Resolvent Operator Research Articles

A study on approximate controllability of linear impulsive equations in Hilbert spaces

In this paper, we study an approximate controllability for the impulsive linear evolution equations in Hilbert spaces. We give a representation of solution in terms of semigroup and impulsive operators. We present the necessary and sufficient conditions for approximate controllability of linear impulsive evolution equation in terms of impulsive resolvent operator. An example is provided to illustrate the application of the obtained results.

On the Controllability of Coupled Nonlocal Partial Integrodifferential Equations Using Fractional Power Operators

In this research paper, we investigate the controllability in the α-norm of a coupled system of integrodifferential equations with state-dependent nonlocal conditions in generalized Banach spaces. We establish sufficient conditions for the system’s controllability using resolvent operator theory introduced by Grimmer, fractional power operators, and fixed-point theorems associated with generalized measures of noncompactness for condensing operators in vector Banach spaces. Finally, we present an application example to validate the proposed methodology in this research.

The resolvent kernel on the discrete circle and twisted cosecant sums

In this paper we present a general unifying principle for computing finite trigonometric sums of types that arise in physics and number theory. We obtain formulas that are more general than previous expressions and deduce linear recursions, which are computationally more efficient than the degree two recursions proved by Zagier. As an application, we provide an answer to a question recently posed by Xie, Zhao and Zhao concerning special values of Dirichlet L-functions. The proofs use the combinatorial Laplacian on cyclic graphs and their twisted coverings. The techniques therefore connect the trigonometric sums to spectral invariants of graphs and open up for future investigations.

Resolvent analysis of turbulent flow laden with low-inertia particles

We extend the resolvent framework to two-phase flows with low-inertia particles. The particle velocities are modelled using the equilibrium Eulerian model. We analyse the turbulent flow in a vertical pipe with Reynolds number of $5300$ (based on diameter and bulk velocity), for Stokes numbers $St^+=0-1$ , Froude numbers $Fr_z=-4,-0.4,0.4,4$ and $1/Fr_z = 0$ (gravity omitted). The governing equations are written in input–output form and a singular value decomposition is performed on the resolvent operator. As for single-phase flows, the operator is low rank around the critical layer, and the true response can be approximated using one singular vector. Even with a crude forcing model, the formulation can predict physical phenomena observed in Lagrangian simulations, such as particle clustering and gravitational effects. Increasing the Stokes number shifts the predicted concentration spectra to lower wavelengths; this shift also appears in the direct numerical simulation spectra and is due to particle clustering. When gravity is present, there are two critical layers, one for the concentration field, and one for the velocity field. For upward flow, the peak of concentration fluctuations shifts closer to the wall, in agreement with the literature. We explain this with the aid of the different locations of the two critical layers. Finally, the model correctly predicts the interaction of near-wall vortices with particle clusters. Overall, the resolvent operator provides a useful framework to explain and interpret many features observed in Lagrangian simulations. The application of the resolvent framework to higher $St^+$ flows in combination with Lagrangian simulations is also discussed.

New Results on the Solvability of Abstract Sequential Caputo Fractional Differential Equations with a Resolvent-Operator Approach and Applications

This paper aims to establish the existence and uniqueness of mild solutions to abstract sequential fractional differential equations. The approach employed involves the utilization of resolvent operators and the fixed-point theorem. Additionally, we investigate a specific example concerning a partial differential equation incorporating the Caputo fractional derivative.

Dependence of eigenvalues for higher odd-order boundary value problems

This paper is concerned with higher odd-order boundary value problems. We first prove that the operators associated with the problems are symmetric and the corresponding eigenvalues are real, and we give the resolvent operators related to the differential operators. Then we obtain that the eigenvalues are not only continuously but also smoothly dependent on the parameters of the problem. Moreover, the differential expressions of the eigenvalues as regards these parameters are given. In particular, we give the Frechet derivatives of the eigenvalues for the leading coefficient function q0 and coefficient functions q1,⋯,qn.

Eddy-viscosity-improved resolvent analysis of compressible turbulent boundary layers

An improved resolvent analysis is proposed in the regime of compressible turbulent boundary layers. To better model nonlinear processes in the input, the resolvent framework is augmented by adding eddy viscosity. To this end, we propose two eddy-viscosity models: a modified Cess eddy-viscosity model coupling the compressibility transformation and outer-layer correction, and a new eddy-viscosity model based on an empirical relationship and mixing-length theory. Both are incorporated into the resolvent operator to examine the performance of the eddy-viscosity-improved resolvent-based reduced-order modelling. Results of the augmented resolvent analysis are compared qualitatively and quantitatively with the first leading mode of spectral proper orthogonal decomposition, by checking the profiles and cross-spectral densities of velocities, density and temperature in two hypersonic turbulent boundary layers under different wall conditions. Higher accuracy of the turbulence prediction is achieved by adding the proposed eddy-viscosity models, particularly for the energetic cycle in the outer-layer region where strong nonlinear energy transfer exists.

Variational-like inclusion problems involving A-maximal m-relaxed η-accretive mappings in the sense of s.i.p. and fixed point problems

This paper is devoted to the introduction of a new class of accretive mappings in the framework of q-uniformly smooth Banach spaces called A-maximal m-relaxed η-accretive mappings in the sense of semi-inner product of type (p) (for short, s.i.p.(p)). The resolvent operator associated with such a mapping is defined; its Lipschitz continuity is proved and an estimate of its Lipschitz constant is computed under some suitable conditions. At the same time, the problem of finding a common element of the set of fixed points of a given nearly asymptotically nonexpansive mapping and the set of solutions of a variational-like inclusion problem involving A-maximal m-relaxed η-accretive mappings in the sense of s.i.p.(p) is investigated. To achieve this goal, a new iterative algorithm is constructed. Using the notions of graph convergence and resolvent operator associated with an A-maximal m-relaxed η-accretive mapping in the sense of s.i.p.(p), a new equivalence relationship between the graph convergence and the resolvent operator convergence of a sequence of A-maximal m-relaxed η-accretive mappings in the sense of s.i.p.(p) is established. As an application of this equivalence, the strong convergence of the sequence generated by our proposed iterative algorithm to a common point of the above two sets is demonstrated under some suitable assumptions imposed on the parameters. The results presented in this paper unify, improve and generalize some recent works in this field.

The harmonic $H^{\infty}$-functional calculus based on the $S$-spectrum

The aim of this paper is to introduce the H^{\infty} -functional calculus for harmonic functions over the quaternions. More precisely, we give meanring to Df(T) for unbounded sectorial operators T and polynomially growing functions of the form Df , where f is a slice hyperholomorphic function and D=\partial_{q_0}+e_{1}\partial_{q_1}+e_{2}\partial_{q_2}+e_{3}\partial_{q_3} is the Cauchy–Fueter operator. The harmonic functional calculus can be viewed as a modification of the well-known S -functional calculus f(T) , with a different resolvent operator. The harmonic H^{\infty} -functional calculus is defined in two steps. First, for functions with a certain decay property, one can make sense of the bounded operator Df(T) directly via a Cauchy-type formula. In a second step, a regularization procedure is used to extend the functional calculus to polynomially growing functions and consequently unbounded operators Df(T) . The harmonic functional calculus is an important functional calculus of the quaternionic fine structures on the S -spectrum, which arise also in the Clifford setting and they encompass a variety of function spaces and the corresponding functional calculi. These function spaces emerge through all possible factorizations of the second map of the Fueter–Sce extension theorem. This field represents an emerging and expanding research area that serves as a bridge connecting operator theory, harmonic analysis, and hypercomplex analysis.

Uniform bounds of families of analytic semigroups and Lyapunov Linear Stability of planar fronts

AbstractWe study families of analytic semigroups, acting on a Banach space, and depending on a parameter, and give sufficient conditions for existence of uniform with respect to the parameter norm bounds using spectral properties of the respective semigroup generators. In particular, we use estimates of the resolvent operators of the generators along vertical segments to estimate the growth/decay rate of the norm for the family of analytic semigroups. These results are applied to prove the Lyapunov linear stability of planar traveling waves of systems of reaction–diffusion equations, and the bidomain equation, important in electrophysiology.

Existence and Stability Results for Time-Dependent Impulsive Neutral Stochastic Partial Integrodifferential Equations with Rosenblatt Process and Poisson Jumps

Abstract In this paper, we discuss the existence, uniqueness and stability of mild solutions of time-dependent impulsive neutral stochastic partial integrodifferential equations with the Rosenblatt process and Poisson jumps. The existence of mild solutions for the equations is discussed by means of the semigroup theory and theory of the resolvent operator. Next, under some sufficient conditions, the results are obtained by using the method of successive approximation and Bihari’s inequality. Finally, an example is provided to illustrate our results.

On the Resolvent Matrix of the Truncated Hausdorff Matrix Moment Problem

On the Resolvent Matrix of the Truncated Hausdorff Matrix Moment Problem

Controllability result in α-norm for some impulsive partial functional integrodifferential equation with infinite delay in Banach spaces

Abstract The purpose of this paper is to study the controllability in the α-norm for some impulsive partial functional integrodifferential equation with infinite delay in Banach space. To do this, we give sufficient conditions ensuring the controllability by assuming that the undelayed part admits a resolvent operator in the sense of Grimmer and that the delayed part is continuous with respect to the fractional power of the generator. The results are obtained by using the Schauder fixed-point theorem.

Study of Uniqueness and Ulam-Type Stability of Abstract Hadamard Fractional Differential Equations of Sobolev Type via Resolvent Operators

This paper focuses on studying the uniqueness of the mild solution for an abstract fractional differential equation. We use Banach’s fixed point theorem to prove this uniqueness. Additionally, we examine the stability properties of the equation using Ulam’s stability. To analyze these properties, we consider the involvement of Hadamard fractional derivatives. Throughout this study, we put significant emphasis on the role and properties of resolvent operators. Furthermore, we investigate Ulam-type stability by providing examples of partial fractional differential equations that incorporate Hadamard derivatives.

Analysis of Abstract Partial Impulsive Integro-Differential System with Delay via Integrated Resolvent Operator

Analysis of Abstract Partial Impulsive Integro-Differential System with Delay via Integrated Resolvent Operator

Final value problem governed by a class of time‐space fractional pseudo‐parabolic equations with weak nonlinearities

We study the final value problem involving a class of semilinear fractional pseudo‐parabolic equations, where the nonlinearity probably takes values in fractional Sobolev spaces. By establishing some estimates for resolvent operators and employing the embedding related to Hilbert scales and fractional Sobolev spaces, we are able to obtain the existence and uniqueness result to the mentioned problem. In addition, the behavior of solutions at initial time is analyzed with respect to the final data. It will be shown that various cases of the nonlinearity functions meet our setting, including functions with gradient term.

Weak Convergence of the Rosenbrock Semi-implicit Method for Semilinear Parabolic SPDEs Driven by Additive Noise

Abstract This paper aims to investigate the weak convergence of the Rosenbrock semi-implicit method for semilinear parabolic stochastic partial differential equations (SPDEs) driven by additive noise. We are interested in SPDEs where the nonlinear part is stronger than the linear part, also called stochastic reaction dominated transport equations. For such SPDEs, many standard numerical schemes lose their stability properties. Exponential Rosenbrock and Rosenbrock-type methods were proved to be efficient for such SPDEs, but only their strong convergence were recently analyzed. Here, we investigate the weak convergence of the Rosenbrock semi-implicit method. We obtain a weak convergence rate which is twice the rate of the strong convergence. Our error analysis does not rely on Malliavin calculus, but rather only uses the Kolmogorov equation and the smoothing properties of the resolvent operator resulting from the Rosenbrock semi-implicit approximation.

Investigation on continuous dependence and regularity solutions of functional integrodifferential equations

In this work, to show the existence and continuous dependence of integral solution for neutral integro-differential equation with finite delay situation. This result established by using integrated resolvent operator with support of Lipschitz continuous and uniqueness part using Banach contraction mapping. Also study the existence of strict solution on reflexive and general Banach spaces. In the last section, an example is provided related to this theory.

Controllability of nonlinear fractional integrodifferential systems involving multiple delays in control

This work studies the existence of solutions and approximate controllability of fractional integrodifferential systems with Riemann-Liouville derivatives and with multiple delays in control. We establish suitable assumptions to prove the existence of solutions. Controllability of the system is shown by assuming a range condition on control operators and Lipschitz condition on non-linear functions. We use the concepts of strongly continuous semigroup rather than resolvent operators. Finally, an example is give to illustrate the theory.

Resolvent operator technique and iterative algorithms for system of generalized nonlinear variational inclusions and fixed point problems: Variational convergence with an application

This article is devoted to investigate the problem of finding a common point of the set of fixed points of a total uniformly L-Lipschitzian mapping and the set of solutions of a system of generalized nonlinear variational inclusions involving P-?-accretive mappings. For finding such an element, a new iterative algorithm is suggested. The concepts of graph convergence and the resolvent operator associated with a P-?-accretive mapping are used and a new equivalence relationship between graph convergence and resolvent operators convergence of a sequence of P-?-accretive mappings is established. As an application of the obtained equivalence relationship, we prove the strong convergence and stability of the sequence generated by our proposed iterative algorithm to a common element of the above two sets. These results are new, and can be viewed as a refinement and improvement of some known results in the literature.