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Keywords:Variable exponent Lebesgue and Sobolev spaces Nonlinear parabolic variational inequality In this article, new properties of variable exponent Lebesgue and Sobolev spaces are examined. Using these properties we prove the existence of the solution of some parabolic variational inequality.

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Some nonlinear integro-differential equations of fourth order with variable exponents of the nonlinearity are considered. The initial-boundary value problem for these equations is investigated and the existence theorem for the problem is proved.