By George A. Anastassiou (auth.), George A. Anastassiou, Oktay Duman (eds.)
Advances in utilized arithmetic and Approximation idea: Contributions from AMAT 2012 is a set of the easiest articles awarded at “Applied arithmetic and Approximation conception 2012,” a world convention held in Ankara, Turkey, could 17-20, 2012. This quantity brings jointly key paintings from authors within the box protecting issues corresponding to ODEs, PDEs, distinction equations, utilized research, computational research, sign thought, confident operators, statistical approximation, fuzzy approximation, fractional research, semigroups, inequalities, distinct services and summability. the gathering may be an invaluable source for researchers in utilized arithmetic, engineering and statistics.
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88) n Notice that Db− f = (−1)n f (n) , n ∈ N. In  we introduced a balanced fractional derivative combining both right and left fractional Caputo derivatives. 22 (). Let f ∈ ACn ([a, b]), α > 0, n ∈ N, n := α , α ≥ ρ + 1, ρ > 0, α / N. Assume f (k) (b) = 0, k = r, r +1, . . , n−1, and Db− f ∈ L∞ ([a, b]). 90) ∀ x ∈ [a, b]. 23. Let ( fi , αi , ρi ), i = 1, . . 22. Set α := ∑ (αi − ρi), γ := ∏ (αi − ρi ), and let p ≥ 1. Here a, b ∈ R, a < b. 91) ⎞ α − mp + 1p 1 p γ (b − a) m ∏ (Γ (αi − ρi + 1)) (α − m + 1) 1 p ⎟ ⎟ ⎠ m ∏ ν i Db− fi i=1 p,(a,b) .
These Φi are convex, increasing, and continuous on R+ . 48), we get b I1 := a α (x − a) αi Ia+ fi (x) (x − a)αi m ∏ i=1 dx ≤ ⎞⎛ ⎛ ⎜ ⎜ ⎝ γ m ∏ (Γ (αi + 1)) b a ⎛ pi (α − m + 1) ∏ (Γ (αi + 1)) ⎟⎜ ⎟ ⎝∏ ⎠ b i=1 a i= j (b − x)α −m+1 f j (x) pj ⎟ | fi (x)| pi dx⎠ · dx ≤ ⎞ γ (b − a)α −m+1 m ⎞ m i=1 ⎜ ⎜ ⎝ pi pi i=1 (α − m + 1) m m i=1 i=1 ⎟ ⎟ ⎠ m b ∏ i=1 a | fi (x)| pi dx . 53) Notice that ∑ αi pi > α ; thus, β := α − ∑ αi pi < 0. Since 0 < x − a < b − a (x ∈ (a, b)), then (x − a)β > (b − a)β . 54), it holds pi dx.
Ii) λm Φ j f j ; Φ1 (| f1 |) , Φ2 (| f2 |) , . . , Φ j integrable functions, where Φ j Let now fj f j , . . , Φm (| fm |) are all Lebesgue means absent item. u (x) = (x − a)α , x ∈ (a, b) . 50) y ∈ (a, b), where α > m − 1. 51) m ∏ i=1 a b Φi (| fi (x)|) dx , αi (| fi |) finite, i = 1, . . 48). 51) turns to b m ∏ a i=1 αi fi (x) dx ≤ Ia+ 34 George A. 52) i=1 αi (| fi |) finite and fi Lebesgue integrable, i = 1, . . , m. where α > m − 1, fi with Ia+ Next let pi > 1, and Φi (x) = x pi , x ∈ R+ .