Semigroups In Hilbert Spaces Research Articles

Qualitative Behaviour of Stochastic Integro-differential Equations with Random Impulses

In this paper, we study the existence and some stability results of mild solutions for random impulsive stochastic integro-differential equations (RISIDEs) with noncompact semigroups in Hilbert spaces via resolvent operators. Initially, we prove the existence of mild solution for the system is established by using Mönch fixed point theorem and contemplating Hausdorff measures of noncompactness. Then, the stability results includes continuous dependence of solutions on initial conditions, exponential stability and Hyers–Ulam stability for the aforementioned system are investigated. Finally, an example is proposed to validate the obtained results.

Harnack inequalities with power $$\pmb {p\in (1,+\infty )}$$ for transition semigroups in Hilbert spaces

Harnack inequalities with power $$\pmb {p\in (1,+\infty )}$$ for transition semigroups in Hilbert spaces

Existence and stability results of mild solutions for random impulsive stochastic partial differential equations with noncompact semigroups

In this paper, we study the existence and some stability results of mild solutions for random impulsive stochastic partial differential equations with noncompact semigroups in Hilbert spaces. Initially, we prove the existence of mild solutions by using Hausdorff measures of noncompactness and the Mönch fixed point theorem. Then, we explore the stability with continuous dependence of initial conditions, Hyers–Ulam stability and mean-square stability of the system by developing some new analysis techniques and establishing an improved inequality. Finally, an example is given to illustrate the abstract results obtained in this paper.

Strong and Weak Convergence Theorems for General Mixed Equilibrium, General Variational Inequality, and Fixed Point Problems for Two Nonexpansive Semigroups in Hilbert Spaces

Strong and Weak Convergence Theorems for General Mixed Equilibrium, General Variational Inequality, and Fixed Point Problems for Two Nonexpansive Semigroups in Hilbert Spaces

Regularizing Properties of (Non-Gaussian) Transition Semigroups in Hilbert Spaces

Let \(\mathcal {X}\) be a separable Hilbert space with norm \(\left \|{\cdot }\right \|\) and let T > 0. Let Q be a linear, self-adjoint, positive, trace class operator on \(\mathcal {X}\), let \(F:\mathcal {X}\rightarrow \mathcal {X}\) be a (smooth enough) function and let W(t) be a \(\mathcal {X}\)-valued cylindrical Wiener process. For α ∈ [0, 1/2] we consider the operator \(A:=-(1/2)Q^{2\alpha -1}:Q^{1-2\alpha }(\mathcal {X})\subseteq \mathcal {X}\rightarrow \mathcal {X}\). We are interested in the mild solution X(t, x) of the semilinear stochastic partial differential equation $$ \left\{\begin{array}{ll} dX(t,x)=\left( AX(t,x)+F(X(t,x))\right)dt+ Q^{\alpha}dW(t), & t\in(0,T];\\ X(0,x)=x\in \mathcal{X}, \end{array}\right. $$ and in its associated transition semigroup $$ P(t)\varphi(x):=\mathbb{E}[\varphi(X(t,x))], \qquad \varphi\in B_{b}(\mathcal{X}),\ t\in[0,T],\ x\in \mathcal{X}; $$ where \(B_{b}(\mathcal {X})\) is the space of the bounded and Borel measurable functions. We will show that under suitable hypotheses on Q and F, P(t) enjoys regularizing properties, along a continuously embedded subspace of \(\mathcal {X}\). More precisely there exists K := K(F, T) > 0 such that for every \(\varphi \in B_{b}(\mathcal {X})\), \(x\in \mathcal {X}\), t ∈ (0, T] and \(h\in Q^{\alpha }(\mathcal {X})\) it holds $$ |P(t)\varphi(x+h)-P(t)\varphi(x)|\leq Kt^{-1/2}\|Q^{-\alpha}h\|. $$

Strong and weak convergence theorems for general mixed equilibrium problems and general variational inequality problems and fixed point problems for two nonexpansive semigroups in Hilbert spaces

In this paper, we introduce some iterative algorithms for finding a common element of the set of solutions of the general mixed equilibrium problem and the set of solutions of a general variational inequality for two cocoercive mappings and the set of common fixed points of two nonexpansive semigroups in Hilbert space. We obtain both strong and weak convergence theorems for the sequences generated by these iterative processes in Hilbert spaces. Our results improve and extend the results announced by many others.

Approximation results for split equilibrium problems and fixed point problems of nonexpansive semigroup in Hilbert spaces

In this paper, we study a modified extragradient method for computing a common solution to the split equilibrium problem and fixed point problem of a nonexpansive semigroup in real Hilbert spaces. The weak and strong convergence characteristics of the proposed algorithm are investigated by employing suitable control conditions in such a setting of spaces. As a consequence, we provide a simplified analysis of various existing results concerning the extragradient method in the current literature. We also provide a numerical example to strengthen the theoretical results and the applicability of the proposed algorithm.

Polynomial decay rate of C 0 semigroup in Hilbert spaces

In the paper, we characterize the polynomial stability of semigroup with generator A on Hilbert space . Let A have compact resolvent and there be a sequence of the eigenvectors of A that forms a Riesz basis for . By the asymptotic relation of the real part and imaginary part of eigenvalues of A, we give the optimal decay rate of polynomial stability of . Moreover, we give the zeros distribution of certain equations. As an application, we illustrate our general results by an acoustical system.

Fixed point, convergence and nonlinear ergodic theorems for some semigroups in Hilbert spaces

In this paper, we first introduce some semigroups of mappings called quasi-nonexpansive, nonspreading, hybrid, TJ-1, TJ-2, and generalized hybrid semigroup. Then, using the theory of invariant means, we prove fixed point theorems, weak convergence theorem of Mann’s type and generalized nonlinear ergodic theorem for such semigroups in a Hilbert space. Furthermore, we prove a strong convergence theorem of Halpern’s type for the proposed semigroups in a Hilbert space. The results presented in this paper mainly extend and improved some well-known results in the literature.

Stability for some impulsive neutral stochastic functional integro-differential equations driven by fractional Brownian motion

The aim of this work is to study the stability for some integro-differential equations with impulses driven by fractional Brownian motion with noncompact semigroup in Hilbert spaces. We assume that...

A spectral theoretical approach for hypocoercivity applied to some degenerate hypoelliptic, and non-local operators

The aim of this paper is to offer an original and comprehensive spectral theoretical approach to the study of convergence to equilibrium, and in particular of the hypocoercivity phenomenon, for contraction semigroups in Hilbert spaces. Our approach rests on a commutation relationship for linear operators known as intertwining, and we utilize this identity to transfer spectral information from a known, reference semigroup $ \widetilde{{P}} = (e^{-t\widetilde{{\mathbf{A}}}})_{t \geqslant 0} $ to a target semigroup $ P $ which is the object of study. This allows us to obtain conditions under which $ P $ satisfies a hypocoercive estimate with exponential decay rate given by the spectral gap of $ \widetilde{{\mathbf{A}}} $. Along the way we also develop a functional calculus involving the non-self-adjoint resolution of identity induced by the intertwining relations. We apply these results in a general Hilbert space setting to two cases: degenerate, hypoelliptic Ornstein-Uhlenbeck semigroups on $ \mathbb{R}^d $, and non-local Jacobi semigroups on $ [0,1]^d $, which have been introduced and studied for $ d = 1 $ in [12]. In both cases we obtain hypocoercive estimates and are able to explicitly identify the hypocoercive constants.

Existence results systems coupled impulsive neutral stochastic functional differential equations with the measure of noncompactness

This paper is devoted to study the existence of solutions for a class of mild solutions for a class of impulsive neutral stochastic functional differential equations driven fractional Brownian motion (fBm) with noncompact semigroup in Hilbert spaces. The new results are obtained by using the Hausdorff measure of noncompactness. The arguments are based upon Monch’s fixed point theorem. Finally, an example is provided to illustrate the developed theory.

General Iterative Methods for Semigroups of Nonexpansive Mappings Related to Optimization Problems

In this article, we introduce two general iterative methods for a certain optimization problem of which the constrained set is the set of the solution set of the variational inequality problem for the fixed point set of nonexpansive semigroups in Hilbert spaces. Under some control conditions, we establish the strong convergence of the proposed methods to the fixed point set, which is the unique solution of a certain optimization problem. Applications to solutions of equilibrium problems are also presented.

Hybrid iterative methods for two asymptotically nonexpansive semigroups in Hilbert spaces

Hybrid iterative methods for two asymptotically nonexpansive semigroups in Hilbert spaces

Existence and exponential stability for impulsive neutral stochastic functional differential equations driven by fBm with noncompact semigroup via Mönch fixed point

In this paper, we investigate the existence and exponential stability of mild solutions for a class of impulsive neutral stochastic functional differential equations driven by fBm with noncompact semigroup in Hilbert spaces. Sufficient conditions for the existence of mild solutions are obtained using the Hausdorff measure of noncompactness and the Mönch fixed point theorem. Further, we establish a new impulsive-integral inequality to prove the exponential stability of mild solutions in the mean square moment. Finally, an example is presented to illustrate our obtained results.

Operator splitting for dissipative delay equations

We investigate Lie–Trotter product formulae for abstract nonlinear evolution equations with delay. Using results from the theory of nonlinear contraction semigroups in Hilbert spaces, we explain the convergence of the splitting procedure. The order of convergence is also investigated in detail, and some numerical illustrations are presented.

Algorithms for finding minimum norm solution of equilibrium and fixed point problems for nonexpansive semigroups in Hilbert spaces

Algorithms for finding minimum norm solution of equilibrium and fixed point problems for nonexpansive semigroups in Hilbert spaces

COMMON SOLUTION TO GENERALIZED MIXED EQUILIBRIUM PROBLEM AND FIXED POINT PROBLEM FOR A NONEXPANSIVE SEMIGROUP IN HILBERT SPACE

In this paper, we introduce and study an explicit hybrid re- laxed extragradient iterative method to approximate a common solution to generalized mixed equilibrium problem and fixed point problem for a nonexpansive semigroup in Hilbert space. Further, we prove that the sequence generated by the proposed iterative scheme converges strongly to the common solution to generalized mixed equilibrium problem and fixed point problem for a nonexpansive semigroup. This common solu- tion is the unique solution of a variational inequality problem and is the optimality condition for a minimization problem. The results presented in this paper are the supplement, improvement and generalization of the previously known results in this area.

Regularization Methods for Nonexpansive Semigroups in Hilbert Spaces

The purpose of this paper is to present a regularization method for finding a common fixed point of a nonexpansive semigroup in a real Hilbert space. A combination of the considered regularization scheme with the proximal point algorithm and another one of the regularization method with an iterative scheme is studied in this research. We also discuss an application of the proposed methods for an initial value problem in Hilbert spaces.

REGIONAL EXPONENTIAL REDUCED OBSERVABILITY IN DISTRIBUTED PARAMETER SYSTEMS

The regional exponential reduced observability concept in the presence for linear dynamical systems is addressed for a class of distributed parameter systems governed by strongly continuous semi group in Hilbert space. Thus, the existence of necessary and sufficient conditions is established for regional exponential reduced estimator in parabolic infinite dimensional systems. More precisely, the introduced approach is developed by using the decomposed system and reduced system in connection with various new concepts of (stability, detectability, estimator, observability and strategic sensors). Finally, we also show that there exists a dynamical system for two-phase exchange system described by the coupled parabolic equations is not exponentially reduced observable in usual sense, but it may be regionally exponentially reduced observable.