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Inverting a nonlinear operator is usually a difficult problem. Consequently, approximation methods are useful. In this paper, we study the relationship between the set of approximate inverses of a given operator N and the invertibility of N.

The objective of this paper is to present certain conditions guaranteeing invertibility of a nonlinear operator between normed linear spaces. The idea is to approximate the given operator by an invertible, possibly linear operator, and reduce the problem to the contraction mapping principle. Several theorems of this kind are given, which appear as generalizations of some early results by I.W. Sandberg, and estimates for an “approximate inverse” are established. Finally, introducing certain “invertibility indices,” further sufficient and necessary conditions for invertibility are given.

We study the behavior of a Hilbert network (i.e., a finite or countably infinite network whose variables are in a Hilbert space and in which the associated total energy is finite) whose elements are affected by perturbations. More specifically, we will give estimates for a change of the current distribution caused by (a) perturbations of the elements of the nominal network when the voltage sources are fixed, and (b) a change of voltage sources in a network whose elements are perturbed. The conditions given in our theorems imply insensitivity and robust stability of the nominal network. The applications of the results are illustrated by an example of an infinite network.

We consider input-output systems (not necessarily of feedback type) on the time domain [0, ∞) which are governed by nonlinear vector integral or differential equations that relate the input and the output. Assuming that these equations depend on a parameterA, describing perturbations, which is allowed to vary in a vicinity of a nominal valueA0 in a linear space, we study how strongly the output is affected by changes of (a)A0 when the input is fixed (insensitivity), and (b) the input whenA is fixed withAA0 being not too large (robust stability).

The concept of a Hilbert network, introduced in another paper, extends network theory to finite or infinite networks whose elements are described by operators on a Hilbert space. The present work investigates a variety of qualitative properties possessed by such networks. In particular, some operators associated with the entire network, such as the driving-point impedances as well as the admittance operator which relates the branch-voltage vector to the branch-current vector, are shown to be—under suitable conditions—either positive, monotonic, or convex. Also, generalized versions of Jeans’ least power theorem and the Shannon–Hagelbarger theorem are proved. Finally, a bound on the power dissipation is determined.

An approximation theorem for a Hammerstein-type equation in a Hilbert,space is proved. Also, applications to the Ritz–Galerkin method and to construction of an approximating feedback system are discussed.

The paper presents a simple yet quite general method for analyzing the sensitivity of a performance characteristic of a system consisting of linear blocks. The analysis is based on properties of a gradient of an abstract function. Applications in network theory and in the theory of feedback systems are discussed.

In this paper we consider a linearization of a nonlinear monotone Hilbert network in a neighborhood of some operating point. Three theorems are established which give estimates for the difference between the exact and approximate current distribution. Two examples of specific networks are considered.

In this paper feedback systems described by nonlinear (possibly multivalued) monotone operators are considered. We establish conditions for the existence and uniqueness of outputs corresponding to a given pair of inputs, conditions for causality, Lipschitz continuity and stability. Also, feedback systems over an extended Hilbert space are discussed. Finally, linear approximations to a nonlinear feedback system are studied.

A local implicit function theorem is proved which assumes certain injectivity rather than differentiability. Moreover, there is given a result on solvability of noncanonic systems of differential equations which arise in optimal control and network theory.

In this paper conditions are given guaranteeing the existence and uniqueness of a current distribution in a Hilbert or algebraic network driven by both EMF and current sources. Moreover, a generalization and converse of Tellegen’s theorem for these networks is discussed.

In this paper, solvability and stability of nonlinear, time-varying semistate equations is discussed. It is shown that generalized solutions exist in a vicinity of an equilibrium. Moreover, results on a Liapunov-like (uniform, asymptotic) stability are given.

In this paper we consider linearizations of a nonlinear equation in a Hilbert space which are close, in a certain sense, to an optimal linearization. We focus our attention on equations involving monotone operators, particularly on those with Hammerstein-type operators. Finally, we discuss a construction of a linearizing operator in the case when a Nemytskii operator is involved.

In this paper we give sufficient conditions for solvability of a Hilbert network some elements of which are described by monotone operators defined only on subsets of the underlying Hilbert space. As a special case we consider a finite nonlinear LRC-network, whose inductors are linear, time-varying.

The paper deals with a qualitative property of linear continuous operators defined on distributions, that is, with a property appearing as a generalization of the traditional causality. The traditional causality of an operator A means that ${\operatorname{supp}}Ax \subset S$ whenever ${\operatorname{supp}}x \subset S$, where A is any interval $[T,\infty )$. Here, causality is defined in the same way, but S is any member from a certain family of subsets of $R^m $, called a scale.A theorem is proved which gives necessary and sufficient conditions for an operator to be causal with respect to a given scale.As an application there is considered a slightly generalized convolution-type operator; it is shown that this operator is causal with respect to a scale which consists of all translations of a fixed cone.

It is shown that every linear, bounded, strictly causal operator on a resolution space $[H,P^t ]$ is in fact contrastive under a certain inner product which generates a norm equivalent to the original norm on H.