Nonoscillatory Solutions of Second-Order Differential Equations without Monotonicity Assumptions

Abstract

The continuability, boundedness, monotonicity, and asymptotic properties of nonoscillatory solutions for a class of second-order nonlinear differential equations ${[p(t)h(x(t))f(x^{\prime }(t))]}^{\prime }=q(t)g(x(t))$ are discussed without monotonicity assumption for function g. It is proved that all solutions can be extended to infinity, are eventually monotonic, and can be classified into disjoint classes that are fully characterized in terms of several integral conditions. Moreover, necessary and sufficient conditions for the existence of solutions in each class and for the boundedness of all solutions are established.