Semilinear Functional Differential Equations Research Articles

Semigroup Approach to Semilinear Partial Functional Differential Equations with Infinite Delay

We describe a semigroup of abstract semilinear functional differential equations with infinite delay by the use of the Crandall Liggett theorem. We suppose that the linear part is not necessarily densely defined but satisfies the resolvent estimates of the Hille-Yosida theorem. We clarify the properties of the phase space ensuring equivalence between the equation under investigation and the nonlinear semigroup.

Controllability Results for Nondensely Defined Semilinear Functional Differential Equations

In this paper we investigate the controllability of first order semilinear functional and neutral functional differential equations in Banach spaces.

On nondensely defined semilinear stochastic functional differential equations with nonlocal conditions

The nonlinear alternative of Leray-Schauder type is used to investigate the existence of solutions for first-order semilinear stochastic functional differential equations in Hilbert spaces.

Existence results for semilinear perturbed functional differential equations with nondensely defined operators

We will establish sufficient conditions for the existence of integral solutions and extremal integral solutions for semilinear functional differential equations with nondensely defined operators in Banach spaces.

The Kneser Property for the Second Order Functional Abstract Cauchy Problem

In this work we establish existence of mild solutions for a class of abstract semi-linear functional differential equations of second order and we prove that the set consisting of the mild solutions for these problems is connected.

Multiple solutions for impulsive semilinear functional and neutral functional differential equations in Hilbert space

The well-known Krasnoselskii twin fixed point theorem is used to investigate the existence of mild solutions for first- and second-order impulsive semilinear functional and neutral functional differential equations in Hilbert spaces.

Non-linear Semigroup of a Class of Abstract Semilinear Functional Differential Equations with a Non-Dense Domain

In this work, we are concerned with a general class of abstract semilinear autonomous functional differential equations with a non-dense domain on a Banach space. Our objective is to study, using the Crandall-Liggett approach, the solutions as a semigroup of non-linear operators.

A note on semilinear abstract functional differential and integrodifferential equations with infinite delay

In this note, we obtain some new existence and uniqueness theorems for mild solutions of the Cauchy problems for semilinear abstract functional differential and integrodifferential equations with infinite delay.

Controllability of stochastic semilinear functional differential equations in Hilbert spaces

In this paper approximate and exact controllability for semilinear stochastic functional differential equations in Hilbert spaces is studied. Sufficient conditions are established for each of these types of controllability. The results are obtained by using the Banach fixed point theorem. Applications to stochastic heat equation are given.

Complete guiding sets for a class of almost-periodic differential equations

A functional differential equation admitting a complete guiding set always has an a priori bounded solution. The introduction of this concept and the idea of reducibility allow the authors to apply a well-known method for the autonomous case to a non-autonomous problem. A previous result of Ortega and Tineo is extended to a non-hyperbolic semilinear functional differential equation with almost-periodic coefficients in its lineal part. The non-hyperbolicity condition is established in terms of the Sacker–Sell spectrum of the associated linear system.

Some Uniqueness Results on Controllability for Functional Semilinear Differential Equations in Fréchet Spaces

In this paper, a recent nonlinear alternative for contraction maps in Frechet spaces, due to Frigon and Granas, is combined with semigroup theory and used for the investigation of the controllability of some classes of semilinear functional and neutral functional differential equations in Banach spaces on a semiinfinite interval.

Solvability of Degenerate Semilinear Parabolic Functional Differential Equations

We obtain local and global theorems on the existence and uniqueness of a solution of the semilinear functional differential equation \(\frac{d}{{dt}}\left[ {Au\left( t \right)} \right] + Bu\left( t \right) = f\left( {t,u_t } \right)\) in a Banach space with a parabolic pencil of operators λA + B, where the operator A can be noninvertible. Abstract results are applied to partial functional differential equations.

EXISTENCE AND STABILITY OF SOLUTIONS OF STOCHASTIC SEMILINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS

In this paper, we study stochastic semilinear functional differential equations in a Hilbert space. First, we prove the existence and uniqueness of a mild solution under two sets of hypotheses. We then consider the exponential stability of the second moment of the solution process of such equations as well as the exponential stability and asymptotic stability in probability of its sample paths. We further consider global stability in the mean. Such results are obtained using both local Lipschitz and non-Lipschitz nonlinearities. Our method is an interplay of the method of successive approximations and a comparison principle. Two applications are included to motivate this study.

Some Uniqueness Results for Impulsive Semilinear Neutral Functional Differential Equations

Abstract The Banach contraction principle is used to investigate the existence and uniqueness of solutions for first and second order impulsive semilinear neutral functional differential equations in Banach spaces.

Impulsive Semilinear Functional Differential Equations

In this paper the Leray–Schauder nonlinear alternative combined with semigroup theory is used to investigate the existence of mild solutions for first-order impulsive semilinear functional differential equations in Banach spaces.

Normal forms for semilinear functional differential equations in Banach spaces and applications. Part II

A normal form theory for functional differential equations in Banach spaces of retarded type is addressed. The theory is based on a formal adjoint theory for the linearized equation at an equilibrium and on the existence of center manifolds for perturbed inhomogeneous equations, established in the first part of this work under weaker hypotheses than those that usually appear in the literature. Based on these results, an algorithm to compute normal forms on finite dimensional invariant manifolds of the origin is presented. Such normal forms are important in obtaining the ordinary differential equation giving the flow on center manifolds explicitly in terms of the original functional differential equation. Applications to Bogdanov-Takens and Hopf bifurcations are presented.

Semilinear functional differential equations in Banach spaces with Hille–Yosida operators

Semilinear functional differential equations in Banach spaces with Hille–Yosida operators

Approximate Controllability for Semilinear Retarded Functional Differential Equations

This paper deals with the approximate controllability of the semilinear functional differential equations with unbounded delays. We will also establish the regularity of the solution of the given system. It is shown the relation between the reachable set of the semilinear system and that of its corresponding linear system by using degree theory. Finally, a simple example to which our main result can be applied is given.

Strict solutions of nonlinear hyperbolic neutral differential equations

In this work, we study a class of abstract semilinear functional differential equations of a neutral type. Our main results concern the existence, uniqueness, and regularity of solutions. We assume that the linear part is nondensely defined, closed, and satisfies the Hille-Yosida condition.

Existence and continuous dependence of mild solutions to semilinear functional-differential equations in Banach spaces

This paper is concerned with a general existence and continuous dependence of mild solutions to semilinear functional differential equations with infinite delay in Banach spaces. In particular, our results are applicable to the equations whose Co-semigroups and nonlinear operators, defined on an open set, are noncompact.