On surjectivity for nonlinear maps in banach spaces

Sunto Sia E uno spazio di Banach e f: E→E un operatore continuo, non necessariamente lineare. Si danno delle condizioni su f affinchè l’equazione x-f(x)=y sia risolubile per ogni y∈E.

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Entrata in Redazione il 30 dicembre 1975.

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This paper was written while the authors where visiting at the Mathematics Department of the University of Bonn in the ? Sonderforschungsbereich 72 ? programm.     

On injections and surjections of continuous function spaces

It is shown that a linear isometry ofC(S) intoC(T) is generated by a linear extension fromS to a quotient space ofT. A dual theorem is proved for a quotient map ofC(S) ontoC(T). An application of the first theorem and a modification of a theorem by Banach and Mazur provide a negative answer to a problem posed by Pelczynski.     

Bounds on nonlinear operators in finite-dimensional banach spaces

Summary We consider Lipschitz-continuous nonlinear maps in finite-dimensional Banach and Hilbert spaces. Boundedness and monotonicity of the operator are characterized quantitatively in terms of certain functionals. These functionals are used to assess qualitative properties such as invertibility, and also enable a generalization of some well-known matrix results directly to nonlinear operators. Closely related to the numerical range of a matrix, the Gerschgorin domain is introduced for nonlinear operators. This point set in the complex plane is always convex and contains the spectrum of the operator's Jacobian matrices. Finally, we focus on nonlinear operators in Hilbert space and hint at some generalizations of the von Neumann spectral theory.     

On banach spaces whose duals areL 1 spaces

A structure theorem for Banach spaces whose duals areL ₁ spaces, is proved. The research of the second named author has been sponsored by the Air Force Office of Scientific Research under Grant AF EOAR 66-18 through the European Office of Aerospace Research (OAR) United States Air Force.     

On banach spaces with unconditional bases

It is shown that for every space with an unconditional basis there exists a uniformly bounded sequence of projectionsP _n;n=1, 2, ... whose ranges are uniformly isomorphic tol _p ⁿ ;n=1, 2, ... either forp=1, orp=2, or forp=∞.     

Embedding unconditional stable banach spaces into symmetric stable banach spaces

In this paper we prove the following result which solves a question raised by A. Pelczynski: “Every stable Banach space with an unconditional basis is isomorphic to a complemented subspace of some stable Banach space with a symmetric basis.” Moreover, we show that all the interpolation spacesl ^p ,l ^q _θ,X,1 1≦p, q<∞ andX stable, are stable.     

On vector-valued amarts and dimension of banach spaces

Probability Theory and Related Fields -     

Existence theorems for nonlinear differential equations having trichotomy in banach spaces

We give existence theorems for weak and strong solutions with trichotomy of the nonlinear differential equation

$$\dot x\left( t \right) = \mathcal{L}\left( t \right)x\left( t \right) + f\left( {t,x\left( t \right)} \right),t \in \mathbb{R}$$

(P)

where {?(t): t ∈ R} is a family of linear operators from a Banach space E into itself and f: R × E → E. By L(E) we denote the space of linear operators from E into itself. Furthermore, for a < b and d > 0, we let C([?d, 0],E) be the Banach space of continuous functions from [?d, 0] into E and f ^d : [a, b] × C([?d, 0],E) → E. Let \(\hat {\mathcal{L}}:[a,b] \to L(E)\) be a strongly measurable and Bochner integrable operator on [a, b] and for t ∈ [a, b] define τ _t x(s) = x(t + s) for each s ∈ [?d, 0]. We prove that, under certain conditions, the differential equation with delay

$$\dot x\left( t \right) = \hat {\mathcal{L}}\left( t \right)x\left( t \right) + {f^d}\left( {t,{\tau _t}x} \right),ift \in \left[ {a,b} \right],$$

(Q)

has at least one weak solution and, under suitable assumptions, the differential equation (Q) has a solution. Next, under a generalization of the compactness assumptions, we show that the problem (Q) has a solution too.

     

On the existence of shortest networks in banach spaces

For dual spaces, and also for L ₁, it is proved that every system of points in such a space admits a shortest network connecting the points. An example of a Banach space is presented in which, for every n ≥ 3, there is a system of n points which cannot be connected by a shortest network.     

On representations,inversions, and approximations of laplace transforms in banach spaces

The paper extend's some of the basic results of classical Laplace transform theory to the Laplace transform of bounded Bochner integrable functions and the Laplace-Stieltjes transform of Lipschitz continuous functions with values in a Banach space.     

Normal structure of banach spaces

GOSSEZ and LAMI DOZO have obtained a sufficient condition for a Banach space with a Schauder basis to have normal structure. This paper generalizes their theorem to obtain a sufficient condition for an arbitrary Banach space to have normal structure. An example is given to show an application of the theorem.Research supported by a Faculty Research Grant of the College of William and Mary.     

Interpolation of several banach spaces

Summary Peetre's K- and J-methods for interpolation are extended to the situation of more than two spaces. The theory developed is applied to interpolation of L_p-spaces with weights and to spaces of Besov and Sobolev type. Entrata in Redazione il 5 luglio 1972.     

EXISTENCE THEOREMS OF SOLUTIONSFOR NONLINEAR IMPULSIVE VOLTERRA INTEGRAL EQUATIONS IN BANACH SPACES

In this paper, we use the Monch fixed-point theorem to prove some existence theoremsof sotuti9ns for the nonlinear impulsive Volterra integral equations in Banach spaces that improveand extend the results in [2].