By George A. Anastassiou

ISBN-10: 0387981276

ISBN-13: 9780387981277

Fractional differentiation inequalities are by way of themselves a massive quarter of study. they've got many functions in natural and utilized arithmetic and lots of different technologies. essentially the most very important functions is in setting up the distinctiveness of an answer in fractional differential equations and structures and in fractional partial differential equations. in addition they supply top bounds to the suggestions of the above equations.

In this e-book the writer provides the Opial, Poincaré, Sobolev, Hilbert, and Ostrowski fractional differentiation inequalities. effects for the above are derived utilizing 3 forms of fractional derivatives, specifically through Canavati, Riemann-Liouville and Caputo. The univariate and multivariate situations are either tested. every one bankruptcy is self-contained. the speculation is gifted systematically in addition to the functions. the applying to info thought is additionally examined.

This monograph is acceptable for researchers and graduate scholars in natural arithmetic. utilized mathematicians, engineers, and different utilized scientists also will locate this e-book useful.

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**Additional resources for Fractional Differentiation Inequalities**

**Example text**

9) we now get that all x ˜ : x0 ≤ x ˜ ≤ x. 29) over the whole interval [x0 , x]. 29) as in (ii). Additionally assume that h ≡ 0 and ai = 0, all i = 0, 1, . . , n − 1. Then the unique solution exists and it is the trivial solution zero. 5 Other Fractional Diﬀerential Equations In this section we review the basic theory of fractional diﬀerential equations based on Miller – Ross fractional derivatives. The exposition follows Podlubny [333]. 30) j=1 Dσk −1 u(t) t=0 = bk , k = 1, . . 31) where Dσk σ k −1 D := := Dαk Dαk−1 · · · Dα1 , Dαk −1 Dαk−1 · · · Dα1 , k σk = αj , k = 1, .

12 with p = 1 and q = ∞ follows. 4. 13. Let ν > γ ≥ 0, and let f ∈ L1 (0, x) have an integrable fractional derivative Dν f ∈ L∞ (0, x) such that Dν−j f (0) = 0 for j = 1, . . , [ν] + 1. 22) t∈[0,x] 0 where Ω5 (x) = xm(ν−γ)+1 . 3 Applications (i) Uniqueness of solution to fractional initial value problem ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ Let γ i ≥ 0, ν > γ i + 1/2, i = 1, . . , r ∈N. Let f ∈ L1 (0, x) have an integrable fractional derivative Dν f ∈ L∞ (0, x) such that Dν−j f (0) = αj ∈R, j = 1, . . , [ν] + 1. ⎪ ⎪ ⎪ Furthermore, let ⎪ ⎪ ⎪ ⎩ Dν f (t) = F (t, {Dγ i f (t)}r ) for all t ∈ [0, x].

2) Let f ∈ Cxν0 ([a, b]), ν ≥ 1 and f (i) (x0 ) = 0, i = 0, 1, . . , n − 1, n := [ν]. 7) f (x) = (Jνx0 Dxν0 f )(x). 9) for all x ∈ [a, b] with x ≥ x0 . 9) is true, also when 0 < ν < 1. We need the following lemma from [17]. 3. Let f ∈ C([a, b]), μ, ν > 0. Then x0 Jμx0 (Jνx0 f ) = Jμ+ν (f ). 10) 26 3. 4. Let ν ≥ 1, γ ≥ 0, be such that ν − γ ≥ 1, so that γ < ν. Call n := [ν], α := ν − n ; m := [γ], ρ := γ − m. Note that ν − m ≥ 1 and n − m ≥ 1. Let f ∈ Cxν0 ([a, b]) be such that f (i) (x0 ) = 0, i = 0, 1, .

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