Fixed point theorems for a class of nonlinear operators in Banach spaces and applications

In this paper, we study a class of nonlinear operator equations with more extensive conditions in ordered Banach spaces. By using the cone theory and Banach contraction mapping principle, the existence and uniqueness of solutions for such equations are investigated without demanding the existence of upper and lower solutions and compactness and continuity conditions. The results in this paper are applied to a class of abstract semilinear evolution equations with noncompact semigroup in Banach spaces and the initial value problems for nonlinear second-order integro-differential equations of mixed type in Banach spaces. The results obtained here improve and generalize many known results.     

Existence of Global Solutions for a Class of Abstract Second-Order Nonlocal Cauchy Problem with Impulsive Conditions in Banach Spaces

In this paper, we are concerned with a class of abstract second-order nonlocal Cauchy problem with impulsive conditions in Banach spaces. First, we study the existence of mild solutions for a class of second-order nonlocal Cauchy problem with impulsive conditions in Banach spaces on an interval [0,a]. Later, we study a couple of cases where we can establish the existence of global solutions for a class of abstract second-order nonlocal Cauchy problem with impulsive conditions in Banach spaces. We apply our theory to study the existence of solutions for impulsive partial differential equations.     

Feasibility and solvability of vector variational inequalities with moving cones in Banach spaces

In this paper we study the feasibility and solvability of vector variational inequalities with moving cones in Banach spaces. We show that the strict feasibility implies solvability of vector variational inequalities with moving cones under suitable conditions. Further we show that under suitable conditions, the homogeneous vector variational inequality with a moving cone is solvable whenever it is feasible. As consequences, we obtain the solvability of vector variational inequalities with feasibility assumptions in Banach spaces.     

A new existence theory for periodic solutions to evolution equations

ABSTRACT

This paper deals with a new existence theory for periodic solutions to a broad class of evolution equations. We first establish new fixed point theorems for affine maps in locally convex spaces and ordered Banach spaces. Our new fixed point results extend, encompass and complement a number of well-known theorems in the literature, including the famous Chow and Hale fixed point theorem. With these obtained fixed point results, we investigate the existence of periodic solutions for some class of nonhomogeneous linear systems in Banach spaces with lack of compactness. Some illustrative examples are also given.     

Properties of the Set of Extreme Solutions for a Class of Nonlinear Second-Order Evolution Inclusions

We establish the existence of extreme solutions for a class of nonlinear second-order evolution inclusions with a nonconvex right-hand side defined on an evolution triple of Banach spaces. Then we show that extreme solutions which belong to the solution set of the original system are in fact dense and codense in the solution set of a system with a convexified right-hand side. The necessary and sufficient conditions for closedness of the solution set for the original system in an appropriate spaces of functions are given as well. Finally, an example of a nonlinear hyperbolic distributed parameter system is worked out in detail.     

Non-zero solutions for a class of generalized variational inequalities in reflexive Banach spaces

In this paper, we study the existence of nonzero solutions for a class of generalized variational inequalities involving set-contractive mappings by using the fixed point index approach in reflexive Banach spaces. Under some suitable assumptions, we show some new existence theorems of nonzero solutions for this class of generalized variational inequalities in reflexive Banach spaces.     

The Bishop–Phelps–Bollobás and approximate hyperplane series properties

We study the Bishop–Phelps–Bollobás property for operators between Banach spaces. Sufficient conditions are given for generalized direct sums of Banach spaces with respect to a uniformly monotone Banach sequence lattice to have the approximate hyperplane series property. This result implies that Bishop–Phelps–Bollobás theorem holds for operators from ${?}_1$ into such direct sums of Banach spaces. We also show that the direct sum of two spaces with the approximate hyperplane series property has such property whenever the norm of the direct sum is absolute.     

Strongly damped wave problems: Bootstrapping and regularity of solutions

The aim of the article is to present a unified approach to the existence, uniqueness and regularity of solutions to problems belonging to a class of second order in time semilinear partial differential equations in Banach spaces. Our results are applied next to a number of examples appearing in literature, which fall into the class of strongly damped semilinear wave equations. The present work essentially extends the results on the existence and regularity of solutions to such problems. Previously, these problems have been considered mostly within the Hilbert space setting and with the main part operators being selfadjoint. In this article we present a more general approach, involving sectorial operators in reflexive Banach spaces.     

Banach spaces with projectional skeletons

A projectional skeleton in a Banach space is a σ-directed family of projections onto separable subspaces, covering the entire space. The class of Banach spaces with projectional skeletons is strictly larger than the class of Plichko spaces (i.e. Banach spaces with a countably norming Markushevich basis). We show that every space with a projectional skeleton has a projectional resolution of the identity and has a norming space with similar properties to Σ-spaces. We characterize the existence of a projectional skeleton in terms of elementary substructures, providing simple proofs of known results concerning weakly compactly generated spaces and Plichko spaces. We prove a preservation result for Plichko Banach spaces, involving transfinite sequences of projections. As a corollary, we show that a Banach space is Plichko if and only if it has a commutative projectional skeleton.     

Vector lattices of weakly compact operators on Banach lattices

A result of Aliprantis and Burkinshaw shows that weakly compact operators from an AL-space into a KB-space have a weakly compact modulus. Groenewegen characterised the largest class of range spaces for which this remains true whenever the domain is an AL-space and Schmidt proved a dual result. Both of these authors used vector-valued integration in their proofs. We give elementary proofs of both results and also characterise the largest class of domains for which the conclusion remains true whenever the range space is a KB-space. We conclude by studying the order structure of spaces of weakly compact operators between Banach lattices to prove results analogous to earlier results of one of the authors for spaces of compact operators.

     

Well-bounded operators on nonreflexive Banach spaces

Every well-bounded operator on a reflexive Banach space is of type (B), and hence has a nice integral representation with respect to a spectral family of projections. A longstanding open question in the theory of well-bounded operators is whether there are any nonreflexive Banach spaces with this property. In this paper we extend the known results to show that on a very large class of nonreflexive spaces, one can always find a well-bounded operator which is not of type (B). We also prove that on any Banach space, compact well-bounded operators have a simple representation as a combination of disjoint projections.

     

Linear evolution equations in scales of Banach spaces

This work is devoted to the study of a class of linear time-inhomogeneous evolution equations in a scale of Banach spaces. Existence, uniqueness and stability for classical solutions is provided. We study also the associated dual Cauchy problem for which we prove uniqueness in the dual scale of Banach spaces. The results are applied to an infinite system of ordinary differential equations but also to the Fokker-Planck equation associated with the spatial logistic model in the continuum.     

Preconditined semilinear stochastic singular and hypersingular integral equations

In this paper, three classes of preconditioners are proposed for solving some stochastic integral equations with the weakly singular kernel and the hypersingular kernel. The first and the second class of preconditioners are based on circulant operators, but the third class of preconditioners is based on iterative substructuring. It is proved that substructuring preconditioners can be better than other preconditioners. Also, the spaces of solutions are discussed such that the solutions of these equations are smooth, therefore, we give special Banach spaces for these integral equations. Finally, numerical results which support our theories are presented     

Genericity and amalgamation of classes of Banach spaces

We study universality problems in Banach space theory. We show that if A is an analytic class, in the Effros-Borel structure of subspaces of C([0,1]), of non-universal separable Banach spaces, then there exists a non-universal separable Banach space Y, with a Schauder basis, that contains isomorphs of each member of A with the bounded approximation property. The proof is based on the amalgamation technique of a class C of separable Banach spaces, introduced in the paper. We show, among others, that there exists a separable Banach space R not containing L¹(0,1) such that the indices β and rND are unbounded on the set of Baire-1 elements of the ball of the double dual R∗∗ of R. This answers two questions of H.P. Rosenthal.We also introduce the concept of a strongly bounded class of separable Banach spaces. A class C of separable Banach spaces is strongly bounded if for every analytic subset A of C there exists Y∈C that contains all members of A up to isomorphism. We show that several natural classes of separable Banach spaces are strongly bounded, among them the class of non-universal spaces with a Schauder basis, the class of reflexive spaces with a Schauder basis, the class of spaces with a shrinking Schauder basis and the class of spaces with Schauder basis not containing a minimal Banach space X.     

Holomorphic automorphisms of complex Banach spaces

We consider holomorphic automorphisms of infinite dimensional complex Banach spaces. First we look at automorphisms with an attracting fixed point to construct Fatou–Bieberbach domains in Banach spaces. Second, we look tame sets in Banach spaces. Recall that a discrete set in X is tame if it can be mapped onto an arithmetic progression via an automorphism of X. We show that bounded discrete sets of Banach spaces allowing a Schauder basis are tame. In contrast, \(l_\infty \) has several bounded discrete sets which are not tame.     

The method of lower and upper solutions for damped elastic systems in Banach spaces

In this paper, we are concerned with the initial value problem of a class of damped elastic systems in an order Banach spaces $E$. By employing the method of lower and upper solutions, we discuss the existence of extremal mild solutions between lower and upper mild solutions for such problem with the associated semigroup is equicontinuous. In addition, two examples are given to illustrate our results.     

On the “bang-bang” principle for nonlinear evolution inclusions

We establish the existence of extreme solutions for a class of nonlinear evolution inclusions with non-convex right-hand side defined on an evolution triple of Banach spaces. Then we show that extreme solutions which belong to the solution set of the original system are in fact dense in the solutions of the system with convexified right-hand side. Subsequently we use this density result to derive nonlinear and infinite-dimensional version of the “bang-bang” principle for control systems. An example of a nonlinear parabolic distributed parameter system is also worked out in detail. Received November 21, 1997     

SHEAVES OF BANACH SPEACES

Abstract

This paper presents a number of results concerning sheaves on a topological space, with values in the category BAN of Banach spaces, over K = R or Ø, and linear contractions. After showing that these sheaves are reflective in the corresponding category of presheaves (Proposition 1) and that the resulting reflection is stalk preserving (Proposition 2), we concentrate on the approximation sheaves, these being BAN-sheaves satisfying a strong patching condition originally due to Auspitz [1]. The interest in these particular sheaves lies in the fact that they are precisely the BAN-sheaves arising as sheaves of continuous sections of the appropriate kind of Banach fibre spaces [1] and thus central to the representation of Banach spaces by continuous sections. Here, we show that the approximation sheaves on any space are characterized as the BAN-presheaves injective relative to certain maps (Proposition 3) and that, for paracompact spaces X, they are exactly those BAN-sheaves S such that each SU, U open in X, admits a suitable C*U-module structure (Proposition 4). Further, we consider the adjointness between the approximation sheaves on a space X and the Banach modules over C*X (Proposition 5) and investigate its special properties for X being Tychonoff (Proposition 6) and Boolean (Proposition 7). We conclude with some observations regarding the failure of the analogues of Swan's Theorem for vector bundles and the Hahn-Banach Theorem in the present context, and some positive facts concerning injectivity for approximation sheaves on Tychonoff spaces.     

Oscillation of differential equations in Banach spaces

Summary In this paper some properties of solutions of the differential equation Y″(t)++P(t) Y(t)=0 in Banach spaces are investigated. In particular, conditions are given for some solutions of such equations to possess an infinite number of zeros as t → ∞ while another condition ensures some solutions possess only a finite number of zeros, Some examples and a theorem show the concept of an oscillatory solution of a differential equation in a Banach space involves pathologies not found in the case of finite dimensional spaces. Upon specialization of the Banach spaces involved the results reduce to known theorems. Entrata in Redazione il 13 maggio 1969.     

Sesquilinear quantum stochastic analysis in Banach space

A theory of quantum stochastic processes in Banach space is initiated. The processes considered here consist of Banach space valued sesquilinear maps. We establish an existence and uniqueness theorem for quantum stochastic differential equations in Banach modules, show that solutions in unital Banach algebras yield stochastic cocycles, give sufficient conditions for a stochastic cocycle to satisfy such an equation, and prove a stochastic Lie–Trotter product formula. The theory is used to extend, unify and refine standard quantum stochastic analysis through different choices of Banach space, of which there are three paradigm classes: spaces of bounded Hilbert space operators, operator mapping spaces and duals of operator space coalgebras. Our results provide the basis for a general theory of quantum stochastic processes in operator spaces, of which Lévy processes on compact quantum groups is a special case.