Semigroups Of Linear Operators Research Articles

The non-homogeneous Cauchy problem of semigroup of linear operators

This paper presents the results of ω-order reversing partial contraction mapping generated by the non-homogeneous Cauchy problem. We investigated several concepts of solution for non-homogeneous Cauchy problem where an analysis was arrived out on the relationship between C' strong and respectively C0-solution. We showed that Z is the infinitesimal generator of a C0-semigroup of contractions. Finally, we obtained that the problem has a unique classical solution and f, a member of L1(a,b; X) is continuous on (a,b).

Results of semigroup of linear operators in extrapolation spaces

Results of an omega-order preserving partial contraction mapping (omega-OCPn) in generalized spaces are presented in this study. Assumed to be a closed linear operator on a Banach space X with a non-empty resolvent set rho(A) is A in omega-OCPn. If A is densely defined, the extrapolation spaces X-1 and X-1 will be associated with A in agreement. However, X-1 is a proper closed subspace of X-1 if A is not densely defined. Then, we demonstrated that the reason these spaces exist is because (X*)-1 and D(A0) are naturally isomorphic to (X*)-1 and (X*)-1, respectively.

Quantitative estimates for bounded holomorphic semigroups

We revisit the theory of one-parameter semigroups of linear operators on Banach spaces in order to prove quantitative bounds for bounded holomorphic semigroups. Subsequently, relying on these bounds we obtain new quantitative versions of two recent results of Xu related to the vector-valued Littlewood–Paley–Stein theory for symmetric diffusion semigroups.

On a composite obtained by a mixture of a dipolar solid with a Moore–Gibson–Thompson media

Our study is dedicated to a mixture composed of a dipolar elastic medium and a viscous Moore–Gibson–Thompson (MGT) material. The mixed problem with initial and boundary data, considered in this context, is approached from the perspective of the existence of a solution to this problem as well as the uniqueness of the solution. Considering that the mixed problem is very complex, both from the point of view of the basic equations and that of the initial conditions and the boundary data, the classical methods become difficult. That is why we preferred to transform it into a problem of Cauchy type on a conveniently constructed Hilbert space. In this way, we immediately proved both the existence and uniqueness of the solution, with techniques from the theory of semigroups of linear operators.

Certain operator algebras of star-like reducible $P\omega_n^*$ transformations

Let $Z_n = 1, 2, 3, \ldots$ denote a distinct non-negative n-order collection of numbers, and $\alpha\omega_n^*$ denote a star-like transformation semigroup. The characterization of $P\omega_n^*$ star-like partial on the $\alpha\omega_n^*$ leads to the semigroup of linear operators. The research produced a completely new classical metamorphosis that was divided into inner product and norm parts. The study demonstrated that any specific star-like transformation $\lambda_i^*, \beta_j^* \in V^*$ is stable and uniformly continuous if there exists $T^{\vartheta^*}:(V^*, \left\langle v - \alpha^* u , u - \alpha^* v \right\rangle) \longrightarrow (V^*, \left\langle u - \alpha^* v , v - \alpha^* u \right\rangle)$ with a star-like polygon $\vartheta^*$ of $\vartheta^*V^*$ such that $T^{\vartheta^*}(v^*) = \vartheta^*V^*.$ Every star-like composite vector space $V^* \in P\omega_n^*$ can be uniquely decomposed as the sum of subspaces $w_i^* \leq W_{i+1}^*$ and $s_j^* \leq S_{j+1}^*$ such that $W_{i+1}^* + S_{j+1}^* \subseteq V^* \in P\omega_n^*.$ The study suggests that the research's findings be used to address issues in the mathematical disciplines of genetics, engineering, code theory, and telecommunications.

On sequences of sectorial forms converging 'from above'

We present a form convergence theorem for sequences of sectorial forms and their associated semigroups in a complex Hilbert space. Roughly speaking, the approximating forms $a_n$ are all `bounded below' by the limiting form $a$, but in contrast to the previous literature there is no monotonicity hypothesis on the sequence. Moreover, the forms are not supposed to be closed or densely defined. For a sectorial form one obtains an associated linear relation, whose negative generates a degenerate strongly continuous semigroup of linear operators. Our hypotheses on the sequence of forms imply strong resolvent convergence of the associated linear relations, which in turn implies convergence of the corresponding semigroups. The result is illustrated by two examples, one of them closely related to the Galerkin method of numerical analysis.

Q-fractional integral operators with two parameters

We use the Poisson kernel of the continuous q-Hermite polynomials to introduce families of integral operators. One of them is semigroups of linear operators. We describe the eigenvalues and eigenfunctions of one family of operators. The action of the semigroups of operators on the Askey–Wilson polynomials is shown to only change the parameters but preserves the degrees, hence we produce transmutation relations for the Askey–Wilson polynomials. The transmutation relations are then used to derive bilinear generating functions involving the Askey–Wilson polynomials.

-coupling mechanisms are sufficient to obtain exponential decay in strain gradient elasticity

AbstractIn this paper, we consider the time decay of the solutions to some problems arising in strain gradient thermoelasticity. We restrict to the two-dimensional case, and we assume that two dissipative mechanisms are introduced, the temperature and the mass dissipation. First, we show that this problem is well-posed proving that the operator defining it generates a contractive semigroup of linear operators. Then, assuming that the function involving the coupling terms is elliptic, the exponential decay of the solutions is concluded as well as the analyticity of the solutions. Finally, we describe how to obtain the exponential stability in the case of hyperbolic dissipation.

Boundary value problems of potential theory for the exterior ball and the approximation and ergodic behaviour of the solutions

The paper is concerned with the interconnection of the boundary behaviour of the solutions of the exterior Dirichlet, Neumann and Robin problems of harmonic analysis for the unit ball in R3 with the corresponding behaviour of the associated ergodic inverse problems for the entire space. Rates of approximation play a basic role.The solutions themselves are evaluated by means of Fourier expansions with respect to spherical harmonics. In case of the first two problems, the basis for the investigation of the approximation and ergodic behaviour is the theory of semigroups of linear operators mapping a Banach space X into itself. The connection between the semigroup property and the major premise of Huygens’ principle is emphasized.Another tool is a Drazin-like inverse operator Aad for the infinitesimal generator A of a semigroup that arises naturally in ergodic theory. This operator is a closed, not necessarily bounded, operator. It was introduced in a paper with U. Westphal (Butzer and Westphal, 1970/71) and extended to a generalized setting with J.J. Koliha (Butzer and Koliha, 2009).Unlike the latter two problems, the solution of Robin’s problem does not have the semigroup property and therefore the semigroup methods applied to Dirichlet’s and Neumann’s problem do not work. The authors give several hints how to overcome these difficulties.

Exponential stabilization for carbon nanotubes modeled as Timoshenko beams with thermoelastic effects

In this article we consider the problem of heat conduction in carbon nanotubes modeled like Timoshenko beams, inspired by the work of Yoonet al. [Compos. Part B: Eng.35(2004) 87–93]. Using the theory of semigroups of linear operators, we prove the well-posedness of the problem and the exponential stabilization of the total energy of the system of differential equations, partially damped, without assuming the known relationship of equality of wave velocities. Furthermore, we analyze the fully discrete problem using a finite difference scheme, introduced by a spatiotemporal discretization that combines explicit and implicit integration methods. We show the construction of numerical energy and simulations that prove our theoretical exponential decay results and display the convergence rates.

Null Controllability of Nonlinear Mixed Integro-differential Systems

The present paper investigates the existence of mild solutions and provides sufficient conditions for the null controllability of nonlinear mixed integrodifferential systems with unbounded linear operators in Banach spaces. It is observed that, if the linear system is controllable, then the nonlinear equivalence is also controllable by subjecting certain smooth conditions on the perturbation function. The results are obtained using semi group of linear operators, fractional powers of operators and the Schauder fixed point theorem.

Stability analysis of laminated beams with Kelvin–Voigt damping and strong time delay

In this paper we consider a model of laminated beams combining viscoelastic damping and strong time-delayed damping. The global well-posedness is proved by using the theory of semigroups of linear operators. We prove the lack of exponential stability when the speed wave propagations are not equal. In fact, we show in this situation, that the system goes to zero polynomially with rate t − 1 / 2 . On the other hand, by constructing some suitable multipliers, we establish that the energy decays exponentially provided the equal-speed wave propagations hold.

Results of Semigroup of Linear Operators Generating a General Class of Semilinear Initial Value Problems

This paper present results of ω-order preserving partial contraction mapping generating a general class of semilinear initial value problems. We consider the use of fractional powers of unbounded linear operators for its application by starting with some results concerning such fractional powers. We assume A to be the infinitesimal generator of an analytic semigroup in a Banach space X, 0 ∈ ρ(A) and defined the fractional powers of A for 0 < α ≤ 1. We also show that Aα is a closed linear operator whose domain D(Aα) ⊃ D(A) is dense in X. Finally we established that the operator is bounded, continuous and Holder continuous.

Results of semigroup of linear operators generating a nonlinear Schrödinger equation

In this paper, we present results of \(\omega\)-order preserving partial contraction mapping generating a nonlinear Schrödinger equation. We used the theory of semigroup to generate a nonlinear Schrödinger equation by considering a simple application of Lipschitz perturbation of linear evolution equations. We considered the space \(L^2(\mathbb{R}^2)\) and of linear operator \(A_0\) by \(D(A_0)=H^2(\mathbb{R}^2)\) and \(A_0u=-i\Delta u\) for \(u\in D(A_0)\) for the initial value problem, we hereby established that \(A_0\) is the infinitesimal generator of a \(C_0\)-semigroup of unitary operators \(T(t)\), \(-\infty<t<\infty\) on \(L^2(\mathbb{R}^2)\).

Results of Semigroup of Linear Operator Generating a Quasilinear Equations of Evolution

In this paper, results of $\omega$-order preserving partial contraction mapping generating a quasilinear equation of evolution were presented. In general, the study of quasilinear initial value problems is quite complicated. For the sake of simplicity we restricted this study to the mild solution of the initial value problem of a quasilinear equation of evolution. We show that if the problem has a unique mild solution $v\in C([0,T]: X)$ for every given $u\in C([0,T]:X)$, then it defines a mapping $u\to v=F(u)$ of $C([0,T]:X)$ into itself. We also show that under the suitable condition, there exists always a $T',\ 0<T'\leq T$ such that the restriction of the mapping $F$ to $C([0,T']:X)$ is a contraction which maps some ball of $C([0,t']:X)$ into itself by proving the existence of a local mild solution of the initial value problem.

Results of Semigroup of Linear Operators Generating a Regular Weak*-continuous Semigroup

This paper present results of $\omega$-order preserving partial contraction mapping generating a regular weak*-continuous semigroup. We consider a semigroup on a Banach space $X$ and $B:X^\odot\rightarrow X^*$ is bounded, then the intertwining formula was used to define a semigroup $T^B(t)$ on $X^*$ which extends the perturbed semigroup $T^B_0(t)$ on $X^\odot$ using the variation of constants formula. We also investigated a certain class of weak*-continuous semigroups on dual space $X^*$ which contains both adjoint semigroups and their perturbations by operators $B:X^\odot\rightarrow X^*$.

Results of Semigroup of Linear Operator Generating a Continuous Time Markov Semigroup

In this paper, we present results of $\omega$-order preserving partial contraction mapping creating a continuous time Markov semigroup. We use Markov and irreducible operators and their integer powers to describe the evolution of a random system whose state changes at integer times, or whose state is only inspected at integer times. We concluded that a linear operator $P:\ell^{1}(X_+)\rightarrow \ell^{1}(X_+)$ is a Markov operator if its matrix satisfies $P_{x,y}\geqslant 0$ and $\sum_{x\in X_+}P_{x,y=1}$ for all $y\in X$.

Results of Hankel Semigroup of Linear Operator as Eventually Uniform Continuous Semigroup

Abstract This paper consists of Hankel results of ω-order reversing partial contraction as a semigroup of linear operator. Spectral mapping theorem was investigated and it was established that ω-order reversing partial contraction mapping is an eventually uniform continuous semigroup in which we showed that the proof of the extension of T (t) to a C 0-semigroup ̂T(t) on a space ̂X containing X isometrically bounded. The space is constructed in such a way that the spectrum of the generator A of T (t) coincides with the spectrum of the generator  of ̂T(t) and the approximate point spectrum of A coincides with the point spectrum of Â.

Bowen–Walters expansiveness for semigroups of linear operators

AbstractWe define positively expansive semigroups of linear operators on Banach spaces. We characterize these semigroups in terms of the point spectrum of the infinitesimal generator. In particular, we prove that a positively expansive semigroup is neither uniformly bounded nor equicontinuous. We apply our results to the Lasota equation.

Exterior Dirichlet and Neumann Problems and the Linked Ergodic Inverse Problems in the Entire Space

The paper is mainly concerned with the interconnection of the boundary behaviour of the solutions of the exterior Dirichlet and Neumann problems of harmonic analysis for the unit disk in {mathbb {R}}^2 and the unit ball in {mathbb {R}}^3 with the corresponding behaviour of the associated ergodic inverse problems for the entire space. The basis is the theory of semigroups of linear operators mapping a Banach space X into itself. The classical one-parameter theory for semigroups applies in the present particular applications, actually for X= L^2_{2pi } in case of the unit disk, and X=L^2(S) in the three dimensional setting, S being the unit sphere in {mathbb {R}}^3. Another tool is a Drazin-like inverse operator B for the infinitesimal generator A of a semigroup that arises naturally in ergodic theory. This operator B is a closed, not necessarily bounded, operator. It was introduced in a paper with Butzer and Westphal (Indiana Univ Math J 20:1163–1174, 1970/1971) and extended to a generalized setting with Butzer and Koliha (J Oper Theory 62:297–326, 2009).