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Fixed point theorems in probabilistic analysis. (English) Zbl 0339.60061

MSC:
60H99 Stochastic analysis
47H10 Fixed-point theorems
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[1] Nonlinear integral equations, Proceedings of an Advanced Seminar Conducted by the Mathematics Research Center, United States Army, at the University of Wisconsin, Madison, April 22-24, vol. 1963, The University of Wisconsin Press, Madison, Wis., 1964.
[2] S. Banach, Sur les opérations dans les ensembles abstraits et leur applications aux équations intégrales, Fund. Math. 3 (1922), 133-181. · JFM 48.0201.01
[3] C. Bessaga, On the converse of the Banach ”fixed-point principle”, Colloq. Math. 7 (1959), 41 – 43. · Zbl 0089.31501
[4] Albert Turner Bharucha-Reid, On random solutions of integral equations in Banach spaces, Trans. 2nd Prague Conf. Information Theory, Publ. House Czehoslovak Acad. Sci., Prague, Academic Press, New York, 1961, pp. 27 – 48.
[5] A. T. Bharucha-Reid, On the theory of random equations, Proc. Sympos. Appl. Math., Vol. XVI, Amer. Math. Soc., Providence, R.I., 1964, pp. 40 – 69. · Zbl 0142.13603
[6] A. T. Bharucha-Reid, Random integral equations, Academic Press, New York-London, 1972. Mathematics in Science and Engineering, Vol. 96. · Zbl 0327.60040
[7] Gheorghe Bocşan, On some fixed point theorems in probabilistic metric spaces, Math. Balkanica 4 (1974), 67 – 70. Papers presented at the Fifth Balkan Mathematical Congress (Belgrade, 1974). · Zbl 0317.60028
[8] H. F. Bohnenblust and S. Karlin, On a theorem of Ville, Contributions to the Theory of Games, Annals of Mathematics Studies, no. 24, Princeton University Press, Princeton, N. J., 1950, pp. 155 – 160. · Zbl 0041.25701
[9] F. F. Bonsall, Lectures on some fixed point theorems of functional analysis, Notes by K. B. Vedak, Tata Institute of Fundamental Research, Bombay, 1962.
[10] William E. Boyce, Random eigenvalue problems, Probabilistic Methods in Applied Mathematics, Vol. 1, Academic Press, New York, 1968, pp. 1 – 73.
[11] Enrique M. Cabaña, Stochastic integration in separable Hilbert spaces, Univ. Repúb. Fac. Ingen. Agrimens. Montevideo Publ. Inst. Mat. Estadí st 4 (1966), 49 – 80 (1966) (English, with Spanish summary).
[12] G. L. Cain Jr. and R. H. Kasriel, Fixed and periodic points of local contraction mappings on probabilistic metric spaces, Math. Systems Theory 9 (1975/76), no. 4, 289 – 297. · Zbl 0334.60004
[13] Ch. Castaing, Sur les multi-applications mesurables, Rev. Française Informat. Recherche Opérationnell 1 (1967), no. 1, 91 – 126 (French). · Zbl 0153.08501
[14] Sherwood C. Chu and J. B. Diaz, Remarks on a generalization of Banach’s principle of contraction mappings, J. Math. Anal. Appl. 11 (1965), 440 – 446. · Zbl 0129.38301
[15] A. I. Dale, Some theoretical aspects of random equations of evolution in theoretical population genetics, Ph. D. Dissertation, Virginia Polytechnic Inst, and State Univ., Blacksburg, Virginia, 1972.
[16] Miloslav Driml and Otto Hanš, Continuous stochastic approximations, Trans. 2nd Prague Conf. Information Theory, Publ. House Czechoslovak Acad. Sci., Prague, Academic Press, New York, 1961, pp. 113 – 122. · Zbl 0098.32201
[17] U. Frisch, Wave propagation in random media, Probabilistic Methods in Applied Mathematics, Vol. 1, Academic Press, New York, 1968, pp. 75 – 198.
[18] H. Furstenberg and H. Kesten, Products of random matrices, Ann. Math. Statist. 31 (1960), 457 – 469. · Zbl 0137.35501
[19] L. A. Gardner Jr., Stochastic approximation and its application to problems of prediction and control synthesis, Internat. Sympos. Nonlinear Differential Equations and Nonlinear Mechanics, Academic Press, New York, 1963, pp. 241 – 258.
[20] Ulf Grenander, Probabilities on algebraic structures, Second edition, Almqvist & Wiksell, Stockholm; John Wiley & Sons Inc., New York-London, 1968. · Zbl 0131.34804
[21] Ioannis Th. Haïnis, Random variables with values in Banach algebras and random transformations in Hilbert spaces, Bull. Soc. Math. Grèce (N.S.) 7 (1966), no. fasc. 2, 179 – 223 (Greek, with French summary).
[22] Otto Hanš, Reduzierende zufällige Transformationen, Czechoslovak Math. J. 7(82) (1957), 154 – 158 (German, with Russian summary). · Zbl 0090.34804
[23] Otto Hanš, Generalized random variables, Transactions of the first Prague conference on information theory, statistical decision functions, random processes held at Liblice near Prague from November 28 to 30, 1956, Publishing House of the Czechoslovak Academy of Sciences, Prague, 1957, pp. 61 – 103.
[24] Otto Hanš, Random fixed point theorems, Transactions of the first Prague conference on information theory, statistical decision functions, random processes held at Liblice near Prague from November 28 to 30, 1956, Publishing House of the Czechoslovak Academy of Sciences, Prague, 1957, pp. 105 – 125.
[25] Otto Hanš, Random operator equations, Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. II, Univ. California Press, Berkeley, Calif., 1961, pp. 185 – 202.
[26] Otto Hanš and Antonín Špaček, Random fixed point approximation by differentiable trajectories, Trans. 2nd Prague Conf. Information Theory, Publ. House Czechoslovak Acad. Sci., Prague, Academic Press, New York, 1960, pp. 203 – 213.
[27] V. Istrǎţescu, An introduction to probabilistic metric spaces with applications, Editura Tehnicǎ, Bucharest, 1974. (Roumanian)
[28] I. Istrăţescu and E. Rovenţa, On fixed point theorems for mappings on probabilistic metric spaces, Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.S.) 19(67) (1975), no. 1-2, 67 – 69 (1976).
[29] Vasile I. Istrăţescu and Ion Săcuiu, Fixed point theorems for contraction mappings on probabilistic metric spaces, Rev. Roumaine Math. Pures Appl. 18 (1973), 1375 – 1380. · Zbl 0293.60014
[30] R. Jajte, Random linear operators, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 21 (1973), 227 – 231 (English, with Russian summary). · Zbl 0258.60007
[31] Norman L. Johnson and Samuel Kotz, Distributions in statistics: continuous multivariate distributions, John Wiley & Sons, Inc., New York-London-Sydney, 1972. Wiley Series in Probability and Mathematical Statistics. · Zbl 0248.62021
[32] Rangachary Kannan and Habib Salehi, Mesurabilité du point fixe d’une transformation aléatoire séparable, C. R. Acad. Sci. Paris Sér. A-B 281 (1975), no. 15, Aiii, A663 – A664 (French, with English summary). · Zbl 0364.60012
[33] R. Kannan and H. Salehi, Random nonlinear equations and monotonic nonlinearities, J. Math. Anal. Appl. 57 (1977), no. 1, 234 – 256. · Zbl 0357.34058
[34] Rangachary Kannan and Habib Salehi, Measurability of solutions of nonlinear equations, Nonlinear functional analysis and differential equations (Proc. Conf., Mich. State Univ., East Lansing, Mich., 1975) Marcel Dekker, New York, 1976, pp. 227 – 244. Lecture Notes in Pure and Appl. Math., Vol. 19. · Zbl 0341.47044
[35] A. N. Kolmogorov and S. V. Fomin, Introductory real analysis, Revised English edition. Translated from the Russian and edited by Richard A. Silverman, Prentice-Hall, Inc., Englewood Cliffs, N.Y., 1970. · Zbl 0213.07305
[36] M. A. Krasnosel\(^{\prime}\)skiĭ, Two remarks on the method of successive approximations, Uspehi Mat. Nauk (N.S.) 10 (1955), no. 1(63), 123 – 127 (Russian).
[37] M. L. Mehta, Random matrices and the statistical theory of energy levels, Academic Press, New York-London, 1967. · Zbl 0925.60011
[38] H. Mittermeier, A note on random preference and equilibrium analysis: Application of a stochastic fixed point theorem, Unpublished manuscript, 1974.
[39] W. L. Morse, Some mathematics in stream temperature modelling (to appear).
[40] A. Mukherjea, Random transformations on Banach spaces, Ph. D. Dissertation, Wayne State Univ., Michigan, 1966. · Zbl 0139.33404
[41] Arunava Mukherjea, Transformations aléatoires séparables: Théorème du point fixe aléatoire, C. R. Acad. Sci. Paris Sér. A-B 263 (1966), A393 – A395 (French). · Zbl 0139.33404
[42] A. Mukherjea, On the measurability of the resolvent of a random kernel, Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.S.) 14(62) (1970), 55 – 59 (1971). · Zbl 0229.60043
[43] A. Mukherjea, On a random integral equation of Uryson type. I (to appear).
[44] A. Mukherjea and A. T. Bharucha-Reid, Separable random operators. I, Rev. Roumaine Math. Pures Appl. 14 (1969), 1553 – 1561. · Zbl 0196.18502
[45] Sam B. Nadler Jr., Sequences of contractions and fixed points, Pacific J. Math. 27 (1968), 579 – 585. · Zbl 0167.44601
[46] M. Z. Nashed and H. Salehi, Measurability of generalized inverses of random linear operators, SIAM J. Appl. Math. 25 (1973), 681 – 692. · Zbl 0245.60004
[47] M. Z. Nashed and J. S. W. Wong, Some varaints of a fixed point theorem of Krasnoselskii and applications to nonlinear integral equations, J. Math. Mech. 18 (1969), 767 – 777. · Zbl 0181.42301
[48] K. G. Oza, Identification problems and random contraction mappings, Ph. D. Dissertation, Univ. of California, Berkeley, 1967.
[49] K. G. Oza and E. I. Jury, System identification and the principle of random contraction mapping., SIAM J. Control 6 (1968), 244 – 257. · Zbl 0182.51504
[50] W. V. Petryshyn, On the approximation-solvability of equations involving \?-proper and psuedo-\?-proper mappings, Bull. Amer. Math. Soc. 81 (1975), 223 – 312. · Zbl 0303.47038
[51] B. L. S. Prakasa Rao, Stochastic integral equations of mixed type. II, J. Mathematical and Physical Sci. 7 (1973), 245 – 260. · Zbl 0273.60038
[52] Louis B. Rall, Computational solution of nonlinear operator equations, With an appendix by Ramon E. Moore, John Wiley & Sons, Inc., New York-London-Sydney, 1969. · Zbl 0175.15804
[53] Thomas L. Saaty, Modern nonlinear equations, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1967. · Zbl 0148.28202
[54] J. Schauder, Der Fixpunktsatz in Funktionenräumen, Studia Math. 2 (1930), 171-182. · JFM 56.0355.01
[55] V. M. Sehgal, Some fixed point theorems in functional analysis and probability, Ph. D. Dissertation, Wayne State Univ., Michigan, 1966.
[56] V. M. Sehgal and A. T. Bharucha-Reid, Fixed points of contraction mappings on probabilistic metric spaces, Math. Systems Theory 6 (1972), 97 – 102. · Zbl 0244.60004
[57] H. Sherwood, Complete probabilistic metric spaces, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 20 (1971/72), 117 – 128. · Zbl 0212.19304
[58] D. R. Smart, Fixed point theorems, Cambridge University Press, London-New York, 1974. Cambridge Tracts in Mathematics, No. 66. · Zbl 0297.47042
[59] T. T. Soong, Random differential equations in science and engineering, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1973. Mathematics in Science and Engineering, Vol. 103. · Zbl 0348.60081
[60] Antonín Špaček, Zufällige Gleichungen, Czechoslovak Math. J. 5(80) (1955), 462 – 466 (German, with Russian summary). · Zbl 0068.32701
[61] Richard B. Thompson , {Collection of articles on fixed point theory}, The Rocky Mountain Mathematics Consortium, Arizona State University, Tempe, Ariz., 1974. Rocky Mountain J. Math. 4 (1974), no. 1.
[62] Chris P. Tsokos and W. J. Padgett, Random integral equations with applications to life sciences and engineering, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Mathematics in Science and Engineering, Vol. 108. · Zbl 0287.60065
[63] M. T. Wasan, Stochastic approximation, Cambridge Tracts in Mathematics and Mathematical Physics, No. 58, Cambridge University Press, London-New York, 1969. · Zbl 0293.62026
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