Abstract
The purpose of this paper is to introduce and study the semi-groups of nonlinear contractions in probabilistic normed spaces and to establish the Crandall-Liggett's exponential formula for some kind of accretive mappings in probabilistic normed spaces. As applications, these results are utilized to study the Cauchy problem for a kind of differential inclusions with accretive mappings in probabilistic normed spaces.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. Barbu,Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International Publishing House (1976).
H. Brezis and A. Pazy, Semigroups of nonlinear contractions on convex sets,J. Funct. Anal.,6 (1970), 367–383.
F. E. Browder, Nonlinear mappings of nonexpansive and accretive type in Banach spaces,Bull. Amer. Math. Soc.,73 (1967), 875–882.
F. E. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces,Proc. Symp. Pure. Math.,18, Part 2 (1976).
S. S. Chang and Y. Q. Chen, On the existence of solution for equations with accretive mappings in probabilistic normed spaces,Applied Mathematics and Mechanics (English Ed.),11, 9 (1990), 821–828.
S. S. Chang, Y. J. Cho and S. M. Kang,Probabilistic Metric Spaces and Nonlinear Operator Theory, Sichuan University Press, P. R. China (1994).
M. G. Crandall and T. Liggett, Generations of semi-groups of nonlinear transformations on general Banach spaces,Amer. J. Math.,93 (1971), 265–298.
M. G. Crandall and A. Pazy, Semigroups of nonlinear contractions and dissipative sets,J. Funct. Anal.,3 (1969), 376–418.
K. S. Ha, K. Y. Shin and Y. J. Cho, Accretive operators in probabilistic normed spaces,Bull. Korean Math. Soc.,31, 1 (1994), 45–54.
T. Kato, Nonlinear semigroups and evolution equations,J. Math. Soc. Japan,19 (1967), 505–520.
T. Kato, Accretive operators and nonlinear evolutions in Banach spaces,Proc. Sym. Pure Math.,18 (1970), 138–161.
Y. Komura, Nonlinear semigroups in Hilbert spaces,J. Math. Soc. Japan,19 (1967), 493–507.
V. Lakshmikantham and S. Leela,Nonlinear Differential Equations in Abstract Spaces, Pergamon Press (1981).
G. Lumer, Semi inner product spaces,Trans. Amer. Math. Soc.,100 (1961), 29–43.
W. Rudin,Functional Analysis, McGraw-Hill Book Company (1973).
B. Schweizer and A. Sklar,Probabilistic Metric Spaces North-Holland (1983).
Author information
Authors and Affiliations
Additional information
Project supported by the National Natural Science Foundation of China
Rights and permissions
About this article
Cite this article
Shisheng, Z., Yuqing, C. Nonlinear semigroups and differential inclusions in probabilistic normed spaces. Appl Math Mech 19, 815–829 (1998). https://doi.org/10.1007/BF02458236
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02458236