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Stability criteria for contractive semigroups via maximality procedures

Published online by Cambridge University Press:  17 April 2009

Mihai Turinici
Mihai Turinici
Affiliation:
Seminarul Matematic “Al. Myller”, University of laşi, 6600 laşi, Romania.
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Abstract

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An abstract metrical version of the well-known Ekeland and Brøndsted maximality principle is used to derive a number of stability criteria for a class of (function) contractive semigroups on (complete) metric spaces, extending a number of classical contributions due to Bre'zis and Browder.

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 24 , Issue 3 , December 1981 , pp. 453 - 469
Copyright
Copyright © Australian Mathematical Society 1981

References

[1]Bhatia, N.P., Szegö, G.P., Stability theory of dynamical systems (Die Grundlehren der mathematischen Wissenschaften, 161. Springer-Verlag, Berlin, Heidelberg, New York, 1970).CrossRefGoogle Scholar
[2]Bishop, Errett and Phelps, R.R., “The support functionals of a convex set”, Proc. Sympos. Pure Math., Volume 7, Convexity, 27–35 (American Mathematical Society, Providence, Rhode Island, 1963).Google Scholar
[3]Bony, Jean-Michel, “Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénéres”, Ann. Inst. Fourier (Grenoble) 19 (1969), 277–304.CrossRefGoogle Scholar
[4]Bourbaki, Nicolas, “Sur le théorèeme de Zorn”, Arch. Math. (Basel) 2 (1949/1950), 434–437.CrossRefGoogle Scholar
[5]Brezis, Haim, “On a characterization of flow-invariant sets”, Comm. Pure Appl. Math. 23 (1970), 261–263.CrossRefGoogle Scholar
[6]Br´ezis, H. and Browder, F.E., “A general principle on ordered sets in nonlinear functional analysis”, Adv. Math. 21 (1976), 355–364.CrossRefGoogle Scholar
[7]Brøndsted, Arne, “On a lemma of Bishop and Phelps”, Pacific J. Math. 55 (1974), 335–341.CrossRefGoogle Scholar
[8]Brøndsted, Arne, “Fixed points and partial orders”, Proc. Amer. Math. Soc. 60 (1976), 365–366.Google Scholar
[9]Brøndsted, Arne, “Common fixed points and partial orders”, Proc. Amer. Math. Soc. 77 (1979), 365–368.CrossRefGoogle Scholar
[10]Browder, Felix E., “Normal solvability and the Fredholm alternative for mappings into infinite dimensional manifolds”, J. Funct. Anal. 8 (1971), 250–274.CrossRefGoogle Scholar
[11]Browder, Felix E., “On a theorem of Caristi and Kirk”, Fixed point theory and its applications, 23–27 (Academic Press [Harcourt Brace Jovanovich], New York, San Francisco, London, 1976).Google Scholar
[12]Caristi, James, “Fixed point theorems for mappings satisfying inwardness conditions”, Trans. Amer. Math. Soc. 215 (1976), 241–251.CrossRefGoogle Scholar
[13]Crandall, Michael G., “A generalization of Peano's existence theorem and flow invariance”, Proc. Amer. Math. Soc. 36 (1972), 151–155.Google Scholar
[14]Daněs, Josef, “A geometric theorem useful in nonlinear functional analysis”, Boll. Un. Mat. Ital. (4) 6 (1972), 369–375.Google Scholar
[15]Ekeland, Ivar, “Sur les problèmes variationnels”, C.R. Acad. Sci. Paris Sér. A 275 (1972), 1057–1059.Google Scholar
[16]Ekeland, I., “On the variational principle”, J. Math. Anal. Appl. 47 (1974), 324–353.CrossRefGoogle Scholar
[17]Ekeland, Ivar, “Nonconvex minimization problems”, Bull. Amer. Math. Soc. N.S. 1 (1979), 443–474.CrossRefGoogle Scholar
[18]Halpern, Benjamin Rigler, “Fixed point theorems for outward maps” (PhD thesis, University of California, Los Angeles, California, 1965).Google Scholar
[19]Holmes, Richard B., Geometric functional analysis and its applications (Graduate Texts in Mathematics, 24. Springer-Verlag, New York, Heidelberg, Berlin, 1975).CrossRefGoogle Scholar
[20]Kasahara, Shouro, “On fixed points in partially ordered sets and Kirk-Caristi theorem”, Math. Sem. Notes Kobe Univ. (1975), no. 2, paper no. 35, 4pp.Google Scholar
[21]Kelley, John L., General topology (Graduate Texts in Mathematics, 27. Springer-Verlag, New York, Heidelberg, Berlin, 1975).Google Scholar
[22]Kirk, W.A., “Caristi's fixed point theorem and metric convexity”, Colloq. Math. 36 (1976), 81–86.CrossRefGoogle Scholar
[23]Martin, R.H. Jr, “Differential equations on closed subsets of a Banach space”, Trans. Amer. Math. Soc. 179 (1973), 399–414.CrossRefGoogle Scholar
[24]Nagumo, Mitio, “Über die Lage der Integralkurven gewöhnlicher Dlfferentialgleichungen”, Proc. Phys.-Math. Soc. Japan (3) 24 (1942), 551–559.Google Scholar
[25]Немыцний, В.В. и Сте⊓анов, В.В. [Nemytskii, V.V. and Stepanov, V.V.], Качесмвенная меорuя ∂uфференцuалЪнЫх уравнеuuˇ [Qualitative theory of differential equations] (OGIZ, Moscow, 1947). See also: V.V. Nemytskii and V.V. Stepanov, Qualitative theory of differential equations (translated by Solomon Lefschetz. Princeton Mathematical Series, 22. Princeton University Press, Princeton, New Jersey, 1960).Google Scholar
[26]Pasicki, L., “A short proof of the Caristi theorem”, Comment. Math. prace Mat. 20 (1977/1978), 427–428.Google Scholar
[27]Pavel, N., “Invariant sets for a class of semi-linear equations of evolution”, Nonlinear Anal. 1 (1976/1977), 187–196.CrossRefGoogle Scholar
[28]Redheffer, R.M., “The theorems of Bony and Brezis on flow-invariant sets”, Amer. Math. Monthly 79 (1972), 740–747.CrossRefGoogle Scholar
[29]Siegel, Jerrold, “A new proof of Caristi's fixed point theorem”, Proc. Amer. Math. Soc. 66 (1977), 54–56.Google Scholar
[30]Turinici, Mihai, “Maximal elements in ordered topological spaces”, Bull. Greek Math. Soc. 20 (1979), 141–148.Google Scholar
[31]Turinici, M., “Maximal elements in a class of order complete spaces”, Math. Japan. 25 (1980), 511–517.Google Scholar
[32]Turinici, Mihai, “Differential inequalities via maximal element techniques”, Nonlinear Anal. 5 (1981), 757–763.CrossRefGoogle Scholar
[33]Turinici, Mihai, “Local and global lipschitzian mappings on ordered metric spaces”, Math. Nachr. (to appear).Google Scholar
[34]Turinici, Mihai, “Flow-invariance theorems via maximal element techniques”, Nederl. Akad. Wetensch. Indag. Math, (to appear).Google Scholar
[35]Turinici, Mihai, “Drop theorems and lipschitzianness tests via maximality procedures”, Acta Math. Acad. Sci. Hungar. (to appear).Google Scholar
[36]Vrabie, Ioan I., “Time optimal control for contingent equations in Hirbert spaces”, An. Ştiinţ. Univ. “Al. I. Cuza” Iasi Sect. I a Mat. (N.S.) 24 (1978), 125–133.Google Scholar
[37]Wallace, A.D., “A fixed-point theorem”, Bull. Amer. Math. Soc. 51 (1945), 413–416.CrossRefGoogle Scholar
[38]Ward, L.E. Jr, “Partially ordered topological spaces”, Proc. Amer. Math. Soc. 5 (1954), 144–161.CrossRefGoogle Scholar
[39]Wong, Chi Song, “On a fixed point theorem of contractive type”, Proc. Amer. Math. Soc. 57 (1976), 283–284.CrossRefGoogle Scholar
[40]Yorke, James A., “Invariance for ordinary differential equations”, Math. Systems Theory 1 (1967), 353–372.CrossRefGoogle Scholar
[41]Забрейно, П.П., Красносельсний, М.А. [Zabreĭko, P.P., Krasnosel'skiĭ], M.A., “О Разрешимости Нелинейных О⊓ераторных Уравнений” [The solvability of nonlinear operator equations], Funkoional. Anal, i Priložen. 5 (1971), no. 3, 42–44. English Transl: Functional Anal. Appl. 5 (1971), 206–208.Google ScholarPubMed