Continuation and limit properties for solutions of strongly nonlinear second order differential equations

Abstract

This paper deals with continuation, boundedness and asymptotic behavior of solutions of second order ordinary differential equations of variational type. The equations considered are nonlinear both in the solution variable and in its derivative. For equations arising from the degenerate Laplace operator, precise asymptotic behavior is established, along with various monotonicity and oscillation properties. Similar results are obtained also for other classes of nonlinear second order differential operators. The results of the paper also include generalizations of theorems of Fowler, Levin and Nohel, and Artstein and Infante. The main tool of the paper is a variational identity which supplies a useful new family of Liapunov functions for nonlinear equations.