Applications of Fixed Point Theorems for Multifunction to Integral Inclusions

Abstract

Dynamical systems described by differential equations with continuous right-hand sides were the areas of vigorous steady in the later half of the twentieth century in applied mathematics, in particular, in the study of viscous fluid motion in a porous medium, propagation of light in an optically nonhomogeneous medium, determining the shape of a solid of revolution moving in a flow of gas with least resistance, etc. Euler’s equation plays a key role in dealing with the existence of the solution of such problems. On the other hand, Filippov, Differentsial’nye Uravneniya, 15:1814–1832, 1979, [232] has developed a solution concept for differential equations with a discontinuous right-hand side. In practice, such dynamical systems do arise and require analysis. Examples of such systems are mechanical systems with Coulomb friction modelled as a force proportional to the sign of a velocity, systems whose control laws have discontinuities.

The most vitally characteristic fact about mathematics, in my opinion, is its quite peculiar relationship to the natural sciences, or, more generally, to any science which interprets experience on a higher more than on a purely descriptive level....

John von Neumann (1947)

Science will not light the lamp in a person whose soul has no fuel.

Meichel de Montaigne

When I was a boy of 14 my father was so ignorant I could hardly stand to have the old man around. But when I got to be 21, I was astonished at how much the old man had learned in 7 years.

Mark Twain