Quasilinearization for initial value problems involving differential equations with “maxima”

An approximate method for solving initial value problems for first order scalar nonlinear differential equations with “maxima” is given. This method uses lower and upper solutions of the given problem, and in a sense is a variant of the method of quasilinearization. The main features of proposed method are that it is constructive in nature, and the obtained successive approximations of the unknown solution converge quadratically. Each successive approximation is a solution of an appropriately chosen initial value problem. In fact, a new scheme for constructing the initial value problems is suggested. An example which illustrates the advantages of the suggested algorithm is also included.