Generalized Laplace transform of locally integrable functions defined on $[0,\infty)$

Bogoljub Stanković

In $[$Bull. Cl. Sci. Math. Nat. Sci. Math. {\bf40} $(2015),\ 99-113]$ we defined the Laplace transform on a bounded interval $[0,b]$, denoted by $^0{\cal L}$, using some ideas of H. Komatsu $[$J. Fac. Sci. Univ. Tokyo, IA, {\bf34} {\rm(1987), 805--820]} and $[$Structure of solutions of differential equations $($Katata/Kyoto, $1995)$, pp. {\rm 227--252}, World Sci. Publishing, River Edge, NJ, {\rm1996]}. %(\cite{Kom} and \cite{Kom1}). We use this definition to extend it to the space of locally integrable functions defined on $[0,\infty)$, which is a wider class then functions $L$ used by G. Doetsch $[$Handbuch der Lalace-Transformation I, Basel -- Stuttgart, $1950-1956$, p.~$32]$. %(\cite{Do}, I, p.~32). As an application we give solutions of integral equations of the convolution type, defined on a bounded interval, or on the half-axis as well, and of equations with fractional derivatives.