Hille and Nehari Type Oscillation Criteria for Conformable Fractional Differential Equations

In this paper, we develop the Hille and Nehari Type criteria for the oscillation of all solutions to the Fractional Differential Equations involving Conformable fractional derivative. Some new oscillatory criteria are obtained by using the Riccati transformations and comparison technique. We show the validity and effectiveness of our results by providing various examples.


Introduction
Fractional calculus turned out to be very attractive to Mathematicians as well as Physicists, Biologists, Engineers and Economists. The first application of fractional calculus was due to Abel in his solution of the Tautocrone problem. It likewise has applications in Biophysics, Quantum mechanics, Wave theory, Polymers, Continuum mechanics, Lie theory, Field theory, Spectroscopy, and group theory, among other applications [1,2,3]. Fractional differential equations are important tools in the modeling for many physical phenomena in many fields of science and engineering, such as electromagnetic waves, viscoelastic system etc, and can be described with very high accuracy. Recently, fractional derivative and associated integral have been freshly defined by Khalil [4,5]. It is a natural extension of usual derivative and it is named as Conformable, because this operator preserves basic properties of classical derivative (see [6][7][8][9]). Since conformable fractional derivative (CFD) is a local and limit based operator, it quickly takes a place in application problems [10][11][12][13][14][15][16]. Comparison principles of Sturm's type will be derived for self-adjoint differential equations. The construction of the main result is given in a very general and novel form in terms of eigenvalues associated with boundary problems for the differential operators. The proof is established as an easy consequence of Courant's variational principle for the quadratic functional associated with an eigen value problem [17], self-adjoint problem [18], differential equations [19,20], non-oscillation theorems [21], oscillation stability [22] and comparison theorems [23]. In 2016, Pospisil and Skripkova [24] introduced the Sturm's comparison principles for conformable fractional differential equations (CFDE's).
Iraqi Journal of Science, 2021, Vol. 62, No. 2, pp: 578-587 579 Motivated by the above papers, the objective of this paper is to establish more general nonoscillation criteria which will contain Nehari criteria as special cases. The essential concept used is the fact that there exists a direct connection between the oscillation problems for the equation , ( )-( ) ( ) (1.1) and the eigen value problem for the equation , ( )-( ) ( ) (1.2) with suitable boundary conditions. Our main concern will be to obtain nonoscillation criteria for the equation ( ). This work is organized as follows: Section 2 is devoted to providing essential preliminaries and properties of CFD. In Section 3, we present Nehari type oscillation criteria by using Courant minimum principle. In Section 4, we consider Hille type oscillation criteria by the method of Riccati technique.

II. PRELIMINARIES
In this section, we introduce some standard definitions and essential lemmas on CFD. First we shall start with the definition. Definition 2.1 Define a function , ) . Then the CFD of of order is defined by We will sometimes write ( ) ( ) for ( )( )to denote the CFD of of order .
where is the improper Riemann integral, and ( ) where is any continuous function in the domain of .

III. MAIN RESULTS
In this section, we prove the Comparison theorems for CFDE's.

Comparison Theorems For Eigenfunctions :
In 1957, Nehari [20] discovered a connection between the oscillatory behavior of the solution of ( ) and the eigenvalue problem , -∫ , ( )- If is the smallest eigenvalue of (3.1), then ,for all real . (3.7) In fact, this is even true for all real , -satisfying the weaker condition ( ) Proof : ) . The proof follows from the identity . .
Integratinglast term of the above inequality by parts valid for all ( ) which do not vanish in the interval ( )and If is an eigenfunction of (3.4) corresponding to the smallest eigenvalue and hence free of zeros in ( ) then the If is an eigenfunction of (3.4) corresponding to the smallest eigen value , then (3.10) can be written as, for Then ( ( )) is never increasing for and the graph of ( ) is concave downwards, it follows that ( ( )) .
Let be a positive eigenfunction of (3.4) in , corresponding to .Then by multiplying ( ) by (3.11), we have By using (3.11), we get By integrating the above inequality from , and using (3.11),we get By taking from to and are positive in ( ) Conversely, if , then (3.13) shows that cannot have a zero to the right of if is the first zero,then (3.13) implies that ( ) an impossibility.

From (3.16) and (3.17), we get
and By substituting (3.19) and (3.20) in (3.18), we obtain and (3.14) follows in the limit , we get Since the left side of inequality (3.13) is nonnegative, the following inequalities are obtained when and respectively:  [23,24] and requires no such restriction on ( )or ( ). This is known as the Hille-Winter comparison theorem.  satisfies (4.9) as pointed out before. Integration of (4.9) from to gives . Since (4.1) is nonoscillatory, Lemma 4.1 shows that the second integral tends to a finite limit as and also that ( ) as . It follows that the last integral also tends to a finite limit as and that ( ) satisfies the integral equation (4.10),which implies that

V.CONCLUSION AND FUTURE WORK
In this investigation, the aim was to present some oscillatory or nonoscillatorybehaviors of the conformable fractional differential equations through the instrument of the Nehari and Hille type theorems. Since the obtained results are general forms of earlier works, they would assist the investigations in future studies.