Arab. J. Math.
Chain rules and inequalities for the BHT fractional calculus on arbitrary timescales
Eze R. Nwaeze 0
Delfim F. M. Torres
[email protected] 0
Mathematics Subject Classification 0
0 E. R. Nwaeze Department of Mathematics, Tuskegee University , Tuskegee, AL 36088 , USA
We develop the Benkhettou-Hassani-Torres fractional (noninteger order) calculus on timescales by proving two chain rules for the α-fractional derivative and five inequalities for the α-fractional integral. The results coincide with well-known classical results when the operators are of (integer) order α = 1 and the timescale coincides with the set of real numbers. The study of fractional (noninteger order) calculus on timescales is a subject of strong current interest [1-4]. Recently, Benkhettou, Hassani and Torres introduced a (local) fractional calculus on arbitrary timescales T (called here the BHT fractional calculus) based on the Tα differentiation operator and the α-fractional integral [5]. The Hilger timescale calculus [6] is then obtained as a particular case, by choosing α = 1. In this paper, we develop the BHT timescale fractional calculus initiated in [5]. Precisely, we prove two different chain rules for the fractional derivative Tα (Theorems 3.1 and 3.3) and several inequalities for the α-fractional integral: Hölder's inequality (Theorem 3.4), Cauchy-Schwarz's inequality (Theorem 3.5), Minkowski's inequality (Theorem 3.7), generalized Jensen's fractional inequality (Theorem 3.8) and a weighted fractional HermiteHadamard inequality on timescales (Theorem 3.9). The paper is organized as follows. In Sect. 2, we recall the basics of the the BHT fractional calculus. Our results are then formulated and proved in Sect. 3.
2 Preliminaries
We briefly recall the necessary notions from the BHT fractional calculus [5]: fractional differentiation and
fractional integration on timescales. For an introduction to the timescale theory we refer the reader to the book
[6].
Theorem 2.2 (See [5]) Let α ∈ (0, 1] and T be a timescale. Assume f : T → R and let t ∈ Tκ . If f is
α-fractional differentiable of order α at t , then
f (σ (t )) = f (t ) + μ(t )t α−1Tα( f )(t ).
ab(γ f )(t ) αt = γ ab f (t ) αt ;
ab f (t ) αt = − ba f (t ) αt ;
ab f (t ) αt = ac f (t ) αt + cb f (t ) αt ;
ab f (t ) αt ≤ ab g(t ) αt .
3 Main results
3.1 Fractional chain rules on timescales
Let t ∈ Tκ and > 0. Since g is α-fractional differentiable at t , we know from Definition 2.1 that there exists
a neighbourhood U1 of t such that
for all s ∈ U1,
f (hg(σ (t )) + (1 − h)g(t )) dh
Moreover, f is continuous on R and, therefore, it is uniformly continuous on closed subsets of R. Observing
that g is also continuous, because it is α-fractional differentiable (see item (i) of Theorem 4 in [5]), there exists
a neighbourhood U2 of t such that
for all s ∈ U2. To see this, note that
∗ =
γ = hg(σ (t )) + (1 − h)g(s) and
β = hg(σ (t )) + (1 − h)g(t ).
Then we have
= t 1−α[g(σ (t )) − g(s)]
f (γ )dh − Tα(g)(t )(σ (t ) − s)
t 1−α[g(σ (t )) − g(s)] − (σ (t ) − s)Tα(g)(t )
f (γ )dh + Tα(g)(t )(σ (t ) − s)
≤ |g(s) − g(t )|
f g(t ) + hμ(t )t α−1Tα(g)(t ) dh Tα(g)(t ).
Proof We begin by applying the ordinary substitution rule from calculus:
∗ + Tα(g)(t ) |σ (t ) − s|
≤ 2 |σ (t ) − s| + 2 |σ (t ) − s|
Let us illustrate Theorem 3.1 with an example.
Example 3.2 Let g : Z → R and f : R → R be defined by
f (g(t ) + hμ(t )t α−1Tα(g)(t ))dh Tα(g)(t )
g(t ) = t 2
f (t ) = et .
= (2t + 1)t 1−αet2
= (2t + 1)t 1−αet2
= t 1−αet2 e2t+1 − 1 .
eh(2t+1)dh
e2t+1 − 1
for all s ∈ U1 and
≤ ∗ 1 + |Tα(ν)(t )| + |T˜α(w)(ν(t ))| |σ (t ) − s|
= |σ (t ) − s|.
This proves the claim.
3.2 Fractional integral inequalities on timescales
where p > 1 and 1p + q = 1.
1
Proof For nonnegative real numbers A and B, the basic inequality
Theorem 3.4 (Hölder’s fractional inequality on timescales) Let α ∈ (0, 1] and a, b ∈ T. If f, g, h : [a, b] →
R are r d-continuous, then
| f (t )g(t )||h(t )| αt ≤
A(t ) =
B(t ) =
and integrating the obtained inequality between a and b, which is possible since all occurring functions are
r d-continuous, we find that
|g(t )||h(t )|1/q
1 |g(t )|q |h(t )|
ατ + q ab |g(τ )|q |h(τ )|
holds. Now, suppose, without loss of generality, that
Applying Theorem 2.3 and the above inequality to
= 0.
= 1.
This directly yields the Hölder inequality (2).
| f (t )g(t )||h(t )| αt ≤
As a particular case of Theorem 3.4, we obtain the following inequality.
Theorem 3.5 (Cauchy–Schwarz’s fractional inequality on timescales) Let α ∈ (0, 1] and a, b ∈ T. If f, g, h :
[a, b] → R are r d-continuous, then
Using Hölder’s inequality (2), we can also prove the following result.
| f (t )g(t )||h(t )| αt ≥
where 1p + q = 1 and p < 0 or q < 0.
1
for any r d-continuous functions F, G : [a, b] → R. The desired result is obtained by taking F (t ) = [ f (t )]−q
and G(t ) = [ f (t )]q [g(t )]q in inequality (3).
Theorem 3.7 (Minkowski’s fractional inequality on timescales) Let α ∈ (0, 1], a, b ∈ T and p > 1 (...truncated)