




Positive continuous solution of a quadratic integral equation of fractional orders 

PP: 1927 

Author(s) 

A. M. A. ElSayed,
H. H. G Hashem,
Y. M. Y. Omar,


Abstract 

We are concerned here with the existence of a unique positive continuous solution
for the quadratic integral of fractional orders
\[
x(t)=a(t)+\lambda\int_{0}^{t}\frac{(ts)^{\alpha1}}{\Gamma(\alpha)}f_1(s,x(s))
~ds.\int_{0}^{t}\frac{(ts)^{\beta1}}{\Gamma(\beta)}f_2(s,x(s))~ds,~~~~~t\in I
\]
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. 




