A note on the approximate controllability of second-order integro-differential evolution control systems via resolvent operators

The approximate controllability of second-order integro-differential evolution control systems using resolvent operators is the focus of this work. We analyze approximate controllability outcomes by referring to fractional theories, resolvent operators, semigroup theory, Gronwall’s inequality, and Lipschitz condition. The article avoids the use of well-known fixed point theorem approaches. We have also included one example of theoretical consequences that has been validated.

On the initial value problem for a class of nonlinear biharmonic equation with time-fractional derivative

In this study, we investigate the intial value problem (IVP) for a time-fractional fourth-order equation with nonlinear source terms. More specifically, we consider the time-fractional biharmonic with exponential nonlinearity and the time-fractional Cahn–Hilliard equation. By using the Fourier transform concept, the generalized formula for the mild solution as well as the smoothing effects of resolvent operators are proved. For the IVP associated with the first one, by using the Orlicz space with the function $\Xi (z)={\textrm {e}}^{|z|^{p}}-1$ and some embeddings between it and the usual Lebesgue spaces, we prove that the solution is a global-in-time solution or it shall blow up in a finite time if the initial value is regular. In the case of singular initial data, the local-in-time/global-in-time existence and uniqueness are derived. Also, the regularity of the mild solution is investigated. For the IVP associated with the second one, some modifications to the generalized formula are made to deal with the nonlinear term. We also establish some important estimates for the derivatives of resolvent operators, they are the basis for using the Picard sequence to prove the local-in-time existence of the solution.

Approximate controllability of multivalued stochastic fractional differential equations with Atangana-Baleanu-Caputo derivatives

This article aims to discuss the approximate controllability of multivalued impulsive stochastic fractional differential equations with ABC derivatives in Hilbert space. Firstly, with the help of stochastic analysis, theory of resolvent operators and the fixed point technique, we confirm the existence of mild solutions for the proposed control system. Secondarily, we show that the proposed system of equations is approximately controllable under a certain hypothesis. To confirm the applicability of the obtained results, an example is provided at the end of this paper.

Existence Results for Fractional Semilinear Integrodifferential Equations of Mixed Type with Delay

In this paper, we discuss a class of fractional semilinear integrodifferential equations of mixed type with delay. Based on the theories of resolvent operators, the measure of noncompactness, and the fixed point theorems, we establish the existence and uniqueness of global mild solutions for the equations. An example is provided to illustrate the application of our main results.

A study on controllability of impulsive fractional evolution equations via resolvent operators

In this article, we study the controllability for impulsive fractional integro-differential evolution equation in a Banach space. The discussions are based on the Mönch fixed point theorem as well as the theory of fractional calculus and the $(\alpha ,\beta )$ ( α , β ) -resolvent operator, we concern with the term $u'(\cdot )$ u ′ ( ⋅ ) and finding a control v such that the mild solution satisfies $u(b)=u_{b}$ u ( b ) = u b and $u'(b)=u'_{b}$ u ′ ( b ) = u b ′ . Finally, we present an application to support the validity study.

Composition Semigroups on Weighted Bergman Spaces Induced by Doubling Weights

We prove that composition semigroups are strongly continuous on weighted Bergman spaces with doubling weights. Point spectra and compact resolvent operators of infinitesimal generators of composition semigroups are characterized.

Iterative Sequences for a Finite Number of Resolvent Operators on Complete Geodesic Spaces

We consider Halpern’s and Mann’s types of iterative schemes to find a common minimizer of a finite number of proper lower semicontinuous convex functions defined on a complete geodesic space with curvature bounded above.