## On delay differential equations with nonlinear boundary conditions

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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)



≡ Fx(t), t ∈ J = [0,T], T > 0,  

0 = g x(0),x(T) ,

(1.1)

where (H1 ) f ∈ C(J × R × R, R), α ∈ C(J,J), α(t) ≤ t, t ∈ J, and g ∈ C(R × R, R). To obtain some existence results for diﬀerential problems, someone can apply the monotone iterative technique, for details see, for example, . 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 (1.1), 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 boundary conditions Copyright © 2005 Hindawi Publishing Corporation Boundary Value Problems 2005:2 (2005) 201–214 DOI: 10.1155/BVP.2005.201

202

Delay diﬀerential equations

of type (1.1) to obtain quite general existence results. It is the first paper where the monotone technique is applied for delay diﬀerential equations when a boundary condition has a nonlinear form. The case when t ≤ α(t) ≤ T, t ∈ J is considered in . In Section 2, delay diﬀerential inequalities are studied. This part is important when the monotone technique is used with problem (1.1). In the next section we study weakly coupled lower and upper solutions of problem (1.1) formulating corresponding results when problem (1.1) has coupled quasisolutions, extremal solutions, or a unique solution. In Section 4, we formulate corresponding existence results for problem (1.1) using the notion of lower and upper solutions of (1.1). In Section 5, some generalizations of the previous results are formulated when we have more delayed arguments. Examples show how to apply the obtained results. 2. Delay diﬀerential inequalities