Oscillation Results for Solutions of Fractional-Order Differential Equations

: This survey paper is devoted to succinctly reviewing the recent progress in the ﬁeld of oscillation theory for linear and nonlinear fractional differential equations. The paper provides a fundamental background for all interested researchers who would like to contribute to this topic.


Introduction
For many years, integer-order differential equations have been used to describe natural or real-life occurrences. However, the factors at play in these situations are extremely complex and diverse. Therefore, it has been realized that integer-order differential equations cannot incorporate all of such features. One can cover up this gap by using fractional-order differential equations which provide better description and interpretation to construct these models. The origin of fractional calculus is the same as that of classical calculus. However, the growth of fractional calculus has stagnated due to insufficient geometrical and unsuitable physical interpretations of fractional derivatives. Researchers came to appreciate the importance of these derivatives with the advent of high-speed computers and precise computational techniques by creating and applying a specific fractional differential operator to a real-life situation. Fractional calculus has become a popular topic in practically every branch of science and engineering. Indeed, it has been expanded rapidly due to the nonlocal character of fractional operators. As a result, fractional calculus and its many applications have piqued the interest of many researchers [1,2].
For specific reasons, most of the real-life phenomena in the world are non-linear. Therefore, it is possible to understand the nonlinear phenomenon of the actual model through the nonlinear equation. Unlike linear equations, it is not always possible to calculate analytical solutions for nonlinear equations. However, one can obtain an approximate solution to the nonlinear equation to better understand how the equation works. The qualitative properties of nonlinear equations such as existence, uniqueness, stability, oscillation, controllability, bifurcation, chaos, etc., can be easily discussed without solving them. Commenting on solutions to equations without solving them can help scientists tackle many research problems.
Nonlinearity is a qualitative property of equations that can be utilized to create or remove oscillation. Torsion oscillations, cardiac oscillations, sinusoidal oscillations, and harmonic oscillations are all examples of practical applications of the theory of oscillation of differential equations. Many academics have developed a systematic examination of the oscillation and non-oscillation of solutions of integer order differential equations. Because of the remarkable interest in the theory of fractional calculus, oscillation of solutions for fractional differential equations has been investigated for the past twenty years. By studying the oscillation of nonlinear fractional differential equations, Grace and other scholars initiated and pioneered this topic. The line has continued to progress, and some notable outcomes have been established and elaborated; the reader can consult the papers cited herein.
This study intends to bring together the recent advances in the field of oscillation theory of linear and nonlinear fractional differential equations, as well as provide researchers with insight into future needs in the field of oscillation of fractional differential equations. The results of this paper will be presented based on different fractional operators.
We use the following notations, definitions and known results of fractional calculus throughout the article. Denote by R the set of all real numbers, and R + the set of all positive real numbers.
A solution of (1) (or (2)) is said to be oscillatory if it has arbitrarily large zeros on (a, ∞); otherwise, it is called non-oscillatory. An equation is said to be oscillatory if all of its solutions are oscillatory.
We find the following popular results of Grace et al. in Reference 20 of [3].
and lim sup where H β is defined as in Theorem 2, then (2) is oscillatory.
and lim sup where H γ is defined as in Theorem 3, then (2) is oscillatory.
In continuation to the above work, Chen et al. [3] established several oscillation theorems for (1) and (2). The authors in [3] observed that the cases β > γ > 1 and 1 > β > γ > 0 were not considered for (1) and (2) in Reference 20 of [3]. The purpose of the paper [3] was to cover this gap and establish new oscillation criteria that improve the results in Reference 20 of [3].
and lim sup for every sufficiently large T, where and lim sup for every sufficiently large T, where G is defined as in Theorem 9, then every bounded solution of (1) is oscillatory.
and lim sup for every sufficiently large T, where G is defined as in Theorem 9, then (2) is oscillatory.
and lim sup for every sufficiently large T, where G is defined as in Theorem 9, then every bounded solution of (2) is oscillatory.
If there exists a positive function v 1 on [a, ∞) such that for some constant K 3 > 0, we have and lim sup ζ→∞ then (27) is oscillatory.
and lim sup then (28) is oscillatory.
In this line, Wang et al. [5] discussed the oscillations of the fractional order differential equation where 0 < κ ≤ 1, q is a positive real-valued function and f 3 : The Riccati transformation technique is used to obtain some sufficient conditions which guarantee that every solution of the equation is oscillatory or the limit inferior converges to zero. Theorem 15 ([5]). If there exists a positive function σ ∈ C (0, ∞) and a sufficiently large ζ 2 ≥ a such that where σ + (s) = max{σ (s), 0}, then either (39) is oscillatory or where σ + (s) = max{σ (s), 0}. If these assumptions hold and lim sup then either (39) is oscillatory or lim inf ζ→∞ u(ζ) = 0.
Theorem 17 ([5]). Assume there is a positive function σ such that σ is continuous on (0, ∞) and a sufficiently large ζ 1 satisfies where m > 1, then either (39) is oscillatory or In this line, Grace established some new criteria for the oscillation of fractional differential equations with the Caputo derivative of the form Moreover, the conditions under which all solutions of this equation are asymptotic to aζ + b as ζ → ∞ for some real numbers a and b, are presented. We find the following results in Reference 10 of [6].
If u is a non-oscillatory solution of (46), then Theorem 21. Let 0 < λ < 1 and condition (56) of Theorem 20 be replaced by and lim inf then the conclusion of Theorem (20) holds.
Using Riccati type transformations, Tunč et al. [8] established some new oscillation criteria for the fractional differential equation where 0 < κ < 1, and there exists a constant K 5 > 0 such that then (67) oscillatory.
Grace dealt with the asymptotic behavior of non-oscillatory solutions of fractional differential equations of the form The following particular cases are considered: where We find the following results in Reference 8 of [9].
Assume that the function f 4 satisfies for some function h 1 : (a, ∞) → R + and real numbers γ > 0 and 0 < β < δ. For the sake of simplification, define where m 2 : (a, ∞) → R + is continuous function. Let q be a conjugate number of p > 1, lim sup lim inf lim sup Then every non-oscillatory solution u of (72) satisfies Theorem 29. Consider (72) with (74). Assume that the function f 4 satisfies for some function h 1 : [a, ∞) → R + and real numbers γ > 0 and 0 < λ < 1. For the sake of simplification, define Then every non-oscillatory solution u of (72) satisfies Then every non-oscillatory solution u of (72) is bounded.
Grace et al. dealt with the boundedness of non-oscillatory solutions of the forced fractional differential equation with positive and negative terms with the particular cases The following conditions are always assumed to hold: We find the following results in Reference 11 of [9].
Seemab et al. [6] established the oscillation criteria and asymptotic behavior of solutions for a class of fractional differential equations by considering equations of the form Then all unbounded solutions of (123) are oscillatory.
Then all bounded solutions of (123) are oscillatory.
Then all bounded solutions of (123) are non-oscillatory.
then no non-oscillatory solution of (123) is bounded away from zero as ζ → ∞.
then no non-oscillatory solution of (123) goes to zero as ζ → ∞.
Graef et al. [9] dealt with the boundedness of non-oscillatory solutions of the forced fractional differential equation with positive and negative terms with the particular cases Here, η ≥ 1 is the ratio of positive odd integers. The following conditions are always assumed to hold: [a, ∞) → (0, ∞) and positive real numbers λ 1 and γ 1 with λ 1 > γ 1 such that for u = 0 and t ≥ a, Theorem 49 ([9]). Assume there exist real number p > 1 such that p(κ − 1) where and then every non-oscillatory solution u of (132), (133) satisfies Theorem 50 ([9]). Assume there exist real number p > 1 such that p(κ − 1) where and then every non-oscillatory solution u of (132), (133) satisfies In this line, Grace dealt with the asymptotic behavior of positive solutions of certain forced fractional differential equations of the form (96) with the particular cases where r 1 : [a, ∞) → (0, ∞) is a continuous function, f 4 : [a, ∞) × R → R is continuous and assume that there exists a continuous function h : [a, ∞) → (0, ∞) and a real number λ with 0 < λ ≤ 1 such that (54) holds. Denote by Here, 0 < λ < 1, ζ ≥ ζ 1 for some t 1 ≥ a, and m 5 : [a, ∞) → (0, ∞) is a given continuous function. We find the following results in Reference 11 of [11].

Oscillation Results via Liouville Operators
The (right-sided) Liouville fractional derivative is defined by Chen [13] obtained several oscillation theorems for the fractional differential equation where 0 < κ < 1, η > 0 is a quotient of odd positive integers, r and q are positive continuous functions on [ζ 0 , ∞) for a certain ζ 0 > 0 and f : R → R is a continuous function such that f(u)/(u η ) ≥ K for a certain constant K > 0 and for all u = 0.
Theorem 60 ( [13]). Suppose that holds. Assume that there exists a positive function b ∈ C 1 [t 0 , ∞) such that (179) holds. Furthermore, assume that for every constant C ≥ t 0 , Then every solution u of (177) is oscillatory or satisfies In [14], Chen discussed the oscillatory behavior of the fractional differential equation with damping where 0 < κ < 1, p ≥ 0 and q > 0 are continuous functions on [ζ 0 , ∞) for a certain ζ 0 > 0 and f : R → R is a continuous function such that f(u)/(u) ≥ K for a certain constant K > 0 and for all u = 0, and where M + (s) = max{b + (s) − b(s)p(s), 0}, and b + is defined as in Theorem 62. Then (186) is oscillatory.

Corollary 9.
Assume that the following condition hold: Then (186) is oscillatory.
Take b(s) = 1. Then from Theorem 62 we obtain the following result.

Corollary 10.
Assume that the following condition hold: Then (186) is oscillatory.
Note that, since Corollary 9 can also be derived from Corollary 10. Obviously, Corollary 10 is better than Corollary 9.
Take b(s) = s. Then from Theorem 63 we obtain the following result.
By the generalized Riccati transformation technique, Han et al. [15] obtained oscillation criteria for a class of nonlinear fractional differential equations of the form where 0 < κ < 1, r and q are positive continuous functions on [ζ 0 , ∞) for a certain ζ 0 > 0; f, g : R → R are continuous functions such that and there exist positive constants k 1 , k 2 such that Moreover, g −1 : R → R is a continuous function such that and there exists some positive constant γ 1 such that Theorem 65 ( [15]). Suppose that holds. Furthermore, assume that there exists a positive function b ∈ C 1 [ζ 0 , ∞) such that Then (196) is oscillatory.
and h + (ζ, s) = max{h(ζ, s), 0}. Then (253) is oscillatory. and hold, and there exist a function b ∈ C 1 ([ζ 0 , ∞), R + ) such that (254) holds. Furthermore, assume that, for every constant T ≥ ζ 0 , Then every solution u of (253) is oscillatory or satisfies and (258) hold. Let b(ζ) and H(ζ, s) be defined as in Theorem 78 such that (255) holds. Further, assume that, for every constant T ≥ ζ 0 , (259) holds. Then every solution u of (253) is oscillatory or satisfies With an appropriate choice of the functions H and b, one can derive from Theorem 77, Theorem 78, Corollary 19 and Corollary 20 a number of oscillation criteria for (253).
By the generalized Riccati transformation technique, Pan et al. [20] obtained oscillation criteria for a class of nonlinear fractional differential equations of the form where 0 < κ < 1, η is a quotient of odd positive integers, g ∈ C 2 (R, R); g is an increasing function and there exists positive k such that Moreover, g −1 : R → R is a continuous function such that and there exists some positive constant γ 1 such that Denote by Theorem 79 ([20]). Assume hold, and there exist two for all sufficiently large T. Then every solution of (264) is oscillatory or satisfies Theorem 80 ([20]). Assume (267)-(268) hold. Furthermore, assume there exist two functions for all sufficiently large T. Then every solution of (264) is oscillatory or satisfies for all sufficiently large T. Then every solution of (264) is oscillatory or satisfies Theorem 82 ([20]). Assume (267)-(268) hold. Furthermore, assume that there exist two functions for all sufficiently large T. Then every solution of (264) is oscillatory or satisfies In Theorems 81 and 82, if we take H(ζ, s) for some special functions such as ln ζ s , then we can obtain the following two corollaries. ([20]). Assume (267)-(268) hold. Furthermore, assume that there exist two functions φ ∈ C 1 ([ζ 0 , ∞), R + ) and ψ ∈ C 1 ([ζ 0 , ∞), [0, ∞)). If

Corollary 21
for all sufficiently large T. Then every solution of (264) is oscillatory or satisfies Corollary 22 ([20]). Assume (267)-(268) hold. Furthermore, assume that there exist two functions for all sufficiently large T. Then every solution of (264) is oscillatory or satisfies

Oscillation Results via Hadamard Operators
, be a finite or infinite interval of the half-axis R + , and let (κ) > 0 and υ ∈ C. The (left-sided) Hadamard fractional integral I κ a of order κ ∈ C, (κ) > 0, is defined by The (left-sided) Hadamard fractional derivative D κ a of order κ ∈ C, (κ) ≥ 0, is defined by The (left-sided) Caputo type Hadamard fractional derivative C D κ a of order κ ∈ C, (κ) ≥ 0, is defined by Abdalla et al. [21] established sufficient conditions for the oscillation of solutions of the following fractional differential equations in the frame of left Hadamard fractional derivatives in the Riemann-Liouville and the Caputo settings. and Theorem 83 ([21]). Let f 2 = 0 and condition (H 1) holds. If and lim sup for every sufficiently large T, then (281) is oscillatory.
Theorem 84 ( [22]). Let the assumptions (H 1) and (H 2) hold with β > γ. If and lim sup for every sufficiently large T, where and lim sup for every sufficiently large T, where H β,γ is defined by (287), then every bounded solution of the problem (281) is oscillatory.

Oscillation Results via Conformable Operators
Definition 5 ( [23][24][25]). The left conformable derivative starting from a of a function f : [a, ∞) → R of order 0 < ρ ≤ 1 is defined by The corresponding left conformable integral is defined as where κ ∈ C, (κ) ≥ 0. Abdalla et al. [26] established sufficient conditions for the oscillation of solutions of the following fractional differential equations in the frame of left Hadamard fractional derivatives in the Riemann-Liouville and the Caputo settings.
The following results improve and generalize the oscillation results in [27]. Remark 5. If we put ρ = 1 in Theorem 103 and Theorem 104, then they reduce to Theorem 95 and Theorem 96, respectively.