Second-order Differential Equations with Deviating Arguments

where f ∈ C(J ×R×R,R) and α∈ C(J , J) (e.g., αmay be defined by α(t)=√t, T ≥ 1 or α(t)= 0.7t, t ∈ J). Moreover, r and γ are fixed real numbers. Differential equations with deviated arguments arise in a variety of areas of biological, physical, and engineering applications, see, for example, [9, Chapter 2]. The monotone iterative method is useful to obtain approximate solutions of nonlinear differential equations, for details see, for example, [10], see also [1–8, 11, 12]. It has been applied successfully to obtain results of existence of quasisolutions for problems of type (1.1), see [4]. In paper [4], it was assumed that function f satisfies a one-sided Lipschitz condition with respect to the last two variables with corresponding functions instead of constants. Note that the special case when f is monotone nonincreasing (with respect to the last two variables) is not discussed in paper [4] and is of particular interest. Moreover, at the end of this paper we formulate conditions under which problem (1.1) has a unique solution. This paper extends some results of [4].