Józef Banaś

On measures of weak noncompactness

SummaryIn this paper an axiomatic approach to the notion of a measure of weak noncompactness is presented. Several properties of the defined measures are given. Moreover, we provide a few concrete realizations of the accepeted axiomatic system in some Banach spaces.

Monotonic solutions of a quadratic integral equation of Volterra type

Abstract We study a nonlinear quadratic integral equation of Volterra type in the Banach space of real functions defined and continuous on a bounded and closed interval. With the help of a suitable measure of noncompactness, we show that the mentioned integral equation has monotonic solutions.

On existence and asymptotic stability of solutions of a nonlinear integral equation

We prove an existence theorem for a nonlinear integral equation being a Volterra counterpart of an integral equation arising in the traffic theory. The method used in the proof allows us to obtain additional characterization in terms of asymptotic stability of solutions of an equation in question.

Solvability of infinite systems of differential equations in Banach sequence spaces

The aim of this paper is to establish sufficient conditions for the solvability of infinite systems of ordinary differential equations in some Banach sequence spaces. The results presented in the paper create mainly the concrete realizations of sufficient conditions for the solvability of ordinary differential equations in Banach spaces formulated with help of the technique of measures of noncompactness. We concentrate on the results being rather convenient and handy in applications.

Sequence Spaces and Measures of Noncompactness with Applications to Differential and Integral Equations

Chapter 1. Introduction to FK spaces.- Chapter 2. Matrix Transformations.- Chapter 3. Some new sequence spaces of non-absolute type.- Chapter 4. Some non-classical sequence spaces.- Chapter 5. Measures of noncompactness.- Chapter 6. Application to compact matrix operators.- Chapter 7. Applications to infinite systems of differential equations.- Chapter 8. Applications to integral equations.

Measures of noncompactness in the study of solutions of nonlinear differential and integral equations

The aim of this paper is to make an overview of some existence results for nonlinear differential and integral equations. Those results were obtained by the author and his co-workers during last years with some help of the technique of measures of noncompactness and a fixed point theorem of Darbo type.

Solvability of Volterra-Stieltjes operator-integral equations and their applications

Abstract We investigate a class of operator-integral equations of Volterra-Stieltjes type and we study the solvability of those equations in the space of continuous functions. Equations in question create a generalization of numerous integral equations considered in nonlinear analysis. The main tool used in our considerations is the technique associated with measures of noncompactness. We show the applicability of our existence result in the study of a few integral equations of Volterra type.

On a Class of Measures of Noncompactness in Banach Algebras and Their Application to Nonlinear Integral Equations

We introduce a class of measures of noncompactness in Banach algebras satisfying certain condition and we prove a flxed point theorem for the product of two operators being contractions with respect to such measures of noncompactess. We also indicate measures of noncompactness in some Banach function algebras which satisfy the mentioned condition. The obtained results are applied to prove a few theorems on the existence of solutions of nonlinear integral equations in Banach algebras. Some characterizations of solutions of considered integral equations are also derived.

On a class of Urysohn—Stieljes quadratic integral equations and their applications

Abstract We investigate a class of nonlinear quadratic integral equation of Urysohn–Stieltjes type. Such a class contains a number of classical nonlinear quadratic integral equations such as the Chandrasekhar quadratic integral equation. We consider the solvability of the equations in question in the space of continuous functions under very general assumptions. Several particular cases of the assumed hypotheses are discussed and numerous applications are indicated.

Compactness Conditions in the Study of Functional, Differential, and Integral Equations

We discuss some existence results for various types of functional, differential, and integral equations which can be obtained with the help of argumentations based on compactness conditions. We restrict ourselves to some classical compactness conditions appearing in fixed point theorems due to Schauder, Krasnosel’skii-Burton, and Schaefer. We present also the technique associated with measures of noncompactness and we illustrate its applicability in proving the solvability of some functional integral equations. Apart from this, we discuss the application of the mentioned technique to the theory of ordinary differential equations in Banach spaces.