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Zbl 1123.34302
Ibrahim, Rabha W.; Momani, Shaher
On the existence and uniqueness of solutions of a class of fractional differential equations.
(English)
[J] J. Math. Anal. Appl. 334, No. 1, 1-10 (2007). ISSN 0022-247X

Summary: We investigate the existence and uniqueness of solutions for the following class of multi-order fractional differential equations $$D_{\beta_1}^{\gamma_1,\delta_1}\dots D_{\beta_n}^{\gamma_n, \delta_n}u(t):= \prod^n_{i=1}D_{\beta_i}^{\gamma_i,\delta_i} u(t):=D_{\beta_i,n}^{\gamma_i, \delta_i}u(t)=f(t,u(t)),\ t\in[0,1],$$ $$u(0)=0, \quad\sum^n_{i=1} \delta_i\le 1,\quad \gamma_i>0,\quad\beta_i>0,\quad 1\le i\le n,$$ where $D_{\beta_i,n}^{\gamma_i,\delta_i}$ denotes the generalized Erdélyi-Kober operator of fractional derivative of order $\delta_i$. Moreover, some properties concerning the positive, maximal, minimal, and continuation of solutions are obtained.
MSC 2000:
*34A12 Initial value problems for ODE
26A33 Fractional derivatives and integrals (real functions)

Keywords: multi-order fractional differential equations; positive solution; fixed point theorem; existence; uniqueness; Erdelyi-Kober operator

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