By George A. Anastassiou

Fractional differentiation inequalities are via themselves an incredible zone of study. they've got many functions in natural and utilized arithmetic and plenty of different technologies. probably the most very important purposes is in developing 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 recommendations of the above equations.

In this publication the writer provides the Opial, Poincaré, Sobolev, Hilbert, and Ostrowski fractional differentiation inequalities. effects for the above are derived utilizing 3 kinds of fractional derivatives, specifically via Canavati, Riemann-Liouville and Caputo. The univariate and multivariate circumstances are either tested. each one bankruptcy is self-contained. the speculation is gifted systematically in addition to the purposes. the applying to details idea can also be examined.

This monograph is appropriate for researchers and graduate scholars in natural arithmetic. utilized mathematicians, engineers, and different utilized scientists also will locate this ebook useful.

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Extra info for Fractional Differentiation Inequalities

Example text

K. Also we assume the order ν > γ1 > γ2 > γ3 > · · · > γk. Let n := [ν]; then n − 1 = [ν − 1]. 29). 29) over the whole interval [x0 , b]. 29). Call g := f1 − f2 ∈ ([x0 , b]). 29) we obtain Cxν0 ([x0 , b]). Clearly g ∈ Cxν−1 0 k γ qj (w) · (Dx0j g)(w) + qk+1 (w) · g(w) = 0, (Dxν0 g)(w) + j=1 all x0 ≤ w ≤ b, g (i) (x0 ) = 0, i = 0, 1, . . , n − 1. Hence k (Dxν0 g)(w) · (Dxν−1 g)(w) 0 γ qj (w) · (Dx0j g)(w) · (Dxν−1 g)(w) 0 + j=1 + qk+1 (w) · g(w) · (Dxν−1 g)(w) = 0. 0 Next we integrate over [x0 , x ˜] ⊆ [x0 , b], k x ˜ x ˜ (Dxν0 g)(w) · (Dxν−1 g)(w) · dw + 0 x0 γ qj (w) · (Dx0j g)(w) · (Dxν−1 g)(w)dw 0 j=1 x0 x ˜ + x0 qk+1 (w) · g(w) · (Dxν−1 g)(w)dw = 0.

F2 n−1 (i) ≤ )| n−1 (i) qi (t)|f1 (t) − f2 (t)| = i=0 qi (t)|g (i) (t)|. i=0 From (n−1) (Daν g(t))2 = (Daν g(t)){F (t, f1 , f1 , . . , f1 (n−1) ) − F (t, f2 , f2 , . . , f2 )}, we obtain n−1 (Daν g(t))2 ≤ |Daν g(t)| qi (t)|g (i) (t)| i=0 n−1 qi (t)(|Daν g(t)| |g (i) (t)|). = i=0 Integrating the last inequality we get n−1 b b (Daν g(t))2 dt ≤ a qi (t)|g (i) (t)| |Daν g(t)|dt a i=0 n−1 ≤ b qi |g (i) (t)| |Daν g(t)|dt. 27) a for i = 0, 1, . . , n − 1. 27) we derive b b ((Daν g)(t))2 dt ≤ φ(b) a b ((Daν g(t))2 dt.

12) 0 Φ(t)q dt. e. e. in (0, x). The function srp+1 (z(s)z (s))1/q is integrable over (0, x) as rp + 1 ≥ 0 and z(s)z (s) is measurable and essentially bounded on (0, x). Applying H¨older’s inequality, we obtain x 0 1/p x srp+1 (z(s)z (s))1/q ds ≤ 0 x(rp+2)/p (z(s))2/q ≤ . 2 Main Results 45 The result then follows when we observe that |Dγ f (s) Dν f (s)| is integrable as Dγ f ∈ AC[0, x] for ν − γ ≥ 1, and Dγ f ∈ C ([0, x]) for ν − γ ∈ (0, 1) , and Dν f ∈ L∞ (0, x). The following result deals with the extreme case of the preceding theorem when p = 1 and q = ∞.