On the approximate controllability of some semilinear partial functional integrodifferential equations with unbonded delay

This work concerns the study of the approximate controllability for some nonlinear partial functional integrodifferential equation with infinite delay arising in the modelling of materials with memory, in the framework of Hilbert spaces. We give sufficient conditions that ensure the approximate controllability of the system by supposing that its linear undelayed is part approximately controllable, admits a resolvent operator in the sense of Grimmer, and by making use of the measure of noncompactness and the Mönch fixed-point Theorem. As a result, we obtain a generalization of several important results in the literature, without assuming the compactness of the resolvent operator. An example of applications is given for illustration

Solvability of nondensely defined partial functional integrodifferential equations using the integrated resolvent operators

In this work, we study the existence and regularity of solutions for a class of nondensely defined partial functional integrodifferential equations. We suppose that the undelayed part admits an integrated resolvent operator in the sense given by Oka [J. Integral Equations Appl. 7(1995), 193–232.]. We give some sufficient conditions ensuring the existence, uniqueness and regularity of solutions. The continuous dependence on the initial data of solutions is also proved. Some examples are provided to illustrate our abstract theory

Functional differential equations with unbounded delay in extrapolation spaces

International audienceWe study the existence, regularity and stability of solutions for nonlinear partial neutral functional differential equations with unbounded de-lay and a Hille-Yosida operator on a Banach space X. We consider two non-linear perturbations: the first one is a function taking its values in X and the second one is a function belonging to a space larger than X, an extrapolated space. We use the extrapolation techniques to prove the existence and regu-larity of solutions and we establish a linearization principle for the stability of the equilibria of our equation

Periodic solutions of abstract neutral functional differential equations and applications

International audienceIn this work, we study the existence of periodic solutions for the following neutral partial functional differential equations of the following form$\frac{d}{dt}[x(t) - L(x_{t})]= A[x(t)- L(x_{t})]+G(x_{t})+f(t)}

Approximate controllability for some integrodifferential measure driven system with nonlocal conditions

In this work, we focus on a specific category of nonlocal integrodifferential equations. The development of a few new sufficient postulates that guarantee solvability and approxi- mative controllability is described here. We apply the theory of the resolvent operator in the sense of Grimmer, as well as the fixed point strategy and the theory of the Lebesgue-Stieljes integral, in the context of the space of regulated functions. In light of this, the prevalence of our findings is greater than that which is found in the literature. At last, and example is comprised that exhibits the significance of developed theory

New Variation of Constants Formula for Some Partial Functional Differential Equations with Infinite Delay

20 pagesIn this work, we give a new variation of constants formula for some partial functional differential equations with infinite delay. We assume that the linear part is not necessarily densely defined and satisfies the known Hille-Yosida condition. When the phase space is a uniform fading memory space, we establish a spectral decomposition of the phase space. This allows us to study the existence of almost periodic solutions when the equation has a bounded solution on the half line $\mathbb{R}^+

Existence, Regularity, and Compactness Properties in the <em>α</em>-Norm for Some Partial Functional Integrodifferential Equations with Finite Delay

The objective, in this work, is to study the alpha-norm, the existence, the continuity dependence in initial data, the regularity, and the compactness of solutions of mild solution for some semi-linear partial functional integrodifferential equations in abstract Banach space. Our main tools are the fractional power of linear operator theory and the operator resolvent theory. We suppose that the linear part has a resolvent operator in the sense of Grimmer. The nonlinear part is assumed to be continuous with respect to a fractional power of the linear part in the second variable. An application is provided to illustrate our results

Center manifold and stability in critical cases for some partial functional differential equations

24 pagesIn this work, we prove the existence of a center manifold for some partial functional differential equations, whose linear part is not necessarily densely defined but satisfies the Hille-Yosida condition. The attractiveness of the center manifold is also shown when the unstable space is reduced to zero. We prove that the flow on the center manifold is completely determined by an ordinary differential equation in a finite dimensional space. In some critical cases, when the exponential stability is not possible, we prove that the uniform asymptotic stability of the equilibrium is completely determined by the uniform asymptotic stability of the reduced system on the center manifold