# Article

 Title: Random fixed point theorems for a certain class of mappings in Banach spaces (English) Author: Jung, Jong Soo Author: Cho, Yeol Je Author: Kang, Shin Min Author: Lee, Byung Soo Author: Thakur, Balwant Singh Language: English Journal: Czechoslovak Mathematical Journal ISSN: 0011-4642 (print) ISSN: 1572-9141 (online) Volume: 50 Issue: 2 Year: 2000 Pages: 379-396 Summary lang: English . Category: math . Summary: Let $(\Omega,\Sigma)$ be a measurable space and $C$ a nonempty bounded closed convex separable subset of $p$-uniformly convex Banach space $E$ for some $p > 1$. We prove random fixed point theorems for a class of mappings $T\: \Omega \times C \rightarrow C$ satisfying: for each $x, y \in C$, $\omega \in \Omega$ and integer $n \ge 1$, $\Vert T^n(\omega , x) - T^n(\omega , y) \Vert \le a(\omega )\cdot \Vert x - y \Vert + b(\omega )\lbrace \Vert x - T^n(\omega ,x) \Vert + \Vert y - T^n(\omega ,y) \Vert \rbrace + c(\omega )\lbrace \Vert x - T^n(\omega ,y) \Vert + \Vert y - T^n(\omega ,x) \Vert \rbrace ,$ where $a,b,c\: \Omega \rightarrow [0, \infty )$ are functions satisfying certain conditions and $T^n(\omega ,x)$ is the value at $x$ of the $n$-th iterate of the mapping $T(\omega ,\cdot )$. Further we establish for these mappings some random fixed point theorems in a Hilbert space, in $L^p$ spaces, in Hardy spaces $H^p$ and in Sobolev spaces $H^{k,p}$ for $1 < p < \infty$ and $k \ge 0$. As a consequence of our main result, we also extend the results of Xu [43] and randomize the corresponding deterministic ones of Casini and Maluta [5], Goebel and Kirk [13], Tan and Xu [37], and Xu [39, 41]. (English) Keyword: $p$-uniformly convex Banach space Keyword: normal structure Keyword: asymptotic center Keyword: random fixed points Keyword: generalized random uniformly Lipschitzian mapping MSC: 47H09 MSC: 47H10 MSC: 47H40 MSC: 60H25 idZBL: Zbl 1048.47043 idMR: MR1761395 . Date available: 2009-09-24T10:33:44Z Last updated: 2016-04-07 Stable URL: http://hdl.handle.net/10338.dmlcz/127577 . Reference: [1] J. Barros-Neto: An Introduction to the Theory of Distribution..Dekker, New York, 1973. MR 0461128 Reference: [2] A. T. Bharucha-Reid: Fixed point theorems in probabilistic analysis.Bull. Amer. Math. Soc. 82 (1976), 641–645. Zbl 0339.60061, MR 0413273, 10.1090/S0002-9904-1976-14091-8 Reference: [3] A. T. Bharucha-Reid: Random Integral Equations.Academic Press, New York and London, 1977. Zbl 0373.60072, MR 0443086 Reference: [4] Gh. Bocsan: A general random fixed point theorem and applications to random equations.Rev. Roumaine Math. Pures Appl. 26 (1981), 375–379. Zbl 0473.60057, MR 0627283 Reference: [5] W. L. Bynum: Normal structure coefficient for Banach space.Pacific J. Math. 86 (1980), 427–436. MR 0590555, 10.2140/pjm.1980.86.427 Reference: [6] E. Casini and E. Maluta: Fixed points of uniformly Lipschitzian mappings in spaces with uniformly normal structure.Nonlinear Anal. TMA 9 (1985), 103–108. MR 0776365 Reference: [7] C. Castaing and M. Valadier: Convex Analysis and Measurable Multifunctions.Springer, Berlin, 1977. MR 0467310 Reference: [8] S. S. Chang: Random fixed point theorems for continuous random operators.Pacific J. Math. 105 (1983), 21–31. Zbl 0512.47044, MR 0688405, 10.2140/pjm.1983.105.21 Reference: [9] J. Daneš: On densifying and related mappings and their applications in nonlinear functional analysis.In: Theory of nonlinear Operators, Proc. Summer School, GDR, Akademie-Verlag, Berlin, Oct.1972 1974, pp. 15–56. MR 0361946 Reference: [10] N. Dunford and J. Schwarz: Linear Operators.Vol I, Interscience, New York, 1958. Reference: [11] W. H. Duren: Theory of $H^p$ Spaces.Academic Press, New York, 1970. MR 0268655 Reference: [12] H. W. Engl: Random fixed point theorems for multivalued mappings.Pacific J. Math. 76 (1978), 351–360. Zbl 0355.47035, MR 0500323, 10.2140/pjm.1978.76.351 Reference: [13] K. Goebel and W. A. Kirk: A fixed point theorem for transformations whose iterates have uniform Lipschitz constant.Studia. Math. 47 (1973), 135–140. MR 0336468 Reference: [14] S. Itoh: A random fixed point theorem for a multivalued contraction.Pacific J. Math 68 (1977), 85–90. Zbl 0335.54036, MR 0451228, 10.2140/pjm.1977.68.85 Reference: [15] S. Itoh: Random fixed point theorems with an application to random differential equations in Banach spaces.J. Math. Anal. Appl. 67 (1979), 261–273. Zbl 0407.60069, MR 0528687, 10.1016/0022-247X(79)90023-4 Reference: [16] C. J. Himmelberg: Measurable relations.Fund. Math. 87 (1975), 53–72. Zbl 0296.28003, MR 0367142 Reference: [17] T. C. Lim: Fixed point theorems for uniformly Lipschitzian mappings in $L^p$ spaces.Nonlinear Anal. 7 (1983), 555–563. MR 0698365 Reference: [18] T. C. Lim: On the normal structure coefficient and the bounded sequence coefficient.Proc. Amer. Math. Soc. 88 (1983), 262–264. Zbl 0541.46017, MR 0695255, 10.1090/S0002-9939-1983-0695255-2 Reference: [19] T. C. Lim, H. K. Xu and Z. B. Xu: An $L^p$ inequalities and its applications to fixed point theory and approximation theory.In: Progress in Approximation Theory, Academic Press, 1991, pp. 609–624. Reference: [20] T. C. Lin: Random approximations and random fixed point theorems for non-self maps.Proc. Amer. Math. Soc. 103 (1988), 1129–1135. Zbl 0676.47041, MR 0954994, 10.1090/S0002-9939-1988-0954994-0 Reference: [21] J. Lindenstrauss and L. Tzafriri: Classical Banach Spaces II—Function Spaces.Springer-Verlag, New York, Berlin, 1979. MR 0540367 Reference: [22] E. A. Lifshitz: Fixed point theorem for operators in strongly convex spaces.Voronez Gos. Univ. Trudy Math. Fak. 16 (1975), 23–28. (Russian) Reference: [23] A. Nowak: Applications of random fixed point theorems in the theory of generalized random differential equations.Bull. Polish Acad. Sci. Math. 34 (1986), 487–494. Zbl 0617.60059, MR 0874895 Reference: [24] N. S. Papageorgiou: Random fixed point theorems for measurable multifunctions in Banach spaces.Proc. Amer. Math. Soc. 32 (1987), 507–514. MR 0840638 Reference: [25] N. S. Papageorgiou: Deterministic and random fixed point theorems for single valued and multivalued functions.Rev. Roumaine Math. Pures Appl. 32 (1989), 53–61. MR 0901435 Reference: [26] S. A. Pichugov: Jung’s constant of the space $L^p$.Mat. Zametki Math. Notes 43 43 (1988 1988), 609–614 348–354. (Russian) MR 0954343 Reference: [27] B. Prus and R. Smarzemski: Strongly unique best approximations and centers in uniformly convex spaces.J. Math. Anal. Appl. 121 (1987), 10–21. MR 0869515, 10.1016/0022-247X(87)90234-4 Reference: [28] S. Prus: On Bynum’s fixed point theorem.Atti. Sem. Mat. Fis. Univ. Modena 38 (1990), 535–545. Zbl 0724.46020, MR 1076471 Reference: [29] S. Prus: Some estimates for the normal structure coefficient in Banach spaces.Rend. Circ. Mat. Palermo XL(2) (1991), 128–135. Zbl 0757.46029, MR 1119750 Reference: [30] L. E. Rybinski: Random fixed points and viable random solutions of functional differential inclusions.J. Math. Anal. Appl. 142 (1989), 53–61. Zbl 0681.60056, MR 1011408, 10.1016/0022-247X(89)90163-7 Reference: [31] V. M. Sehgal and C. Waters: Some random fixed point theorems for condensing operators.Proc. Amer. Math. Soc. 90 (1984), 425–429. MR 0728362, 10.1090/S0002-9939-1984-0728362-7 Reference: [32] V. M. Sehgal and S. P. Singh: On random approximations and a random fixed point theorem for set valued mappings.Proc. Amer. Math. Soc. 95 (1985), 91–94. MR 0796453, 10.1090/S0002-9939-1985-0796453-1 Reference: [33] R. Smarzewski: Strongly unique best approximations in Banach spaces II.J. Approx. Theory 51 (1987), 202–217. MR 0913618, 10.1016/0021-9045(87)90035-9 Reference: [34] R. Smarzewski: On the inequality of Bynum and Drew.J. Math. Anal. Appl. 150 (1990), 146–150. MR 1059576, 10.1016/0022-247X(90)90201-P Reference: [35] K. K. Tan and X. Z. Yuan: Some random fixed point theorems.In Fixed Point Theory and Applications, K. K. Tan (ed.), World Scientific, Singapore, 1992, pp. 334–345. MR 1190049 Reference: [36] K. K. Tan and X. Z. Yuan: On deterministic and random fixed points.Proc. Amer. Math. Soc 119 (1993), 849–856. MR 1169051, 10.1090/S0002-9939-1993-1169051-2 Reference: [37] K. K. Tan and H. K. Xu: Fixed point theorems for Lipschitzian semigroups in Banach spaces.Nonlinear Anal. 20 (1993), 395–404. MR 1206429, 10.1016/0362-546X(93)90144-H Reference: [38] D. H. Wagner: Survey of measurable selection theorems.SIAM J. Control Optim. 15 (1977), 859–903. Zbl 0407.28006, MR 0486391, 10.1137/0315056 Reference: [39] H. K. Xu: Fixed point theorems for uniformly Lipschitzian semigroups in uniformly convex Banach spaces.J. Math. Anal. Appl. 152 (1990), 391–398. MR 1077935, 10.1016/0022-247X(90)90072-N Reference: [40] H. K. Xu: Some random fixed point theorems for condensing and nonexpansive operators.Proc. Amer. Math. Soc. 110 (1990), 395–400. Zbl 0716.47029, MR 1021908, 10.1090/S0002-9939-1990-1021908-6 Reference: [41] H. K. Xu: Inequalities in Banach spaces with applications.Nonlinear Anal. 16 (1991), 1127–1138. Zbl 0757.46033, MR 1111623, 10.1016/0362-546X(91)90200-K Reference: [42] H. K. Xu: A random fixed point theorem for multivalued nonexpansive operators in a uniformly convex Banach space.Proc. Amer. Math. Soc. 117 (1993), 1089–1092. MR 1123670, 10.1090/S0002-9939-1993-1123670-8 Reference: [43] H. K. Xu: Random fixed point theorems for nonlinear uniformly Lipschitzian mappings.Nonlinear Anal. 26 (1996), 1302–1311. Zbl 0864.47051, MR 1376105 Reference: [44] C. Zalinescu: On uniformly convex functions.J. Math. Anal. Appl. 95 (1988), 344–374. MR 0716088, 10.1016/0022-247X(83)90112-9 .

## Files

Files Size Format View
CzechMathJ_50-2000-2_13.pdf 376.3Kb application/pdf View/Open

Partner of