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functional-differential equation of hyperbolic type; Cauchy problem; Fredholm alternative; well-posedness; existence of solutions
We study the question of the existence, uniqueness, and continuous dependence on parameters of the Carathéodory solutions to the Cauchy problem for linear partial functional-differential equations of hyperbolic type. A theorem on the Fredholm alternative is also proved. The results obtained are new even in the case of equations without argument deviations, because we do not suppose absolute continuity of the function the Cauchy problem is prescribed on, which is rather usual assumption in the existing literature.
[1] Bielawski, D.: On the set of solutions of boundary value problems for hyperbolic differential equations. J. Math. Anal. Appl. 253 (2001), 334-340. DOI 10.1006/jmaa.2000.7118 | MR 1804580 | Zbl 0966.35080
[2] Carathéodory, C.: Vorlesungen über Reelle Funktionen. Leipzig und Berlin: B. G. Teubner (1918), German.
[3] Deimling, K.: Absolutely continuous solutions of Cauchy's problem for $u_{xy}=f(x,y,u, u_x,u_y)$. Ann. Mat. Pura Appl., IV. Ser. 89 (1971), 381-391. DOI 10.1007/BF02414955 | MR 0330809
[4] Deimling, K.: Das Picard-Problem für $u_{xy}=f(x, y, u, u_x, u_y)$ unter Carathéodory-Voraussetzungen. Math. Z. 114 (1970), 303-312 German. DOI 10.1007/BF01112700 | MR 0262643
[5] Dunford, N., Schwartz, J. T.: Linear Operators. I. General Theory. New York and London: Interscience Publishers. XIV (1958). MR 0117523 | Zbl 0084.10402
[6] Grigolia, M.: On the existence and uniqueness of solutions of the Goursat problem for systems of functional partial differential equations of hyperbolic type. Mem. Differ. Equ. Math. Phys. 16 (1999), 154-158. MR 1691354 | Zbl 0940.35201
[7] Grigolia, M. P.: On a generalized characteristic boundary value problem for hyperbolic systems. Differ. Uravn. 21 (1985), 678-686.
[8] Kharibegashvili, S.: Goursat and Darboux type problems for linear hyperbolic partial differential equations and systems. Mem. Differ. Equ. Math. Phys. 4 (1995), 127 p. MR 1415805 | Zbl 0870.35001
[9] Kiguradze, T.: Some boundary value problems for systems of linear partial differential equations of hyperbolic type. Mem. Differ. Equ. Math. Phys. 1 (1994), 144 p. MR 1296228 | Zbl 0819.35003
[10] Kiguradze, T. I.: On periodic boundary value problems for linear hyperbolic equations. I. Differ. Equations 29 (1993), 231-245, translation from Differ. Uravn. 29 281-297 (1993). MR 1236111
[11] Kiguradze, T. I.: Periodic boundary value problems for hyperbolic equations. II. Differ. Equations 29 (1993), 542-549, translation from Differ. Uravn. 29 637-645 (1993). MR 1250721 | Zbl 0849.35067
[12] Lakshmikantham, V., Pandit, S. G.: The method of upper, lower solutions and hyperbolic partial differential equations. J. Math. Anal. Appl. 105 (1985), 466-477. DOI 10.1016/0022-247X(85)90062-9 | MR 0778480 | Zbl 0569.35056
[13] Mitropol'skij, Yu. A., Urmancheva, L. B.: On two-point problem for systems of hyperbolic equations. Ukr. Math. J. 42 (1990), 1492-1498, translation from Ukr. Mat. Zh. 42 1657-1663 (1990). MR 1098465
[14] Natanson, I. P.: Theory of Functions of Real Variable. Nauka, Moscow (1974), Russian. MR 0354979
[15] Schaefer, H. H.: Normed tensor products of Banach lattices. Proc. internat. Sympos. partial diff. Equ. Geometry normed lin. Spaces II. Isr. J. Math. 13 (1972), 400-415. MR 0333754
[16] Šremr, J.: Absolutely continuous functions of two variables in the sense of Carathéodory. Electron J. Differ. Equ. 2010 (2010) 11 p. MR 2740595 | Zbl 1200.26016
[17] Tolstov, G. P.: On the mixed second derivative. Mat. Sb., N. Ser. 24 (1949), 27-51 Russian. MR 0029971
[18] Tricomi, F. G.: Lezioni Sulle Equazioni a Derivate Parziali. Corso di Analisi Superiore. Editrice Gheroni, Torino (1954), Italian. MR 0067293 | Zbl 0057.07502
[19] Vejvoda, O.: Periodic solutions of a linear and weakly nonlinear wave equation in one dimension. I. Czech. Math. J. 14 (1964), 341-382. MR 0174872
[20] al., O. Vejvoda et: Partial Differential Equations. SNTL, Praha (1981).
[21] Walczak, S.: Absolutely continuous functions of several variables and their application to differential equations. Bull. Pol. Acad. Sci., Math. 35 (1987), 733-744. MR 0961712 | Zbl 0691.35029
[22] Walter, W.: Differential and Integral Inequalities. Translated by Lisa Rosenblatt and Lawrence Shampine. Ergebnisse der Mathematik und ihrer Grenzgebiege. Band 55. Berlin-Heidelberg-New York: Springer-Verlag (1970). MR 0271508 | Zbl 0252.35005
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