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Abstract : The paper shows that several well-known results about continuous linear operators on Banach spaces can be generalized to the wider class of convex processes, as defined by Rockafellar. In particular, the open mapping theorem and the standard bound for the norm of the inverse of a perturbed linear operator can be extended to convex processes. In the last part of the paper, these theorems are exploited to prove results about the stability of solution sets of certain operator inequalities and equations in Banach spaces. These results yield quantitative bounds for the displacement of the solution sets under perturbations in the operators and/or in the right-hand sides. They generalize the standard results on stability of unique solutions of linear operator equations.
On the relationship between bounded solutions of linear and quasilinear differential equations in Banach spaces
Article On the relationship between bounded solutions of linear and quasilinear differential equations in Banach spaces was published on July 1, 1990 in the journal Demonstratio Mathematica (volume 23, issue 3).
Read more2Chebyshev Polynomial Iterations and Approximate Solutions of Linear Operator Equations
An iteration scheme for the approximate solution of a linear operator equation in a Banach space is discussed from the viewpoint of Chebyshev polynomials. The optimal rate of convergence is described by numerical characteristics which are similar to (but different from) the classical Chebyshev constants. The abstract results are illustrated by some examples which frequently arise in applications.
Read more12Modified Noor iterations for nonlinear equations in Banach spaces
Modified Noor iterations for nonlinear equations in Banach spaces
10Bishop–Phelps cones given by an equation in Banach spaces
In this work, we study Bishop–Phelps cones (briefly, BP cones) given by an equation in Banach spaces. Due to the special form, these cones enjoy interesting properties. We show that nontrivial BP cones have given by an equation form a ‘large family’ in some sense in any Banach space and they can be used to characterize the strict convexity of the space. We obtain conditions for a convex cone to be included in or to be itself a BP cone given by an equation. Furthermore, we give an affirmative answer to the open question of whether these cones may possess a nonempty interior in infinite-dimensional spaces and introduce Lorentz cones in some classical Banach spaces of sequences as illustrations. We also obtain explicit representations for all BP cones given by an equation in some classical Banach spaces. Finally, we present some short applications in optimal control and approximation.
Read more36Existence and regularity of solutions for some partial functional integrodifferential equations in Banach spaces
Existence and regularity of solutions for some partial functional integrodifferential equations in Banach spaces
6Product integral approximations of solutions to linear operator equations
In the paper we develop a class of iterative methods for approximating solutions of linear operator equations in a Banach space. The main techniques involve a product integral representation of solutions to linear Stieltjes integral equations, a variation of parameters formula, and the asymptotic convergence of solutions to the homogeneous integral equation.
Read more16Special solutions for linear functional differential equations and asymptotic behaviour
The present paper deals with special solutions for linear functional differential equations with small delays. The importance of special solutions relies on the fact that, under some conditions, they can be used to describe the asymptotic behaviour of all solutions. Sufficient conditions for the existence of special solutions for nonautonomous equations in $\mathbb R^n$ are given. For autonomous equations in $\mathbb R^n$, results relating special solutions and characteristic values are presented. We also consider linear autonomous functional differential equations in Banach spaces, and use special solutions to study the asymptotic behaviour of their solutions. Several applications are given.
Read moreStochastic differential equations in a Banach space driven by the cylindrical Wiener process
Stochastic differential equations in a Banach space driven by the cylindrical Wiener process
Hypergeometric series solutions of linear operator equations
Hypergeometric series solutions of linear operator equations
11Perturbed projection methods for split equations of the first kind
The paper investigates the approximate solution of linear operator equations of type (J-K)u=v, with J assumed invertible, obtained by applying a linear projection on both sides of the equation, together with a linear perturbation operator on the left side and a perturbation element on the right side.
Read more20Regularization and convergence for ill-posed backward evolution equations in Banach spaces
Regularization and convergence for ill-posed backward evolution equations in Banach spaces
4A Projective Newton Method for Semilinear Operator Equations in Banach Spaces
A projective Newton method is presented for the numerical solution of semilinear operator equations in Banach spaces and its convergence properties are studied. Applications to boundary value problems for nonlinear ordinary and partial differential equations are discussed
Read more41Decay mild solutions for two-term time fractional differential equations in Banach spaces
In this paper, we deal with the nonlocal Cauchy problem for a class of two-term time fractional differential equations in Banach spaces. By constructing a suitable measure of noncompactness on the space of solutions, we prove the existence of a compact set containing decay mild solutions to the mentioned problem.
Read more130General local convergence theory for a class of iterative processes and its applications to Newton’s method
General local convergence theory for a class of iterative processes and its applications to Newton’s method
7Das Tschebyscheff-Verfahren in Banach-Räumen
Two theorems on an iteration method of order three -- usually referred to as Chebyshev's method -- for the approximate solution of nonlinear operator equations in Banach spaces are given. Very simple and weak assumptions are obtained as well as error estimates which are much better compared with known results. Some worked examples of operator equations of different types are given.
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