Accretive (*)-extensions and Realization Problems

On the extension problem for accretive differential operators

Convergence of composite iterative methods for finding zeros of accretive operators

We investigate the correspondence between operator monotone functions and Stieltjes functions. Then we shall prove that the class of nonnegative operator monotone functions is closed under certain operations, and applying those we decide the case when some binary operations become operator means.

Boomerang uniformity of some classes of functions over finite fields

The problem of determining source term in a semilinear wave equation is considered. The source term is represented as the product ƒ(u(x, t))p(x), where ƒ(s) is a given function, u(x, t) is a solution to Cauchy problem for wave equation, p(x) is an unknown function. To determine p(x) the additional information on the solution of the Cauchy problem u(α(t), t) = g(t), u(β(t), t) = h(t) is used. Theorems of existence and uniqueness of solution to an inverse problem in the class of continuous functions p(x) and in the class of functions p(x) = po + xq(x), where q(x) is continuous, are proved.

The problem of determining source term in a semilinear wave equation is considered. The source term is represented as the product ƒ(u(x, t))p(x), where ƒ(s) is a given function, u(x, t) is a solution to Cauchy problem for wave equation, p(x) is an unknown function. To determine p(x) the additional information on the solution of the Cauchy problem u(α(t), t) = g(t), u(β(t), t) = h(t) is used. Theorems of existence and uniqueness of solution to an inverse problem in the class of continuous functions p(x) and in the class of functions p(x) = po + xq(x), where q(x) is continuous, are proved.

We consider the approximation of the inverse square root of regularly accretive operators in Hilbert spaces. The approximation is of rational type and comes from the use of the Gauss–Legendre rule applied to a special integral formulation of the fractional power. We derive sharp error estimates, based on the use of the numerical range, and provide some numerical experiments. For practical purposes, the finite-dimensional case is also considered. In this setting, the convergence is shown to be of exponential type. The method is also tested for the computation of a generic fractional power.

This paper investigates a generalized fractional coupled transport system in a Hilbert space, in which such system contains Caputo-Katugampola differential terms and Caputo-Katugampola integral terms, as well as fully nonlinear coupling functions. First, we transform this system into an iterative system by using a semi-discrete method in time based on the backward Eulerian difference format (i.e., Rothe method). Second, employing Gronwall's inequality and the properties of $ m $-accretive operators, we establish the unique solvability and several priori estimates of solutions for the iterative system. Finally, we present two illustrative examples to demonstrate the obtained theoretical results.

In this paper, we prove a strong convergence theorem for finding a common solution of a general system of finite variational inequalities for finite different inverse-strongly accretive operators and solutions of fixed point problems for a nonexpansive semigroup in a Banach space based on a viscosity approximation method by using weak contraction mappings. Moreover, we can apply the above results to find the solutions of the class of k-strictly pseudocontractive mappings and apply a general system of finite variational inequalities into a Hilbert space. The results presented in this paper extend and improve the corresponding results of Ceng et al. (2008), Katchang and Kumam (2011), Wangkeeree and Preechasilp (2012), Yao et al. (2010) and many other authors. MSC:47H05, 47H10, 47J25.

The sum $A+B$ of two nonnegative selfadjoint relations (multi-valued operators) $A$ and $B$ is a nonnegative relation. The class of all extremal extensions of the sum $A+B$ is characterized as products of relations via an auxiliary Hilbert space associated with $A$ and $B$. The so-called form sum extension of $A+B$ is a nonnegative selfadjoint extension, which is constructed via a closed quadratic form associated with $A$ and $B$. Its connection to the class of extremal extensions is investigated and a criterion for its extremality is established, involving a nontrivial dependence on $A$ and $B$.

The investigation has defined the characteristic criterion (and its modification) of solvability of the problem of differential realization of the bundle of controlled trajectory curves of determined chaotic dynamic processes in the class of bilinear non-autonomous ordinary second- and higher-order differential equations (with and without delay) in the separable Hilbert space. The problem statement under consideration belongs to the type of converse problems for the additive combination of nonstationary linear and bilinear operators of the evolution equation in the Hilbert space. The constructions of tensor products of the Hilbert spaces, structures of lattices with an orthocomplement, the theory of extension of M2 -operators and the functional apparatus of the entropy Relay Ritz operator represent the basis of this theory. It has been shown that in the case of the finite bundle of the controlled trajectory curves the existence of the property of sub-linearity of the given operator allows one to obtain sufficient conditions of existence of such realizations. Side by side with solving the main problems, grounded are topological-group conditions of continuity of projectivization of the Relay Ritz operator with computing the fundamental group (Poincare group) of its compact image. The results obtained give incentives for the development of the quantitative theory of converse problems of higher-order multilinear evolution equations with the operators of generalized delay describing, for example, differential modeling of nonlinear Van der Pol oscillators or Lorentz strange attractors.

We consider the class of abstract kinetic equations (Tψ)′(x)=−Aψ(x) on the half-line x∈(0,∞) where T is an injective self-adjoint operator and A is an accretive compact perturbation of the identity, both of them defined on a Hilbert space. Half-range boundary conditions are imposed. If Re A≥δ1I for some δ>0, we establish the unique solvability of the problem. If Re A≥0 and Ker A=Ker (Re A), we prove that the problem has at least one bounded (in norm) solution and give a complete description of its measure of nonuniqueness. The result is applied to derive the well-posedness of the equation of transfer of polarized light and some multigroup neutron transport equations.

In this paper we consider a problem that follows directly from realization problem: how to find Markovian representations , even minimall, for a given family of Hilbert spaces, understood as outputs of a stochastic dynamic system S1, provided it is in a certain causality relationship with another family of Hilbert spaces , i. e. with some informations about states of a stochastic dynamic system S2. This paper is continuation of the papers Gill and Petrovic [7] and Petrovic [16,17].

We consider a problem (that follows directly from realization problem) on finding Markovian representations for a given family of Hilbert spaces such that each of these two families provides exactly the same amount of information about some other family of Hilbert spaces.

We prove a "quantified" version of the Weyl-von Neumann theorem, more precisely, we estimate the ranks of approximants to compact operators appearing in the Voiculescu's theorem applied to commutative algebras. This allows considerable simplifications in uniform K-homology theory, namely it shows that one can represent all the uniform K-homology classes on a fixed Hilbert space with a fixed *-representation of C_0(X), for a large class of spaces X.