By Yong Zhou
This worthwhile e-book is dedicated to a quickly constructing quarter at the examine of the qualitative idea of fractional differential equations. it's self-contained and unified in presentation, and offers readers the mandatory heritage fabric required to move extra into the topic and discover the wealthy study literature.
The instruments used contain many classical and glossy nonlinear research equipment similar to fastened aspect thought, degree of noncompactness process, topological measure strategy, the Picard operators process, serious aspect thought and semigroups concept. according to examine paintings conducted via the writer and different specialists prior to now 4 years, the contents are very new and complete. it really is invaluable 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 resources for Basic Theory of Fractional Differential Equations
17. (Lakshmikantham, Wen and Zhang, 1994) Suppose that p(t, θ) is a p-function. For A > 0, τ ∈ (σ, ∞) (τ may be σ if σ > −∞), let x ∈ C([p(τ, −1), τ + A], Rn ) and ϕ ∈ C([−1, 0], Rn ). Then we have (i) xt is continuous in t on [τ, τ +A] and p˜(t, θ) = p(τ +t, θ)−τ is also a p-function; (ii) if p(τ + t, −1) < τ for t > 0, then there exists −1 < s(τ, t) < 0 such that p(τ + t, s(τ, t)) = τ and p(τ + t, −1) ≤ p(τ + t, θ) ≤ τ, for − 1 ≤ θ ≤ s(τ, t), τ ≤ p(τ + t, θ) ≤ τ + t, for s(τ, t) ≤ θ ≤ 0. Moreover, s → 0 uniformly in τ as t → 0; (iii) there exists a function η ∈ C([p(τ, −1), τ ], Rn ) such that η(p(τ, θ)) = ϕ(θ) for − 1 ≤ θ ≤ 0.
Direct 1 calculation gives that (t − s)q−1 ∈ L 1−q1 [t0 , t], for t ∈ [t0 , t0 + α] and q1 ∈ (0, q). In light of H¨older inequality, we obtain that (t − s)q−1 f (s, xs ) is Lebesgue integrable with respect to s ∈ [t0 , t] for all t ∈ [t0 , t0 + α], and t t0 |(t − s)q−1 f (s, xs )|ds ≤ (t − s)q−1 1 L 1−q1 [t0 ,t] m 1 L q1 [t0 ,t0 +α] . Eqn. ws-book975x65 Fractional Functional Differential Equations or setting u = t0 + s, 1 g(t0 + t, xt0 +t ) − g(t0 , xt0 ) = Γ(q) xt0 = ϕ. 16), we have xt0 +t (θ) = x(p(t0 + t, θ)) = x(˜ p(t, θ) + t0 ) = η(˜ p(t, θ)) + z(˜ p(t, θ)) = η˜t (θ) + z˜t (θ), for t ∈ [0, α].
Note that the frequently used symbol “xt ” (in Hale, 1977; Lakshmikantham, 2008; Lakshmikantham, Wen and Zhang, 1994, xt (θ) = x(t + θ), where −τ ≤ θ ≤ 0, r > 0, r = const) in the theory of functional differential equations with bounded delay is a partial case of the above definition. Indeed, in this case we can put p(t, θ) = t + rθ, θ ∈ [−1, 0]. 16. 14) holds almost everywhere on [t0 , t0 + α]. We need the following lemma relative to p-function before we proceed further, which is taken from Lakshmikantham, Wen and Zhang, 1994.