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Regularity for variational evolution integrodifferential inequalities. (English) Zbl 1253.45007
Summary: 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.
MSC:
45J05 Integro-ordinary differential equations
49J40 Variational inequalities
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[1] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Nordhoff Leiden, The Netherlands, 1976. · Zbl 0343.49010
[2] V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, vol. 190 of Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1993. · Zbl 0840.01033
[3] 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. · Zbl 0993.34060
[4] 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. · Zbl 0806.34051
[5] 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. · Zbl 1032.35116
[6] 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. · Zbl 0962.93013
[7] 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. · Zbl 1123.34044
[8] 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. · Zbl 0794.49002
[9] H. Tanabe, Equations of Evolution, vol. 6 of Monographs and Studies in Mathematics, Pitman, London, UK, 1979. · Zbl 0417.35003
[10] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problmes and Applications, Springer, Berlin, Germany, 1972. · Zbl 0227.35001
[11] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, vol. 18 of North-Holland Mathematical Library, North-Holland, Amsterdam, The Netherlands, 1978. · Zbl 0387.46032
[12] G. Di Blasio, K. Kunisch, and E. Sinestrari, “L2-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. · Zbl 0538.45007
[13] J. L. Lions and E. Magenes, Problemes Aux Limites Non Homogenes Et Applications, vol. 3, Dunod, Paris, France, 1968. · Zbl 0165.10801
[14] J. P. Aubin, “Un Théorème de Compacité,” Comptes Rendus de l’Académie des Sciences, vol. 256, pp. 5042-5044, 1963. · Zbl 0195.13002
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