M. Ruiz Galán

Biorthogonal systems for solving Volterra integral equation systems of the second kind

With the aid of biorthogonal systems in adequate Banach spaces, the problem of approximating the solution of a system of nonlinear Volterra integral equations of the second kind is turned into a numerical method that allows it to be solved numerically.

Linear Volterra integro-differential equation and Schauder bases

In this paper we present a numerical method to approximate the solution of the linear Volterra integro-differential equation. By making use of the expression of a function of the Banach space C([0,1]) in terms of a basis Schauder, we are able to define a sequence of functions which approximate (in the uniform sense) the solution of such equation. Likewise, we study the error committed in each approximation. Some advantages that this method possess are that it is not necessary to solve linear equation systems and moreover, the involved integrals are immediate.

ISOMORPHISMS, SCHAUDER BASES IN BANACH SPACES, AND NUMERICAL SOLUTION OF INTEGRAL AND DIFFERENTIAL EQUATIONS

ABSTRACT The authors determine the inverse of an isomorphism between two Banach spaces, the first of them with a Schauder basis, by means of certain best approximation sequences. As a consequence, they state an algorithm that allows one to solve numerically some linear integral and differential equations.

Numerical Treatment of Fixed Point Applied to the Nonlinear Fredholm Integral Equation

The authors present a method of numerical approximation of the fixed point of an operator, specifically the integral one associated with a nonlinear Fredholm integral equation, that uses strongly the properties of a classical Schauder basis in the Banach space .

Biorthogonal Systems Approximating the Solution of the Nonlinear Volterra Integro-Differential Equation

This paper deals with obtaining a numerical method in order to approximate the solution of the nonlinear Volterra integro-differential equation. We define, following a fixed-point approach, a sequence of functions which approximate the solution of this type of equation, due to some properties of certain biorthogonal systems for the Banach spaces and .

Analytical Techniques for a Numerical Solution of the Linear Volterra Integral Equation of the Second Kind

In this work we use analytical tools—Schauder bases and Geometric Series theorem—in order to develop a new method for the numerical resolution of the linear Volterra integral equation of the second kind.

About some numerical approaches for mixed integral equations

Abstract In this paper we tackle on mixed Volterra–Fredholm integral equations, as in linear as in non linear cases. To the aim to obtain numerical solutions of these models, the authors propose in the linear case the direct collocation method using a p-order quasi interpolating spline class and in the nonlinear case the fixed point method based on polynomial approximation built by Schauder tensor bases. The advantages of both methods are outlined and their convergence is studied. Numerical results confirm the theoretical statements.

A new minimax theorem and a perturbed James's theorem

The main result of this paper is a sufficient condition for the minimax relation to hold for the canonical bilinear form on X × Y , where X is a nonempty convex subset of a real locally convex space and Y is a nonempty convex subset of its dual. Using the known “converse minimax theorem”, this result leads easily to a nonlinear generalisation of Jamess (“sup”) theorem. We give a brief discussion of the connections with the “sup-limsup theorem” and, in the appendix to the paper, we give a simple, direct proof (using Goldstines theorem) of the converse minimax theorem referred to above, valid for the special case of a normed space.

Compactness, Optimality, and Risk

This is a survey about one of the most important achievements in optimization in Banach space theory, namely, James’ weak compactness theorem, its relatives, and its applications. We present here a good number of topics related to James’ weak compactness theorem and try to keep the technicalities needed as simple as possible: Simons’ inequality is our preferred tool. Besides the expected applications to measures of weak noncompactness, compactness with respect to boundaries, size of sets of norm-attaining functionals, etc., we also exhibit other very recent developments in the area. In particular we deal with functions and their level sets to study a new Simons’ inequality on unbounded sets that appear as the epigraph of some fixed function f. Applications to variational problems for f and to risk measures associated with its Fenchel conjugate f ∗ are studied.

Numerical approaches for systems of Volterra-Fredholm integral equations

In this work we introduce two numerical methods for solving systems of Volterra-Fredholm integral equations. In the nonlinear case we suggest a fixed point method, where the iterations are perturbed in a suitable way according to a Schauder basis in the Banach space of continuous functions C[a,b]^2. In the linear case we propose a collocation method based on a particular class of approximating functions. In both methods, convergence analysis and/or low computational cost are analysed, taking into account the properties of the basis under consideration. Numerical results confirm the theoretical study.