Solvability of an Integral Equation of Volterra-Wiener-Hopf Type

Abstract

The paper presents results concerning the solvability of a nonlinear integral equation of Volterra-Stieltjes type. We show that under some assumptions that equation has a continuous and bounded solution defined on the interval0,∞and having a finite limit at infinity. As a special case of the mentioned integral equation we obtain an integral equation of Volterra-Wiener-Hopf type. That fact enables us to formulate convenient and handy conditions ensuring the solvability of the equation in question in the class of functions defined and continuous on the interval0,∞and having finite limits at infinity.

A Unified Approach to Some Classes of Nonlinear Integral Equations

“…with the function ( , ) of the form (24). Then, we can formulate the following existence result concerning (55) [21] (cf. also [36]).…”

“…For example, we formulate below the assumption of such a type which is connected with Theorem 10 (see [21]). Then, we have the following lemma [21]. Lemma 11.…”

“…Then, satisfies assumption (vi) of Theorem 10. It can be shown that Lemma 11 enables us to formulate convenient requirements concerning, for example, the function = ( ) appearing in (6), which guarantee that the Volterra-Wiener-Hopf counterpart of (55) satisfies assumptions imposed in Theorem 10. We omit details which can be found in [21].…”

“…Such equations are a special case of integral equations with kernels depending on the difference of arguments and they also play very important role in applications (cf. [3,12,[21][22][23]).…”