Functional Evolution Equations Research Articles

Total controllability of nonlocal semilinear functional evolution equations with non-instantaneous impulses

Total controllability of nonlocal semilinear functional evolution equations with non-instantaneous impulses

Nonlocal controllability of mild solutions for neutral evolution equations with state-dependent delay in Fréchet spaces

In this paper, we prove the controllability of mild solutions of neutral functional evolution equations with state-dependent delay and nonlocal conditions. We establish the non local controllability of mild solutions under certain conditions by combining Avramescu’s nonlinear alternative for the sum of compact and contraction operators in Fréchet spaces with semigroup theory.

Retracted: Asymptotic Behavior of Solution for Functional Evolution Equations with Stepanov Forcing Terms

Retracted: Asymptotic Behavior of Solution for Functional Evolution Equations with Stepanov Forcing Terms

Poisson stable solutions for stochastic functional evolution equations with infinite delay

This work is devoted to Poisson stable (including stationary, periodic, quasi-periodic, Bohr almost periodic, Bohr almost automorphic, Birkhoff recurrent, almost recurrent in the sense of Bebutov, Levitan almost periodic, pseudo-periodic, pseudo-recurrent) solutions for stochastic functional evolution equations (SFEEs) with infinite delay. First, we prove the existence of bounded mild solutions for SFEEs with infinite delay. Then, according to the relationship between the solution and (drift and diffusion) coefficients, we obtain such Poisson stable solutions. Because the solutions of the delay equations are not Markov, we employ the solution maps in some phase space as a viable alternative for studying further asymptotic properties, and we also discuss Poisson stable solution maps and their asymptotic stability.

Global functional calculus, lower/upper bounds and evolution equations on manifolds with boundary

Given a smooth manifold M (with or without boundary), in this paper we establish a global functional calculus, without the standard assumption that the operators are classical pseudo-differential operators, and the Gårding inequality for global pseudo-differential operators associated with boundary value problems. The analysis that we follow is free of local coordinate systems. Applications of the Gårding inequality to the global solvability for a class of evolution problems are also considered.

Ulam–Hyers–Rassias Stability of Neutral Functional Integrodifferential Evolution Equations with Non-instantaneous Impulses on an Unbounded Interval

Ulam–Hyers–Rassias Stability of Neutral Functional Integrodifferential Evolution Equations with Non-instantaneous Impulses on an Unbounded Interval

Asymptotic behavior of stochastic functional differential evolution equation

In this work we study the long time behavior of nonlinear stochastic functional-differential equations in Hilbert spaces. In particular, we start with establishing the existence and uniqueness of mild solutions. We proceed with deriving a priory uniform in time bounds for the solutions in the appropriate Hilbert spaces. These bounds enable us to establish the existence of invariant measure based on Krylov-Bogoliubov theorem on the tightness of the family of measures. Finally, under certain assumptions on nonlinearities, we establish the uniqueness of invariant measures.

Approximate Controllability of Non-autonomous Second Order Impulsive Functional Evolution Equations in Banach Spaces

This article investigates the approximate controllability of second order non-autonomous functional evolution equations involving non-instantaneous impulses and nonlocal conditions. First, we discuss the approximate controllability of second order linear system in detail, which lacks in the existing literature. Then, we derive sufficient conditions for approximate controllability of our system in separable reflexive Banach spaces via linear evolution operator, resolvent operator conditions, and Schauder’s fixed point theorem. Moreover, in this paper, we define proper identification of resolvent operator in Banach spaces. Finally, we provide two concrete examples to validate our results.

Existence and compactness of solution of semilinear integro-differential equations with finite delay

UDC 517.9 We present some existence and uniqueness results for a class of functional integro-differential evolution equations generated by the resolvent operator for which the semigroup is not necessarily compact. It is proved that the set of solutions is compact. Our approach is based on fixed point theory. Finally, some examples are given to illustrate the results.

Approximate controllability of fractional order non-instantaneous impulsive functional evolution equations with state-dependent delay in Banach spaces

Abstract This paper deals with the control problems governed by fractional impulsive functional evolution equations with state-dependent delay involving Caputo fractional derivatives in Banach spaces. The main objective of this work is to formulate sufficient conditions for the approximate controllability of the considered system in separable reflexive Banach spaces. We have exploited the resolvent operator technique and Schauder’s fixed point theorem in the proofs to achieve this goal. The approximate controllability of linear system is discussed in detail, which lacks in the existing literature. Moreover, we point out some shortcomings of the existing works in the context of characterization of mild solution, phase space, and approximate controllability of fractional order impulsive systems in Banach spaces. Finally, we investigate the approximate controllability of the fractional order heat equation with non-instantaneous impulses and delay by using the developed results.

Local and Global Existence and Uniqueness of Solution for Class of Fuzzy Fractional Functional Evolution Equation

For fuzzy fractional functional evolution equations, the concept of global and local existence and uniqueness will be presented in this work. We employ the contraction principle and successive approximations for global and local existence and uniqueness, respectively, as given 0 c D q H x I = f I , x I + ∫ 0 I g I , s , x s ds , I ≥ I 0 , I ∈ 0 , T , x I = ψ I − I 0 = ψ 0 ∈ C σ , I 0 ≥ I ≥ I 0 − σ , x ′ I = ψ ′ I = ψ 1 , where C σ denotes the set of fuzzy continuous mapping defined on I 0 − σ , T and σ > 1 . We also use this method to solve fuzzy fractional functional evolution equations with fuzzy population models and distributed delays using fuzzy fractional functional evolution equations. To explain these results, some theorems are given. Finally, certain fuzzy fractional functional evolution equations are illustrated.

Analytical Approaches on the Attractivity of Solutions for Multiterm Fractional Functional Evolution Equations

The most important objective of the current research is to establish some theoretical existence and attractivity results of solutions for a novel nonlinear fractional functional evolution equations (FFEE) of Caputo type. In this respect, we use a familiar Schauder’s fixed-point theorem (SFPT) related to the method of measure of noncompactness (MNC). Furthermore, we consider the operator E and show that it is invariant and continuous. Moreover, we provide an application to show the capability of the achieved results.

On a study of Sobolev‐type fractional functional evolution equations

Sobolev‐type fractional functional evolution equations have many applications in the modeling of many physical processes. Therefore, we investigate the existence of mild solutions for fractional‐order time‐delay evolution equation of Sobolev type with multiorders in a Banach space with the help of two various methods. First, we study abstract problem by assuming the existence and compactness of E−1 and introduce a closed‐form representation of a mild solution via a newly defined delayed Mittag–Leffler‐type function which is generated by nonpermutable and permutable linear‐bounded operators. Our second method is based on the theory of Sobolev‐type delayed resolvent families generated by the pair (A, E) and a subordination principle, and in this case, any restrictive assumptions are not required on E. Furthermore, we derive an exact analytical representation of solutions for multidimensional fractional functional dynamical systems with nonpermutable and permutable matrices. As an example, the Sobolev‐type delayed wave equation with a damping term is provided to illustrate the applications of the abstract results.

Asymptotic Behavior of Solution for Functional Evolution Equations with Stepanov Forcing Terms

Through the use of the measure theory, evolution family, “Acquistapace–Terreni” condition, and Hölder inequality, the core objective of this work is to seek to analyze whether there is unique μ -pseudo almost periodic solution to a functional evolution equation with Stepanov forcing terms in a Banach space. Certain adequate conditions are derived guaranteeing there is unique μ -pseudo almost periodic solution to the equation by Lipschitz condition and contraction mapping principle. Finally, an example is used to demonstrate our theoretical findings.

Order topology in minkowski space and applications

In this article, we consider and study the order topology in Minkowski spacetime of the special theory of relativity , i.e. the finest locally convex topology τ on spacetime for which every order bounded subset of spacetime is τ -bounded. This order topology that is introduced into spacetime as an ordered vector space proves to be Hausdorff and differs from Zeeman's order topology. Applying the order topology we obtain new results by applying and extending previous results on the mean ergodic theorem and functional differential evolution equations in the Minkowski space .

Quantum dynamics under continuous projective measurements: Non-Hermitian description and the continuum-space limit

The problem of the time of arrival of a quantum system in a specified state is considered in the framework of the repeated measurement protocol and in particular the limit of continuous measurements is discussed. It is shown that for a particular choice of system-detector coupling, the Zeno effect is avoided and the system can be described effectively by a non-Hermitian effective Hamiltonian. As a specific example we consider the evolution of a quantum particle on a one-dimensional lattice that is subjected to position measurements at a specific site. By solving the corresponding non-Hermitian wave function evolution equation, we present analytic closed-form results on the survival probability and the first arrival time distribution. Finally we discuss the limit of vanishing lattice spacing and show that this leads to a continuum description where the particle evolves via the free Schrodinger equation with complex Robin boundary conditions at the detector site. Several interesting physical results for this dynamics are presented.

Neutral multivalued integro-differential evolution equations with infinite state-dependent delay

Our problem through this work is to give the existence of mild solutions for the first order class of neutral functional multivalued integro-differential evolution equations with infinite state-dependent delay using the nonlinear alternative of Frigon for multivalued contraction maps in Fréchet spaces combined with the semi-group theory.

Helicity-dependent generalization of the JIMWLK evolution

We generalize the JIMWLK evolution equation to include the small-$x$ evolution of helicity distributions. To derive helicity JIMWLK, we use the method devised by A.H. Mueller for the usual unpolarized JIMWLK, and apply it to the helicity evolution equations derived earlier. We obtain a functional evolution equation for the generalized JIMWLK weight functional. The functional now is a sum of a helicity-independent term and a helicity-dependent term. The kernel of the new evolution equation consists of the standard eikonal leading-order JIMWLK kernel plus new terms containing derivatives with respect to the sub-eikonal quark and gluon fields. The new kernel we derive can be used to generate the standard JIMWLK evolution and to construct small-$x$ evolution equations for flavor-singlet operators made of any number of light-cone Wilson lines along with one Wilson line containing sub-eikonal helicity-dependent local operator insertions (a "polarized Wilson line").

Neutral stochastic functional differential evolution equations driven by Rosenblatt process with varying-time delays

Hermite processes are self-similar processes with stationary increments, the Hermite process of order 1 is fractional Brownian motion and the Hermite process of order 2 is the Rosenblatt process. In this paper we consider a class of time-dependent neutral stochastic functional differential equations with finite delay driven by Rosenblatt process with index H ∈ ( 1/2 , 1) which is a special case of a self-similar process with long-range dependence. More precisely, we prove the existence and uniqueness of mild solutions by using stochastic analysis and a fixed-point strategy. Finally, an illustrative example is provided to demonstrate the effectiveness of the theoretical result.

Fractional differential equations of Sobolev type with sectorial operators

This paper treats the asymptotic behavior of resolvent operators of Sobolev type and its applications to the existence and uniqueness of mild solutions to fractional functional evolution equations of Sobolev type in Banach spaces. We first study the asymptotic decay of some resolvent operators (also called solution operators) and next, by using fixed point results, we obtain the existence and uniqueness of solutions to a class of Sobolev type fractional differential equation. We notice that, the existence or compactness of an operator $$E^{-1}$$ is not necessarily needed in our results.