A note on a class of convolution integral equations (Q2788833)

The aim of the paper is to study the convolution integral equation given by NEWLINE\[NEWLINE\int_a^x(x-t)^{\lambda-1}{\mathcal F}_{\rho,\lambda}^\sigma[\omega(x-t)^\rho]\varphi(t)\,dt=g(x),NEWLINE\]NEWLINE where the kernel function \({\mathcal F}_{\rho,\lambda}^\sigma(x)\) is defined by NEWLINE\[NEWLINE{\mathcal F}_{\rho,\lambda}^\sigma(x)=\sum_{k=0}^\infty\frac{\sigma(k)}{\Gamma(\rho k+\lambda)}x^kNEWLINE\]NEWLINE with \(\rho,\lambda\in{\mathbb C}\) and suitable sequence \(\{\sigma(k)\}\). The latter function includes the generalized Mittag-Leffler function and extended hypergeometric functions as special cases. Some conditions for the boundedness of the corresponding integral operator \({\mathcal J}_{\rho,\lambda,a+;\omega}^\sigma\) on \(L^1(a,b)\) are obtained. Relations of \({\mathcal J}_{\rho,\lambda,a+;\omega}^\sigma\) and the Riemann-Liouville operator are studied. These relations are used for solving the above integral equation on \(L^1(a,b)\) under certain conditions.