Oscillation Criteria for a Class of Discrete Nonlinear Fractional Equations

The theory of fractional derivatives goes back to Leibniz's note to L'Hospital dated 30 September 1695 about the meaning of the derivative of non integer order and this led to the appearance of the theory of derivatives and integrals of arbitrary order. Fractional differential equations are generalizations of classical differential equations of integer order. In recent days, oscillatory behavior of fractional differential/difference equations has been investigated by authors, see papers [2]-[12]. Formal treatment on the subject of fractional derivatives and fractional integrals are presented in the books, see [16]-[19]. In the last few years, many authors found that fractional derivatives and fractional integrals were applied in widespread fields of science and engineering, especially in mathematical modeling real world phenomenon and simulation of systems and processes and control systems. Nowadays, many authors have investigated some qualitative aspects of fractional differential equations, such as the existence, the uniqueness and stability of solutions. But the discrete analog of fractional difference equations are studied by very few authors, see [13][15]. Now we study the oscillatory behavior of the following fractional difference equation of the form


INTRODUCTION
The theory of fractional derivatives goes back to Leibniz's note to L'Hospital dated 30 September 1695 about the meaning of the derivative of non integer order and this led to the appearance of the theory of derivatives and integrals of arbitrary order. Fractional differential equations are generalizations of classical differential equations of integer order. In recent days, oscillatory behavior of fractional differential/difference equations has been investigated by authors, see papers [2]- [12]. Formal treatment on the subject of fractional derivatives and fractional integrals are presented in the books, see [16]- [19]. In the last few years, many authors found that fractional derivatives and fractional integrals were applied in widespread fields of science and engineering, especially in mathematical modeling real world phenomenon and simulation of systems and processes and control systems. Nowadays, many authors have investigated some qualitative aspects of fractional differential equations, such as the existence, the uniqueness and stability of solutions. But the discrete analog of fractional difference equations are studied by very few authors, see [13]- [15]. Now we study the oscillatory behavior of the following fractional difference equation of the form (9) t  t  0,is the ratio of two odd positive integers and  denotes the Riemann-Liouville difference operator of order 0 1. A solution x(t) of (1) is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is nonoscillatory. Equation (1) is said to be oscillatory if all its solutions are oscillatory.

Preliminaries and Basic Lemmas
In this section, we provide preliminary results of discrete fractional calculus, which will be used throughout this paper. Definition 2.1. (see [13]) Let  0. The  -th fractional sum of f is defined by where f is defined for s a mod(1) and   is defined for t (a  )mod(1) and The fractional sum   f maps functions defined on N a to functions defined on N a  . Letting t, we obtain x(t)  which is a contradiction, since   x(t)  0. This completes the proof. (1)

Where
Then every solution of equation (1) is oscillatory.
Proof. Suppose to the contrary that equation (1) has a nonoscillatory solution x(t) . Without loss of generality, we may assume that x(t)0 on 1 t t . We define the function w(t) by Riccati substitution The Bulletin of Society for Mathematical Services and Standards Vol. 9 27 (7) Then we have w(t)0for t t 1 . Also, we have Now using the inequality (see [1]) (8) we have (9) Using the above inequality, we obtain (10) where Taking Where Then every solution of (1) is oscillatory.
Proof. Suppose the contrary that x(t) is a nonoscillatory solution of (1). Without loss of generality, we may assume that x(t) is an eventually positive solution of (1). We proceed as in the proof of Theorem (3.4) to get (10). Multiplying (10) by H(t, s) and summing from t 1 to t 1, we obtain (12) Using summation by parts formula, we get Letting t, we have which is a contradiction to (11). The proof is complete.
Example 3.6. Consider the following fractional difference equation (15) Here, We will apply Theorem (3.4) and it remains show that condition (6) is satisfied. Taking (s) = s , we obtain which implies that (6) holds. Therefore, by Theorem (3.4) every solution of (15) is oscillatory.