Resolvent Operator Research Articles

Generalized variational inclusion: graph convergence and dynamical system approach

<p>This work focused on the investigation of a generalized variation inclusion problem. The resolvent operator for generalized $ \eta $-co-monotone mapping was structured, the Lipschitz constant was estimated and its relationship with the graph convergence was accomplished. An Ishikawa type iterative algorithm was designed by incorporating the resolvent operator and total asymptotically non-expansive mapping. By employing the novel implication of graph convergence and analyzing the convergence of the considered iterative method, the common solution of the generalized variational inclusion and the set of fixed points of a total asymptotically non-expansive mapping was obtained. Moreover, a generalized resolvent dynamical system was investigated. Some of its attributes were discussed and implemented to examine the considered generalized variation inclusion problem.</p>

A Study of The Scattering Properties of Eigenparameter- Dependent Matrix Difference Operator With Transmission Condition

In this paper, we set a transmission boundary value problem for a matrix valued difference equation on the semi axis. The main purpose of this study is to examine the properties of scattering solutions and scattering functions of this problem. Firstly, by giving the Jost solution and scattering solutions of this problem, we obtain the Jost function and the scattering function of the problem. We also investigate eigenvalues, spectral singularities, resolvent operator and continuous spectrum of this problem.

Existence and Successive Approximations of Mild Solution for Integro-differential Equations in Banach Spaces

Abstract This paper discusses the global convergence of successive approximations methods for solving integro-differential equation via resolvent operators in Banach spaces. We prove a theorem on the global convergence of successive approximations to the unique solution of the problems. An example is given to show the application of our result.

Pseudo-resolvent energy of weighted graphs

Let G be a finite order graph. Its resolvent matrix and its resolvent energy are respectively given by Rξ(A(G)) = (A(G) – ξI)–1 and ER(G) = Σn i = 1 (n – λi)–1. For some classes of weighted graphs, we know that λ1 = n which means that the formula for ER(G) is not applicable. In this work, the pseudo-resolvent energy for those classes of graphs are obtained where λ1 = n.

2×2 operator matrix with real parameter and its spectrum

In the present paper we consider a linear bounded self-adjoint 2×2 block operator matrix Aμ (so called generalized Friedrichs model) with real parameter μ ∈ R. It is associated with the Hamiltonian of a system consisting of at most two particles on a d -dimensional lattice Zd, interacting via creation and annihilation operators. Aμ is linear bounded self-adjoint operator acting in the two-particle cut subspace of the Fock space, that is, in the direct sum of zero-particle and one-particle subspaces of a Fock space. We find the essential and discrete spectra of the block operator matrix Aμ. The Fredholm determinant and resolvent operator associated to Aμ are constructed. The spectrum of Aμ plays an important role in the study of the spectral properties of the Hamiltonians associated with the energy operator of a lattice system describing two identical bosons and one particle, another nature in interactions, without conservation of the number of particles on a lattice.

Approximate controllability of neutral Hilfer fractional differential equations of Sobolev-type in a Hilbert Space

In this paper, our main purpose is to establish the controllability results for nonlocal neutral Hilfer fractional differential equations of Sobolev-type in a Hilbert Space as well as to generalize the results that existed in the literature on this topic. We present three types of conditions on the nonlocal initial condition's function to prove the existence of a mild solution for nonlocal neutral Hilfer fractional differential equations of Sobolev-type, and we then derive the approximate controllability results for the system. With help of an approximate technique, we establish the existence and controllability results under the weaker hypothesis (continuous only) on the nonlocal initial condition's function. The main tools applied in our analysis are semigroup theory, fractional calculus, resolvent operator theory, the theory of fractional powers of operators, Krasnoselskii's fixed point theorem, Schauder's fixed point theorem, and approximating technique. Finally, we provide two examples as applications to illustrate our main results.

Solving delay integro-differential inclusions with applications

<abstract><p>This work primarily delves into three key areas: the presence of mild solutions, exploration of the topological and geometrical makeup of solution sets, and the continuous dependency of solutions on a second-order semilinear integro-differential inclusion. The Bohnenblust-Karlin fixed-point method has been integrated with Grimmer's theory of resolvent operators. Ultimately, the study delves into a mild solution for a partial integro-differential inclusion to showcase the achieved outcomes.</p></abstract>

Null controllability of Hilfer fractional stochastic differential equations with nonlocal conditions

<p style='text-indent:20px;'>This paper assesses a class of nonlocal Hilfer fractional stochastic differential equations (NHFSDEs) governed by fractional Brownian motion (fBm). New sufficient condition of exact null controllability for this stochastic setting has been investigated using fractional calculus, fixed point theory and a theory of resolvent operator. The derived result is new because it generalizes several of previously published results. However, the obtained results are proven by an illustration using stochastic partial differential equations to demonstrate the present application characteristic of null controllability. A filter example is demonstrated to make use of resolvent operator in a practical way.</p>

Approximate controllability of non-instantaneous impulsive stochastic integrodifferential equations driven by Rosenblatt process via resolvent operators

In this work, we investigate the existence of a mild solution and the approximate controllability of non-instantaneous impulsive stochastic integrodifferential equations driven by the Rosenblatt process in Hilbert space with the Hurst parameter \(\mathsf{H} \in (1/2, 1)\). We achieve the result using the semigroup theory of bounded linear operators, Grimmer's resolvent operator theory, and stochastic analysis. Using Krasnoselskii's and Schauder's fixed point theorems, we demonstrate the existence of mild solutions and the approximate controllability of the system. Finally, an example shows the potential for significant results.

Boundary value problems with displacement for one mixed hyperbolic equation of the second order

The paper studies two nonlocal problems with a displacement for the conjugation of two equations of second-order hyperbolic type, with a wave equation in one part of the domain and a degenerate hyperbolic equation of the first kind in the other part. As a non-local boundary condition in the considered problems, a linear system of FDEs is specified with variable coefficients involving the first-order derivative and derivatives of fractional (in the sense of Riemann-Liouville) orders of the desired function on one of the characteristics and on the line of type changing. Using the integral equation method, the first problem is equivalently reduced to a question of the solvability for the Volterra integral equation of the second kind with a weak singularity; and a question of the solvability for the second problem is equivalently reduced to a question of the solvability for the Fredholm integral equation of the second kind with a weak singularity. For the first problem, we prove the uniform convergence of the resolvent kernel for the resulting Volterra integral equation of the second kind and we prove that its solution belongs to the required class. As to the second problem, sufficient conditions are found for the given functions that ensure the existence of a unique solution to the Fredholm integral equation of the second kind with a weak singularity of the required class. In some particular cases, the solutions are written out explicitly.

CONVERGENCE ANALYSIS FOR APPROXIMATING SOLUTION OF VARIATIONAL INCLUSION PROBLEM

This article aims to define a new resolvent operator for variational inclusion problems in the framework of Banach spaces. We design a rapid algorithm using the resolvent operator to approximate the solution of the variational inclusion problem in Banach spaces. Additionally, we show that the algorithm articulated in this article converges faster than the well-known and notable algorithm due to Fang and Huang. To show the superiority and prevalence of the obtained results, we propound a numerical and computational example upholding our claim. Lastly, a minimization problem is solved with the help of the proposed algorithm, which is the first attempt in the current context of the study.

Square dA-integrable solutions of differential systems with measures on the semiaxis

This paper extends the Weyl-Titchmarsh theory to the case of generalized differential systems. It is that proved the characteristic matrix of the resolvent kernel of a differential system with measures belongs to a locus, which is a matrix analog of the Weyl disk. It is established that such matrix disks are nested and converge to the limiting disk or point depending on the limiting behavior of the radii. This limiting set plays an important role in discussing of the number of solutions to such system that are square dA-integrated on the semiaxis.

The linearized Poisson–Nernst–Planck system as heat flow on the interval under non‐local boundary conditions

The linearized Poisson–Nernst–Planck (PNP) equation under closed ends around a neutral state is studied. It is reduced to a damped heat equation under non‐local boundary conditions, which leads to a stochastic interpretation of the linearized equation as a Brownian particle which jump and is reflected, at Poisson distributed time, to one of the end points of the channel, with a probability which is proportional to its distance from this end point. An explicit expansion of the heat kernel reveals the eigenvalues and eigenstates of both the PNP equation and its adjoint. For this, we take advantage of the representation of the resolvent operator and recover the heat kernel by applying the inverse Laplace transform.

Customized Douglas-Rachford splitting methods for structured inverse variational inequality problems

Recently, structured inverse variational inequality (SIVI) problems have attracted much attention. In this paper, we propose new splitting methods to solve SIVI problems by employing the idea of the classical Douglas-Rachford splitting method (DRSM). In particular, the proposed methods can be regarded as a novel application of the DRSM to SIVI problems by decoupling the linear equality constraint, leading to smaller and easier subproblems. The main computational tasks per iteration are the evaluations of certain resolvent operators, which are much cheaper than those methods without taking advantage of the problem structures. To make the methods more implementable in the general cases where the resolvent operator is evaluated in an iterative scheme, we further propose to solve the subproblems in an approximate manner. Under quite mild conditions, global convergence, sublinear rate of convergence, and linear rate of convergence results are established for both the exact and the inexact methods. Finally, we present preliminary numerical results to illustrate the performance of the proposed methods.

Well-posedness results for a new class of stochastic spatio-temporal SIR-type models driven by proportional pure-jump Lévy noise

This paper provides a first attempt to incorporate the massive discontinuous changes in the spatio-temporal dynamics of epidemics. Namely, we propose an extended class of epidemic models, governed by coupled stochastic semilinear partial differential equations, driven by pure-jump Lévy noise. Based on the considered type of incidence functions, by virtue of semigroup theory, a truncation technique and Banach fixed point theorem, we prove the existence and pathwise uniqueness of mild solutions, depending continuously on the initial datum. Moreover, by means of a regularization technique, based on the resolvent operator, we acquire that mild solutions can be approximated by a suitable converging sequence of strong solutions. With this result at hand, for positive initial states, we derive the almost-sure positiveness of the obtained solutions. Finally, we present the outcome of several numerical simulations, in order to exhibit the effect of the considered type of stochastic noise, in comparison to Gaussian noise, which has been used in the previous literature. Our established results lay the ground-work for investigating other problems associated with the new proposed class of epidemic models, such as asymptotic behavior analyses, optimal control as well as identification problems, which primarily rely on the existence and uniqueness of biologically feasible solutions.

Existence results for some stochastic functional integrodifferential systems driven by Rosenblatt process

Abstract This work investigates the existence and uniqueness of mild solutions to a class of stochastic integral differential equations with various time delay driven by the Rosenblatt process. We can obtain alternative conditions that guarantee mild solutions by using the resolvent operator in the Grimmer sense, stochastic analysis, fixed-point methods, and noncompact measures. We give an example to illustrate the theory.

Approximation of solutions to integro-differential time fractional order parabolic equations in L^{p}-spaces

In this paper we study the initial boundary value problem for a class of integro-differential time fractional order parabolic equations with a small positive parameter ε. Using the Laplace transform, Mittag-Leffler operator family, C_{0}-semigroup, resolvent operator, and weighted function space, we get the existence of a mild solution. For suitable indices pin [1,+infty ) and sin (1,+infty ), we first prove that the mild solution of the approximating problem converges to that of the corresponding limit problem in L^{p}((0,T), L^{s}(Omega )) as varepsilon rightarrow 0^{+}. Then for the linear approximating problem with ε and the corresponding limit problem, we give the continuous dependence of the solutions. Finally, for a class of semilinear approximating problems and the corresponding limit problems with initial data in L^{s}(Omega ), we prove the local existence and uniqueness of the mild solution and then give the continuous dependence on the initial data.

Equilibrium problems and proximal algorithm using tangent space products

This paper presents equilibrium problems and their regularized problems in the framework of Hadamard spaces. Using the concept of tangent space products, we introduce a resolvent operator and deduce essential properties in relation to equilibrium problem. Furthermore, we analyze the regularized problems involving resolvent operators. Finally, we establish two convergence results concerning proximal algorithms

The final value problem for anomalous diffusion equations involving weak-valued nonlinearities

We are concerned with the final value problem governed by a class of semilinear anomalous diffusion equations, where the nonlinearity may take values in Hilbert scales with negative orders. The solvability and Hölder regularity results are proved by establishing estimates in Hilbert scales for resolvent operators in connection with nonlinearity function. The obtained results are applicable for some concrete problems modeling subdiffusion phenomena.

Stochastic McKean–Vlasov equations with Lévy noise: Existence, attractiveness and stability

This work considers McKean–Vlasov equations driven by Lévy noise in which the coefficients rely not only on the state of the unknown process but also on its probability distribution, and researches the well-posed, attractiveness and stability of solutions with resolvent operator theory. Under global Lipschitz condition, the well-posed of mild solutions is first studied for McKean–Vlasov integro-differential equations by using contraction mapping principle. Then according to Cauchy–Schwartz inequality and Itô isometry formula, one gains global attracting set and quasi-invariant set of solutions of McKean–Vlasov integro-differential equations and also obtained sufficient conditions of stability with continuous dependence on the coefficient of solutions. Moreover, one attains mean-square stability of solutions by Gronwall inequality and almost sure stability by applying Borel–Cantelli lemma and Chebyshev inequality for the aforementioned equation. Finally two examples are presented to verify our results.