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Oscillation of solutions of some generalized nonlinear αdifference equations
Advances in Difference Equations volume 2014, Article number: 109 (2014)
Abstract
In this paper, the authors discuss the oscillation of solutions of some generalized nonlinear αdifference equation
k\in [a,\mathrm{\infty}), where the functions p, q, f and τ are defined in their domain of definition and \alpha >1, ℓ is a positive real. Further, uf(u)>0 for u\ne 0, p(k)>0 and {lim}_{k\to \mathrm{\infty}}(k\tau (k))=\mathrm{\infty}, where {R}_{k}={\sum}_{r=0}^{[\frac{k\ell}{\ell}]}\frac{1}{{\alpha}^{r}p(r\ell )}\to \mathrm{\infty} as k\to \mathrm{\infty} and u(k) is defined for k\ge {min}_{i\ge 0}(i\tau (i)) for all k\in [a,\mathrm{\infty}) for some a\in [0,\mathrm{\infty}).
MSC:39A12.
1 Introduction
The basic theory of difference equations is based on the operator Δ defined as \mathrm{\Delta}u(k)=u(k+1)u(k), k\in \mathbb{N}=\{0,1,2,3,\dots \}. Even though many authors [1–4] have suggested the definition of Δ as
there was no significant progress in this area. But recently, [5] considered the definition of Δ as given in (2) and developed the theory of difference equations in a different direction. For convenience, the operator Δ defined by (2) is labeled as {\mathrm{\Delta}}_{\ell}, and by defining its inverse {\mathrm{\Delta}}_{\ell}^{1}, many interesting results and applications in number theory (see [5–7]) were obtained. By extending the study related to the sequences of complex numbers and ℓ being real, some new qualitative properties of the solutions like rotatory, expanding, shrinking, spiral and weblike were obtained for difference equation involving {\mathrm{\Delta}}_{\ell}. The results obtained using {\mathrm{\Delta}}_{\ell} can be found in [5–7]. Popenda and Szmanda [8, 9] defined Δ as
and based on this definition, they studied the qualitative properties of a particular difference equation, and no one else has handled this operator. Recently Manuel et al. [6, 10] considered the definition of {\mathrm{\Delta}}_{\ell} as given in (3), and by defining its inverse, some interesting results on number theory were obtained.
In [11], Szafranski and Szmanda obtained sufficient conditions for the oscillation of a similar difference equation involving Δ. In this paper the theory is extended from Δ to {\mathrm{\Delta}}_{\alpha (\ell )} for all real k\in [a,\mathrm{\infty}), and we discuss the oscillatory behavior of solutions of generalized nonlinear αdifference equation (1).
Throughout this paper, we make use of the following assumptions.

(a)
\mathbb{N}=\{0,1,2,3,\dots \}, \mathbb{N}(a)=\{a,a+1,a+2,\dots \};

(b)
{\mathbb{N}}_{\ell}(a)=\{a,a+\ell ,a+2\ell ,\dots \};

(c)
\lceil x\rceil and [x] denote upper integer and integer part of x, respectively;

(d)
j=k{k}_{i}[\frac{k{k}_{i}}{\ell}]\ell, {k}_{i}\in [0,\mathrm{\infty}).
2 Preliminaries
In this section, we present some preliminaries which will be useful for future discussion.
Definition 2.1 [12]
The inverse of the generalized αdifference operator denoted by {\mathrm{\Delta}}_{\alpha (\ell )}^{1} on u(k) is defined as follows. If {\mathrm{\Delta}}_{\alpha (\ell )}v(k)=u(k), then
where k\in {\mathbb{N}}_{\ell}(j), j=k[\frac{k}{\ell}]\ell.
Lemma 2.2 [13]
If the realvalued function u(k) is defined for all k\in [a,\mathrm{\infty}) and \alpha >1, then
for all k\in {\mathbb{N}}_{\ell}(j), j=ka[\frac{ka}{\ell}]\ell.
Definition 2.3 [7]
The solution u(k) of (1) is called oscillatory if for any {k}_{1}\in [a,\mathrm{\infty}) there exists {k}_{2}\in {\mathbb{N}}_{\ell}({k}_{1}) such that u({k}_{2})u({k}_{2}+\ell )\le 0. The difference equation itself is called oscillatory if all its solutions are oscillatory. If the solution u(k) is not oscillatory, then it is said to be nonoscillatory (i.e., u(k)u(k+\ell )>0 for all k\in [{k}_{1},\mathrm{\infty})).
3 Main results
In this section we present conditions for the oscillation of equation (1).
Theorem 3.1 Assume that

(i)
q(k)\ge 0 and {\sum}_{r=0}^{\mathrm{\infty}}{\alpha}^{r}q(r\ell )=\mathrm{\infty},

(ii)
{lim\hspace{0.17em}inf}_{u(k)\to \mathrm{\infty}}f(u(k))>0.
Then every solution of Equation (1) is oscillatory.
Proof Assume that Equation (1) has a nonoscillatory solution u(k), and we assume that u(k) is eventually positive. Then there is a positive integer {k}_{1} such that
From Equation (1) we have
and so p(k){\mathrm{\Delta}}_{\alpha (\ell )}u(k) is eventually nonincreasing. We first show that
In fact, if there is {k}_{2}\ge {k}_{1} such that p({k}_{2}){\mathrm{\Delta}}_{\alpha (\ell )}u({k}_{2})=c<0 and p(k){\mathrm{\Delta}}_{\alpha (\ell )}u(k)\le c for k\ge {k}_{2}, that is,
hence by Lemma 2.2,
which contradicts the fact that u(k)>0 for k\ge {k}_{2}. Hence, p(k){\mathrm{\Delta}}_{\alpha (\ell )}u(k)\ge 0 for k\ge {k}_{1}. Therefore we obtain
Let L={lim}_{k\to \mathrm{\infty}}u(k).
Then L>0 is finite or infinite.
Case 1. L>0 is finite.
From the function f(k) defined in its domain of definition, we have
Thus, we may choose a positive integer {k}_{4} (\ge {k}_{1}) such that
By substituting (7) in Equation (1) we obtain
By Lemma 2.2, we obtain
and so
which contradicts (i).
Case 2. L=\mathrm{\infty}. For this case, from condition (ii) we have
and so we may choose a positive constant c and a positive integer {k}_{5} sufficiently large such that
Substituting (9) into Equation (1) we have
Using a similar argument as in Case 1, we obtain a contradiction to condition (i). This completes the proof. □
Example 3.2 For the generalized αdifference equation
all the conditions of Theorem 3.1 hold and hence all the solutions are oscillatory. In fact u(k)={(\alpha )}^{\lceil \frac{k}{\ell}\rceil}k is one such solution.
Theorem 3.3 Assume that

(iii)
q(k)\ge 0 and {\sum}_{r=0}^{\mathrm{\infty}}{\alpha}^{r}R(r\ell )q(r\ell )=\mathrm{\infty}.
Then every bounded solution of (1) is oscillatory.
Proof Proceeding as in the proof of Theorem 3.1, with the assumption that u(k) is a bounded nonoscillatory solution of (1), we get inequality (8), and so we obtain
It is easy to see that
Using (10) in (11), (11) reduces to
which implies
Hence, there exists a constant c such that
which is a contradiction to condition (iii) which completes the proof. □
Example 3.4 For the generalized αdifference equation
all the conditions of Theorem 3.3 hold and hence all the solutions are oscillatory. In fact u(k)=\frac{{(1)}^{\lceil \frac{k}{\ell}\rceil}}{{2}^{k}} is one such solution.
Theorem 3.5 Assume that

(iv)
(k\tau (k)) is nondecreasing, where \tau (k)\in [0,\mathrm{\infty}),

(v)
there exists p({k}_{n}) such that p({k}_{n})\le 1 for {k}_{n}\in [0,\mathrm{\infty}),

(vi)
{\sum}_{r=0}^{\mathrm{\infty}}{\alpha}^{r}q(r\ell )=\mathrm{\infty},

(vii)
f is nondecreasing and there is a nonnegative constant M such that
\underset{s\to 0}{lim\hspace{0.17em}sup}\frac{s}{f(s)}=M.(12)
Then the difference {\mathrm{\Delta}}_{\alpha (\ell )}u(k) of every solution u(k) of Equation (1) oscillates.
Proof If not, then Equation (1) has a solution u(k) such that its difference {\mathrm{\Delta}}_{\alpha (\ell )}u(k) is nonoscillatory. Assume first that the sequence {\mathrm{\Delta}}_{\alpha (\ell )}u(k) is eventually negative. Then there is a positive integer {k}_{1} such that
and so u(k) is decreasing for k\ge {k}_{1}, which implies that u(k) is also nonoscillatory. Set
Then
By Lemma 2.2, we have
and by (vi) we get
which implies that eventually
By (15), we can choose {k}_{3} (\ge {k}_{2}) such that
That is,
Set {lim}_{k\to \mathrm{\infty}}u(k)=L. Then L\ge 0. Now we prove that L=0. If L>0, then we have
since f(k) is defined in its domain of definition. Choosing {k}_{4} sufficiently large such that
and substituting (19) into (17), we obtain
From Lemma 2.2, we have
which implies that {lim}_{k\to \mathrm{\infty}}u(k)=\mathrm{\infty}.
This contradicts (16). Hence {lim}_{k\to \mathrm{\infty}}u(k)=0.
By the assumptions we have
From this we can choose {k}_{4} such that
That is, u(k\tau (k))<(M+\ell )f(u(k\tau (k))), k\ge {k}_{5}, and so from (17) we get
In particular, for a function p({k}_{n}) satisfying condition (v), we have
for k sufficiently large, which implies that
for all large k. This is a contradiction. The case that {\mathrm{\Delta}}_{\alpha (\ell )}u(k) is eventually positive can be treated in a similar fashion and this completes the proof of the theorem. □
Example 3.6 For the generalized αdifference equation
all the conditions of Theorem 3.5 hold and hence all the solutions are oscillatory. In fact u(k)={(\alpha )}^{\lceil \frac{k}{\ell}\rceil}{2}^{k} is one such solution.
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Acknowledgements
The authors express their sincere thanks to the referees for the careful and detailed reading of the manuscript and very helpful suggestions. Research supported by the National Board for Higher Mathematics, Department of Atomic Energy, Government of India, Mumbai. The second authors also gratefully acknowledges that part of this research was partially supported by the University Putra Malaysia under the GPIBT Grant Scheme having project no. GPIBT/2013/9420100.
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Manuel, M.M.S., Kılıçman, A., Srinivasan, K. et al. Oscillation of solutions of some generalized nonlinear αdifference equations. Adv Differ Equ 2014, 109 (2014). https://doi.org/10.1186/168718472014109
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DOI: https://doi.org/10.1186/168718472014109