Sufficient conditions for oscillation of a nonlinear fractional nabla difference system

In this paper, we study the oscillation of nonlinear fractional nabla difference equations of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \left\{ \begin{array}{ll}\nabla _a^{\alpha }x(t)+q(t)f(x(t))=g(t), &{}\quad t\in {\mathbb {N}}_{a+1},\\ \nabla _a^{-(1-\alpha )}x(t)|_{t=a}=c, \end{array}\right. $$\end{document}∇aαx(t)+q(t)f(x(t))=g(t),t∈Na+1,∇a-(1-α)x(t)|t=a=c,where c and α are constants, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\alpha <1,\nabla _a^{\alpha }$$\end{document}0<α<1,∇aα is the Riemann–Liouville fractional nabla difference operator of order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha , a\ge 0$$\end{document}α,a≥0 is a real number, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {N}}_{a+1}=\{a+1,a+2,\ldots \}$$\end{document}Na+1={a+1,a+2,…}. Some sufficient conditions for oscillation are established.

In Atici and Eloe (2012), Atici and Eloe considered the following initial value problem for a nonlinear fractional difference equation where 0 < ν ≤ 1 and a is any real number. The authors obtained that x(t) is a solution of (E4) if and only if Motivated by the papers (Atici and Eloe 2012;Alzabut and Abdeljawad 2014;Kisalar et al. 2015;Li 2016), in this paper, we investigate the oscillation of a nonlinear fractional nabla difference system of the form where c and α are constants, 0 < α < 1, ∇ α a is the Riemann-Liouville fractional nabla difference operator of order α, a ≥ 0 is a real number, and N a+1 = {a + 1, a + 2, . . .}.
In this paper, we always assume that A solution x(t) of the system (1) is said to be oscillatory if it is neither eventually positive nor eventually negative, otherwise it is nonoscillatory. (1)

Preliminaries
In this section, we collect some basic definitions and lemmas that will be important to us in what follows. For an excellent introduction to the discrete fractional calculus, we refer the reader to papers Eloe 2009a, 2012;Abdeljawad and Atici 2012;Anastassiou 2010Anastassiou , 2011Abdeljawad 2011Abdeljawad , 2013b. ( Proof Using Definition 1, it follows from (8) that Using Definition 2, it follows from (10) that The proof of Lemma 5 is complete.
Lemma 6 Let a ≥ 0 and 0 < α < 1 be real number, u, v : N a → R. If then Proof It follows from (11) that Summing both sides of (13) from a to t, we have Using Definition 1, from (14) we easily obtain (12). This completes the proof of Lemma 6.

Main results
In this section, we establish the oscillation results of system (1).

Theorem 7 Assume that and
Then every solution x(t) of the system (1) is oscillatory. (10) .
Proof Suppose to the contrary that there is a nonoscillatory solution x(t) of system (1). It is obvious that there exists t 0 ∈ N a+1 such that x(t) > 0 or x(t) < 0, t ≥ t 0 .
Case 1 x(t) > 0, t ≥ t 0 . Noting the assumption (A), from the system (1) , we have Using Lemma 6, from (17), we have Using Definition 2, Lemma 3 in the left-hand side of (18) and noting the initial condition of system (1) , we obtain Using Definition 1, it follows from the right-hand side of (18) Noting (23) and taking t → ∞ in (22), we have which contradicts with x(t) > 0.
Case 2 x(t) < 0, t ≥ t 0 . Noting the assumption (A), from the system (1) , we have By Lemma 6, from (24), we obtain Using the procedure of Case 1, it follows from (25) that Noting (23) and taking t → ∞ in (26), we have which contradicts with x(t) < 0. This completes the proof of Theorem 7.
Theorem 8 Assume that there exists t 0 ∈ N a+1 such that and Then every solution x(t) of the system (1) is oscillatory.
Proof Suppose to the contrary that there is a nonoscillatory solution x(t) of system (1). It is obvious that there exists t 0 ∈ N a+1 such that x(t) > 0 or x(t) < 0, t ≥ t 0 .
Case 2 x(t) < 0, t ≥ t 0 . As in the proof of Theorem 7, we obtain (24). Then, using the above mentioned method, we easily obtain a contradiction. This completes the proof of Theorem 8.

Remarks
In our Definition 1, the fractional sum in (2) starts at a. In Abdeljawad and Atici (2012), Abdeljawad and Atici introduced the following fractional sum.
Let ν > 0. The ν-th fractional sum f is defined by Obviously, the fractional sum in (31) starts at a + 1. In Abdeljawad and Atici (2012), the authors established the relation between the operators ∇ −ν a and ∇ −ν a and considered the following initial value problem for a nonlinear fractional difference equation where 0 < α < 1 and a is any real number. The authors obtained that x(t) is a solution of (E5) if and only if Using the idea in Abdeljawad and Atici (2012), we can try to investigate the oscillation of the following nonlinear fractional nabla difference system

Examples
In this section, we give some examples to illustrate our main results.