On Weighted Cauchy-Type Problem of Riemann-Liouville Fractional Differential Equations in Lebesgue Spaces with Variable Exponent

Year 2024, Volume: 7 Issue: 2, 93 - 101, 23.05.2024

Mokhtar Mokhtari , Ahmed Refice , Mohammed Said Souıd , Ali Yakar

https://doi.org/10.32323/ujma.1409291

Abstract

This paper aims to investigate the existence, uniqueness, and stability properties for a class of fractional weighted Cauchy-type problem in the variable exponent Lebesgue space $L^{p(.)}$. The obtained results are set up by employing generalized intervals and piece-wise constant functions so that the $L^{p(.)}$ is transformed into the classical Lebesgue spaces.
Moreover, the usual Banach Contraction Principle is utilized, and the Ulam-Hyers (UH) stability is studied. At the final stage, we provide an example to support the accuracy of the obtained results.

Keywords

Fixed point theorem , Fractional differential equations , Ulam-Hyers stability , Variable exponent Lebesgue spaces

References

• [1] W. Orlicz, Über konjugierte exponentenfolgen, Studia Mathematica, 3(1) (1931), 200-211.
• [2] H Nakano, Modular Semi-Ordered Spaces, Maruzen Co. Ltd., Tokyo, Japan, 1950.
• [3] H. Nakano, Topology and Topological Linear Spaces, Maruzen Co., Ltd., Tokyo, 1951.
• [4] I. I. Sharapudinov, Topology of the space Lp(t)(0;1), Matematicheskie Zametki, 26(4) (1979), 613-632.
• [5] O. Kovavcik, J. Rakosnik, On spaces l p(x) and wk;p(x), Czechoslovak Math. J., 41(116) (1991), 592-618.
• [6] X.L. Fan, D. Zhao, On the spaces l p(x)(w) and wk;p(x)(w), J. Math. Anal. Appl., 263(2) (2001), 424-446.
• [7] R. Aboulaich, D. Meskine, A. Souissi, New diffusion models in image processing, Comput. Math. Appl., 56(4) (2008), 874-882.
• [8] H. Attouch, G. Buttazzo, G. Michaille, Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization, MOS-SIAM series on optimization, 2014.
• [9] E. M. Bollt, R. Chartrand, S. Esedoğlu, P. Schultz, K. R. Vixie, Graduated adaptive image denoising: Local compromise between total variation and isotropic diffusion, Adv. Comput. Math., 31 (2009), 61-85.
• [10] Y. Chen, W. Guo, Q. Zeng, Y. Liu, A nonstandard smoothing in reconstruction of apparent diffusion coefficient profiles from diffusion weighted images, Inverse Probl. Imaging, 2(2) (2008), 205-224.
• [11] Y. Chen, S. Levine, M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66(4) (2006), 1383-1406.
• [12] C. Derbazi, Z. Baitiche, M. Benchohra, A. Cabada, Initial value problem for nonlinear fractional differential equations with y-caputo derivative via monotone iterative technique, Axioms, 9(2) (2020), 57.
• [13] J. G. Abdulahad, S. A. Murad, Local existence theorem of fractional differential equations in Lp space, Raf. J. Comp. Maths, 9(2) (2012), 71-78.
• [14] R. P. Agarwal, V. Lupulescu, D. O’Regan, Lp solutions for a class of fractional integral equations, J. Integral Equ. Appl., 29(2) (2017), 251-270.
• [15] S. Arshad, V. Lupulescu, D. O’Regan, Lp solutions for fractional integral equations,, Fract. Calc. Appl., 17(1) (2014), 259-276.
• [16] B. Dong, Z. Fu, and J. Xu, Riesz-Kolmogorov theorem in variable exponent Lebesgue spaces and its applications to Riemann-Liouville fractional differential equations,, Sci. China Math., 61(10) (2018), 1807-1824.
• [17] A. Refice, M. Inc, M. S. Hashemi, M. S. Souid, Boundary value problem of Riemann-Liouville fractional differential equations in the variable exponent Lebesgue spaces Lp (.), J. Geom. Phys., 178 (2022), 104554.
• [18] A. Refice, M. S. Souid, Juan L.G. Guirao, H. Günerhan, Terminal value problem for Riemann-Liouville fractional differential equation in the variable exponent Lebesgue space Lp(:) , Math. Meth. Appl. Sci., (2023), 1–19.
• [19] M. S. Souid, A. Refice, K. Sitthithakerngkiet, Stability of p (.)-integrable solutions for fractional boundary value problem via piecewise constant functions, Fractal Fract., 7(2) (2023), 198.
• [20] K. Benia, M.S. Souid, F. Jarad, M. Alqudah, T. Abdeljawad, Boundary value problem of weighted fractional derivative of a function with a respect to another function of variable order, J. Inequal. Appl., 2023(2023), 127.
• [21] D. O’Regan, R. Agarwal, S. Hristova, M. Abbas, Existence and stability results for differential equations with a variable-order generalized proportional Caputo fractional derivative, Mathematics, 12(2) (2024), 233.
• [22] D. O’Regan, S. Hristova, R. Agarwal, Ulam-type stability results for variable orderY-tempered Caputo fractional differential equations, Fractal Fract., 8(1) (2023), 11.
• [23] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Elsevier Science Tech, 2006.
• [24] D.V. Cruz-Uribe, A. Fiorenza, Variable Lebesgue Spaces: Foundations and Harmonic Analysis, Applied and Numerical Harmonic Analysis, Springer Basel, 2013.
• [25] H. Royden, P. Fitzpatrick, Real Analysis. Pearson Modern Classics for Advanced Mathematics Series, Pearson, 2017.
• [26] V. S. Guliyev, S. G. Samko, Maximal, potential, and singular operators in the generalized variable exponent Morrey spaces on unbounded sets, J. Math. Sci., 193(2) (2013), 228-248.
• [27] H. Rafeiro, S. Samko, Variable exponent campanato spaces, J. Math. Sci., 172(1) (2011), 143-164.
• [28] A. Benkerrouche, M. S. Souid, K. Sitthithakerngkiet, A. Hakem, Implicit nonlinear fractional differential equations of variable order, Bound. Value Probl., 2021 (2021), 64.
• [29] M. Benchohra, J. E. Lazreg, Existence and Ulam stability for nonlinear implicit fractional differential equations with Hadamard derivative, Stud. Univ. Babes-Bolyai Math., 62(1) (2017), 27-38.