Class Of Operator Equations Research Articles

Generalized Functions in the Study of Signals and Systems

We collect three instances where the theory of generalized functions may still make contributions to the study of signals and systems. In the first, a purely algebraic approach is presented for LTI-ODE's, in terms of two operators, D and T, respectively the differentiation operator and the multiplication-by-the-independent-variable operator. This formalism adds simplicity, a duality theory, and nicely generalizes to other classes of operator equations and their solutions. In the second part we extend the classical bilateral Laplace transform to include Bohl functions with support in ℝ by invoking Sato's hyperfunctions. Finally, in the third case we use the Colombeau algebra to allow for products of generalized functions. This is important in the study of (smooth) nonlinear systems driven by impulsive inputs, and hybrid system theory.

Iterative Solution for Systems of a Class of Abstract Operator Equations in Banach Spaces and Application

In this paper, by using the partial order method, the existence and uniqueness of a solution for systems of a class of abstract operator equations in Banach spaces are discussed. The result obtained in this paper improves and unifies many recent results. Two applications to the system of nonlinear differential equations and the systems of nonlinear differential equations in Banach spaces are given, and the unique solution and interactive sequences which converge the unique solution and the error estimation are obtained.

An exact solution method for Fredholm integro-differential equations

Introduction: Linear boundary value problems for Fredholm ordinary integro-differential equations are seldom consideredwith integral boundary conditions in the literature. In our case, integro-differential equations are subject to multipoint or nonlocalintegral boundary conditions. It should be noted that finding exact solutions even for multipoint problems or problems with nonlocalintegral boundary conditions with a differential equation is a difficult task. Purpose: Finding the uniqueness and existencecriterion of solutions for Fredholm ordinary integro-differential equations with multipoint or nonlocal integral boundary conditionsand obtaining exact solutions in closed form of such problems. Results: Within the class of abstract operator equations, for thespecial case of Fredholm integro-differential equations with multipoint or nonlocal integral boundary conditions, a criterion for theexistence and uniqueness of an exact solution is proved and the analytical representation of the solution is given. A direct methodanalytically solving such problems is proposed, in which all calculations are reproducible in any program of symbolic calculations.If the user sets the input parameters and the initial conditions of the problem, the computer codes check the conditions of existenceand uniqueness and of solution generate the analytical solution. The stages of the solution method are illustrated by twoexamples. The article uses computer algebra system Mathematica to demonstrate the results.

On a Class of Operator Equations in Locally Convex Spaces

We consider a general method of solving equations whose left-hand side is a series by powers of a linear continuous operator acting in a locally convex space. Obtained solutions are given by operator series by powers of the same operator as the left-hand side of the equation. The research is realized by means of characteristics (of order and type) of operator as well as operator characteristics (of operator order and operator type) of vector relatively of an operator. In research we also use a convergence of operator series on equicontinuous bornology.

Vector-Valued Functions Generated by the Operator of Finite Order and Their Application to Solving Operator Equations in Locally Convex Spaces

This work is devoted to solving some classes of operator equations, based on the solution of auxiliary one-parameter family of equations, which is obtained fromthe original operator equation by formal replacement of the operator of the integrated parameter. Solutions are vector-valued functions represented by power series or integral. We investigate some properties of these solutions, namely, growth characteristics, the domain of analyticity. The investigation is realized by means of order and type of operator, operator order and operator type of the vector relative to the operator.

Existence of solutions for a class of operator equations

In this paper we deal with the existence and multiplicity of nontrivial solutions to a class of operator equation. By using infinite dimensional Morse theory, we establish some conditions which guarantee that the equation has many nontrivial solutions.

Optimization on class of operator equations in the probabilistic case setting

In this paper, we introduce a problem of the optimization of approximate solutions of operator equations in the probabilistic case setting, and prove a general result which connects the relation between the optimal approximation order of operator equations with the asymptotic order of the probabilistic width. Moreover, using this result, we determine the exact orders on the optimal approximate solutions of multivariate Freldholm integral equations of the second kind with the kernels belonging to the multivariate Sobolev class with the mixed derivative in the probabilistic case setting.

On a Class of Operator Equations

We assume that E1 and E2 are Banach spaces, a:E1→ E2 is a continuous linear surjective operator, f:E1→ E2 is a nonlinear completely continuous operator. In this paper, we study existence problems for the equation a(x)=f(x) and estimate the topological dimension dim of the set of solutions.

On information complexity of certain classes of operator equations

For some classes of operator equations of the second kind, we obtain an estimate of information complexity exact in order. We construct a new projection-type method that realizes the optimal estimate.

On the optimal rate of convergence of the projection-iterative method and some generalizations of it on a class of equations with smoothing operators

For some classes of operator equations of the second kind with smoothing operators, we find the exact order of the optimal rate of convergence of generalized projection-iterative methods.

Range Inclusion and Operator Equations

LetAandDbe positive operators on a complex Hilbert spaceH. In this work we show that the operator equationlDX+X*A=A(l>0) has a unique solutionClsatisfying certain conditions, if there exists a positive operatorBsuch thatAB+(D−B)A≥0. This implies several previously known results in special cases on this class of operator equations. Also, along with some definitions, a counterexample to a previous conjecture will be given.

On a certain class of operator equations with small parameter and regularization of ill-posed problems

On a certain class of operator equations with small parameter and regularization of ill-posed problems

A method of studying the existence of nontrivial solutions to some classes of operator equations with an application to resonance problems in mechanics

A method of studying the existence of nontrivial solutions to some classes of operator equations with an application to resonance problems in mechanics

Bifurcation from characteristic values of arbitrary multiplicity

Existence theorems are proved for bifurcation points of the system F(λ,x)∊C, where C is a closed cone in a Banach space. A parameter family of nontrivial solutions in a neighborhood of a bifurcation point are described in an analytic form. Different forms of linearization are utilized, which are weaker than continuous Frechet differentiability. The multiplicity of the characteristic value of the linearized part is not restricted. A principal tool is the use of generalized implicit function theorems for operators whose derivative has no bounded inverse or whose range is not necessarily closed. Existence of bifurcation points is established for a class of operator equations of the form T = l(7lambda;,v)+H(λ,v)+K(λ,v), where T and L(λ,.) are linear (not necessarily bounded) mappings, and H, K are nonlinear mappings that satisfy certain assumptions.

On the exact degree of complexity of a class of operator equations of the second kind in a Hilbert space

The exact exponent of complexity is found for approximate solutions of a certain class of operator equations in a Hilbert space. A method for information setup and the algorithm for realization of this optimal degree are presented. As a consequence, we find the exact exponent of complexity for approximate solutions of Fredholm integral equations of the second kind whose kernels and free terms include square integrable ψ-derivatives.

Existence and gap-bifurcation of multiple solutions to certain nonlinear eigenvalue problems

IN THIS PAPER we study: (i) a class of operator equations in an abstract Hilbert space; and (ii) the L2-theory of certain nonlinear Schrodinger equations which can be viewed as special cases of (i). In order to describe the type of abstract nonlinear eigenvalue problems to be discussed, consider a real Hilbert space H with scalar product (* , *) and norm II.11 and let S be a (not necessarily bounded) positive self-adjoint linear operator in li. We write S in the form

Optimal rate of convergence of V. I. Lebedev's method on some classes of operator equations

Upper and lower estimates are obtained for the rate of convergence of V. I. Lebedev's method on certain classes of operator equations of the second kind.

On a class of nonlocal nonlinear elliptic problems

We give a direct approach to the solvability of a class of nonlocal problems which admit a formulation in terms of quasi- variational inequalities. We are motivated by nonlinear elliptic boundary value problems in which certain coefficients depend, in a rather general way, on the solution itself through global quantities like the total mass, the total flux or the total energy. We illustrate the existence results with several applications, including an implicit Signorini problem for steady diffusion of biological populations and a class of operator equations in nonlinear mechanics. We also discuss the non- uniqueness of the solutions.

Galerkin information, the hyperbolic cross, and the complexity of operator equations

We obtain an exact power order of the complexity of the approximate solution of a certain class of operator equations in a Hibert space. We show that the optimal power order is realized by an algorithm that uses Galerkin information associated with the hyperbolic cross. As a corollary we derive an exact power order of the complexity of the approximate solution of Volterra integral equations whose kernels and free terms belong to Sobolev classes.

Approximate Solutions of a Certain Class of Operator Equations

Approximate Solutions of a Certain Class of Operator Equations