Nonlinear Operator Theory Research Articles

Convergence of the Kato approximants for evolution equations involving functional perturbations

The existence of a unique strong solution of the nonlinear abstract functional differential equation u′(t) + A(t)u(t) = F(t,u t), u 0 = φεC 1(¦−r,0¦,X),tε¦0, T¦ , (E) is established. X is a Banach space with uniformly convex dual space and, for tϵ ¦0, T¦, A(t) is m-accretive and satisfies a time dependence condition suitable for applications to partial differential equations. The function F satisfies a Lipschitz condition. The novelty of the paper is that the solution u( t) of (E) is shown to be the uniform limit (as n → ∞) of the sequence u n ( t), where the functions u n ( t) are continuously differentiate solutions of approximating equations involving the Yosida approximants. Thus, a straightforward approximation scheme is now available for such equations, in parallel with the approach involving the use of nonlinear evolution operator theory.

Initial boundary value problems for the Carleman equation

We consider initial boundary value problems for the Carleman equations. The theory of nonlinear accretive operators is applied to provide generalized solutions and to consider well-posedness of the system in the L 1(0, 1) sense. The solutions are represented by product integrals, an abstract backward difference scheme.

Unrenormalized Schwinger–Dyson equations and dynamical mass generation

In dynamical mass generation of fermion fields by fermion-spin-1-boson interactions, anomalous magnetic moment-like terms in three-point functions play an essential role, so that the Schwinger–Dyson equations do not admit a trivial solution, i.e., a solution without dynamically generated mass. Applicability of Altman’s and Lika’s versions of nonlinear operator theory to unrenormalized Schwinger–Dyson equations is discussed and algorithms for construction of approximate solutions are proposed.

Non-linear uncertain feedback systems with initial state values†

A single input-output non-linear plant w is considered which, due to uncertainty, is known only to be a member of a set . At t = 0, the plant has initial state values given by a vector Y 0, member of a defined set 0. At this instant it is switched into a feedback structure with compensation function G, which has zero initial state values. For certain classes of  it is shown how G may be designed to achieve arbitrary fast attenuation of the plant output for all we Y 0e0. The design philosophy is to convert this problem into one of disturbance attenuation in a linear time-invariant plant P with zero initial states ; Pe𝒫 a set of linear time-invariant plants and the disturbance de a set of disturbances. ,𝒫 are determined by 0,  and the performance specifications. Schauder's fixed point theorem is used to rigorously justify this technique. The design procedure may be readily used by engineers who are quite ignorant of the mathematical theory of non-linear operators. Ordinary, familiar frequency-respo...

Multiplicity and stability of burner stabilized premixed laminar flames: A general proof

The stability of a one-dimensional burner-stabilized premixed flame is investigated for an arbitrary Lewis number and general N-species mass action kinetic laws. The flame is assumed to be radially adiabatic. The existence of unsteady solutions is proved via fixed point theory for nonlinear operators acting on closed bounded spaces. The stability and multiplicity of steady states is studied with the help of bifurcation theory. The major result is that for an N-species system, 2 N+1 steady states are possible, N of which are unstable. This result is recast in experimental terms for burner-stabilized flames. There exist critical values of experimentally controlled parameters—e.g. incoming flow velocity, heat transfer rate at the burner plugs, and species mass fraction in the feed—for which the time-asymptotic solution bifurcates from the nontrivial steady state solution. These bifurcations lead to burner extinction but periodic solutions cannot be excluded. The present conclusions are, believed to be the first generalization of recent specific results obtained by Margolis, Joulin and Clavin, and Hennemann et al, on bifurcation in premixed laminar flames. A discussion of the extension to multidimensional flames is presented, together with the limitations of the present technique.

Some considerations to the fundamental theory of infinite delay equations

In recent years we see an increasing interest in infinite delay equations. The main reason is that equations of this type become more and more important for different applications. Regardless of the specific problem one has to deal with it is in most cases necessary to establish some fundamental theory as, for instance, existence, uniqueness of solutions and continuous dependence on initial data. Also in the last few years we find an increasing number of papers where the general theory of linear and nonlinear semigroups or evolution operators is applied to functional differential equations. Of fundamental importance for all approaches is the right choice of the state space which in most cases is a Banach space of functions or of equivalence classes of functions. For equations with bounded delays this in general is not a difficult problem. But for infinite delay equations the choice of an appropriate state space is no more trivial. It was natural to investigate which properties of the state space are sufficient in order to establish the fundamental theory for infinite delay equations. Up to now the most thorough discussion of this problem is contained in [12]. The set of axioms given there seems to be in more or less final form. Other systems of axioms are just slight modifications of those given in [12] (cf., for instance, [18, 211) or consider more special cases (as in [6, 17). Section 1 of this paper can be considered as a discussion of the axioms given in [12] which results in a somewhat streamlined version of these axioms. It should be mentioned here that state spaces for equations with bounded retardation are included as special case, of course. In Section 2 we prove existence, uniqueness and continuous dependence of solutions concentrating on the nonstandard situation where the right-hand side of the equation is not defined for elements in the state space but only for representatives of elements in the state space. Such situations occur for instance if difference-differential equations are considered in a space like UP

Existence and local uniqueness of solutions of the Percus-Yevick equation

By means of nonlinear operator theory, the existence and local uniqueness of solutions of the Percus-Yevick equation are established under certain conditions.

Contributions to the spectral theory for nonlinear operators in Banach spaces

Si introduce una definizione di spettro σ(f) per applicazioni continue definite in uno spazio di Banach. Tale definizione coincide con quella classica nel caso in cui f sia lineare e continua. Alcuni dei risultati piu noti della teoria spettrale lineare vengono estesi al caso non lineare. In particolare si dimostra che σ(f) e chiuso e che la sua frontiera e contenuta nello spettro puntuale approssimato σπ(f).

On the constructive solution of nonlinear functional equations

In the recent development of the theory of nonlinear operators in Banach spaces, a number of general existence theorems have been established for various classes of mappings by the use of compactness, convexity, or topological arguments that do not give a constructive procedure for the generation of the solutions thus proved to exist. Even in cases when the solutions are unique and one obtains them as limits of a precisely determined sequence of approximants, they often fail to satisfy an important and useful principle of constructivity in that the procedures have no effective control provided for the error at each state of the approximation. This is particularly the case for the Galerkin approximations used in the theory of operators of monotone type and for various fixed-point methods used in the theory of nonlinear accretive operators. About a decade ago, the writer developed existence results which satisfied these principles of constructivity for the case of a broad class of continuous monotone mappings in Hilbert spaces, and more generally, for continuous accretive mappings in Banach spaces X whose conjugate spaces X* are uniformly convex. These results were stated in the rather inaccessible paper [l] and developed in detail in the middle of a lengthy treatment of accretive operators in the writer’s paper [2]. It is our object here to develop these results explicitly and in detail in connection with the problem of constructivity. Our renewed interest in this question was stimulated by the recent paper by Bruck [3], who has developed an iteration procedure to obtain the solution of the equation (I + T)(u) = 0 for a continuous, bounded monotone operator T on a Hilbert space H with explicit control of the error. Bruck’s result has the curious feature that it depends on the assumption of the prior existence of the solution. Although the procedure we give is not an iteration procedure it corresponds to a simpler intuitive picture of the situation in the general context of accretive operators and the error control does not depend upon the assumption of the existence of a solution.

On spectral theory for nonlinear operators

Let E be an infinite-dimensional Banach space and II’: E E be a (possibly nonlinear) quasi-bounded compact map. Recently Furi and Vignoli [3] introduced a notion of a spectrum X(T) to obtain criteria for the surjectivity of I T. Z(T) turns out to be the approximate point spectrum for linear T. In this paper some further properties of Z( 2’) are given. Among other results we prove that either 0 E Z(T) or it is surrounded by Z(T). Both cases arc possible. As an illustration we show that the famous Birkhoff-Kellogg theorem is a consequence of this fact. We prove also that if T is asymptotically odd then 0 E Z(T). This is a generalization of the well-known fact that 0 E o(T) for linear T (dim E = co). Consequently a compact odd map defined on the unit sphere S of E contains zero in the closure of its range. We thank Furi and Vignoli for some helpful discussions.

Observer theory for distributed-parameter systems

This paper examines the problem of the approximate reconstruction of the unknown state variables in distributed-parameter systems. New results on the observer theory for important classes of linear and non-linear operator, partial differential, and partial differential-integral equations in describing distributed-parameter systems are presented. The specific developments employ the recent results on Lyapunov stability theory, along with the theory of linear and non-linear semigroup operators, and their infinitesimal generators. The questions of observability, stability of the state reconstruction error dynamics associated with the proposed observer structure are discussed. The theoretical results are illustrated with some applications to problems of the kinetic lumping of complex distributed-parameter chemical reaction systems, as well as the observer design for linear and non-linear distributed-parameter diffusion systems.

Polyhomogeneous maps and best local approximations of degree ?

Contemporary investigations in the theory of nonlinear integral operators ([9], [10]) and in the differential calculus ([5], [11]) have led to generalizations of the notion of a polynomial map between two vector spaces. This article studies basic properties of such so-called polyhomogeneous maps. Our initial point of reference is the recent study of polynomial maps by Bochnak and Siciak [2]. Our examination of continuity properties leads to new characterizations of braked spaces and sequential spaces. Then, we turn to the polyhomogeneous approximating maps studied by Melamed and Perov [9] and Moore and Nashed [10]. We present some generalizations of the results of [9] and [10] and then go on to study permanence properties of such approximations.

Book review. [Theory of nonlinear operators

Book review. [Theory of nonlinear operators

An existence theorem for the v. Kármán equations under the condition of free boundary

The paper concerns the v. Kármán equations governing the bending of a thin elastic plate under the condition of free boundary. Starting from the definition of a variational solution, the boundary value problem considered is replaced by an equivalent abstract operator equation to which the known theorems of the nonlinear operator theory apply. The main result consists in an existence theorem of a variational solution for the problem under consideration.

On a problem in the theory of nonlinear Fredholm operators

We prove that for any Fredholm operator A(x) of class C1 and zero index in a Hilbert space, in a neighborhood of any star compactum T lying in the domain of A we can define a completely continuous and continuously differentiable operator C, so that the linear operator A′(x)+ C′(x) has a bounded inverse for all x∃T.

A Leray-Schauder theorem for a class of nonlinear operators

One of the most useful tools for dealing with the theory of nonlinear operators in concrete analytical problems is the Leray-Schauder fixed point theorem [5]. A disadvantage is that it is only available for the completely continuous (that is, compact and continuous) operators, and moreover the proof of the theorem uses the theory of the topological degree of a mapping. However, in a recent publication, Browder [1] has given an elementary proof without recourse to degree theory, tt was observed by the authors that this method of proof is applicable to a wider class of operators, namely those we shall call strongly P-compact. In this note we shall show that for a restricted class of Banach spaces (including, however, all separable Hilbert spaces) the class of strongly P-compact operators is quite extensive, and shall prove a fixed point theorem like that of Leray-Schauder for this class. Throughout this paper X will denote a real Banach space which has the following property: (re)l: There exists a sequence {F.} of finite dimensional subspaces of X, with F.CF,+I for each n, and U F, dense in X, and a corresponding sequence

On the theory of non-linear operators

On the theory of non-linear operators