Generalized Laplace transform of locally integrable functions defined on [0,∞)

Abstract

In [Bull. Cl. Sci. Math. Nat. Sci. Math. 40 (2015), 99 − 113] we defined the Laplace transform on a bounded interval [0, b], denoted by 0L, using some ideas of H. Komatsu [J. Fac. Sci. Univ. Tokyo, IA, 34 (1987), 805–820] and [Structure of solutions of differential equations (Katata/Kyoto, 1995), pp. 227–252, World Sci. Publishing, River Edge, NJ, 1996]. We use this definition to extend it to the space of locally integrable functions defined on [0,1), which is a wider class then functions L used by G. Doetsch [Handbuch der Lalace-Transformation I, Basel – Stuttgart, 1950 − 1956, 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.