Oscillation for a Class of Fractional Differential Equation

Fractional differential equations have been of great interest recently. Apart from diverse areas of mathematics, fractional differential equations arise in rheology, dynamical processes in self-similar and porous structures, fluid flows, electrical networks, viscoelasticity, chemical physics, and many other branches of science. There have appeared lots of works in which fractional derivatives are used for a better description of considered material properties; mathematical modelling based on enhanced rheological models naturally leads to differential equations of fractional order and to the necessity of the formulation of initial conditions to such equations. It is caused both by the intensive development of the theory of fractional calculus itself and by the applications; see [1–6]. It should be noted that most of the papers and books on fractional calculus are devoted to the solvability of linear fractional differential equations. Recently, there are many papers dealing with the qualitative theory, especially the existence of solutions (or positive solutions) of nonlinear initial (or boundary) value problems for fractional differential equation (or system) by the use of techniques of nonlinear analysis (fixed-point theorems, Leray-Schauder theory, Adomian decomposition method, etc.); see [7–11]. The oscillation theory as a part of the qualitative theory of differential equations has been developed rapidly in the last decades, and there has been a great deal of work on the oscillatory behavior of integer order differential equations. However, there are only very few papers dealing with the oscillation of fractional differential equation; see [12–15]. Grace et al. [12] initiated the oscillatory theory of fractional differential equations


Introduction
Fractional differential equations have been of great interest recently.Apart from diverse areas of mathematics, fractional differential equations arise in rheology, dynamical processes in self-similar and porous structures, fluid flows, electrical networks, viscoelasticity, chemical physics, and many other branches of science.There have appeared lots of works in which fractional derivatives are used for a better description of considered material properties; mathematical modelling based on enhanced rheological models naturally leads to differential equations of fractional order and to the necessity of the formulation of initial conditions to such equations.It is caused both by the intensive development of the theory of fractional calculus itself and by the applications; see [1][2][3][4][5][6].
It should be noted that most of the papers and books on fractional calculus are devoted to the solvability of linear fractional differential equations.Recently, there are many papers dealing with the qualitative theory, especially the existence of solutions (or positive solutions) of nonlinear initial (or boundary) value problems for fractional differential equation (or system) by the use of techniques of nonlinear analysis (fixed-point theorems, Leray-Schauder theory, Adomian decomposition method, etc.); see [7][8][9][10][11].
The oscillation theory as a part of the qualitative theory of differential equations has been developed rapidly in the last decades, and there has been a great deal of work on the oscillatory behavior of integer order differential equations.However, there are only very few papers dealing with the oscillation of fractional differential equation; see [12][13][14][15].Grace et al. [12] initiated the oscillatory theory of fractional differential equations where    denotes the Riemann-Liouville differential operator of order  with 0 <  < 1 and the functions  1 ,  2 , and V are continuous.By the expression of solution and some inequalities, oscillation criteria are obtained for a class of nonlinear fractional differential equations.The results are also stated when the Riemann-Liouville differential operator is replaced by Caputo's differential operator.
Chen [13] considered the oscillation of the fractional differential equation where   −  is the Liouville right-sided fractional derivative of order  ∈ (0, 1) of ,  > 0 is a quotient of odd positive integers,  and  are positive continuous functions on [ 0 , ∞) for a certain  0 > 0, and  : R → R is a continuous function such that ()/(  ) >  for a certain constant  > 0 and for all  ̸ = 0.They established some oscillation criteria for the equation by using a generalized Riccati transformation technique and an inequality.
In 2013, Chen [15] studied oscillatory behavior of the fractional differential equation with the form where   −  is the Liouville right-sided fractional derivative of order  ∈ (0, 1) of .
By a solution of (4), we mean a nontrivial function 4) for  > 0. Our attention is restricted to those solutions of (4) which exist on R + and satisfy sup{|()| :  >  * } > 0 for any  * ≥ 0. A solution  of ( 4) is said to be oscillatory if it is neither eventually positive nor eventually negative.Otherwise it is nonoscillatory.Equation ( 4) is said to be oscillatory if all its solutions are oscillatory.

Preliminaries
For the convenience of the reader, we give some background materials from fractional calculus theory.These materials can be found in the recent literature; see [12,13,16,17].
Definition 1 (see [16]).The Liouville right-sided fractional integral of order  > 0 of a function  : R + → R on the half-axis R + is given by provided that the right side is pointwise defined on R + , where Γ(⋅) is the gamma function.
Definition 2 (see [16]).The Liouville right-sided fractional derivative of order  > 0 of a function  : R + → R on the half-axis R + is given by where ⌈⌉ := min{ ∈ Z :  ≥ }, provided that the right side is pointwise defined on R + .
The following lemma is fundamental in the proofs of our main results.

Main Results
Theorem 5. Suppose that (H 1 )-(H 3 ) and hold.Furthermore, assume that there exists a positive function where  1 ,  2 are defined as in ( 3 ).Then every solution of (4) is oscillatory.
Proof.Suppose that  is a nonoscillatory solution of (4).Without loss of generality, we may assume that  is an eventually positive solution of (4).Then there exists where  is defined as in (7).Therefore, it follows from (4) that Thus, ()((  − )()) is strictly increasing on [ 1 , ∞) and is eventually of one sign.Since () > 0 for  ∈ [ 0 , ∞) and (H 2 ), we see that (  − )() is eventually of one sign.We now claim that If not, then (  − )() is eventually positive, and there exists Therefore, from (8), we have Then, we get Integrating the above inequality from  2 to , we have Letting  → ∞, we see This contradicts (10).Hence, ( 14) holds.Define the function  by the generalized Riccati substitution Then we have () > 0 for  ∈ [ 1 , ∞).From ( 19), ( 4), (8) Taking from Lemma 4 and (20) we get Integrating both sides of the inequality (22) from  0 to , we obtain Taking the limit supremum of both sides of the above inequality as  → ∞, we get lim sup which contradicts (11).The proof is complete.
where  1 ,  2 , and  are defined as in Theorem 5. Then all solutions of (4) are oscillatory.
Proof.Suppose that  is a nonoscillatory solution of (4).Without loss of generality, we may assume that  is an eventually positive solution of (4).We proceed as in the proof of Theorem 5 to get (22), that is, Multiplying the previous inequality by (, ) and integrating from  0 to  − 1, for  ∈ [ 1 + 1, ∞), we obtain Therefore, which is a contradiction to (26).The proof is complete.
Next, we consider the case which yields that (10) does not hold.In this case, we have the following results.
Theorem 7. Suppose that (H 1 )-(H 3 ) and (30) hold,  is an increasing function, and that there exists a positive function  ∈  1 [ 0 , ∞) such that (11) holds.Furthermore, assume that for every constant  ≥  0 , Then every solution  of (4) is oscillatory or satisfies Proof.Assume that  is a nonoscillatory solution of (4).Without loss of generality, assume that  is an eventually positive solution of (4).Proceeding as in the proof of Theorem 5, there are two cases for the sign of (  − )().The proof when (  − )() is eventually negative is similar to that of Theorem 5 and hence is omitted.