Some properties of Urysohn‐Stieltjes integral operators

Abstract

We investigate some properties of Urysohn-Stieltjes integral operators. The assumptions will be formulated under which the operators in question transform the space of coatinuous functions into itself or in the space of functions of bounded variation, are continuous or are compact. The solvability of an integral equation of Urysohn-Stieltjes type will be also discusse

Nondecreasing solutions of a quadratic integral equation of Urysohn type

“…Especially, that technique turns out to be very useful tool in the existence theory for several types of integral equations [1][2][3][4].…”

On the Volterra-Stieltjes Integral Equation and Axiomatic Measures of Weak Noncompactness

“…In last years the study of the integral equations of Volterra-Stieltjes type in a Banach space has been developed extensively. However almost all of the work was done using the strong topology (see [3]- [9]), while the study of our problem involving the weak topology is lagging behind. The main tools were De Blasi measure of weak noncompactness and Kubiaczyk's fixed point theorem.…”

On the Weak Solutions of the Urysohn-Stieltjes Functional Integral Equations

“…where g : I × I → R and the symbol d s indicates the integration with respect to s. Equations of type (1) and some of their generalizations were considered in paper (see [3]), for the properties of the Urysohn-Stieltjes integral (see Banaś [1]).…”

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The analysis of Urysohn-Stieltjes integral operators has been studied in [1]. Here we study the existence of weakly solution of functional integral equations of Urysohn-Stieltjes type and Hammerstien-Stieltjes type in the reflexive Banach space E. Also, we prove the existence of the weak maximal and weak minimal solutions.

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