About a generalization of Bellman–Bihari type inequalities for discontinuous functions and their applications

In the present paper we introduce the conditions of solvability for Chaplygin’s problem with discontinuous functions in two independent variables, satisfying integro-sum inequalities. The new type of nonlinear integral and Wendroff’s inequality for discontinuous functions are investigated. As applications, the conditions of boundedness solutions of partial differential equations of hyperbolic type with impulse influence on some hypersurfaces {Γj}⊂R+2 are obtained. Some historical aspects of the theory of integro-sum inequalities are presented.