Miheţ, Dorel On set-valued nonlinear equations in Menger probabilistic normed spaces. (English) Zbl 1116.47062 Fuzzy Sets Syst. 158, No. 16, 1823-1831 (2007). Summary: We introduce the concept of probabilistic \((\varphi,\psi)\)-contractor couple, which allows us to consider set-valued nonlinear operator equations in a large class of probabilistic normed spaces. Our results complete those from Y.J.Cho, N. J. Huang and S. M. Kang [Fuzzy Sets Syst. 110, No. 1, 115–122 (2000; Zbl 0949.47059)] and J.-X. Fang [ibid. 131, No. 3, 357–364 (2002; Zbl 1027.47081)]. Cited in 2 ReviewsCited in 7 Documents MSC: 47S50 Operator theory in probabilistic metric linear spaces 47J05 Equations involving nonlinear operators (general) 47H04 Set-valued operators Keywords:Menger probabilistic normed spaces; probabilistic \(\varphi\)-contractor couple; nonlinear operator equations; fuzzy mappings Citations:Zbl 0949.47059; Zbl 1027.47081 PDFBibTeX XMLCite \textit{D. Miheţ}, Fuzzy Sets Syst. 158, No. 16, 1823--1831 (2007; Zbl 1116.47062) Full Text: DOI References: [1] Altman, M., Contractors and contractor directions. Theory and applications. A new approach to solving equations, Lecture Notes in Pure and Applied Mathematics, vol. 32 (1977), Marcel Dekker: Marcel Dekker New York · Zbl 0363.65045 [2] Balakrishna Reddy, B.; Subrahmanyam, P. V., Altman’s contractors and fixed points of multivalued mappings, Pacific J. Math., 99, 1, 127-136 (1982) · Zbl 0528.47043 [3] Chang, S. S., Probabilistic contractors and the solutions of nonlinear equations in probabilistic normed spaces, Chinese Sci. Bull., 15, 1451-1454 (1990) [4] Chang, S. S.; Cho, Y. J.; Jung, J. S.; Wang, F., Nonlinear operator equations with probabilistic contractors in probabilistic normed spaces, Commun. Korean Math. Soc., 11, 117-129 (1996) · Zbl 0945.47054 [5] Chang, S. S.; Cho, Y. J.; Kang, S. M., Probabilistic Metric Spaces and Nonlinear Operator Theory (1994), Sichuan University Press: Sichuan University Press Chengdu [6] Chang, S. S.; Cho, Y. J.; Wang, F., On the existence and the uniqueness problem of solutions for set-valued and single-valued nonlinear operator equations in probabilistic normed spaces, Internat. J. Math. Math. Sci., 17, 2, 389-396 (1994) · Zbl 0801.47043 [7] Chitra, A.; Subrahmanyam, P. V., A note on contractors in random normed spaces, Demonstratio Math., XXVIII, 3, 603-609 (1995) · Zbl 0847.46048 [8] Cho, Y. J.; Huang, H. J.; Kang, S. M., Nonlinear equations for fuzzy mappings in probabilistic normed spaces, Fuzzy Sets and Systems, 110, 115-122 (2000) · Zbl 0949.47059 [9] Fang, J. X., \(( \Phi, \Delta )\)-type probabilistic contractor and the solutions for nonlinear operator equations in Menger PN-spaces, Appl. Math. Mech., 6, 17, 491-496 (1996) [10] Fang, J. X., On nonlinear equations for fuzzy mappings in probabilistic normed spaces, Fuzzy Sets and Systems, 131, 357-364 (2002) · Zbl 1027.47081 [11] Fang, J. X.; Song, G. A., \( \Psi \)-contractor and the solutions for nonlinear operator equations in fuzzy normed spaces, Fuzzy Sets and Systems, 121, 267-273 (2001) · Zbl 1008.47070 [12] Hadžić, O.; Pap, E., Fixed point theory in probabilistic metric spaces, in Mathematics and its Applications, vol. 536 (2001), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht, Boston, London [13] Hadžić, O.; Pap, E., New classes of probabilistic contractions and applications to random operators, (Cho, Y. J.; Kim, J. K.; Kang, S. M., Fixed Point Theory and Applications, vol. 40 (2003), Nova Science Publishers: Nova Science Publishers Hauppauge, NY), 97-119 · Zbl 1069.54026 [14] Huang, N. J.; Lan, H. Y., A couple of nonlinear equations with fuzzy mappings in fuzzy normed spaces, Fuzzy Sets and Systems, 152, 209-222 (2005) · Zbl 1101.47060 [15] Lee, A. C.; Padgett, W. J., Random contractors and the solutions of random nonlinear equations, Nonlinear Anal. TMA, 1, 173-185 (1977) · Zbl 0347.60045 [16] V. Radu, Lectures on Probabilistic Analysis, West University of Timişoara Surveys, Lectures Notes and Monographs Series on Probability, Statistics and Applied Mathematics, vol. 2, 1994.; V. Radu, Lectures on Probabilistic Analysis, West University of Timişoara Surveys, Lectures Notes and Monographs Series on Probability, Statistics and Applied Mathematics, vol. 2, 1994. [17] Singh, S. L.; Mishra, S. N., Contractors on random normed spaces, Bull. Cal. Math. Soc., 87, 107-112 (1995) · Zbl 0829.47064 [18] Zeng, W. Z., Probabilistic contractor and nonlinear equations in Menger PN-spaces, J. Math. Res. Expos., 11, 47-51 (1991) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.