Abstract
Asymptotic behavior of positive solutions of nonlinear fractional differential equations with Caputo-type Hadamard derivative
Highlights
IntroductionFor at least n−times differentiable function f : (a, +∞) → R the Caputo-type Hadamard derivative of fractional order α is defined as
Consider the initial value problemC,H Dar x(t) = e(t) + f (t, x(t)), a > 1, δkx(a) = bk, bk, k = 1, 2, · · · n are real constants, [1]and the integral equation x(t) = e(t) + tr−1k(t, s) > 1, [2]where r = n + α − 1, α ∈ (0, 1), n ∈ Z+, C,H Dar x(t) is the Caputo-type Hadamard modification of fractional derivative of a Cn− scalar valued function x(t) defined on the interval [a, ∞), which was recently proposed by Jarad et al [1]
We introduce some notations and definition of fractional calculus [3,4]
Summary
For at least n−times differentiable function f : (a, +∞) → R the Caputo-type Hadamard derivative of fractional order α is defined as. All conditions of Theorem 6 are satisfied and every positive solution x of equation [1] is bounded. In addition we assume that the function b(t) ln t a n−1 is bounded and conditions [7], [9] and [15] hold. Q ds Clearly, the conclusion of Theorem 6 holds This together with [7] and [8] imply that the second integral in this inequality is bounded and one can see that x(t) ≤ M + e(t),.
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