Existence Results for BVP of a Class of Generalized Fractional-Order Implicit Differential Equations

Year 2022, Volume: 5 Issue: 3, 114 - 123, 30.09.2022

Kadda Maazouz , Dvivek Vivek , Elsayed Elsayed

https://doi.org/10.33434/cams.1069182

Cited By: 1

Abstract

In this paper, we study the existence of solutions to boundary value problem for implicit differential equations involving generalized fractional derivative via fixed point methods.

Keywords

Generalized fractional derivative, Boundary value problem, Integral equation, Fixed point

Supporting Institution

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Project Number

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References

• [1] D. R. Anderson, D. J. Ulness, Properties of Katugampola fractional derivative with potential application in quantum mechanics, J. Math. Phys., 56 (2015).
• [2] M. S. Abdo, Further results on the existence of solutions for generalized fractional quadratic functional integral equations, J. Math. Anal & Model, 1(1) (2020), 33-46.
• [3] A. Bashir, S. Sivasundaram, Some existence results for fractional integro-differential equations with nonlinear conditions, Comm. Appl. Analysis, 12 (2) (2008), 107-112.
• [4] B. N. Abood, S. S. Redhwan, O. Bazighifan, K. Nonlaopon, Investigating a generalized fractional quadratic integral equation, Fractal Fract., 6 (2022), 251.
• [5] M. Benchohra, K. Maazouz, Existence and uniqueness results for implicit fractional differential equations with integral boundary conditions, Comm. App. Analysis, 20 (2016), 355-366.
• [6] A. Cabada, K. Maazouz, Results for Fractional Differential Equations with Integral Boundary Conditions Involving the Hadamard Derivative, In: Area I. et al. (eds) Nonlinear Analysis and Boundary Value Problems, NABVP 2018, Springer Proceedings in Mathematics & Statistics, vol 292. Springer, Cham, 2019.
• [7] G. J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003.
• [8] M. Janaki, K. Kanagarajan, E. M. Elsayed, Existence criteria for Katugampola fractional type impulsive differential equations with inclusions, J. Math. Sci. Model., 2(1) (2019), 51-63.
• [9] M. Janaki, K. Kanagarajan, D. Vivek, Analytic study on fractional implicit differential equations with impulses via Katugampola fractional Derivative, Int. J. Math. Appl., 6(2-A) (2018), 53-62.
• [10] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
• [11] U. N. Katugampola, New approach to generalized fractional integral, Appl. Math. Comput., 218(3) (2011), 860-865.
• [12] U. N Katugampola, New approach to generalized fractional derivative, Bull. Math. Anal. Appl., 6 (4) (2014), 1-15.
• [13] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differenatial Equations, North-Holland Mathematics Studies, 204, Elsevier Science B. V. Amsterdam, 2006.
• [14] V. Lakshmikantham, S. Leela, J. Vasundhara, Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, 2009.
• [15] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity. An Introduction to Mathematical Models, Imperial College Press, London, 2010.
• [16] K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993. [17] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
• [18] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Yverdon, 1993.
• [19] D. R. Smart, Fixed Point Theorems, Cambridge University Press, Cambridge 1974, Bull. Math. Anal. Appl., 6 (4) (2014), 1-15.
• [20] S. S. Redhwan, S. L. Shaikh, M. S. Abdo, Caputo-Katugamola type implicit fractional differential equation with two-point anti-periodic boundary conditions, Results in Nonlinear Analysis, 5(1) (2022), 12-28.
• [21] S. S. Redhwan, S. L. Shaikh, M. S. Abdo, W. Shatanawi, K. Abodayeh, M. A. Almalahi, T. Aljaaidi, Investigating a generalized Hilfer-type fractional differential equation with two-point and integral boundary conditions, AIMS Math., 7(2) (2021), 1856-187.
• [22] D. Vivek, E. M. Elsayed, K. Kanagarajan, Dynamics and stability results for impulsive type integro-differential equations with generalized fractional derivative, Math. Nat. Sci., 4 (2019), 1-12.
• [23] D. Vivek, K. Kanagarajan, S. Harikrishnan, Theory and analysis of impulsive type pantograph equations with Katugampola fractioanl derivative, J. Vabration Testing and System Dynamic, 2 (1) 2018, 9-20.
• [24] D. Vivek, E. M. Elsayed, K. Kanagarajan, Theory of fractional implicit differential equations with complex order, J. Uni. Math., 2(2) (2019), 154-165.
• [25] H. Wang, Existence results for fractional functional differential equations with impulses, J. Appl. Math. Comput., 38 (2012), 85-101.
• [26] E. Zeidler, Nonlinear functional Analysis and its Applications-I Fixed Point Theorem, Springer, New-York, 1993.