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On the Uniqueness and Boundedness of Solutions of Hyperbolic Differential Equations†

Published online by Cambridge University Press:  24 October 2008

V. Lakshmikantham
V. Lakshmikantham
Affiliation:
Osmania University, Hyderabad University of California, Los Angeles
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Extract

Consider a characteristic initial value problem of partial differential equations

where the functions E (x) and F (y) are real valued, uniformly Lipschitz continuous on 0 ≤ x ≤ a, 0 ≤ y ≤ b, respectively. Suppose f (x, y, u, p, q) is a real-valued and continuous function defined on 0 ≤ ≤ b. By a solution of (1), we mean a real-valued continuous function u (x, y), having partial derivatives ux (x, y), uy (x, y) and ux, y (x, y) in the domain 0 ≤ x ≤ a, 0 ≤ y ≤ b almost everywhere.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 58 , Issue 4 , October 1962 , pp. 583 - 587
Copyright
Copyright © Cambridge Philosophical Society 1962

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References

REFERENCES

(1)Diaz, J. B., and Walter, W. L., On the uniqueness theorems for ordinary differential equations and for partial differential equations of hyperbolic type. Trans. American Math. Soc. 96 (1960), 90–100.Google Scholar
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