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Generalized Laplace transform of locally integrable functions defined on $$[0,\infty)$$. (English) Zbl 1469.44001
Summary: In [the author, Bull. Cl. Sci. Math. Nat. Sci. Math. 40, 99–113 (2015; Zbl 1456.44002)] we defined the Laplace transform on a bounded interval $$[0,b]$$, denoted by $$^0\mathcal{L}$$, using some ideas of H. Komatsu [J. Fac. Sci., Univ. Tokyo, Sect. I A 34, 805–820 (1987; Zbl 0644.44001); in: Structure of solutions of differential equations. Proceedings of the Taniguchi symposium, Katata, Japan, June 26–30, 1995 and the RIMS symposium, Kyoto, Japan, July 3–7, 1995. Singapore: World Scientific. 227–252 (1996; Zbl 0894.35005)]. We use this definition to extend it to the space of locally integrable functions defined on $$[0,\infty)$$, which is a wider class than functions $$L$$ used by G. Doetsch [Handbuch der Laplace-Transformation. I: Die theoretischen Grundlagen der Laplace-Transformation. Basel: Birkhäuser (1950; Zbl 0040.05901); Handbuch der Laplace-Transformation. II: Anwendungen der Laplace-Transformation. Basel, Stuttgart: Birkhäuser (1955; Zbl 0065.34001); Handbuch der Laplace-Transformation. III: Anwendungen der Laplace-Transformation. 2. Abteilung. Basel: Birkhäuser (1956; Zbl 0070.33102)]. 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.
##### MSC:
 44A10 Laplace transform
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##### References:
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