ON DELAY DIFFERENTIAL EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS TADEUSZ JANKOWSKI Received 21 July 2004 The monotone iterative method is used to obtain sufficient conditions which guarantee that a delay differential equation with a nonlinear boundary condition has quasisolutions, extremal solutions, or a unique solution. Such results are obtained using techniques of weakly coupled lower and upper solutions or lower and upper solutions. Corresponding results are also obtained for such problems with more delayed arguments. Some new interesting results are also formulated for delay differential inequalities. 1. Introduction In this paper we discuss the boundary value problem x (t) = f t,x(t),x α(t) ≡. | ON DELAY DIFFERENTIAL EQUATIONS WITH NONLINEAR BOUNDARY CoNdITIONS TADEUSZ JANKOWSKI Received 21 July 2004 The monotone iterative method is used to obtain sufficient conditions which guarantee that a delay differential equation with a nonlinear boundary condition has quasisolutions extremal solutions or a unique solution. Such results are obtained using techniques of weakly coupled lower and upper solutions or lower and upper solutions. Corresponding results are also obtained for such problems with more delayed arguments. Some new interesting results are also formulated for delay differential inequalities. 1. Introduction In this paper we discuss the boundary value problem x t f t x t x a t Fx t t e J 0 T T 0 0 g x 0 x T where Hl f e C J X R X R R a e C J J a t t t e J and g e C R X R R . To obtain some existence results for differential problems someone can apply the monotone iterative technique for details see for example 8 . In recent years much attention has been paid to the study of ordinary differential equations with different conditions but only a few papers concern such problems with nonlinear boundary conditions see for example 1 2 3 4 . The monotone technique can also be successfully applied to ordinary delay differential problems which are special cases of see for example 5 7 9 10 11 . It is known that the monotone method works when a function appearing on the right-hand side of a differential problem satisfies a one-sided Lipschitz condition with a corresponding constant or constants . It is important to indicate that also the authors of the above-mentioned papers obtained their results under such an assumption. In this paper we consider a more general case when constants are replaced by functions. This remark is important when we have differential problems with deviated arguments since in such cases we can obtain less restrictive conditions from corresponding differential inequalities. In this paper we discuss delay problems with nonlinear .
