Necessary and Sufficient Conditions for Positive Solutions of Second-Order Nonlinear Dynamic Equations on Time Scales

This paper is concerned with the existence of nonoscillatory solutions for the nonlinear dynamic equation on time scales. By making use of the generalized Riccati transformation technique, we establish some necessary and sufficient criteria to guarantee the existence. The last examples show that our results can be applied on the differential equations, the difference equations, and the -difference equations. 1. Introduction In the recent decade there have been many literatures to study the oscillatory properties for second-order dynamic equations on time scales; see, for example, [1–11] and the references therein. In particular, the dynamic equation of the form has been attracting one’s interesting; see, for example, [3, 5, 8]. Motivated by the papers mentioned as above, in this paper we consider the existence of nonoscillatory solutions for nonlinear dynamic equation on a time scale , where . Referring to [12, 13], a time scale can be defined as an arbitrary nonempty subset of the set of real numbers, with the properties that every Cauchy sequence in converges to a point of with the possible exception of Cauchy sequences which converge to a finite infimum or finite supremum of . On any time scale , the forward and backward jump operators are defined, respectively, by where and . A point is said to be right-scattered if , right-dense if , left-scattered if , and left-dense if . A derived set from is defined as follows: when has a left-scattered maximum , otherwise . Definition 1.1. For a function and , we define the delta-derivative of to be the number (provided it exists) with the property that, for any , there is a neighborhood of (i.e., for some ) such that We say that is delta-differentiable (or in short: differentiable) on provided exists for all . For two differentiable functions and at with , it holds that Definition 1.2. A function is called an antiderivative of provided holds for all . By the antiderivative, the Cauchy integral of is defined as , and . Definition 1.3. Let be a function, where is called right-dense continuous (rd-continuous) if it is right continuous at right-dense points in and its left-sided limits exist (finite) at left-dense points in . To distinguish from the traditional interval in , we define the interval in by Let (or ) denote the set of all rd-continuous functions defined on , and (or ) denote the set of all differentiable functions whose derivative is rd-continuous. Since we are interested in the existence of nonoscillatory solutions of (1.2), we make the blanket assumption that and . As defined in [1], by a solution