Geodesic
Continuous and generalized solutions of singular integro-differential equations in Banach spaces
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, no. 11 (2012), pp. 62-74
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Continuous and generalized solutions of singular equations in Banach spaces are studied. We apply Lyapunov–Schmidt's ideas and the generalized Jordan sets techniques and reduce partial differential-operator equations with the Fredholm operator in the main expression to regular problems. In addition the left and right regularizators of singular operators in Banach spaces and fundamental operators in the theory of generalized solutions of singular equations are constructed.
Keywords: singular PDE, regularizators, fundamental operator-function.
Mots-clés : distributions 
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N. A. Sidorov; M. V. Falaleev. Continuous and generalized solutions of singular integro-differential equations in Banach spaces. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, no. 11 (2012), pp. 62-74. http://geodesic.mathdoc.fr/item/VYURU_2012_11_a6/

[1] R. Cassol, R. Schowalter, Singular and Degenerate Cauchy Problems, Academ. Press, N.Y.–San Francisco–London, 1976 | MR

[2] Falaleev M. V., “Fundamental Operator-functions of the Singular Differential Operators in the the Banach Spaces”, Sib. Math. J., 41:5 (2000), 1167–1182 | MR | Zbl

[3] J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford University Press, Inc., N.Y., 1985 | MR | Zbl

[4] Grazhdantseva E. Yu., “The Fundamental Operator Function of an Incomplete Singular Differential-difference Operator in Banach Spaces”, J. of Optimization, Control and Intelligence, 7 (2003)

[5] Kato T., The Theory of Perturbations of Linear Operators, Mir Publ., M., 1972 | Zbl

[6] Korpusov M. O., Pletnev Y. D., Sveshnikov A. G., “On Quasi-steddy Process in the Conducting Medium Without Dispersion”, Comput. Math. Math. Phys., 40:8 (2000), 1237–1249 | MR | Zbl

[7] Krein S. G., Chernyshov N. I., Singularly Disturbed Differential Equations in Banach Spaces, Preprint, Institute of Mathematics, Siberian Branch, USSR, Acad. Sci., 1979

[8] I. Petrovsky, “Über das Cauchysche problem für system von partiellen Differentialgleichungen”, Math. Sb., 2:5 (1937), 815–870

[9] L. Schwartz, Theorie des distributions, v. I, Paris, 1950; v. II, 1951

[10] Sidorov N. A., “The Branching of the Solutions of Differential Equations with a Degeneracy”, Differential Equations, 1973, no. 9, 1464–1481 | MR | Zbl

[11] Sidorov N. A., General Regularization Questions in Problems of Branching Theory, Irkutsk Gos. Univ., Irkutsk, 1982 | MR

[12] Sidorov N. A., “The Initial-value Problem for Differential Equations with the Fredholm Operator in the Main Part”, Vestnik of Chelyabinsk State University, Ser. 3. Mathematics. Mechanics, 1999, no. 2, 103–112 | MR

[13] Sidorov N. A., Blagodatskaya E. B., “Differential Equations with the Fredholm Operator in the Leading Differential Expression”, Soviet Math. Dokl., 44:1 (1992), 302 – 305 | MR

[14] N. Sidorov, B. Loginov, A. Sinithyn, M. Falaleev, Lyapunov–Schmidt methods in Nonlinear Analysis and Applications, Kluwer Academic Publishers, Dordrecht, 2002 | MR | Zbl

[15] G. Sviridyuk, V. Fedorov, Linear Sobolev Type Equations and Degenerate Semigroups of Operators, VSP, Utrecht, 2003, 228 pp. | MR | Zbl

[16] Trenogin V. A., “Branching of Solutions of Nonlinear Equations in Banach Spaces”, Uspekhy Mathemat. Sciences, 13:4 (1958), 197–203 | MR

[17] M. M. Vainberg, V. A. Trenogin, The Theory of Branching of Solutions of Nonlinear Equations, Wolters-Noordhoff, Groningen, 1974

[18] Vladimirov V. S., Generalized Functions in Mathematical Physics, Nauka, M., 1979 | MR

[19] Whitham G. B., Linear and Non-Linear Wales, Mir, M., 1977 | MR

[20] M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. Ferreira, “Existence and Uniform Decay for a Non-Linear Viscoelastic Equation with Strong Damping”, Math. Meth. Appl. Sci., 24 (2001), 1043–1053 | DOI | MR | Zbl