M-accretive Operators Research Articles

Adjoints of Sums of M-Accretive Operators and Applications to Non-Autonomous Evolutionary Equations

Adjoints of Sums of M-Accretive Operators and Applications to Non-Autonomous Evolutionary Equations

Rates of Convergence and Metastability for Chidume’s Algorithm for the Approximation of Zeros of Accretive Operators in Banach Spaces

In this paper we give a quantitative analysis of an explicit iteration method due to C.E. Chidume for the approximation of a zero of an m-accretive operator A : X → 2 X in Banach spaces which does not involve the computation of the resolvent of A.

Enhanced dissipation by circularly symmetric and parallel pipe flows

We study enhanced dissipation due to the combined effect of diffusion or hyperdiffusion and advection by an incompressible flow with circular or cylindrical symmetry in 2 and 3 space dimensions, respectively. By using resolvent estimates for m-accretive operators (Wei, 2021), under a suitable condition on the velocity adapted from Gallay and Coti Zelati (2021), we establish enhanced dissipation for the advection-(hyper)diffusion equation and quantify it in terms of rates of decay in time for the solution, suitably projected, with an improved explicit dependence on the diffusivity. Our results extend prior results in Coti Zelati and Dolce (2020).

Evolution Problems with m-Accretive Operators and Perturbations

This paper is devoted to the study of perturbation evolution problems involving time-dependent m-accretive operators. We present for a specific class of m-accretive operators with convex weakly compact-valued perturbation, some results about the existence of absolutely continuous solutions and BRVC solutions. We finish by giving several applications to various domains such as relaxation results, second-order evolution inclusions, fractional-order equations coupled with m-accretive operators and Skorohod differential inclusions.

Invariance and strict invariance for nonlinear evolution problems with applications

Sufficient conditions for the invariance of evolution problems governed by perturbations of (possibly nonlinear) m-accretive operators are provided. The conditions for the invariance with respect to sublevel sets of a constraining functional are expressed in terms of the Dini derivative of that functional, outside the considered set in directions determined by the considered m-accretive operator. An approach for non-reflexive Banach spaces is developed and some result improving a recent paper (Cannarsa et al., 2018) is presented. Applications to nonlinear obstacle problems and age-structured population models are presented in spaces of continuous functions where an advantage of such approach is taken. Moreover, some new abstract criteria for the so-called strict invariance are derived and their direct application to problems with barriers is discussed.

The Moore-Penrose inverse of accretive operators with application to quadratic operator pencils

We establish some relationships between an m-accretive operator and its Moore-Penorse inverse. We derive some perturbation result the Moore-Penorse inverse of a maximal accretive operator. As an application we give a factorization theorem for a quadratic pencil of accretive operators. Also, we study a result of existence, uniqueness, and maximal regularity of the strict solution for complete abstract second order differential equation. Illustrative examples are also given.

ABSTRACT FRACTIONAL CALCULUS FOR m-ACCRETIVE OPERATORS

In this paper we aim to construct an abstract model of a differential operator with a fractional integro-differential operator composition in final terms, where modeling is understood as an interpretation of concrete differential operators in terms of the infinitesimal generator of a corresponding semigroup. We study such operators as a Kipriyanov operator, Riesz potential, difference operator. Along with this, we consider transforms of m-accretive operators as a generalization, introduce an operator class $\mathfrak{G_{\alpha}}$ and provide a description of its spectral properties.

On the strong convergence of the proximal point algorithm with an application to Hammerstein euations

Let E be a real normed space. A new notion of quasi-boundedness for operators $$A:E\rightarrow 2^E$$ is introduced and the following general important result for accretive operators is proved: an accretive operator with zero in the interior of its domain is quasi-bounded. Using this result, a new strong convergence theorem for approximating a zero of an m-accretive operator is proved in a uniformly smooth real Banach space. This result complements the celebrated proximal point algorithm for approximating solutions of $$0\in Au$$ in a real Hilbert space where A is a maximal monotone operator. Furthermore, as an application of our theorem, a new strong convergence theorem for approximating a solution of a Hammerstein equation is proved. Finally, several numerical experiments are presented to illustrate the strong convergence of the sequence generated by our algorithm and the results obtained are compared with those obtained using some recent important algorithms.

Approximation of zeros of m-accretive mappings, with applications to Hammerstein integral equations

"An algorithm for approximating zeros of m-accretive operators is constructed in a uniformly smooth real Banach space. The sequence generated by the algorithm is proved to converge strongly to a zero of an m-accretive operator. In the case of a real Hilbert space, our theorem complements the celebrated proximal point algorithm of Martinet and Rockafellar for approximating zeros of maximal monotone operators. Furthermore, the convergence theorem proved is applied to approximate a solution of a Hammerstein integral equation. Finally, numerical experiments are presented to illustrate the convergence of our algorithm."

Diffusion and mixing in fluid flow via the resolvent estimate

In this paper, we first present a Gearhardt-Pr\"uss type theorem with a sharp bound for m-accretive operators. Then we give two applications: (1) give a simple proof of the result proved by Constantin et al. on relaxation enhancement induced by incompressible flows; (2) show that shear flows with a class of Weierstrass functions obey logarithmically fast dissipation time-scales.

Convergence of a hybrid viscosity approximation method for finding zeros of m-accretive operators

For finding a zero of an m-accretive operator in a real Banach space, we consider an implicit hybrid viscosity approximation method and show that its explicit variant converges strongly to a zero under weaker assumptions than the ones used recently by Ceng et al. (Numer. Func. Anal. Opt. 35(2), 142–165, 2012). First, we examine the case of spaces which are uniformly smooth or reflexive and strictly convex with a uniformly Gâteaux differentiable norm. Afterwards, we improve our convergence results when the space is uniformly convex. Furthermore, we show that the strong convergence of some modified Halpern and Krasnosel’skii–Mann type methods can also be deduced from our results. Finally, a numerical example is also given to illustrate the convergence analysis of the considered method.

Splitting-midpoint method for zeros of the sum of accretive operator and μ-inversely strongly accretive operator in a q-uniformly smooth Banach space and its applications

Combining the implicit midpoint method and the splitting method, we present a new iterative algorithm with errors to solve the problems of finding zeros of the sum of m-accretive operators and μ-inversely strongly accretive operators in a real q-uniformly smooth and uniformly convex Banach space. We obtain some strong convergence theorems, which demonstrate the relationship between the zero of the sum of m-accretive operator and μ-inversely strongly accretive operator and the solution of one kind variational inequality. Moreover, the applications of the main results on the nonlinear problems with Neumann boundaries and Signorini boundaries are demonstrated.

A generalisation of the form method for accretive forms and operators

The form method as popularised by Lions and Kato is a successful device to associate m-sectorial operators with suitable elliptic or sectorial forms. McIntosh generalised the form method to an accretive setting, thereby allowing to associate m-accretive operators with suitable accretive forms. Classically, the form domain is required to be densely embedded into the Hilbert space. Recently, this requirement was relaxed by Arendt and ter Elst in the setting of elliptic and sectorial forms. Here we study the prospects of a generalised form method for accretive forms to generate accretive operators. In particular, we work with the same relaxed condition on the form domain as used by Arendt and ter Elst. We give a multitude of examples for many degenerate phenomena that can occur in the most general setting. We characterise when the associated operator is m-accretive and investigate the class of operators that can be generated. For the case that the associated operator is m-accretive, we study form approximation and Ouhabaz type invariance criteria.

ON STABILITY OF SQUARE ROOT DOMAINS FOR NON‐SELF‐ADJOINT OPERATORS UNDER ADDITIVE PERTURBATIONS

Assuming $T_0$ to be an m-accretive operator in the complex Hilbert space $\mathcal{H}$, we use a resolvent method due to Kato to appropriately define the additive perturbation $T = T_0 + W$ and prove stability of square root domains, that is, $$ dom\big((T_0 + W)^{1/2}\big) = dom\big(T_0^{1/2}\big). $$ Moreover, assuming in addition that $dom\big(T_0^{1/2}\big) = dom\big((T_0^*)^{1/2}\big)$, we prove stability of square root domains in the form $$dom\big((T_0 + W)^{1/2}\big) = dom\big(T_0^{1/2}\big) = dom\big((T_0^*)^{1/2}\big) = dom\big(((T_0 + W)^*)^{1/2}\big), $$ which is most suitable for PDE applications. We apply this approach to elliptic second-order partial differential operators of the form $$ - div(a\nabla \, \cdot \,) + \big(\mathbf{B}_1\cdot \nabla \cdot \big) + div \big(\mathbf{B}_2 \cdot \big) + V $$ in $L^2(\Omega)$ on certain open sets $\Omega \subseteq \mathbb{R}^n$, $n \in \mathbb{N}$, with Dirichlet, Neumann, and mixed boundary conditions on $\partial \Omega$, under general hypotheses on the (typically, nonsmooth, unbounded) coefficients and on $\partial\Omega$.

A new iterative algorithm for the sum of two different types of finitely many accretive operators in Banach space and its connection with capillarity equation

AbstractIn this paper, we present a new iterative algorithm with errors to solve the problems of finding zeros of the sum of finitely many m-accretive operators and finitely many α-inversely strongly accretive operators in a real smooth and uniformly convex Banach space. Strong convergence theorems are established, which extend the corresponding works given by some authors. Moreover, the relationship among the zero of the sum of m-accretive operator and α-inversely strongly accretive operator, the solution of one kind variational inequality, and the solution of the capillarity equation is investigated.

On the Asymptotic Stability and Contraction of an Evolution Equation

An Evolution Equation on a Banach Space with m-accretive operator is considered. The solution of this Evolution Equation is shown to exist, asymptotically stable andcontract. Keywords: Evolution Equation; Asymptotic stability; Contraction; Banach Space; Existence. 2010 Mathematics Subject Classification:35B40

Convergence rates in regularization for a system of nonlinear ill-posed equations with m-accretive operators

In this paper, we study an operator version of the modified Browder-Tikhonov regularization method for finding a common solution for a system of ill-posed operator equations involving m-accretive operators Ai, i =0 , ... , N, in a reflexive Banach space. The convergence rates of the regularized solutions are estimated not only in the infinite-dimensional space, but also in connection with its finite-dimensional approximations without the weakly sequential continuity of the dual mapping. MSC: 47H17; 47H20

Strong convergence of a parallel iterative algorithm in a reflexive Banach space

In this paper, a parallel iterative algorithm is investigated for common zeros of a family of m-accretive operators. Strong convergence theorems are established in a reflexive Banach space.

Some results on zero points of m-accretive operators in reflexive Banach spaces

Abstract A modified proximal point algorithm is proposed for treating common zero points of a finite family of m-accretive operators. A strong convergence theorem is established in a reflexive, strictly convex Banach space with the uniformly Gâteaux differentiable norm.

Existence results for second-order monotone differential inclusions on the positive half-line

Consider in a real Hilbert space H the differential equation (inclusion) (E): p(t)u″(t)+q(t)u′(t)∈Au(t)+f(t) a.e. in (0,∞), with the condition (B): u(0)=x∈D(A)¯, where A:D(A)⊂H→H is a (possibly set-valued) maximal monotone operator whose range contains 0; p,q∈L∞(0,∞), with essinfp>0 and q+∈L1(0,∞). More than four decades ago, V. Barbu established the existence of a unique bounded solution to (E), (B), in the particular case p≡1, q≡0 and f≡0. Subsequently the existence and uniqueness of bounded solutions in the homogeneous case (f≡0) have been further investigated by H. Brezis (1972), N. Pavel (1976), L. Véron (1974–1976), and by E.I. Poffald and S. Reich (1986) when A is an m-accretive operator in a Banach space. The non-homogeneous case has received less attention from this point of view. In this paper, we prove existence and uniqueness of bounded solutions to (E), (B) in the general case of non-constant functions p, q satisfying the mild conditions above, thus compensating for the lack of existence theory for such kind of second order problems. Note that our results open up the possibility to apply Lions' method of artificial viscosity towards approximating the solutions of some nonlinear parabolic and hyperbolic problems, as shown in the last section of the paper.