Laplace transform $\dot{\mathcal L}$ for functions belonging to $L[0,b], \; 0< b < \infty$ is defined. This definition is given by using the idea 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]}. for Laplace hyperfunctions. As an application of $\dot{\mathcal L}$ we solve an equation with fractional derivative and an integral equation of the first kind of convolution type.