Manuel Ruiz Galán

Characterization of the Solvability of Generalized Constrained Variational Equations

In a general context, that of the locally convex spaces, we characterize the existence of a solution for certain variational equations with constraints. For the normed case and in the presence of some kind of compactness of the closed unit ball, more specifically, when we deal with reflexive spaces or, in a more general way, with dual spaces, we deduce results implying the existence of a unique weak solution for a wide class of linear elliptic boundary value problems that do not admit a classical treatment. Finally, we apply our statements to the study of linear impulsive differential equations, extending previously stated results.

A Version of James' Theorem for Numerical Radius

We prove that if all the rank-one bounded operators on a Banach space X attain their numerical radii, then X must be reflexive, but the converse does not hold. In fact, every reflexive space with basis can be renormed in such a way that there is a rank-one operator not attaining the numerical radius. The classical James’ Theorem [12] characterizes reflexive spaces as those Banach spaces such that every functional attains its norm. Recently, some attention has been paid to the study of denseness of numerical radius attaining operators (see, for instance, [4, 9, 3, 14, 2, 10]), a problem analogous to the well-known parallel question for the norm. The general idea gained from looking at the results is that, even if some more tricky work is required, almost any result that is true for norm does also hold for numerical radius. Since James’ Theorem can be looked at as, equivalently, a Banach space is reflexive if and only if every rank-one operator on it attains its norm, we raised the parallel version for the numerical radius. In order to pose the problem in a more precise way, let us introduce some notation. Given a Banach space X over the scalar field K (R or C), we shall denote by BX and SX , respectively, the closed unit ball and the unit sphere of X; X ∗ will be the dual space of X, and L(X) will be the Banach space of all (bounded and linear) operators from X into itself, both spaces endowed with their usual norms. Let us recall that the numerical radius of an operator T is the real number v(T ) given by v(T ) = sup{|x∗Tx| : (x, x∗) ∈ Π(X)}, where Π(X) := {(x, x∗) ∈ SX × SX∗ : x∗(x) = 1}, and such an operator T attains its numerical radius if there exists an element (x0, x ∗ 0) ∈ Π(X) such that |x∗0Tx0| = v(T ). Instead of the set Π(X), it will be useful for us to consider the set Π̂(X) given by Π̂(X) := {(x, x∗) ∈ SX × SX∗ : |x∗(x)| = 1}. It is clear that v(T ) = sup{Re x∗Tx : (x, x∗) ∈ Π̂(X)}, and T attains its numerical radius if and only if there is some element (x0, x ∗ 0) ∈ Π̂(X) with Re x∗0Tx0 = v(T ). For a complete survey about the theory of numerical range of an element in a Banach algebra, we recommend the monographs [7] and [8]. This topic has been Received 26 June 1997. 1991 Mathematics Subject Classification 47A12, 46B10. Research partially supported by DGICYT project no. PB96-1906. Bull. London Math. Soc. 31 (1999) 67–74 68 maría d. acosta and manuel ruiz galán found to be a useful tool for dealing with several questions in Banach algebras and in the geometry of Banach spaces [13]. First, we shall prove that a Banach space X on which every rank-one operator attains its numerical radius must be reflexive. Then we shall exhibit counterexamples to show that the converse does not hold. For operators T far from being compact, of course, there is no hope of finding any relationship between reflexivity and the assertion that T attains its numerical radius. For instance, if X has 1-unconditional basis, then it is easy to construct a diagonal operator not attaining its numerical radius even if the space is reflexive. To prove the announced result, we shall make use of the following maximinimax principle due to S. Simons [15, Theorem 5]. Proposition 1. Let X0 and Y0 be non-void sets, and let f : X0 × Y0 → R be a bounded function such that for every sequence {zn} of elements in co f(·, Y0) (where co denotes the convex hull ) and any sequence {tn} of positive real numbers with ∑∞ n=1 tn = 1, there exists a bounded real function z on X0 satisfying lim inf n zn 6 z 6 lim sup n zn, and the function ∑∞ n=1 tn(zn − z) attains its supremum on X0. Then inf ∅6=F⊂Y0 finite sup x∈X0 inf f(x, F) 6 sup ∅6=G⊂X0 finite inf y∈Y0 sup f(G, y). The above inequality has been used successfully to obtain new proofs of James’ Theorem (see [15]). In order to work with the above proposition, we first prove the following technical result. Lemma 1. Let X be a Banach space, and let F be a finite subset of L(X) such that ‖T‖ 6 1 for all T ∈ F . Then sup (x,x∗)∈Π̂(X) inf T∈F Re x ∗Tx = sup (x∗ ,x∗∗)∈Π̂(X∗) inf T∈F Re x ∗∗T ∗x∗. Proof. The left-hand side of the equation is clearly less than or equal to the right-hand side. In order to check the reverse inequality, let us fix ε > 0, (x∗0, x∗∗ 0 ) ∈ Π̂(X∗), and write λ := x∗∗ 0 (x∗0)−1, a scalar with |λ| = 1. Since F is finite and BX is w∗-dense in BX∗∗ , we can find z ∈ BX such that |x∗0(x∗∗ 0 − z)| < ε 2 36 and |(x∗∗ 0 − z)T ∗x∗0| < ε 3 , for all T ∈ F. (1)

Variational equations with constraints

Abstract In this work we deal with a constrained variational equation associated with the usual weak formulation of an elliptic boundary value problem in the context of Banach spaces, which generalizes the classical results of existence and uniqueness. Furthermore, we give a precise estimation of the norm of the solution.

Inverse Problems: Theory and Application to Science and Engineering 2015

1 Department of Mathematics and Statistics, University of Guelph, Guelph, ON, Canada N1G 2WA 2Department of Economics, Management, and Quantitative Methods, University of Milan, 20122 Milan, Italy 3 Department of Applied Mathematics and Sciences, Khalifa University, 127788 Abu Dhabi, UAE 4Department of Mathematics and Statistics, Acadia University, Wolfville, NS, Canada B4P 2R6 5Department of Applied Mathematics, University of Granada, 18071 Granada, Spain

Fractal-Based Methods and Inverse Problems for Differential Equations: Current State of the Art

We illustrate, in this short survey, the current state of the art of fractal-based techniques and their application to the solution of inverse problems for ordinary and partial differential equations. We review several methods based on the Collage Theorem and its extensions. We also discuss two innovative applications: the first one is related to a vibrating string model while the second one considers a collage-based approach for solving inverse problems for partial differential equations on a perforated domain.

Some Recent Developments in Applied Functional Analysis

From its early stages, the intensive development of functional analysis and the remarkable advances of its methods cannot be explained without its link with other areas of mathematics and, above all, its role as an essential framework for numerical analysis and computer simulation, PDEs, modeling realworld phenomena, variational inequalities, or optimization, just to name a few. In this special issue we highlight some aspects of functional analysis which are used in connection with other branches of mathematics or science, either as a direct application or as a theoretical result which is essential for such an application. Although it is not possible to collect here the huge production of the research activity on this vast field of modern mathematics, the selected works gather together a range of topics which reflect some of the current research on applied functional analysis: bases in Banach spaces, wavelet transforms, fixed point theory, and applications to ODEs, electronic circuit simulation, or numerical solution of PDEs, integral equations, or problems on option pricing in mathematical finance. In this way, we have achieved one of our purposes, which is the exchange of ideas among researchers working both in abstract and applied functional analysis.

Ranges of operators and convex variational inequalities

The main topic in this work is the analysis of the range of a continuous and linear operator between Banach spaces, approached here through non-linear techniques, more precisely, by means of the classical Fan minimax theorem. We characterise the elements in the range of such an operator as those satisfying a certain variational inequality and provide a numerical scheme of Galerkin type to determine approximately the preimage of an element that lies in that range. The passage from the theoretical setting to its numerical realisation is done by means of the use of bases in adequate spaces. In addition we deal with the extension of the previous results to the case of systems of variational inequalities.