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Mathematical Sciences Letters
An International Journal


Volumes > Vol. 2 > No. 1


Positive continuous solution of a quadratic integral equation of fractional orders

PP: 19-27
A. M. A. El-Sayed, H. H. G Hashem, Y. M. Y. Omar,
We are concerned here with the existence of a unique positive continuous solution for the quadratic integral of fractional orders \[ x(t)=a(t)+\lambda\int_{0}^{t}\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}f_1(s,x(s)) ~ds.\int_{0}^{t}\frac{(t-s)^{\beta-1}}{\Gamma(\beta)}f_2(s,x(s))~ds,~~~~~t\in I \] where $f_1,~f_1$ are Carath\{e}odory functions. As an application the Cauchy problems of fractional order differential equation \[ *D^\alpha\sqrt{x(t)}=f(t,x(t)),~~t>0 \] with one of the two initial values $~x(0) =0$ or $I^{1-\alpha}\sqrt{x(t)}=0$ will be studied.\Some examples are considered as applications of our results.

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