Leszek Olszowy

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.

Fixed point theorems in the Fréchet space C(R+) and functional integral equations on an unbounded interval

Abstract The purpose of this paper is to establish fixed point theorems for operators in the Frechet space of continuous functions on the real half-axis. In our considerations we apply the technique of measures of noncompactness in conjunction with the Tikchonov fixed point theorem. The obtained results are applied in the proof of the solvability of a nonlinear functional integral equation with the initial value. Moreover, we show that solutions of that equation are uniformly globally asymptotically attractive. The results presented in the paper allow to improve existence theorems for integral and functional equations obtained earlier in many research papers.

Solvability of infinite systems of singular integral equations in Fréchet space of continuous functions

The aim of this paper is to show how some measures of noncompactness in the Frechet space of continuous functions defined on an unbounded interval can be applied to an infinite system of singular integral equations. The results obtained generalize and improve several ones.

Existence of Mild Solutions for Semilinear Nonlocal Cauchy Problems in Separable Banach Spaces

In the paper we study the existence of mild solutions of a semilinear evolution equation with nonlocal initial conditions under the assumptions of the Hausdorff measure of noncompactness in separable Banach space.

On existence of solutions of a neutral differential equation with deviating argument

We establish theorems on the existence and asymptotic characterization of solutions of a differential equation of neutral type with deviated argument on neutral type. The mentioned differential equation admits both delayed and advanced arguments. In our considerations we use technique linking measures of noncompactness with the Tikhonov fixed point principle in suitable Frechet space. This approach admits us to improve and extend some results.

Continuous Dependence of the Solutions of Nonlinear Integral Quadratic Volterra Equation on the Parameter

We prove results on the existence and continuous dependence of solutions of a nonlinear quadratic integral Volterra equation on a parameter. This dependence is investigated in terms of Hausdorff distance. The considerations are placed in the Banach space and the Frechet space.

Existence of mild solutions for semilinear differential equations with nonlocal and impulsive conditions

This paper is concerned with the existence of mild solutions for impulsive semilinear differential equations with nonlocal conditions. Using the technique of measures of noncompactness in Banach and Fréchet spaces of piecewise continuous functions, existence results are obtained both on bounded and unbounded intervals, when the impulsive functions and the nonlocal item are not compact in the space of piecewise continuous functions but they are continuous and Lipschitzian with respect to some measure of noncompactness, and the linear part generates only a strongly continuous evolution system.

Moduli of near convexity of the Baernstein space

In this paper we give the exact formulas for the so-called moduli of near convexity of the Baernstein space. This space is frequently used in the geometric theory of Banach spaces and in other branches of nonlinar functional analysis.