Abstract
We consider initial-value problems for infinite systems of first-order partial functional differential equations. The unknown function is the functional argument in equations and the partial derivations appear in the classical sense. A theorem on the existence of a solution and its continuous dependence upon initial data is proved. The Cauchy problem is transformed into a system of functional integral equations. The existence of a solution of this system is proved by using integral inequalities and the iterative method. Infinite differential systems with deviated argument and differential integral systems can be derived from the general model by specializing given operators.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
P. Brandi and R. Ceppitelli, “Existence, uniqueness and continuous dependence for a first-order nonlinear partial differential equations in a hereditary structure,” Ann. Pol. Math., 47, 121-136 (198).
Z. Kamont, “Existence of solutions of first-order partial differential functional equations,” Ann. Soc. Math. Pol., Comm. Math., 25, 249-263 (1985).
J. Szarski, “Generalized Cauchy problem for differential functional equations with first-order partial derivatives,” Bull. Acad. Pol. Sci., Sér. Sci. Math., Astron. Phys., 24, 575-580 (1976).
T. Wazżewski, “Sur le probléme de Cauchy relatif á un systéme d'équations aux dériveés partielles,” Ann. Soc. Pol. Math., 15, 101-127 (1936).
M. I. Umanaliev and J. A. Vied, “On integral differential equations with first-order partial derivatives,” Differents. Uravn., 25, 465-477 (1989).
T. Czlapiński and Z. Kamont, “Generalized solutions for quasilinear hyperbolic systems of partial differential functional equations,” J. Math. Anal. Appl., 172, 353-370 (1993).
P. Besala, “Observations on quasi-linear partial differential equations,” Ann. Pol. Math., 53, 267-283 (1991).
T. Czlapiński, “On the existence of generalized solutions of nonlinear first-order partial differential functional equations in two independent variables,” Czech. Math. J., 41, 490-506 (1991).
Z. Kamont and S. Zacharek, “On the existence of weak solutions of nonlinear first-order partial differential equations in two independent variables,” Boll. Unione Mat. Ital., 6, No. 5-B, 851-879 (1986).
P. Brandi, Z. Kamont, and A. Salvadori, Existence of Weak Solutions for Partial Differential Functional Equations (Preprint).
Z. Kamont, Hyperbolic Functional Differential Inequalities and Applications, Kluwer, Dordrecht (1999).
M. Cinquini and S. Cibrario, “Nuove ricerche sui sistemi di equazioni non lineari a derivate parziali in piÚ variabili indipendenti,” Rend. Semin. Mat. Fis. Univ. Milano., 52, 531-550 (1982).
M. Cinquini and S. Cibrario, “Sopra una classe di sistemi di equazioni non lineari a derivate parziali in piÚ variabili indipendenti,” Ann. Math. Pura Appl., 140, 223-253 (1985).
S. Brzychczy, “Approximate iterative method and the existence of solutions of nonlinear parabolic differential functional equations,” Ann. Pol. Math., 42, 37-43 (1983).
S. Brzychczy, “Chaplygin method for a system of nonlinear parabolic differential functional equations,” Differents. Uravn., 22, 705-708 (1986).
S. Brzychczy, “Existence of solutions for nonlinear systems of differential functional equations of parabolic type in arbitrary domains,” Ann. Pol. Math., 47, 309-317 (1987).
Z. Kamont, “Initial-value problems for hyperbolic differential functional systems,” Boll. Unione Mat. Ital., 7, No. 8-B, 965-984 (1994).
J. S zarski, “Cauchy problem for an infinite system of differential functional equations with first-order partial derivatives,” Ann. Soc. Math. Pol., Comm. Math., Spec. I, 293-300 (1978).
J. Szarski, “Comparison theorems for infinite systems of differential functional equations and strongly coupled infinite systems of first-order partial differential equations,” Rocky Mountain J. Math., 10, 237-246 (1980).
D. Jaruszewska-Walczak, “Generalized solutions of the Cauchy problem for infinite systems of functional differential equations,” Funct. Different. Equat., 6, No. 3-4, 305-326 (1999).
S. Brzychczy, “Existence of solutions and monotone iterative method for infinite systems of parabolic differential functional equations,” Ann. Pol. Math. (to appear).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kamont, Z. Infinite Systems of Hyperbolic Functional Differential Equations. Ukrainian Mathematical Journal 55, 2006–2030 (2003). https://doi.org/10.1023/B:UKMA.0000031662.80755.f9
Issue Date:
DOI: https://doi.org/10.1023/B:UKMA.0000031662.80755.f9