By Sagan H.

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**Extra resources for Advanced calculus**

**Example text**

From the properties of the function V , it follows that there exists a constant δ = δ (t0 , ε ) > 0 such that if ||x|| ≤ δ , then sup V (t0+ , x) < w1 (ε ). 4). We shall prove that ||x(t;t0 , x0 )|| < ε for t ≥ t0 . Suppose that this is not true. 5) for which ||x0 || < δ and t ∗ > t0 , tk < t ∗ ≤ tk+1 , for some fixed integer k such that ||x(t ∗ )|| ≥ ε and ||x(t;t0 , x0 )|| < ε , t ∈ [t0 ,tk ]. 23) and the properties of E + Ik , for any k, we can find t 0 , tk < t 0 ≤ t ∗ , such that ||x(t 0 )|| > ε and x(t 0 ;t0 , x0 ) ∈ Ω .

8) are equi-bounded. 2 and we omit the details here. 1 holds, and a constant H > 0 such that: 1. 1) hold for (t, x) ∈ [t0 , ∞) × Rn , where w1 , w2 ∈ K and w1 (u) → ∞ as u → ∞. 2. 2) is valid whenever ||φ (0)|| ≥ H, V (t + θ , φ (θ )) ≤ p(V (t, φ (0))) for −r ≤ θ ≤ 0, where t ∈ [t0 , ∞), φ ∈ C , β ∈ (0, 1), p is continuous and non-decreasing on R+ , and p(u) > u as u > 0. 8) are uniformly bounded. 8) are uniformly ultimately bounded. 17 is related to a result of Hale (see [Hale 1977]) for integer-order functional differential equations.

All comparison results will be given in terms of Caputo fractional derivatives. Applying some relations between Caputo and Riemann–Liouville fractional derivatives, we shall use these lemmas in the case of Riemann–Liouville fractional derivatives. 1). 11) t0 Dt u = g(t, u), where 0 < β < 1, g : [t0 , ∞) × R+ → R. β Let u0 ∈ R+ . 11), which satisfies the initial condition u+ (t0 ;t0 , u0 ) = u0 . e. 12) is given by β −1 + u (t0 ;t0 , x0 ) = u0 . t0 Dt Note that in this case, the above initial condition can be replaced by the condition u0 u+ (t0 ;t0 , u0 ) = (t − t0 )β −1 .