Sufficient conditions are given for the existence of multiple positive solutions of a boundary value problem of the form x (t)+q(t)f (x(t)) = 0, t ∈ [0, 1], x(0) = 0 and x(1) = R β α x(s)dg(s), where 0 < α < β < 1. A weaker boundary value problem is used to get information on the corresponding integral operator. Then the results follow by applying the Krasnosel'skiȋ fixed point theorem on a suitable cone. 2000 Mathematics Subject Classification. 34B18.

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By using the Schauder fixed point theorem, we establish a result for the existence of solutions of a boundary value problem on the half-line to second order nonlinear delay differential equations. We also present the application of our result to the special case of second order nonlinear ordinary differential equations as well as to a specific class of second order nonlinear delay differential equations. Moreover, we give a general example which demonstrates the applicability of our result. Introduction and statement of the main result.The asymptotic theory of delay differential equations, and especially of ordinary differential equations, is an area in which there is great activity among a large number of investigators. In this theory, an interesting problem is the study of the existence of solutions with prescribed asymptotic behavior for delay (and, in particular, ordinary) differential equations. Among numerous articles dealing with this problem, we choose to refer to the papers by Kusano and Trench [5], [6], Liu [7], Mustafa and Rogovchenko [8], Philos [9], Philos, Sficas and Staikos [10], Philos and Staikos [11], Yan [14], Yan and Liu [15], Yin [16], and Zhao [17]. It is of great interest to investigate, in particular, the existence of solutions with prescribed asymptotic behavior, which are global in the sense that they are solutions on the whole given interval. The results of the papers [5]-[8] and [14]-[17] are concerned with this particular problem. In the case of second order nonlinear ordinary or delay differential equations, the existence of global solutions with prescribed asymptotic behavior is usually formulated as the existence of solutions of boundary value problems on the half-line (see, for example, [7] and [14]-[17]).This paper is concerned with the existence of solutions of a boundary value problem on the half-line to second order nonlinear delay differential equations. By specializing our main result to the case of second order nonlinear ordinary differential equations, we are led to a result that is closely related to some recent results given in [7] and [14]- [17].