William T. Patula

A second-order nonlinear difference equation: Oscillation and asymptotic behavior

Discrete analogues are investigated for well-known results on oscillation, growth, and asymptotic behavior of solutions of y″ + q(t) yγ = 0, for q(t) ⩾ 0 and for q(t) ⩽ 0. The analogue of Atkinsons oscillation criterion is shown to be true for Δ2yn − 1 + qn ynγ = 0, but the analogue for Atkinsons nonoscillation criterion is shown to be false.

An existence theorem for a nonlinear difference equation

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.

Growth and Oscillation Properties of Second Order Linear Difference Equations

This paper establishes the existence of recessive and dominant solutions for nonoscillatory and certain types of oscillatory second order homogeneous linear difference equations. Growth properties concerning these solutions are established. This information is then used to obtain two limit point results. Several sufficient conditions for oscillation are also presented.

Bounded and zero convergent solutions of second-order difference equations

Abstract In this paper we study boundedness and monotonicity properties of a homogeneous second-order linear difference equation. Under certain conditions it is shown that every solution must eventually be monotonic. The question of when such monotonic solutions diverge to infinity or converge to zero and the corresponding impact this behavior has on recessive and dominant solutions is investigated. A necessary and sufficient condition for all solutions to be bounded is presented. Comparison theorems are also discussed.

Growth, Oscillation and Comparison Theorems for Second Order Linear Difference Equations

This paper studies homogeneous and nonhomogeneous second order linear difference equations. Comparison theorems based on the coefficients are proven for the homogeneous equation. Existence of so called r-type solutions for the forced equation is established. Oscillation and nonoscillation properties of the forced equation based on the forcing term and the associated homogeneous equation are discussed.

Oscillatory second order linear difference equations and Riccati equations

Oscillation criteria are established for the equation

Growth and oscillation properties of solutions of a fourth order linear difference equation

c_n x_{n + 1} + c_{n - 1} x_{n - 1} = b_n x_n

On a Max Type Recurrence Relation with Periodic Coefficients

,

Recessive solutions of block tridiagonal nonhomogeneous systems

c_n > 0

Nonoscillatory solutions of forced second-order linear equations

, involving asymptotic behavior of the quantity