Beata Rzepka

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.

Nondecreasing solutions of a quadratic integral equation of Urysohn type

Applying the technique associated with measures of noncompactness, we prove the existence of nondecreasing solutions of a quadratic integral equation of Urysohn type in the space of real functions defined and continuous on a closed bounded interval.

Nondecreasing solutions of a quadratic singular Volterra integral equation

We study the existence of nondecreasing solutions of a quadratic singular Volterra integral equation in the space of continuous functions on bounded interval. The main tool utilized in our considerations is the technique associated with certain measure of noncompactness related to monotonicity. The results obtained in the paper may be applied to a wide class of singular Volterra integral equations.

On solutions of an infinite system of singular integral equations

Using the classical Schauder fixed point principle we prove an existence result concerning an infinite system of singular integral equations. The obtained result is applied to establish the solvability of an infinite system of differential equations of fractional order.

On local attractivity and asymptotic stability of solutions of a quadratic Volterra integral equation

The existence of solutions of a nonlinear quadratic Volterra integral equation is studied. In our considerations we apply the technique of measures of noncompactness in conjunction with the classical Schauder fixed point principle. Such an approach allows us to obtain a result on the existence of solutions of an equation in question which are uniformly locally attractive or asymptotically stable.

Functions related to convexity and smoothness of normed spaces

We introduce a few functions related to convexity and smoothness of normed spaces. Those functions turn out to be moduli of convexity or smoothness or play an intermediate role. We calculate the exact formulas for introduced functions in some classical Banach spaces. An application to geometry of normed spaces is also indicated.

Measures of Noncompactness in the Space of Continuous and Bounded Functions Defined on the Real Half-Axis

In this chapter, we present a review of results concerning measures of noncompactness in the space of real functions defined, continuous and bounded on the real half-axis and furnished with the supremum norm. We will also investigate measures of noncompactness in a more general space of functions defined and continuous on the real half-axis and tempered by a given function. Moreover, we show the applicability of those measures of noncompactness in the theory of nonlinear functional integral equations.

Compactness Conditions in the Theory of Nonlinear Differential and Integral Equations

and Applied Analysis Compactness Conditions in the Theory of Nonlinear Differential and Integral Equations Guest Editors: Jozef Banaś, Mohammad Mursaleen, Beata Rzepka, and Kishin Sadarangani