Riesz-Kolmogorov theorem in variable exponent Lebesgue spaces and its applications to Riemann-Liouville fractional differential equations |
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Authors: | Baohua Dong Zunwei Fu Jingshi Xu |
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Institution: | 1.Department of Mathematics,Nanjing University of Information Science and Technology,Nanjing,China;2.Department of Mathematics,Linyi University,Linyi,China;3.Department of Computer Science,The University of Suwon,Hwaseong,Korea;4.Department of Mathematics,Hainan Normal University,Haikou,China |
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Abstract: | In this paper, we obtain the necessary and sufficient condition of the pre-compact sets in the variable exponent Lebesgue spaces, which is also called the Riesz-Kolmogorov theorem. The main novelty appearing in this approach is the constructive approximation which does not rely on the boundedness of the Hardy-Littlewood maximal operator in the considered spaces such that we do not need the log-Hölder continuous conditions on the variable exponent. As applications, we establish the boundedness of Riemann-Liouville integral operators and prove the compactness of truncated Riemann-Liouville integral operators in the variable exponent Lebesgue spaces. Moreover, applying the Riesz-Kolmogorov theorem established in this paper, we obtain the existence and the uniqueness of solutions to a Cauchy type problem for fractional differential equations in variable exponent Lebesgue spaces. |
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