Oscillations of a Class of Forced Second-Order Differential Equations with Possible Discontinuous Coefficients

We study the oscillation of all solutions of a general class of forced second-order differential equations, where their second derivative is not necessarily a continuous function and the coefficients of the main equation may be discontinuous. Our main results are not included in the previously published known oscillation criteria of interval type. Many examples and consequences are presented illustrating the main results. 1. Introduction Let and let denote the set of all real functions absolutely continuous on every bounded interval . We study the oscillatory behaviour of all solutions of the following class of forced second-order differential equations: where the functions , , , and satisfy some general conditions given in Section 2. A continuous function is said to be oscillatory if there is a sequence , such that for all and as . A differential equation is oscillatory if all its solutions are oscillatory. The forcing term is a sign-changing function (possibly discontinuous). This can be formulated by the following hypothesis: for every there exist two intervals and , , such that The coefficient may be a discontinuous function on and the case occurs in our main results and examples too. Two important classes of functions are included in the differential operator as The first one is the classic second-order differential operator which is linear in and the second one is the so-called one-dimensional mean curvature differential operator; see Examples 1 and 2. Depending on , we propose the following four simple models for (1): (i) is strictly positive and continuous on as (ii) is nonnegative and continuous on as (iii) is nonnegative and discontinuous on as (iv) is sign changing and discontinuous on as where and is an arbitrary function such that for all , for instance, or . According to Corollaries 7 and 10, we will show that (4)–(7) are oscillatory provided the function satisfies for all ; see Examples 8–13. It is interesting that in particular for and , (4) allows an explicit oscillatory solution as shown in Figures 1 and 2. Figure 1: Function is a solution of ( 4). Figure 2: and hence is not a continuous function. Moreover, as a consequence of Corollary 7, one can show that all solutions of (4) are oscillatory; for details see Example 8. The main goal of this paper is to give some sufficient conditions on functions , and the coefficients , , and such that (1) is oscillatory; see Theorems 3 and 4. It will also cover the model equations (4)–(7) as well as some other examples presented in Section 2. To the best of our knowledge, it seems that there are