogeneous and recurrent solution of the same Neumann problem in the C1 ( Ω )-topology. In contrast, if the lower order terms are not recurrent, the asymptotic states need not be solutions to the original problem and we prove that every solution stabilizes around such an asymptotic state again in the C1 ( Ω )-topology. In all cases the dynamics of the asymptotic solutions are governed by a compact and connected set of scalar ordinary differential equations, which are thereby asymptotically equivalent to the original Neumann problem for large times. A major difficulty to be bypassed in the proofs of our theorems stems from the fact that we allow the nonlinearitics to depend explicitly on the gradient of the unknown function. Our method of proof rests upon the use of comparison principles and upon the existence of exponential dichotomies for the family of evolution operators associated with the principal part of the equations. It is also based on ideas that stem for the classic reduction methods for non-autonomous finite-dimentional dynamical systems originally devised by Miller, Strauss-Yorke and Sell .

Monotone Methods for Nonlinear Boundary Value Problems in Banach Spaces

MONOTONE methods have been used to generate multiple solutions of nonlinear boundary value problems for both ordinary and partial differential equations. Keller [I] and Sattinger [2], extending the chord method, considered nonlinear partial differential equations containing no gradient term. The inclusion of the gradient term was first introduced by Chandra and Davis [3] who considered the ordinary boundary value problem

A VARIATION OF RAZUMIKHIN'S METHOD FOR RETARDED FUNCTIONAL DIFFERENTIAL EQUATIONS

Publisher Summary This chapter discusses a variation of Razumikhins method for retarded functional differential equations. Although Liapunov functionals are theoretically perhaps the natural way to study various stability properties of functional differential equations, Liapunov functions have played a prominent role in such investigations. The idea of employing Liapunov functions to functional differential equations was apparently first conceived by Razumikhin in 1960. In 1962, Driver helped to clarify some of the ideas of the Razumikhin method and introduced them to the non-Russian reader. One of the basic techniques involved in this classical method has been to examine inequalities in connection with the derivative of a Liapunov function. This derivative is often examined with respect to a certain subset C0 of the general solution space C. This chapter presents a variation of the Razumikhin method to gain information concerning the behavior of solutions of functional differential equations.

MONOTONE METHOD FOR NONLINEAR BOUNDARY VALUE PROBLEMS BY LINEARIZATION TECHNIQUES

Publisher Summary This chapter discusses monotone method for nonlinear boundary value problems by linearization techniques. Monotone methods have been used to generate multiple solutions of nonlinear boundary value problems for ordinary and partial differential equations. Keller and Sattinger, extending the chord method, considered nonlinear partial differential equations containing no gradient term. The inclusion of the gradient term was first introduced by Chandra and Davis who considered the boundary value problem. The chapter discusses the modification of a nonlinear method by providing a linear procedure. It also presents the monotone method and explains the existence of minimal and maximal solutions of a stated boundary value problem.

Properties of the Radii of Stability and Instability

The radius of stability and the radius of instability of the zero solution of the differential equation x’ = f (t, x) were introduced by Salvadori and Visentin [9], [10]. These radii in some sense provide a measure of the “region” of stability or instability of the zero solution. This knowledge has been used in the study of small solutions x p (t)of perturbations of the differential equation x’ = f (t, x) given by x’ = f(t, x p ) + h(t, x p ). In particular a relationship between the radius of stability of the zero solution of x’ = f (t, x) and its total stability was also introduced in [9] and [10]. Having been motivated by mechanical systems subject to conservative perturbations these authors analyzed the total stability of x’ = f (t, x) using the perturbed differential equation x’ = g (t, x, λ,) where g (t, x, 0) = f (t, x) and λ is a parameter in some Banach Space β. In this paper we often will assume β is the real line.

Discrete dynamical systems and bifurcation for periodic differential equations

On etudie les problemes de bifurcation et de stabilite pour une famille a 1 parametre de systemes differentiels periodiques dans R n

EXCHANGE OF STABILITY AND HOPF BIFURCATION

Publisher Summary This chapter focuses on the exchange of stability and Hopf bifurcation. In Hopf bifurcation theory, for a one parameter family of differential equations, there is the evolution of periodic orbits emerging from an equilibrium point. In most applications, this phenomenon is because of a sudden change in the stability properties of the equilibrium point as the parameter crosses a critical value. The chapter discusses the measurement of change of stability of the origin to analyze the number of bifurcating periodic orbits. To do this, it is required to assume a structural property on the asymptotic stability of the origin of (l0). This type of structural property was used by Andronov in his work on bifurcation theory. The chapter also reviews generalized transversality condition.

Generalized Hopf Bifurcation in Rn and h-Asymptotic Stability

Publisher Summary This chapter discusses the generalized Hopf bifurcation in R n and h-asymptotic stability. The concept and analysis of structural stability of two dimensional autonomous differential systems was provided by Andronov and Pontryagin. Within the next 15 years, Andronov and his group developed a major part of the theory of two dimensional structurally stable and unstable system of differential equations. Motivated by problems in classical and celestial mechanics, mathematicians and physicists have been very concerned with providing a qualitative description of the trajectories of perturbed systems in a neighborhood of a structurally unstable equilibrium point in R n (n ≥ 2 ).