Abstract and Applied Analysis

Regularity for Variational Evolution Integrodifferential Inequalities

Yong Han Kang and Jin-Mun Jeong

Full-text: Open access

Abstract

We deal with the regularity for solutions of nonlinear functional integrodifferential equations governed by the variational inequality in a Hilbert space. Moreover, by using the simplest definition of interpolation spaces and the known regularity result, we also prove that the solution mapping from the set of initial and forcing data to the state space of solutions is continuous, which very often arises in application. Finally, an example is also given to illustrate our main result.

Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 797516, 18 pages.

Dates
First available in Project Euclid: 5 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1365174076

Digital Object Identifier
doi:10.1155/2012/797516

Mathematical Reviews number (MathSciNet)
MR2984019

Zentralblatt MATH identifier
1253.45007

Citation

Kang, Yong Han; Jeong, Jin-Mun. Regularity for Variational Evolution Integrodifferential Inequalities. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 797516, 18 pages. doi:10.1155/2012/797516. https://projecteuclid.org/euclid.aaa/1365174076


Export citation

References

  • V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Nordhoff Leiden, The Netherlands, 1976.
  • V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, vol. 190 of Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1993.
  • J.-M. Jeong, D.-H. Jeong, and J.-Y. Park, “Nonlinear variational evolution inequalities in Hilbert spaces,” International Journal of Mathematics and Mathematical Sciences, vol. 23, no. 1, pp. 11–20, 2000.
  • N. U. Ahmed and X. Xiang, “Existence of solutions for a class of nonlinear evolution equations with nonmonotone perturbations,” Nonlinear Analysis. Series A, vol. 22, no. 1, pp. 81–89, 1994.
  • J.-M. Jeong and J.-Y. Park, “Nonlinear variational inequalities of semilinear parabolic type,” Journal of Inequalities and Applications, vol. 6, no. 2, Article ID 896837, pp. 227–245, 2001.
  • J. M. Jeong, Y. C. Kwun, and J. Y. Park, “Approximate controllability for semilinear retarded functional-differential equations,” Journal of Dynamical and Control Systems, vol. 5, no. 3, pp. 329–346, 1999.
  • Y. Kobayashi, T. Matsumoto, and N. Tanaka, “Semigroups of locally Lipschitz operators associated with semilinear evolution equations,” Journal of Mathematical Analysis and Applications, vol. 330, no. 2, pp. 1042–1067, 2007.
  • N. U. Ahmed, “Optimal control of infinite-dimensional systems governed by integrodifferential equations,” in Differential Equations, Dynamical Systems, and Control Science, vol. 152 of Lecture Notes in Pure and Applied Mathematics, pp. 383–402, Dekker, New York, NY, USA, 1994.
  • H. Tanabe, Equations of Evolution, vol. 6 of Monographs and Studies in Mathematics, Pitman, London, UK, 1979.
  • J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problmes and Applications, Springer, Berlin, Germany, 1972.
  • H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, vol. 18 of North-Holland Mathematical Library, North-Holland, Amsterdam, The Netherlands, 1978.
  • G. Di Blasio, K. Kunisch, and E. Sinestrari, “${L}^{2}$-regularity for parabolic partial integro-differential equations with delay in the highest-order derivatives,” Journal of Mathematical Analysis and Applications, vol. 102, no. 1, pp. 38–57, 1984.
  • J. L. Lions and E. Magenes, Problemes Aux Limites Non Homogenes Et Applications, vol. 3, Dunod, Paris, France, 1968.
  • J. P. Aubin, “Un Théorème de Compacité,” Comptes Rendus de l'Académie des Sciences, vol. 256, pp. 5042–5044, 1963.