Theorems about existence and uniqueness of solutions to differential equations

In this thesis, Cauchy problem is investigated for ordinary differential equations system with discontinuous right-hand sides. The theorems about the existence and uniqueness of the solution of this problem in the generalized sense are expressed and proved. Specifically, the existence and uniqueness of the solution that satisfies the Lipschitz condition is examined. This thesis has four parts which the first two are related to the preliminary process. The first part consists of the basic concepts, the Cauchy problem in the classical sense and the generalized solution concepts and auxiliary theorems. The second part consists of theorems about the existence and uniqueness of the classical solution of the Cauchy problem. In the proofs in this section, Banach fixed point theorem, Arzelá-Ascoli theorem is used and the Euler fracture line method and Picard approach method are used. In the third part, the existence and uniqueness of the solution of the Cauchy problem for ordinary differential equations with discontinuous right-hand sides are studied. In the last part of the study, the existence and uniqueness of the solution of the problem of a higher order ordinary differential equation is examined. Also in this section, a generalized state of the solution of the Cauchy problem is investigated.