Hille-Yosida Operators Research Articles

On impulsive functional differential inclusions with Hille–Yosida operators in Banach spaces

We consider a semilinear functional differential inclusion with infinite delay and impulse characteristics in a Banach space assuming that its linear part is a non-densely defined Hille–Yosida operator. We assume that the multivalued nonlinearity of upper Carathèodory or almost lower semicontinuous type satisfies a regularity condition expressed in terms of the measures of noncompactness. We apply the theory of integrated semigroups and the theory of condensing multivalued maps to obtain local and global existence results. The application to an optimization problem for an impulsive feedback control system is given.

Semilinear differential and integrodifferential equations with Hille-Yosida operators

Existence and uniqueness of weak and strong solutions for semilinear differential and integrodifferential abstract equations with a Hille-Yosida operator are proved together with criteria for the global existence of the solutions. An application is given to a semilinear one-dimensional wave equation.

Relatively Bounded Extensions of Generator Perturbations

The Miyadera perturbation theorem provides as a by-product that operators defined on a core for the generator of a C0-semigroup and satisfying the Miyadera condition have a relatively bounded extension to the domain of the generator. We show that a weakening of the Miyadera condition characterizes relative boundedness with respect to the generator. We also investigate extensions of these results to Hille-Yosida operators. The various conditions we use in the abstract part are illustrated by several examples. 2000 MSC: 47D06, 47D62, 47A55, 47B60

Almost Automorphic and Pseudo-Almost Automorphic Solutions to Semilinear Evolution Equations with Nondense Domain

We study the existence and uniqueness of almost automorphic (resp., pseudo-almost automorphic) solutions to a first-order differential equation with linear part dominated by a Hille-Yosida type operator with nondense domain.

On integrated semigroups and age structured models in {$L^p$} spaces

In this paper, we first develop some techniques and results for integrated semigroups when the generator is not a Hille-Yosida operator and is non-densely defined. Then we establish a theorem of Da Prato and Sinestrari's type for the nonhomogeneous Cauchy problem and prove a perturbation theorem. In particular, we obtain necessary and sufficient conditions for the existence of mild solutions for non-densely defined non-homogeneous Cauchy problems. Next we extend the results to $L^{p}$ spaces and consider the semilinear and non-autonomous problems. Finally we apply the results to studying age-structured models with dynamic boundary conditions in $L^{p}$ spaces. We also demonstrate that neutral delay differential equations in $L^{p}$ can be treated as special cases of the age-structured models in an $L^{p}$ space.

Local existence for retarded Volterra integrodifferential equations with Hille–Yosida operators

This paper is devoted to finding some existence and uniqueness theorems for classical solutions of the integrodifferential equations with infinite delay. We give some sufficient conditions ensuring the existence and uniqueness of solutions. We assume the linear part is not necessary to be densely defined and satisfies a Hille–Yosida condition so that it is the generator of a nondegenerated, locally Lipschitz continuous integrated semigroup. By using matrix operators and fixed point theorems, we obtain new results for the retarded integrodifferential equations.

An integrodifferential wave equation

This paper is devoted to the study of the integrodifferential equation $$ u'(t) =Au (t) +\int^t_0 a(t-s) A_1 u(s) ds +f (t) , \quad t\ge 0, $$ where $A$ is a Hille-Yosida operator in a Banach space $X$, $A_1 \in {\mathcal L} (D (A);$ $ X)$ and $a$ has bounded variation. Existence, uniqueness and estimates of strict and weak solutions are proved by extrapolation methods and the Miller scheme. Applications are given to the Cauchy-Dirichlet problem for the integrodifferential wave equation $$ w_{tt} (t,x) =w_{xx} (t,x)+\int^t_0 a (t-s) w_{xx} (s,x) ds +f(t,x), \quad t\ge 0, \quad x\in [0, \ell]. $$

Product Semigroups and Extrapolation Spaces

A new formulation of the theory of extrapolation spaces is used to recast a theorem on the resolvent of the sum of a generator of a semigroup and a Hille–Yosida operator and to prove a regularity result related to an abstract equation.

Asynchronous exponential growth for an age dependent population equation with delayed birth process

We show that the solutions of a population equation with age structure and delayed birth process have asynchronous exponential growth (AEG). We use operator matrices and Hille-Yosida operators as well as Perron-Frobenius techniques.

Stability of asymptotic properties of Hille-yosida operators under perturbations and retarded differential equations

We give conditions on the strongly continuous semigroup (T 0(t)) t≥0, generated by the part of a Hille-Yosida operator A on X 0 := D(A), and a non-autonomous family of operators (B(t)) t≥0 such that the evolution family (U(t,s)) t≥s≥0, generated by the part of (A + B(t)) t≥0 in X 0, inherits some asymptotic properties of (T 0(t)) t≥0 as boundedness, asymptotic almost periodicity, uniform ergodicity and total uniform ergodicity. Our main result is obtained via an extended result of Batty-Chill [7], which we prove here. We illustrate our abstract hypotheses and results by an example from population dynamics. Also, an application to non-autonomous retarded differential equations is given.

Critical spectrum and stability for population equations with diffusion in unbounded domains

In this work we study the asymptotic behavior of the solutions of some population equations with diffusion in unbounded domains by using the notion of critical spectrum introduced recently by R. Nagel and J. Poland [9]. To do this, we extend the abstract results of Brendle-Nagel-Poland [4], concerning the persistence under perturbations of the critical spectrum of a semigroup, to Hille-Yosida operators.

Existence and stability of solutions for a class of partial neutral functional differential equations

In this work, we consider a class of nonlinear partial neutral functional di¤erential equations with a nondensely defined Hille-Yosida operator. We first prove the local existence, uniqueness and regularity of solutions. Second, we study the global existence and stability. In the end, we extend in the autonomous case, results of Hale ([21], [23]) concerning dissipativeness and existence of a global attractor to our situation.

The Miller scheme in semigroup theory

In this paper we apply a set-up introduced by R. K. Miller to transform a linear, inhomogeneous Cauchy problem for the generator of a semigroup on a Banach space into a homogeneous one for a matrix operator, which is the generator of a semigroup on a suitable product space. By using restriction theorems and extrapolation spaces we obtain new results for the inhomogeneous Cauchy problem for Hille-Yosida operators in Favard spaces.

Generation theorems for $\varphi $ Hille-Yosida operators

This paper introduces the concept of $\varphi$ Hille-Yosida operators and studies several generation theorems. We show that if a once-integrated semigroup $\{S(t) \}_{t \geq 0}$ satisfies $\Phi (t) := limsup_{h \rightarrow 0^{+}} \frac {1}{h} ||S(t + h) - S(t)|| < \infty$ for all $t > 0 \ a. e.$, then $\Phi (\cdot )$ is locally bounded on $(0, \infty )$ and exponentially bounded. In addition, some other interesting results are presented.

Regularity properties of perturbed Hille-Yosida operators and retarded differential equations

In this work we give sufficient conditions on the semigroup (T0(t))t\geq0, generated by the part of a Hille—Yosida operator in the closure of its domain, in order that certain perturbations preserve some regularity properties of (T0(t))t\geq0, like the norm continuity, compactness and differentiability. An application of our abstract results to retarded differential equations is given.

Asymptotic behavior of hyperbolic Cauchy problems for Hille-Yosida operators with an application to retarded differential equations

In this work, we use the extrapolation methods to study the existence and the uniqueness of bounded and almost periodic, asymptotic almost periodic, pseudo almost periodic and Eberlein-weak almost periodic solutions to hyperbolic Cauchy problems for Hille-Yosida operators d/dtx(t) = Ax(t) + f(t), t ∈ R, when the inhomogeneities f take their values in the extrapolated Favard class corresponding to A.

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

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

Linearized stability for semilinear non-autonomous evolution equations with applications to retarded differential equations

We consider the semilinear non-autonomous evolution equation $\frac{d}{dt}u(t)=Au(t)+G(t,u(t))$, $t\geq s\geq 0,$ where $(A,D(A))$ is a Hille-Yosida operator on a Banach space $X$ and $G$ is a continuous function on $\mathbb R_+\times \overline{D(A)}$ with values in the extrapolated Favard class corresponding to $A$. In our main results we present principles of linearized stability and instability for a solution of such an equation. Our approach is based on the theory of extrapolation spaces. We apply the results to non-autonomous semilinear retarded differential equations.