An existence theorem for a nonlinear difference equation

Abstract

where p > 1 and {s,)‘p is a real sequence. We establish conditions under which (1.1) has a positive nondecreasing solution. Here a solution of (1.1) is a real sequence y = (_Y~); which satisfies (1.1). Since (1.1) is a recurrence relation, given real initial values y, and y, , it is clear that we can inductively obtain y, , y, , . . . . Since p is a real number greater than 1, the sequence (yk] may not be real (that is, not a solution). Note, however, that any constant multiple of a solution is again a solution. The existence question originated in [l], where it was shown that Hardy’s inequality for a series [2] can be viewed as a necessary condition for the existence of a positive nondecreasing solution of (1.1). Our existence theorem reduces to a nonoscillation criterion in the case p = 2, since then (1.1) changes to a second order linear difference equation (see, e.g. [3]). A brief outline of the paper is as follows. Section 2 introduces a Riccati-type transformation and then develops various necessary conditions for the existence of a positive nondecreasing solution. Section 3 discusses some of the functional analytic background which is needed in order to apply Schauder’s theorem. Section 4 contains the proof of our existence theorem, and Section 5 describes a comparison theorem which allows us to drop the assumption that the coefficients sk in (1.1) are nonnegative.