Oscillation Theory for Superlinear Differential Equations

This chapter presents oscillation and nonoscillation theory for solutions of second order nonlinear differential equations of superlinear type. Section 4.1 deals with the oscillation of superlinear equations with sign changing coefficients. Here, first we shall discuss some results which involve integrals and weighted integrals of the alternating coefficients, and then provide several criteria which use average behaviors of these integrals. More general averages such as ‘weighted averages’ and ‘iterated averages’ are also employed. In Section 4.2, we impose some additional conditions on the superlinear terms which allow us to proceed further and extend and improve some of the results established in Section 4.1. In fact, an asymptotic study has been made and interesting oscillation criteria have been proved. In Section 4.3, first we shall provide sufficient conditions which guarantee the existence of nonoscillatory solutions, and then present necessary and sufficient conditions for the oscillation of superlinear equations. Oscillation results via comparison of nonlinear equations of the same form as well as with linear ones of the same order are also established. In Section 4.4, we shall extend some of the results of the previous sections and establish several new oscillation criteria for more general superlinear equations. Necessary and sufficient conditions for such equations to be oscillatory are also given. Section 4.5 deals with the oscillation of forced-superlinear differential equations with alternating coefficients. Finally, Section 4.6 presents the oscillation and nonoscillation criteria for second order superlinear equations with nonlinear damping terms.