Let us consider a functional equation Ax=y characterized by an ill-posed linear operator A acting between two Banach spaces X and Y. In this talk, we propose an extension of the Tikhonov regularization approach to the (unconventional) setting where X and Y are both two variable exponent Lebesgue spaces. Basically, a variable exponent Lebesgue space is a (non-Hilbertian) Banach space where the exponent used in the definition of the norm is not constant, but rather is a function of the domain. This way, we can automatically assign different ``amount’’ of regularization, related to different values of the exponent function, on different regions of the domain. In the case of image deblurring problems, different pointwise regularization is useful because background, low intensity, and high intensity values of the image to restore require different filtering (i.e., regularization) levels, depending on the local signal to noise ratios in all the different portions of the image domain. A numerical evidence will be also discussed. References Diening, L., Harjulehto, P., Hästö, P., Ruzicka, M., Lebesgue and Sobolev spaces with variable exponents. Lecture Notes in Mathematics. vol. 2017, Springer, 2011. Schuster, T., Kaltenbacher, B., Hofmann, B., and Kazimierski, K. S., Regularization Methods in Banach Spaces. Radon Series on Computational and Applied Mathematics, vol. 10, De Gruyter, 2012 C. Estatico, S. Gratton, F. Lenti, D. Titley-Peloquin, “A conjugate gradient like method for p-norm minimization in functional spaces”, Numerische Mathematik,