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On the existence of continuous solutions for a singular system of non-linear fractional differential equations. (English) Zbl 1153.26004

The author applies Krasnoselskii’s fixed point theorem to prove the existence continuous solution of the system of fractional order integral equations

${x}_{i}\left(t\right)={h}_{i}\left(t\right)+{\lambda }_{i}{I}^{{\alpha }_{i}}\left[{f}_{i}\left(x\left(t\right)\right)+{g}_{i}\left(x\left(t\right)\right)\right],\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}t\in \left[0,1\right],\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\alpha }_{i}\in \left(0,1\right),\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}1\le i\le n$

under the conditions of continuity and monotonicity (the ${f}_{i}$ are nondecreasing and the ${g}_{i}$ are nonincreasing). The existence of the maximal and minimal solutions has been proved when ${g}_{i}\left(x\right)=c$. These results generalize that of the author, A. M. A. El-Sayed and O. L. Moustafa [Ann. Soc. Math. Pol., Ser. I, Commentat. Math. 42, No. 2, 209–220 (2002; Zbl 1033.45003)].

##### MSC:
 26A33 Fractional derivatives and integrals (real functions) 35A35 Theoretical approximation to solutions of PDE