Random semilinear evolution equations in Banach spaces

Kravvaritis, Dimitrios

doi:10.1090/S0002-9939-1991-1056680-8

Random semilinear evolution equations in Banach spaces
HTML articles powered by AMS MathViewer

by Dimitrios Kravvaritis PDF

Proc. Amer. Math. Soc. 113 (1991), 715-722 Request permission

Abstract:

In this paper we prove the existence of mild solutions for random, semilinear evolution equations involving a random, linear, unbounded $m$-dissipative operator and a random single valued or multivalued perturbation. Finally, an application to a random semilinear partial differential equation is given.

References

Evgenios P. Avgerinos and Nikolaos S. Papageorgiou, Random nonlinear evolution inclusions in reflexive Banach spaces, Proc. Amer. Math. Soc. 104 (1988), no. 1, 293–299. MR 958086, DOI 10.1090/S0002-9939-1988-0958086-6
Viorel Barbu, Nonlinear semigroups and differential equations in Banach spaces, Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden, 1976. Translated from the Romanian. MR 0390843
Georges A. Bécus, Random generalized solutions to the heat equation, J. Math. Anal. Appl. 60 (1977), no. 1, 93–102. MR 442471, DOI 10.1016/0022-247X(77)90051-8
A. T. Bharucha-Reid, Random integral equations, Mathematics in Science and Engineering, Vol. 96, Academic Press, New York-London, 1972. MR 0443086
Shigeru Itoh, Random fixed-point theorems with an application to random differential equations in Banach spaces, J. Math. Anal. Appl. 67 (1979), no. 2, 261–273. MR 528687, DOI 10.1016/0022-247X(79)90023-4
Shigeru Itoh, Random differential equations associated with accretive operators, J. Differential Equations 31 (1979), no. 1, 139–154. MR 524822, DOI 10.1016/0022-0396(79)90157-8
J. Kampé de Fériet, Random solutions of partial differential equations, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. III, University of California Press, Berkeley-Los Angeles, Calif., 1956, pp. 199–208. MR 0084927
Dimitrios Kravvaritis and Nikolaos Stavrakakis, Perturbations of maximal monotone random operators, Proceedings of the symposium on operator theory (Athens, 1985), 1986, pp. 301–310. MR 872291, DOI 10.1016/0024-3795(86)90322-8
Nikolaos S. Papageorgiou, Random differential inclusions in Banach spaces, J. Differential Equations 65 (1986), no. 3, 287–303. MR 865064, DOI 10.1016/0022-0396(86)90021-5
Nikolaos S. Papageorgiou, On measurable multifunctions with applications to random multivalued equations, Math. Japon. 32 (1987), no. 3, 437–464. MR 914749
Nikolaos S. Papageorgiou, Convergence theorems for Banach space valued integrable multifunctions, Internat. J. Math. Math. Sci. 10 (1987), no. 3, 433–442. MR 896595, DOI 10.1155/S0161171287000516
N. H. Pavel, Differential equations, flow invariance and applications, Research Notes in Mathematics, vol. 113, Pitman (Advanced Publishing Program), Boston, MA, 1984. MR 774954
Nicolae H. Pavel, Nonlinear evolution operators and semigroups, Lecture Notes in Mathematics, vol. 1260, Springer-Verlag, Berlin, 1987. Applications to partial differential equations. MR 900380, DOI 10.1007/BFb0077768
A. Pazy, A class of semi-linear equations of evolution, Israel J. Math. 20 (1975), 23–36. MR 374996, DOI 10.1007/BF02756753
M.-F. Sainte-Beuve, On the extension of von Neumann-Aumann’s theorem, J. Functional Analysis 17 (1974), 112–129. MR 0374364, DOI 10.1016/0022-1236(74)90008-1
T. T. Soong, Random differential equations in science and engineering, Mathematics in Science and Engineering, Vol. 103, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1973. MR 0451405
I. I. Vrabie, Nonlinear evolution equations: existence via compactness, Semigroups, theory and applications, Vol. I (Trieste, 1984) Pitman Res. Notes Math. Ser., vol. 141, Longman Sci. Tech., Harlow, 1986, pp. 242–248. MR 876948
Ioan I. Vrabie, The nonlinear version of Pazy’s local existence theorem, Israel J. Math. 32 (1979), no. 2-3, 221–235. MR 531265, DOI 10.1007/BF02764918