Caputo fractional $q$-difference equations in Banach spaces
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Abstract
This paper aims to explore the existence results of a certain type of Caputo fractional qq-difference equations in Banach spaces. To achieve this goal, we employ a fixed point theorem that relies on the concept of measure of noncompactness and the convex-power condensing operator. We give an illustrative example in the last section.
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References
S. Abbas, M. Benchohra, J. R. Graef and J. Henderson, Implicit fractional differential and integral equations: existence and stability, De Gruyter Series in Nonlinear Analysis and Applications, Vol. 26, De Gruyter, Berlin, 2018.
S. Abbas, M. Benchohra, J. E. Lazreg, J. J. Nieto and Y. Zhou, Fractional differential equations and inclusions: classical and advanced topics, Series on Analysis, Applications and Computation, Vol. 10, World Scientific, Singapore, 2023.
S. Abbas, M. Benchohra and G. M. N’Guérékata, Advanced fractional differential and integral equations, Mathematics research developments series, Nova Science Publishers, New York, 2015.
C. R. Adams, On the linear ordinary q-difference equation, Annals of Mathematics, Second Series, 30(1/4) (1928), 195–205.
R. Agarwal, Certain fractional q-integrals and q-derivatives, Mathematical Proceedings of the Cambridge Philosophical Society, 66(2) (1969), 365-370.
B. Ahmad, Boundary value problem for nonlinear third order q-difference equations, Electronic Journal of Differential Equations, 2011(94) (2011), 1–7.
B. Ahmad, S. K. Ntouyas and L. K. Purnaras, Existence results for nonlocal boundary value problems of nonlinear fractional q-difference equations, Advances in Difference Equations, 140(2012) (2012), 1–15.
J. C. Alvàrez, Measure of noncompactness and fixed points of nonexpansive condensing mappings in locally convex spaces, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales, Serie A. Matemáticas Madrid, 79(1-2) (1985), 53–66.
J. P. Aubin, I. Ekeland, Applied Nonlinear Analysis, John Wiley Sons, New York, 1984.
J. Banas and ` K. Goebel, Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Maths, New York-Basel, Marcel Dekker Inc, New York, 1980.
A. Boutiara, M. Benbachir, Existence and uniqueness results to a fractional q-difference coupled system with integral boundary conditions via topological degree theory, International Journal of Nonlinear Analysis and Applications, 13(1) (2022), 3197–3211.
F. Browder, On the convergence of successive approximations for nonlinear functional equations, Indagationes Mathematicae, 30(1) (1968), 27–35.
R. D. Carmichael, The general theory of linear q-difference equations, American journal of mathematics, 34(2) (1912), 147–168.
C. Derbazi, H. Hammouche, A. Salim and M. Benchohra, Weak solutions for fractional Langevin equations involving two fractional orders in banach spaces, Afrika Matematika, 34(1) (2023), 1.
M. El-Shahed, H. A. Hassan, Positive solutions of q-difference equation, Proceedings of the American Mathematical Society, 138(5) (2010), 1733–1738.
S. Etemad, S. K. Ntouyas and B. Ahmad, Existence theory for a fractional q-integrodifference equation with q-integral boundary conditions of different orders, Mathematics, 7(8) (2019), 1-15.
H. R. Heinz, On the behavior of measure of noncompactness with respect to differentiation and integration of vector-valued functions, Nonlinear Analysis: Theory, Methods and Applications, 7(12) (1983), 1351-1371.
R. Herrmann, Fractional calculus: An Introduction for Physicists, World Scientific Publishing Company: Singapore, 2011.
R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific: Singapore, 2000.
V. Kac and P. Cheung, Quantum Calculus, Springer, New York, 2002.
A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, , Vol. 204, Elsevier Science B.V, Amsterdam, 2006.
N. Laledj, A. Salim, J-E. Lazreg, S. Abbas, B. Ahmad and M. Benchohra, On implicit fractional q-difference equations: Analysis and stability. Mathematical Methods in the Applied Sciences, 45(17) (2022), 10775–10797.
L. Liu, F. Guo, C. Wu and Y. Wu, Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces, Journal of Mathematical Analysis and Applications, 309(2) (2005), 638–649.
H. Mönch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Analysis: Theory, Methods and Applications, 4(5) (1980), 985–999.
D. O’Regan, Fixed point theory for weakly sequentially continuous mapping, Mathematical and Computer Modelling, 27(5) (1998), 1–14.
W. Rahou, A. Salim, J. E. Lazreg and M. Benchohra, Existence and stability results for impulsive implicit fractional differential equations with delay and Riesz–Caputo derivative, Mediterranean Journal of Mathematics, 20(3) (2023), 143.
P. M. Rajkovic, S. D. Marinkovic and M-S. Stankovic, Fractional integrals and derivatives in q-calculus, Applicable analysis and discrete mathematics,1(1) (2007), 311–323, University of Belgrade, Serbia.
P. M. Rajkovic, S. D. Marinkovic and M. S. Stankovic, On q-analogues of Caputo derivative and Mittag-Leffler function, Fractional calculus and applied analysis, 10(4) (2007), 359–373.
A. Salim, S. Abbas, M. Benchohra and E. Karapinar, A Filippov’s theorem and topological structure of solution sets for fractional q-difference inclusions, Dynamic Systems and Applications, 31 (2022), 17–34.
A. Salim and M. Benchohra, Existence and uniqueness results for generalized Caputo iterative fractional boundary value problems, Fractional Differential Calculus, 12(1) (2022), 197–208.
S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Amsterdam, 1987.
V. E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, Heidelberg; Higher Education Press, Beijing, 2010.
Y. Zhou, J. R. Wang and L. Zhang, Basic Theory of Fractional Differential Equations, Second edition. World Scientific Publishing Co. Pte. Ltd, Hackensack, NJ, 2017.