Existence Results for BVP of a Class of Generalized Fractional-Order Implicit Differential Equations
Yıl 2022,
Cilt: 5 Sayı: 3, 114 - 123, 30.09.2022
Kadda Maazouz
,
Dvivek Vivek
,
Elsayed Elsayed
Öz
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.
Kaynakça
- [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.
Yıl 2022,
Cilt: 5 Sayı: 3, 114 - 123, 30.09.2022
Kadda Maazouz
,
Dvivek Vivek
,
Elsayed Elsayed
Kaynakça
- [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.