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where q(t) is nonnegative and continuous on [0, °°) and y is any real number satisfying 0 < y < 1. (These assumptions on q(t) and y will be implicitly assumed throughout.) There are necessary and sufficient conditions for the existence of nonoscillatory solutions y(t) of (1.1) satisfying either lim(_.„ y(t) = a ^ Oorlim(-,„ (y(t)/(t) = a ^ 0 (cf. Sec. 2). Our main purpose, then, is to discuss other possible types of nonoscillatory solutions, for example, solutions which grow like a fractional power of t. Very little is known about such solutions. We will give criteria for the existence of such solutions and also a criterion for a certain type of "dichotomy" which may occur. All solutions of (1.1) exist on [0, <») as follows from a theorem of Wintner (Hartman [5]). By definition, a solution of (1) is said to be oscillatory if it has arbitrarily large zeros; i.e., if T > 0, then there is a t > T such that y{t) = 0. If there is a T > 0 such that y(t) 5^ 0 for t > T, then y(t) is called nonoscillatory. Note that if y is the quotient of odd integers, Eq. (1.1) takes the form

Rate of growth of nonoscillatory solutions for the differential equation 𝑦̈+𝑞(𝑡)𝑦^{𝛾}𝑠𝑔𝑛𝑦=0, 0<𝛾<1