Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-24T02:34:46.676Z Has data issue: false hasContentIssue false
  • English
  • Français

An Intermediate Value Property for Operators with Applications to Integral and Differential Equations

Published online by Cambridge University Press:  20 November 2018

J. S. Muldowney  and
D. Willett
J. S. Muldowney
Affiliation:
University of Alberta, Edmonton, Alberta
D. Willett
Affiliation:
University of Alberta, Edmonton, Alberta
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

It is well known that a real valued continuous function f on a closed interval S assumes every value between its maximum and minimum on S, i.e. if ξ is such that f(α) ≦ ξ ≦ f(β) then there exists γ between α and β such that f(γ) = ξ. The purpose of this paper is to develop the existence theory associated with differential and integral inequalities in the context of an intermediate value property for operators on partially ordered spaces. This has the advantage of allowing rather simple proofs of known results while in most cases giving slight improvements, and in some cases substantial improvements, in these results. Classical and recent results from different areas are unified under one principle.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 26 , Issue 1 , February 1974 , pp. 27 - 41
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Azbelev, N. V. and Tsalyuk, Z. B., A contribution on the problem of differential inequalities, Differencial'nye Uravnenija 1 (1965), 431–438.Google Scholar
2. Beckenbach, Edwin F. and Bellman, Richard, Inequalities (Springer-Verlag, New York, 1965).Google Scholar
3. Bihari, I., A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations, Acta Math. Acad. Sci. Hungar. 7 (1956), 81–94.Google Scholar
4. Coppel, W. A., `Stability and asymptotic behaviour of differential equations (D. C. Heath, Boston, 1965).Google Scholar
5. Coppel, W. A., Disconjugacy (Springer-Verlag, Berlin, 1971).Google Scholar
6. Dunkel, G. M., On nested functional differential equations, SIAM J. Appl. Math. 18 (1970), 514–525.Google Scholar
7. Erbe, Lynn H., Nonlinear boundary value problems for second order differential equations, J. Differential Equations 7 (1970), 459–472.Google Scholar
8. Gollwitzer, H. E., A note on a functional inequality, Proc. Amer. Math. Soc. 23 (1969), 642–647.Google Scholar
9. Grimmer, R. C. and Waltman, P., A comparison theorem for a class of nonlinear differential inequalities, Monatsh. Math. 72 (1968), 133–136.Google Scholar
10. Grimm, L. J. and Schmitt, K., Boundary value problems for differential equations with deviating arguments, Aequationes Math. 3 (1969), 24–38.Google Scholar
11. Gronwall, T. H., Note on the derivatives with respect to a parameter of the solutions of a system of differential equations, Ann. of Math. 20 (1918), 292–296.Google Scholar
12. Jackson, Lloyd K., Subfunctions and second-order ordinary differential inequalities, Advances in Math. 2 (1968), 307–363.Google Scholar
13. Jackson, Lloyd K. and Schrader, Keith W., Comparison theorems for nonlinear differential equations, J. Differential Equations 3 (1967), 248–255.Google Scholar
14. Kamke, E., Zür Theorie der Systeme gewöhnlicher Differentialgleichungen. II, Acta. Math. 58 (1932), 57–85.Google Scholar
15. Kantorovitch, L., The method of successive approximations for functional equations, Acta Math. 71 (1939), 63–97.Google Scholar
16. Klassen, Gene A., Differential inequalities and existence theorems for second and third order boundary value problems, J. Differential Equations 10 (1971), 529–537.Google Scholar
17. Levin, A. Ju., Some problems bearing on the oscillation of solutions of linear differential equations, Dokl. Akad. Nauk SSSR 148 (1963), 512–515 (Soviet Math. Dokl. 4 (1963), 121–124).Google Scholar
18. Li, Yue-Shang, The bound, stability and error estimates for the solution of nonlinear differential equations, Chinese Math. Acta 3 (1963), 34–41.Google Scholar
19. Maroni, P., Une generalisation non lineare de l'inéqualité de Gronwall, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 16 (1968), 703–709.Google Scholar
20. Perron, O., Ein neuer Existenzbeweis fur die Intégrale der Differentialgleichung y′ = f(x, y), Math. Ann. 76 (1915), 471–484.Google Scholar
21. Pokornyi, Ju. V., Some estimates of the Green's function of a multi-point boundary value problem, Mat. Zametki 4 (1968), 533–540.Google Scholar
22. Reid, W. T., Properties of solutions of an infinite system of ordinary linear differential equations … , Trans. Amer. Math. Soc. 32 (1930), 284–318.Google Scholar
23. Talpalaru, Pavel, Inégalités différentielles pour une certaine équation fonctionelle de Volterra, An. Sti. Univ. “Al. I. Cuza” Jasi Sect. I a Mat. 14 (1968), 313–319.Google Scholar
24. Schmitt, K., Periodic solutions of nonlinear second order differential equations, Math. Z. 98 (1967), 200–207.Google Scholar
25. Schmitt, K., Boundary value problems for differential equations with deviating arguments, Aequationes Math. 3 (1969), 126–190.Google Scholar
26. Schmitt, K., On solutions of nonlinear differential equations with deviating arguments, SIAM J. Appl. Math. 17 (1969), 1171–1176.Google Scholar
27. Schmitt, K., Periodic solutions of linear second order differential equations with deviating argument, Proc. Amer. Math. Soc. 26 (1970), 282–285.Google Scholar
28. Schmitt, K., Periodic solutions of systems of second-order differential equations, J. Differential Equations 11 (1972), 180–192.Google Scholar
29. Wajewski, T., Sur un système des inégalités ordinaires non linéaires, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 17 (1969), 225–229.Google Scholar
30. Walter, Wolfgang, Differential and integral inequalities, Erg. Math. Grenzgebiete 55 (Springer-Verlag, Berlin, 1970).Google Scholar
31. Waltman, P., and Hanson, D. L., A note on a functional equation, J. Math. Anal. Appl. 10 (1965), 330–333.Google Scholar