Journal of Applied Mathematics and Physics, 2016, 4, 787-795 Published Online April 2016 in SciRes. http://www.scirp.org/journal/jamp http://dx.doi.org/10.4236/jamp.2016.44089 How to cite this paper: Abdalmonem, A., Abdalrhman, O. and Tao, S.P. (2016) The Boundedness of Fractional Integral with Variable Kernel on Variable Exponent Herz-Morrey Spaces. Journal of Applied Mathematics and Physics, 4, 787-795. http://dx.doi.org/10.4236/jamp.2016.44089 The Boundedness of Fractional Integral with Variable Kernel on Variable Exponent Herz-Morrey Spaces Afif Abdalmonem 1,2* , Omer Abdalrhman 1,3 , Shuangping Tao 1 1 College of Mathematics and Statistics, Northwest Normal University, Lanzhou, China 2 Faculty of Science, University of Dalanj, Dalanj, Sudan 3 College of Education, Shendi University, Shendi, Sudan Received 19 March 2016; accepted 24 April 2016; published 28 April 2016 Copyright © 2016 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/ Abstract In this paper, we study the boundedness of the fractional integral with variable kernel. Under some assumptions, we prove that such kind of operators is bounded from the variable exponent Herz-Morrey spaces to the variable exponent Herz-Morrey spaces. Keywords Fractional Integral, Variable Kernel, Variable Exponent, Herz-Morrey Spaces 1. Introduction Let 0 n µ < < , ( ) ( ) ( ) 1 1 n r n L L S r ∞ − Ω∈ × ≥  is homogenous of degree zero on n  , 1 n S − denotes the unit sphere in n  . If i) For any , n xz ∈  , one has ( ) ( ) , , x z xz λ Ω =Ω ; ii) ( ) ( ) ( ) ( ) ( ) 1 1 1 : sup , d n r n n n r r L L S s x xz z σ ∞ − − × ∈ ′ ′ Ω = Ω <∞ ∫   The fractional integral operator with variable kernel , T µ Ω is defined by ( ) ( ) ( ) , , d, n n xx y T f x f y y x y µ µ Ω − Ω − = − ∫  * Corresponding author.