Quasi-autonomous Evolution Inclusions

In this paper, sufficient conditions are given to investigate the existence of mild solutions on a semi-infinite interval for first order semi linear impulsive neutral functional differential evolution inclusions with infinite delay using a recently developed nonlinear alternative for contractive multivalued maps in Frechet spaces due to Frigon combined with semigroup theory. The existence result has been proved without assumption of compactness of the semigroup. We introduced a new phase space for impulsive system with infinite delay and claim that the phase space considered by different authors are not correct.

Bounded and periodic solutions of infinite delay evolution equations

This work concerns the study of the exact boundary controllability for some non-densely defined semilinear partial functional differential equations with infinite delay, in the framework of general Banach spaces.We establish sufficient conditions for the exact controllability by using the theory of integrated semigroup and the measure of noncompactness.The existence of solutions is achieved by using the Mnch fixed point theorem without assuming compactness of the C 0 -semigroup on the closure of the domain.As a result we obtain a generalization of several important results in the literature.An example is given to illustrate the theory.

The weak solution of a functional differential equation in a general Banach space

ABSTRACTIn this work, we determine a mild solution for the Hilfer differential equation, which comprises two characteristic operators related to the semigroup and certain probability density functions. By using Mönch fixed‐point theorem and the noncompactness measure, we provide a set of sufficient conditions for the controllability of ‐Hilfer fractional differential equation with infinite delay.

On a nonlocal problem for semilinear differential equations with upper semicontinuous nonlinearities in general Banach spaces

This paper deals with the nonlocal problems for a class of nonlinear first-order evolution inclusions. Some existence results are established for the cases of a convex and of a nonconvex valued perturbation terms. Also, the existence of extremal solutions and a strong relaxation theorem are obtained. Subsequently a nonlinear hyperbolic optimal control problem is considered and the existence theorems based on the proven results are obtained. Then the nonlinear version of "bang---bang" principle for control systems is given as well by utilizing the relaxation theorem.

The goal of this review article is to provide a survey about the foundations of semilinear stochastic partial differential equations. In particular, we provide a detailed study of the concepts of strong, weak, and mild solutions, establish their connections, and review a standard existence and uniqueness result. The proof of the existence result is based on a slightly extended version of the Banach fixed point theorem.

Chapter 6. Differential Inclusions with Infinite Delay

We consider a model of infinite dimensional differential variational inequalities formulated by a parabolic differential inclusion and an elliptic variational inequality. The existence of global solution and global attractor for the semiflow governed by our system is proved by using measure of noncompactness. Copyright © 2017 John Wiley & Sons, Ltd.

Nonlinear functional-integral equations contain the unknown function nonlinearly, occurring extensively in theory developed to a certain extent toward different applied problems and solutions thereof. Nonlinear science serves to reveal the nonlinear descriptions of widely different systems, has had a fundamental impact on complex dynamics. Accordingly, this study provides the basic definitions and results regarding Banach spaces with the proof as well as applications. Besides, proving the existence and uniqueness of the solutions by the Banach theorem is carried out. The existence results of the equations are generally obtained based on fundamental methods by which the fixed-point theorems are frequently applied. Subsequently, the definition of the measure pertaining to noncompactness on Banach space along with properties is presented. Hence, the aim is to study nonlinear functional-integral equations in the Banach space [Formula: see text] using the measure of noncompactness. In addition, the proof of the existence of global attractors and asymptotic stabilities, having at least one solution which pertains to [Formula: see text] is looked into. As the last stage, the definitions of global attractivity and asymptotical stability for equations on Banach space [Formula: see text] are given while proving the existence of solutions and global attractivity. Besides these, the existence and asymptotical stability are proven. Furthermore, the proof of the existence of global attractors and asymptotic stabilities, having at least one solution which pertains to [Formula: see text] for an example nonlinear integral equation has been demonstrated by one example premediated and the existence of blow-up point has been examined in one and more than one points regarding the nonlinear integral equation. The example nonlinear integral equation as addressed in this study has been examined in terms of its blow-up point and asymptotic stability with computational complexity for the first time. By doing so, it has been demonstrated that the existence of blow-up point has been the case in one and more than one points and besides this, the examination of its asymptotic stability has shown that it is asymptotically stable at least in one point, yet has no blow-up point pertaining to [Formula: see text]. This has enabled thorough analysis of computational complexity of the related nonlinear functional-integral equations within the example both addressing dimension of input in Banach space and that of dataset with [Formula: see text] dimension. This applicable approach assures the accurate transformation of the complex problems towards optimized solutions through the generation of advanced mathematical models in nonlinear dynamic settings characterized by complexity.

Remarks on semilinear partial functional-differential equations with infinite delay

Stabilizing nonlinear distributed parameter systems within the framework of the late lumping approach is a challenging problem. Diffusion‐reaction systems, described by semilinear partial differential equations, represent a common class of distributed parameter systems encountered in real applications. In this work, the zeroing dynamics method is used to design a stabilizing infinite‐dimensional controller for this important class of systems. Both distributed and boundary actuations are investigated. The design of the controller relies on the concept of the characteristic index. In the case of distributed control, as the characteristic index is finite, the design is straightforward. However, in the case of boundary control, the characteristic index is infinite, thus an equivalent pointwise control form is derived to reduce it to a special case of distributed control, which enables the controller design. In the disturbance‐free case, the developed controllers ensure exponential stability. In the presence of disturbances, the closed‐loop system remains stable, and the steady‐state error can be made arbitrarily small by tuning the controller parameters. Four common application examples from the literature are provided to demonstrate the effectiveness of the developed controllers: catalytic rod, monoenzyme system, heated rod, and FitzHugh‐Nagumo equation.

Asymptotic stability of infinite-dimensional semilinear systems: Application to a nonisothermal reactor

In this work, we prove some fixed point results in generalized Banach spaces ( G B S {\mathfrak{G}}{\mathfrak{B}}{\mathfrak{S}} s) in the sense of Perov using the vector degree of nondensifiability tools. The given result generalizes Darbo’s and Krasnoselskii’s theorems, which are connected with the vector measure of noncompactness. An existence result for coupled Caputo fractional delay differential equations in the G B S C ( [ − τ , T ] , R ) × C ( [ − τ , T ] , R ) {\mathfrak{G}}{\mathfrak{B}}{\mathfrak{S}}\hspace{0.33em}{\mathcal{C}}\left(\left[-\tau ,T],{\mathbb{R}})\times {\mathcal{C}}\left(\left[-\tau ,T],{\mathbb{R}}) is given to show the significance and the applicability of our theoretical results.