Derivation and Integration

Abstract

Below we develop a theory directed to the study of properties of solutions of ordinary differential equations y′ = f (t, y). It will be helpful in the investigation of equations with complicated discontinuities in their right hand sides. A natural approach to the definition of a solution of an equation is to consider a function as a solution, if at every point of its domain the function has a derivative satisfying the equation. This plan may be realized, for instance, when we study equations with continuous right hand sides. But there are many very simple cases when solutions in the sense stated are absent; however, there exist solutions in some near but different meaning. Moreover, these solutions correspond to the contents of the (physical, geometric, etc.) problem solved with the help of the equation under consideration. Our aim in this chapter is to introduce and to discuss a notion of the derivative which suits for large spectrum of such situations. A natural restriction is that if a problem may be solved with the help of classical methods then no new ‘solution’ appears on account of the play on definitions.