Oscillation Theory of q-Difference Equation

In this research article, the authors present the oscillation theory of the q-difference equation k(t)y(qt) + k ( t q ) y ( t q ) = r(t)y(t), where r(t) = k(t) + k ( t q ) − q(t). In particular we prove that this q-difference equation is oscillatory or non-oscillatory for different conditions.


Introduction
The numerical and analytical solutions of q-difference operator has an important role in different fields such as science and engineering, whose solution has a better understanding of the physical features of the problem [6,5]. The authors in [1] introduced a ∆ q operator and then derived many results using the generalized q-difference equation ∆ t q v(k) = u(k), q = 1 and for any real k. The authors in [2,3] developed the q-alpha multi-series formula for finite and higher order q-alpha formula for infinite series.
The branch of differential equation theory is widely used in oscillation theory [4,9]. The existence or non-existence of oscillatory solutions to a given equation or system are contained in the basic problem of classical theory of oscillation [7,8].Till recently no special importance was given to the study of oscillations using q-difference equation. Hence in this article, we are primarily interested in the oscillation theory of q-difference equation.

Preliminaries
Here, we present some preliminaries which will be used for further discussion.
Definition 2.1 [1] Let f (k) be the real valued funtion on [0, ∞) and q = 1 be a fixed real number. Then the q-difference operator, denoted by ∆ q , on f(k) is defined as (1) Definition 2.2 Let 1 = q and m be any positive integers and where k(t) is a sequence defined for t ∈ Z + . Then y(t) is called an oscillatory solution of (2) if y(t) y(qt) ≤ 0. Otherwise it is called non oscillatory solution.
2 q−self adjoint second order q−difference equation Here, we developed the q-self adjoint second order q-difference equation where k(t) > 0, t ∈ Z + . By Definition 2.1, the above equation becomes which implies where Hence any equation of the form with k 0 (t) > 0 and k 2 (t) > 0, can be put in the q-self adjoint form (3) or (4). Now multiplying both sides of (6) by a positive sequence h(t) yields Comparing (4) and (7), we get and Then Putting the value of h(t/q) by replacing t by t/q in the above equation repeatedly, we obtain and hence is a solution of (6).

Main Results
This section deals with the oscillatory and nonoscillatory solution of the q-difference equation k(t)y(qt) + k t q y t q = r(t)y(t), where r(t) = k(t) + k t q − q(t) based on the given conditions. Lemma 4.1 If there exists a subsequence r(t m ) ≤ 0 with t m → ∞ as m → ∞, then every solution of (4) is oscillatory. proof On contrary, suppose there exists a non oscilltory solution y(t) > 0 for t ≥ N of (4). Then we obtain which is a contradiction and hence the proof.
Lemma 4.2 Suppose that r(t) > 0 for t ∈ Z + . Then every solution y(t) of (4) is non oscillatory if and only if every solution z(t) of (9) is positive for t ≥ N, for some N > 0. proof Suppose that (4) has a non oscillatory solution y(t), which yields y(t)y(qt) > 0 for t ≥ N. Also from (4), z(t) > 0 for t > N. Conversely assuming z(t) is a positive solution of (9). Then we construct inductively y(t) as y(N ) = 1 and y(qt) = k(t) r(qt) z(t)y(t) with t ≥ N, which gives y(qN ) = k(N ) r(qN ) z(N )y(N ) > 0. Similarly, we can prove y(qN ) > 0 for n ≥ 2 and so y(t) > 0 for n ≥ N. Thus y(t) is a non oscillatory solution of equation (4).

Theorem 4.3
If r(t)r(qt) ≤ (4 − ) k 2 (t) for some > 0 and for all t ≥ N , then every solution of (4) oscillates. proof If r(t)r t q ≤ (4 − ) k 2 (t) for some ≥ 4, then r(t)r t q ≤ 0. By lemma 4.1, every solution of (4) is oscillatory. Hence we may assume that 0 < < 4. Now let us assume the contrary. Then by lemma 4.2, (9) has a positive solution z(t) for t ≥ N.
Using the assumption in (10) yields c(t) ≥ 1 (4− ) . Again using result 3.2, s(t), s(t) ≥ z(t) > 1 for all t ≥ N, is a solution of Now, we define a positive sequence y(t) inductively as follows: Substituting s(t) in (11), we get which yields y(qt) − √ 4 − y(t) + y( t q ) = 0, t ≥ N, whose characteristic roots are Hence the solutions are non oscillatory, which is a contradiction.
Hence y(t) is a nonoscillatory solution of (12).

Conclusion
n our research work, we discussed about the oscillation theory of the q-difference equation k(t)y(qt) + k t q y t q = r(t)y(t), where r(t) = k(t) + k t q − q(t). Some theorems are also proved using the concept of oscillation theory.