Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Compactness of the solution operator for a linear evolution equation with distributed measures
HTML articles powered by AMS MathViewer

by Ioan I. Vrabie PDF
Trans. Amer. Math. Soc. 354 (2002), 3181-3205 Request permission

Abstract:

The main goal of the present paper is to define the solution operator $(\xi ,g)\mapsto u$ associated to the evolution equation $du=(Au)dt+dg$, $u(0)=\xi$, where $A$ generates a $C_0$-semigroup in a Banach space $X$, $\xi \in X$, $g\in BV([ a,b ];X)$, and to study its main properties, such as regularity, compactness, and continuity. Some necessary and/or sufficient conditions for the compactness of the solution operator extending some earlier results due to the author and to Baras, Hassan, Veron, as well as some applications to the existence of certain generalized solutions to a semilinear equation involving distributed, or even spatial, measures, are also included. Two concrete examples of elliptic and parabolic partial differential equations subjected to impulsive dynamic conditions on the boundary illustrate the effectiveness of the abstract results.
References
  • N. U. Ahmed, Some remarks on the dynamics of impulsive systems in Banach Spaces, J. of Dynamics of Continuous, Discrete and Impulsive Systems Ser. A Math. Anal. 8 (2001), 261–274.
  • Pierre Baras, Jean-Claude Hassan, and Laurent Véron, Compacité de l’opérateur définissant la solution d’une équation d’évolution non homogène, C. R. Acad. Sci. Paris Sér. A-B 284 (1977), no. 14, A799–A802. MR 430864
  • V. Barbu, Personal communication.
  • V. Barbu and Th. Precupanu, Convexity and optimization in Banach spaces, 2nd ed., Mathematics and its Applications (East European Series), vol. 10, D. Reidel Publishing Co., Dordrecht; Editura Academiei Republicii Socialiste România, Bucharest, 1986. MR 860772
  • I. Bejenaru, J. I. Diaz, I. I. Vrabie, An abstract approximate controllability result and applications to elliptic and parabolic systems with dynamic boundary conditions, Electronic Journal of Differential Equations, 2001, no. 50, 1-19.
  • S. Bochner, A. E. Taylor, Linear functionals on certain spaces of abstractly-valued functions, Ann. of Math. 39(1938), 913-944.
  • H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Mathematics Studies, No. 5, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973 (French). MR 0348562
  • Haïm Brézis and Avner Friedman, Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pures Appl. (9) 62 (1983), no. 1, 73–97. MR 700049
  • J. Diestel and J. J. Uhl Jr., Vector measures, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis. MR 0453964
  • Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
  • Bekkai Messirdi, Résonances des systèmes perturbeés d’opérateurs de Schrödinger, Maghreb Math. Rev. 5 (1996), no. 1-2, 113–121 (French, with English, French and Arabic summaries). MR 1801043
  • A. R. Collar, On the reciprocation of certain matrices, Proc. Roy. Soc. Edinburgh 59 (1939), 195–206. MR 8
  • S. Gutman, Compact perturbations of $m$-accretive operators in general Banach spaces, SIAM J. Math. Anal. 13 (1982), no. 5, 789–800. MR 668321, DOI 10.1137/0513054
  • Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
  • Norimichi Hirano, Local existence theorems for nonlinear differential equations, SIAM J. Math. Anal. 14 (1983), no. 1, 117–125. MR 686238, DOI 10.1137/0514008
  • Norimichi Hirano, Existence of periodic solutions for nonlinear evolution equations in Hilbert spaces, Proc. Amer. Math. Soc. 120 (1994), no. 1, 185–192. MR 1174494, DOI 10.1090/S0002-9939-1994-1174494-8
  • Norimichi Hirano and Noriko Mizoguchi, Existence of periodic solutions for semilinear parabolic equations, Topology in nonlinear analysis (Warsaw, 1994) Banach Center Publ., vol. 35, Polish Acad. Sci. Inst. Math., Warsaw, 1996, pp. 39–49. MR 1448425
  • Chaim Samuel Hönig, Volterra Stieltjes-integral equations, Notas de Matemática, No. 56. [Mathematical Notes, No. 56], North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. Functional analytic methods; linear constraints; Mathematics Studies, No. 16. MR 0499969
  • A. Pazy, A class of semi-linear equations of evolution, Israel J. Math. 20 (1975), 23–36. MR 374996, DOI 10.1007/BF02756753
  • A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486, DOI 10.1007/978-1-4612-5561-1
  • Jacques Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4) 146 (1987), 65–96. MR 916688, DOI 10.1007/BF01762360
  • Naoki Shioji, Local existence theorems for nonlinear differential equations and compactness of integral solutions in $L^p(0,T;X)$, Nonlinear Anal. 26 (1996), no. 4, 799–811. MR 1362753, DOI 10.1016/0362-546X(94)00320-H
  • Naoki Shioji, Periodic solutions for nonlinear evolution equations in Banach spaces, Funkcial. Ekvac. 42 (1999), no. 2, 157–164. MR 1718787
  • 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
  • Ioan I. Vrabie, Periodic solutions for nonlinear evolution equations in a Banach space, Proc. Amer. Math. Soc. 109 (1990), no. 3, 653–661. MR 1015686, DOI 10.1090/S0002-9939-1990-1015686-4
  • Ioan I. Vrabie, A compactness criterion in $C(0,T;X)$ for subsets of solutions of nonlinear evolution equations governed by accretive operators, Rend. Sem. Mat. Univ. Politec. Torino 43 (1985), no. 1, 149–157. MR 859853
  • I. I. Vrabie, Compactness methods for nonlinear evolutions, 2nd ed., Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 75, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1995. With a foreword by A. Pazy. MR 1375237
  • Ioan I. Vrabie, Compactness in $L^p$ of the set of solutions to a nonlinear evolution equation, Qualitative problems for differential equations and control theory, World Sci. Publ., River Edge, NJ, 1995, pp. 91–101. MR 1372742
Similar Articles
Additional Information
  • Ioan I. Vrabie
  • Affiliation: Faculty of Mathematics, “Al. I. Cuza" University of Iaşi, Iaşi 6600, Romania
  • Address at time of publication: P. O. Box 180, Ro, Iş 1, Iaşi 6600, Romania
  • Email: ivrabie@uaic.ro
  • Received by editor(s): March 19, 2001
  • Received by editor(s) in revised form: September 21, 2001
  • Published electronically: April 1, 2002
  • Additional Notes: This research was supported in part by the CNCSU/CNFIS Grant C120(1998) of the World Bank and the Romanian Government
  • © Copyright 2002 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 354 (2002), 3181-3205
  • MSC (2000): Primary 47D06, 46G10, 47B07; Secondary 35A05, 35J99, 35K99
  • DOI: https://doi.org/10.1090/S0002-9947-02-02997-5
  • MathSciNet review: 1897396