Abstract
We prove that, for the validity of a certain theorem on differential inequalities for a linear functional differential equation of hyperbolic type {fx327-01} with a negative linear operator {fx327-02}, it is necessary that ℓ be an (a, c)-Volterra operator.
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References
S. Bernfeld and V. Lakshmikantham, “Monotone methods for nonlinear boundary-value problems in Banach spaces,” Nonlin. Analysis, 3, 303–316 (1979).
K. Deimling and V. Lakshmikantham, “Existence and comparison theorems for differential equations in Banach spaces,” Nonlin. Analysis, 3, 569–575 (1979).
V. Lakshmikantham and S. G. Pandit, “The method of upper, lower solutions and hyperbolic partial differential equations,” J. Math. Anal. Appl., 105, 466–477 (1985).
J. Mawhin, R. Ortega, and A. M. Robles-Pérez, “A maximum principle for bounded solutions of the telegraph equations and applications to nonlinear forcings,” J. Math. Anal. Appl., 251, 695–709 (2000).
R. Ortega and A. M. Robles-Pérez, “A maximum principle for periodic solutions of the telegraph equation,” J. Math. Anal. Appl., 221, 625–651 (1998).
D. Sattinger, “Monotone methods in nonlinear elliptic and parabolic boundary-value problems,” Indiana Univ. Math. J., 21, 979–1000 (1972).
A. Lomtatidze, S. Mukhigulashvili, and J. Šremr, “Nonnegative solutions of the characteristic initial-value problem for linear partial functional differential equations of hyperbolic type,” Math. Comput. Modelling (to appear). Draft version: Preprint No. 160, Mathematical Institute, Academy of Sciences of the Czech Republic (2005).
J. Šremr, “On the characteristic initial-value problem for linear partial functional differential equations of hyperbolic type,” Proc. Edinburgh Math. Soc. (to appear). Draft version: Preprint No. 161, Mathematical Institute, Academy of Sciences of the Czech Republic (2005).
K. Deimling, “Absolutely continuous solutions of Cauchy problem for u xy = f(x, y, u, u x, uy),” Ann. Mat. Pura Appl., 89, 381–391 (1971).
O. Dzagnidze, “Some new results on the continuity and differentiability of functions of several real variables,” Proc. A. Razm. Math. Inst., 134, 1–144 (2004).
G. P. Tolstov, “On the mixed second derivative,” Mat. Sb., 24(66), 27–51 (1949).
C. Carathéodory, Vorlesungen über relle funktionen, Teubner, Leipzig-Berlin (1918).
S. Lojasiewicz, An Introduction to the Theory of Real Functions, Wiley-Interscience, Chichester (1988).
S. Walczak, “Absolutely continuous functions of several variables and their application to differential equations,” Bull. Polish Acad. Sci. Math., 35, No. 11–12, 733–744 (1987).
E. Bravyi, A. Lomtatidze, and B. Půža, “A note on the theorem on differential inequalities,” Georgian Math. J., 7, No. 4, 627–631 (2000).
H. Štěpánková, “A note on the theorem on differential inequalities,” Electron. J. Qual. Theory Differ. Equat., 7, 1–8 (2005).
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Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 2, pp. 283–292, February, 2008.
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Šremr, J. Some remarks on linear functional differential inequalities of hyperbolic type. Ukr Math J 60, 327–337 (2008). https://doi.org/10.1007/s11253-008-0061-4
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DOI: https://doi.org/10.1007/s11253-008-0061-4