Hille-Yosida Operators Research Articles

Anti-periodic problem for semilinear differential inclusions involving Hille-Yosida operators

In this paper we are interested in the anti-periodic problem governed by a class of semilinear differential inclusions with linear parts generating integrated semigroups. By adopting the Lyapunov-Perron method and the fixed point argument for multivalued maps, we prove the existence of anti-periodic solutions. Furthermore, we study the long-time behavior of mild solutions in connection with anti-periodic solutions. Consequently, as the nonlinearity is of single-valued, we obtain the exponential stability of anti-periodic solutions. An application of theoretical results to a class of partial differential equations will be given.

Bifurcation analysis of an age‐structured epidemic model with two staged‐progressions

In this paper, we develop an age‐structured model with two infectious stages to study the effect of infection age on dynamic properties. By reformulating the model as a non‐densely defined Cauchy problem and applying theorem related with Hille–Yosida operator, the threshold dynamics are investigated. Theoretical analysis shows that destabilization of the age‐dependent endemic equilibrium can occur due to the perturbation of critical infection age that separates the two infectious stages. Furthermore, numerical simulations are conducted to illustrate the dynamical behaviors of stability switching.

On solvability of the impulsive Cauchy problem for integro-differential inclusions with non-densely defined operators.

We prove the existence of at least one integrated solution to an impulsive Cauchy problem for an integro-differential inclusion in a Banach space with a non-densely defined operator. Since we look for integrated solution we do not need to assume that A is a Hille Yosida operator. We exploit a technique based on the measure of weak non-compactness which allows us to avoid any hypotheses of compactness both on the semigroup generated by the linear part and on the nonlinear term. As the main tool in the proof of our existence result, we are using the Glicksberg-Ky Fan theorem on a fixed point for a multivalued map on a compact convex subset of a locally convex topological vector space. This article is part of the theme issue 'Topological degree and fixed point theories in differential and difference equations'.

Existence of local stable manifolds for some nondensely defined nonautonomous partial functional differential equations

In this paper, we establish the existence of local stable manifolds for a semi-linear differential equation, where the linear part is a Hille–Yosida operator on a Banach space and the nonlinear forcing term [Formula: see text] satisfies the [Formula: see text]-Lipschitz conditions, where [Formula: see text] belongs to certain classes of admissible function spaces. The approach being used is the fixed point arguments and the characterization of the exponential dichotomy of evolution equations in admissible spaces of functions defined on the positive half-line.

Regularity properties of some perturbations of non-densely defined operators with applications

This paper is to study some conditions on semigroups, generated by some class of non-densely defined operators in the closure of its domain, in order that certain bounded perturbations preserve some regularity properties of the semigroup such as norm continuity, compactness, differentiability and analyticity. Furthermore, we study the critical and essential growth bound of the semigroup under bounded perturbations. The main results generalize the corresponding results in the case of Hille-Yosida operators. As an illustration, we apply the main results to study the asymptotic behaviors of a class of age-structured population models in $ L^p $ spaces ($ 1 \leq p < \infty $).

Intermediate and extrapolated spaces for bi-continuous operator semigroups

We discuss the construction of the full Sobolev (H\"older) scale for non-densely defined operators on a Banach space with rays of minimal growth. In particular, we give a construction for extrapolation- and Favard spaces of generators of (bi-continuous) semigroups, or which is essentially the same, Hille-Yosida operators on Saks spaces.

Delay independent stability for some partial functional differential equations with infinite delay under the light of positivity

In this work, we deal with the positivity and stability study for some partial functional differential equations with infinite delay and Hille–Yosida operators. We first show the positivity of solutions in a suitable phase space. This enables us to establish a stability criterion independently of delay.

Approximate controllability results for non-densely defined fractional neutral differential inclusions with Hille–Yosida operators

ABSTRACTIn this paper, we mainly focused the approximate controllability results for a class of non-densely defined fractional neutral differential control systems. First, we establish a set of sufficient conditions for the approximate controllability for a class of fractional differential inclusions with infinite delay where the linear part is non-densely defined and satisfies the Hille–Yosida condition. The main techniques rely on Bohnenblust– Karlin's fixed point theorem, operator semigroups and fractional calculus. Further, we extend the result to study the approximate controllability concept with nonlocal conditions. Finally, an example is also given for the illustration of the obtained theoretical results.

Asymptotic behaviors of a size-structured population model

In this paper we devote ourselves to the study of the asymptotic behavior of a size-structured population dynamics with random diffusion and delayed birth process. Within a semigroup framework, we discuss the local stability and asynchrony respectively for the considered population system under some conditions. We use for our discussion the techniques of operator matrices, Hille-Yosida operators, positivity, spectral analysis as well as Perron-Frobenius theory.

Asymptotic analysis of a spatially and size-structured population model with delayed birth process

This paper is devoted to the study of a spatially and size-structured population dynamics model with delayed birth process. Our focus is on the asymptotic behavior of the system, in particular on the effect of the spatial location and the time lag on the long-term dynamics. To this end, within a semigroup framework, we derive the locally asymptotic stability and asynchrony results respectively for the considered population system under some conditions. For our discussion, we use the approaches concerning operator matrices, Hille-Yosida operators, spectral analysis as well as Perron-Frobenius theory. We also do two numerical simulations to illustrate the obtained stability and asynchrony results.

The Asymptotic Behavior of an Age-Cycle Structured Cell Model with Delay

This paper is devoted to the study of an age-cycle structured proliferating cell population with delayed boundary condition. Individual cells are distinguished by age and cell cycle length. We consider a general biological rule corresponding to delayed boundary condition. Our focus is on the asymptotic behavior of the system, in particular on the effect of the time lag on the long-term dynamics. To this end, within a semigroup framework, we derive the locally asymptotic stability and asynchrony results, respectively, for the considered population system under some conditions. For our discussion, we use operator matrices, Hille-Yosida operators, spectral analysis, as well as Perron-Frobenius theory.

Existence and controllability for nondensely defined partial neutral functional differential inclusions

We give sufficient conditions for the existence of integral solutions for a class of neutral functional differential inclusions. The assumptions on the generator are reduced by considering nondensely defined Hille-Yosida operators. Existence and controllability results are established by combining the theory of addmissible multivalued contractions and Frigon’s fixed point theorem. These results are applied to a neutral partial differential inclusion with diffusion.

RANDOM ATTRACTOR FOR STOCHASTIC PARTIAL FUNCTIONAL DIFFERENTIAL EQUATIONS WITH INFINITE DELAY

In this paper we are concerned with a class of stochastic partial functional differential equations with infinite delay. Supposing that the linear part is a Hille-Yosida operator but not necessarily densely defined and employing the integrated semigroup and random dynamics theory, we present some appropriate conditions to guarantee the existence of a random attractor.

Normal forms for semilinear equations with non-dense domain with applications to age structured models

Normal form theory is very important and useful in simplifying the forms of equations restricted on the center manifolds in studying nonlinear dynamical problems. In this paper, using the center manifold theorem associated with the integrated semigroup theory, we develop a normal form theory for semilinear Cauchy problems in which the linear operator is not densely defined and is not a Hille–Yosida operator and present procedures to compute the Taylor expansion and normal form of the reduced system restricted on the center manifold. We then apply the main results and computation procedures to determine the direction of the Hopf bifurcation and stability of the bifurcating periodic solutions in a structured evolutionary epidemiological model of influenza A drift and an age structured population model.

Global attractor for a class of functional differential inclusions with Hille–Yosida operators

We study the dynamics for a class of functional differential inclusions whose linear part generates an integrated semigroup. Some techniques of measure of noncompactness are deployed to prove the global solvability and the existence of a compact global attractor for the m-semiflow generated by our system. The obtained results generalize recent ones in the same direction.

On semilinear differential inclusions in Banach spaces with nondensely defined operators

We consider a semilinear differential inclusion in a Banach space assuming that its linear part is a nondensely defined Hille–Yosida operator whereas Caratheodory-type multivalued nonlinearity satisfies a regularity condition expressed in terms of the Hausdorff measure of noncompactness. We apply the theory of integrated semigroups and the fixed point theory of condensing multivalued maps to obtain local and global existence results and to prove the continuous dependence of the solutions set on initial data. An application to an optimization problem for a feedback control system is given.

Global Attractor for Some Partial Functional Differential Equations with Infinite Delay

This paper deals with a class of partial functional differential equations with infinite delay. Supposing that the linear part is a Hille-Yosida operator but not necessarily densely defined, and employing the integrated semigroups and dissipative dynamics theory, we present some appropriate conditions to guarantee existence of a global attractor.

Hille-Yosida operators and Cauchy problems

A new proof of a temporal regularity result for an abstract Cauchy problem with a Hille-Yosida operator is given having consequences on the existence of other types of solutions. New types of weak solutions are introduced.

Hopf bifurcation for non-densely defined Cauchy problems

In this paper, we establish a Hopf bifurcation theorem for abstract Cauchy problems in which the linear operator is not densely defined and is not a Hille–Yosida operator. The theorem is proved using the center manifold theory for non-densely defined Cauchy problems associated with the integrated semigroup theory. As applications, the main theorem is used to obtain a known Hopf bifurcation result for functional differential equations and a general Hopf bifurcation theorem for age-structured models.

Global solvability for abstract semilinear evolution equations

The Cauchy problem for the abstract semilinear evolution equation u′(t) = Au (t) + B (u (t)) + C (u (t)) is discussed in a general Banach space X. Here A is the so-called Hille-Yosida operator in X, B is a differentiable operator from D (A) into X, and C is a locally Lipschitz continuous operator from D (A) into itself. A vectorvalued functional defined only on X is used and appropriate conditions on the nonlinear operators B and C are imposed so that a vector-valued functional defined on the domain of the operator A may be constructed in order to specify the growth of a global solution. The advantage of our formulation lies in the fact that it is possible to obtain a global solution by checking some energy inequalities concerning only low order derivatives (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)