Solutions of the Cauchy problem for factorized Sturm-Liouville equation in a Banach algebra

N.S.Trush

Let ${\mathcal A}$ be a Banach algebra with the unit, $a\in{\mathcal A}$, and $\tau\in L_p((0,1),{\mathcal A}), \,\,\,p\in[1,\infty).$ We derive integral representations of solutions of the initial value problem for the differential equation $-(\frac{d}{dx}+\tau)(\frac{d}{dx}-\tau)u=ua^2$ on the interval $[0,1]$.

1. Hryniv, R.O., Mykytyuk, Ya.V. {\it Transformation operators for Sturm--Liouville operators with singular potentials} \sep Math. Phys. Anal. Geom. -- 2004. -- V.7. -- P.119--149.

2. Marchenko V. Sturm--Liouville Operators and Their Applications, Naukova Dumka Publ., Kiev, 1977 (in Russian); Engl. transl.: Birkhauser Verlag, Basel, 1986.

3. Carlson R. An inverse problem for the matrix Schr\"{o}dinger equation. J. Math. Anal. Appl. -- 2002. -- 267(2). -- P.564--575.

4. Chelkak D., Korotyaev E. {\it Parametrization of the isospectral set for the vector-valued Sturm--Liouville problem. // J. Funct. Anal. -- 2006. -- V.241. --P.359--373.

5. Albeverio S., Hryniv R., Mykytyuk Ya. Inverse spectral problems for Dirac operators with summable potentials // Russian Journal of Math. Physycs. -- 2005. -- V.12, no.4 -- P.406--424.

6. R.~E.~Edwards, Fourier Series. A Modern Introduction, $2^{\mathrm{nd}}$ ed., Vol.{1}, {Graduate Texts in Mathematics}, {64}, Springer-Verlag, New York-Berlin, 1979.

	Solutions of the Cauchy problem for factorized Sturm-Liouville equation in a Banach algebra
Author	N.S.Trush nctrush@mail.ru Lviv Ivan Franko National University
Abstract	Let ${\mathcal A}$ be a Banach algebra with the unit, $a\in{\mathcal A}$, and $\tau\in L_p((0,1),{\mathcal A}), \,\,\,p\in[1,\infty).$ We derive integral representations of solutions of the initial value problem for the differential equation $-(\frac{d}{dx}+\tau)(\frac{d}{dx}-\tau)u=ua^2$ on the interval $[0,1]$.
Keywords	Cauchy problem, factorized Sturm-Liuville equation, Banach algebra
DOI	doi:10.30970/ms.31.1.75-82
Reference	1. Hryniv, R.O., Mykytyuk, Ya.V. {\it Transformation operators for Sturm--Liouville operators with singular potentials} \sep Math. Phys. Anal. Geom. -- 2004. -- V.7. -- P.119--149. 2. Marchenko V. Sturm--Liouville Operators and Their Applications, Naukova Dumka Publ., Kiev, 1977 (in Russian); Engl. transl.: Birkhauser Verlag, Basel, 1986. 3. Carlson R. An inverse problem for the matrix Schr\"{o}dinger equation. J. Math. Anal. Appl. -- 2002. -- 267(2). -- P.564--575. 4. Chelkak D., Korotyaev E. {\it Parametrization of the isospectral set for the vector-valued Sturm--Liouville problem. // J. Funct. Anal. -- 2006. -- V.241. --P.359--373. 5. Albeverio S., Hryniv R., Mykytyuk Ya. Inverse spectral problems for Dirac operators with summable potentials // Russian Journal of Math. Physycs. -- 2005. -- V.12, no.4 -- P.406--424. 6. R.~E.~Edwards, Fourier Series. A Modern Introduction, $2^{\mathrm{nd}}$ ed., Vol.{1}, {Graduate Texts in Mathematics}, {64}, Springer-Verlag, New York-Berlin, 1979.
Pages	75-82
Volume	31
Issue	1
Year	2009
Journal	Matematychni Studii
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