Complementary equations: a fractional differential equation and a Volterra integral equation
It is shown that a continuous, absolutely integrable function satisfies the initial value problem \[ D^{q}x(t) = f(t,x(t)), \qquad \lim_{t \to 0^+} t^{1-q}x(t) = x^{0} \qquad (0 < q < 1) \] on an interval $(0, T]$ if and only if it satisfies the Volterra integral equation \[ x(t) = x^...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
2015-03-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Subjects: | |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=3782 |
| Summary: | It is shown that a continuous, absolutely integrable function satisfies the initial value problem
\[
D^{q}x(t) = f(t,x(t)), \qquad \lim_{t \to 0^+} t^{1-q}x(t) = x^{0} \qquad (0 < q < 1)
\]
on an interval $(0, T]$ if and only if it satisfies the Volterra integral equation
\[
x(t) = x^{0}t^{q-1}+\frac{1}{\Gamma (q)}\int_{0}^{t}(t-s)^{q-1}f(s, x(s))\,ds
\]
on this same interval. In contradistinction to established existence theorems for these equations, no Lipschitz condition is imposed on $f(t,x)$. Examples with closed-form solutions illustrate this result. |
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| ISSN: | 1417-3875 |