By Yong Zhou
This useful booklet is dedicated to a speedily constructing zone at the examine of the qualitative concept of fractional differential equations. it's self-contained and unified in presentation, and gives readers the required heritage fabric required to move extra into the topic and discover the wealthy examine literature.
The instruments used contain many classical and sleek nonlinear research equipment corresponding to fastened aspect idea, degree of noncompactness strategy, topological measure procedure, the Picard operators process, severe element conception and semigroups concept. in accordance with study paintings performed by way of the writer and different specialists prior to now 4 years, the contents are very new and accomplished. it's priceless to researchers and graduate scholars for learn, 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 similar components of analysis.
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Extra resources for Basic Theory of Fractional Differential Equations
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. e. 1) on J if the function [b(t) − k(t, bt )] is absolutely continuous on I, and [b(t) − k(t, bt )] is absolutely continuous on I, and C α t0 Dt (b(t) bt0 ≥ ϕ. e. 1) on J. 10. 1), one has x(t) ≤ xM (t) for all t ∈ J. 1) on J. We need the following hypotheses in the sequel. 1) has a lower solution a and an upper solution b with a ≤ b; (k0) k(t, z) is continuous with respect to t on I for any z ∈ C; (k1) |k(t, z) − k(t, y)| ≤ k0 z − y ∗ , for z, y ∈ C, t ∈ I, where k0 > 0; (k2) k(t, z) is nondecreasing with respect to z for any z ∈ C and almost all t ∈ I; 1 1 (f1) f is L δ -Lipschitz, and there exists η ∈ [0, α) such that |f (t, 0)| ∈ L η (I, R+ ); (f2) f (t, z) is nondecreasing with respect to z for any z ∈ C and almost all t ∈ I; 1 (g1) g is L β -Carath´eodory; (g2) g(t, z) is nondecreasing with respect to z for any z ∈ C and almost all t ∈ I; 1 (h1) h is L γ -Chandrabhan.
1) has a minimal and a maximal solution in [a, b] defined on J provided that Lσ (1+q1 )(1−δ) < 1. 12) Proof. t 1 (Ax)(t) = k(t, xt ) − k(t0 , ϕ) + (t − s)α−1 f (s, xs )ds, for t ∈ I, Γ(α) t0 (Ax)(t0 + θ) = 0, for θ ∈ I0 , t 1 (Bx)(t) = ϕ(0) + (t − s)α−1 g(s, xs )ds, for t ∈ I, Γ(α) t0 (Bx)(t0 + θ) = ϕ(θ), for θ ∈ I0 , and t 1 (t − s)α−1 h(s, xs )ds, for t ∈ I, Γ(α) t0 (Cx)(t0 + θ) = 0, for θ ∈ I0 , (Cx)(t) = where x ∈ C(J, Rn ). Obviously, Ax+ Bx+ Cx ∈ C(J, Rn ) for every x ∈ C(J, Rn ).
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.
Basic Theory of Fractional Differential Equations by Yong Zhou
Categories: Mathematical Physics