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Stability Theorems for Solutions to the Optimal Inventory Equation

Published online by Cambridge University Press:  14 July 2016

S. Edward Boylan
S. Edward Boylan*
Affiliation:
Rutgers, The State University, Newark, N.J.
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Extract

In a previous paper, [1] it was shown that a solution, f(x) will exist for the optimal inventory equation (where f(y − z) = f(0), y < z) provided:

  1. 1. g(x) ≧ 0, x ≧ 0;

  2. 2. 0 < a < 1;

  3. 3. h(x) is monotonically nondecreasing, h(0) = 0;

  4. 4. F is a distribution function on [0, ∞);

    (In [1], 1–4 were denoted collectively as (A).)

    and either

  5. 5a. g(x) is continuous for all x ≧ 0;

  6. 5b. limx→∞g(x) = ∞;

  7. 5c h(x) is continuous for all x > 0 (Theorem 2 of [1]);

    or

  8. 6. g(x) is uniformly continuous for all x ≧ 0 (Theorem 3 of [1]).

Type
Short Communications
Information
Journal of Applied Probability , Volume 6 , Issue 1 , April 1969 , pp. 211 - 217
Copyright
Copyright © Applied Probability Trust 

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References

[1] Boylan, E. (1966) Existence and uniqueness theorems for the optimal inventory equation. SIAM J. Appl. Math. 14, 961–969.Google Scholar
[2] Feller, W. (1966) An Introduction to Probability Theory and its Applications Vol. II. Wiley, New York.Google Scholar
[3] Rudin, W. (1964) Principles of Mathematical Analysis. McGraw Hill, New York. 2nd edition.Google Scholar
[4] Bellman, R. (1957) Dynamic Programming. Princeton University Press, Princeton, N. J. Google Scholar