Introduction - Cornell University.pdfVIP

Georgian Mathematical Journal Volume 9 (2002), Number 2, 287–294 ORDINARY DIFFERENTIAL EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS TADEUSZ JANKOWSKI Abstract. The method of lower and upper solutions combined with the monotone iterative technique is used for ordinary differential equations with nonlinear boundary conditions. Some existence results are formulated for such problems. 2000 Mathematics Subject Classification: 34A45, 34B15, 34A40. Key words and phrases: Lower and upper solutions, monotone iterations, convergence, existence of solutions. 1. In this paper, we shall consider the following differential problem () = () = [0 ] 0 (1) (0) = (( )) where ( ) ( ) It is well known that the monotone iterative technique is a powerful method used to approximate solutions of several problems (see, for example [5]). The purpose of this paper is to show that it can be applied successfully to problems of type (1). Assuming one-sided Lipschitz condition on (with respect to the second variable) combined with the corresponding monotonicity conditions on , it is shown that linear iterations converge to a solution of problem (1). Some comparison results are also formulated. In many papers, the monotone iterative technique was applied to problem (1) when () = + If = 0 then (1) reduces to the initial value problem for differential equations and this case is considered, for example, in [6], [7], [9], [10]. If = 1 and = 0 then we have a periodic boundary problem considered, for example, in [8], [10], while if = 1 and = 0