An Extension of Darbo’s Theorem and Its Application

Abstract

Here, some extensions of Darbo fixed point theorem associated with measures of noncompactness are proved. Then, as an application, our attention is focused on the existence of solutions of the integral equation $x(t)=F(t,f(t,x({\alpha }_{\mathrm{1}}(t))$, $x({\alpha }_{\mathrm{2}}(t)))$, $((Tx)(t)/\Gamma (\alpha ))\times {\int }_{\mathrm{0}}^t‍(u(t,s,{\max }_{[\mathrm{0},r(s)]}\left|x({\gamma }_{\mathrm{1}}(\tau ))\right|$, ${\max }_{[\mathrm{0},r(s)]}\left|x({\gamma }_{\mathrm{2}}(\tau ))\right|)$/$(t-s{)}^{\mathrm{1}-\alpha })ds$, ${\int }_{\mathrm{0}}^{\infty }v(t,s,x(t))ds)$, in the space of real functions defined and continuous on the interval $[\mathrm{0,1}]$.