By Yong Zhou
This valuable booklet is dedicated to a swiftly constructing zone at the study of the qualitative idea of fractional differential equations. it's self-contained and unified in presentation, and gives readers the mandatory historical past fabric required to head additional into the topic and discover the wealthy examine literature.
The instruments used contain many classical and smooth nonlinear research equipment resembling mounted element concept, degree of noncompactness procedure, topological measure technique, the Picard operators method, severe aspect conception and semigroups conception. in response to examine paintings conducted via the writer and different specialists in past times 4 years, the contents are very new and finished. it's valuable to researchers and graduate scholars for examine, seminars, and complex graduate classes, in natural and utilized arithmetic, physics, mechanics, engineering, biology, and comparable disciplines.
Readership: Graduate scholars and researchers within the fields of fractional differential equations, fractional calculus and comparable parts of analysis.
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Extra info for Basic Theory of Fractional Differential Equations
Eqn. ws-book975x65 Fractional Functional Differential Equations q−1 1−q2 where b1 = ∈ (−1, 0), K = ℓ1 1 L q2 [t0 , t0 +A] 49 . 37) ∗ with k → 0 as xt − yt ∗ → 0. By the hypothesis that g(t, ϕ) is atomic at 0 on Ω, there exist a nonsingular continuous matrix function A(t, yt ) and a function L(t, yt , ψ) which is linear in ψ such that gϕ′ (t, yt )ψ = A(t, yt )ψ(0) + L(t, yt , ψ). 39) ∗ if ψ(θ) = 0 for −1 ≤ θ ≤ s. 17, there is s(t1 , t−t1 ) ∈ [−1, 0] with s(t1 , t − t1 ) → 0 as t → t1 such that |L(t, yt , xt − yt )| ≤ γ(t, yt , −s(t1 , t − t1 )) xt − yt ∗.
T ∈ I for all z, y ∈ C. 6. A mapping g : I × C → Rn is said to be Carath´eodory if (i) t → g(t, z) is Lebesgue measurable for each z ∈ C; (ii) z → g(t, z) is continuous almost everywhere for t ∈ I. e. t ∈ I for all z ∈ C with z ∗ ≤ r. Eqn. 7. A mapping h : I × C → Rn is said to be Chandrabhan if (i) t → h(t, z) is Lebesgue measurable for each z ∈ C; (ii) z → h(t, z) is nondecreasing almost everywhere for t ∈ I. e. t ∈ I for all z ∈ C with z ∗ ≤ r. 8. 1). 9. 1) on J if the function [a(t) − k(t, at )] is absolutely continuous on I, and C α t0 Dt (a(t) at0 ≤ ϕ.
12) are satisfied. 13) has a minimal and a maximal solution in [0, b] defined on [−1, σ]. Eqn. ws-book975x65 Basic Theory of Fractional Differential Equations p-Type Neutral Equations Introduction Let C = C([−1, 0], Rn ) denote the space of continuous functions on [−1, 0]. For any element ϕ ∈ C, define the norm ϕ ∗ = supθ∈[−1,0] |ϕ(θ)|. 15) q where C t0 Dt is Caputo fractional derivative of order 0 < q < 1, Ω is an open subset of [0, ∞) × C and g, f : Ω → Rn are given functionals satisfying some assumptions that will be specified later.