Stability Theory and the Existence of Hilfer Type Fractional Implicit Differential Equations with Boundary Conditions

Year 2019, Volume: 7 Issue: 2, 279 - 287, 15.10.2019

D. Vivek K. Kanagarajan Elsayed Elsayed

Abstract

In this paper, we consider the existence and Ulam stability of solutions to the boundary value problem for implicit differential equations involving Hilfer fractional derivative. With the help of properties of Hilfer fractional calculus and fixed point methods, we derive existence and stability results.

Keywords

Fractional implicit differential equations, Boundary value problem, Hilfer fractional derivative

References

• [1] S. Abbas, M. Benchohra, S. Sivasundaram, Dynamics and Ulam stability for Hilfer type fractional differential equations, Nonlinear Stud., 23(4) 627-637, (2016).
• [2] B. Ahmad, S. K. Ntouyas, Initial value problems for hybrid Hadamard fractional differential equations, Electron. J. Differential Equations, 161 (2014),1-8.
• [3] B. Ahmad, J. J. Nieto, Existence of solutions for nonlocal boundary value problems of higher-order nonlinear fractional differential equations, Abstr. Appl. Anal., vol. 2009, Article ID 494720, 9 pages, 2009. doi:10.1155/2009/494720.
• [4] B. Ahmad, J. J. Nieto, Riemann-Liouville fractional differential equations with fractional boundary conditions, Fixed point Theory, 13(2) (2012), 329-336.
• [5] Sz.Andras, J.J.Kolumban, On the Ulam-Hyers stability of first order differential systems with nonlocal initial conditions, Nonlinear Anal. Theory Methods Appl., 82 (2013),1-11.
• [6] M. Benchohra, J. E. Lazreg, Existence and unqiueness results for nonlinear implicit fractional differential equations with boundary conditions, Romanian Journal of Mathematics and Computer science, 4 (2014), 60-72.
• [7] P. L. Butzer, A. A. Kilbas, J. J. Trujillo, Compositions of Hadamard-type fractional integration operators and the semigroup property, J. Math. Anal. Appl., 269(2) (2002), 387-400.
• [8] P. L. Butzer, A. A. Kilbas, J. J. Trujillo, Mellin transform analysis and integration by parts for Hadamard-type fractional integrals, J. Math. Anal. Appl., 270(1) (2002), 1-15.
• [9] K.M.Furati, M.D.Kassim, N.e-.Tatar, Existence and uniqueness for a problem involving Hilfer fractional derivative, Comput. Math. Appl., 64(6) (2012), 1616-1626.
• [10] K.M.Furati, M.D.Kassim, N.e-.Tatar, Non-existence of global solutions for a differential equation involving Hilfer fractional derivative, Electron. J. Differential Equations, 235 (2013), 1-10.
• [11] H. Gu, J.J. Trujillo, Existence of mild solution for evolution equation with Hilfer fractional derivative, Appl. Math. Comput., 257 (2014), 344-354.
• [12] R. W. Ibrahim, Generalized Ulam-Hyers stability for fractional differential equations, Int. J. Mat.,23 (2012),doi:10.1142/S0129167X12500565.
• [13] S. M. Jung, Hyers-Ulam stability of linear differential equations of first order, Appl. Math. Lett., 17 (2004),1135-1140.
• [14] R. Kamocki, C. Obcznnski, On fractional Cauchy-type problems containing Hilfer derivative, Electron J. Qual. Theory Differ. Equ., 50 (2016) 1-12.
• [15] K.M.Furati, M.D.Kassim, N.e-.Tatar, Well-psedness and stability for a differential problem with Hilfer-Hadamard fractional derivative,Abst. Appl. Anal, 1 (2013), 1-12.
• [16] Z. Gao, X. Yu, Existence results for BVP of a class of Hilfer fractional differential equations, J. Appl. Math. Comput., (2016), 17.
• [17] P. Muniyappan, S. Rajan, Hyers-Ulam-Rassias stability of fractional differential equation, Int. J. Pure Appl. Math., 102 (2015),631-642.
• [18] J. Wang, L. Lv, Y. Zhou, Boundary value problems for fractional differential equations involving Caputo derivative in Banach spaces, J. Appl.Math. Comp., 38(2012), 209-224.
• [19] I.A. Rus, Ulam stabilities of ordinary differential equations in a Banach space, Carpathian J. Math., 26, (2010),103-107.
• [20] Z. Shuqin, Existence of solutions for a boundary value problems of fractional order, Acta Math. Sci., 26B(2) (2006),220-228.
• [21] R.Hilfer, Application of fractional Calculus in Physics, World Scientific, Singapore, 1999.
• [22] R.Hilfer, Y.Luchko, Z. Tomovski, Operational method for the solution of fractional differential equations with generalized Riemann-Lioville fractional derivative, Frac. cal. Appl. Anal., 12 (2009), 289-318.
• [23] J. Wang, Y. Zhang, Nonlocal initial value problems for differential equations with Hilfer fractional derivative, Appl. Math. Comput., 266 (2015), 850-859.
• [24] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, in: Mathematics Studies, vol. 204, Elsevier, 2006.
• [25] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Amsterdam, 1987, Engl. Trans. from the Russian.
• [26] I. Podlubny, Fractional Differential Equations, in: Mathematics in Science and Engineering, vol. 198, Acad. Press, 1999.
• [27] D. Vivek, K. Kanagarajan, S. Sivasundaram, Dynamics and stability of pantograph equations via Hilfer fractional derivative, Nonlinear Stud., 23(4) (2016), 685-698.
• [28] D. Vivek, K. Kanagarajan, E. M. Elsayed, Some existence and stability results for Hilfer-fractional implicit differential equations with nonlocal Conditions, Mediterranean Journal of Mathematics, 15(2018), 15.
• [29] D. Vivek, K. Kanagarajan, Seenith Sivasundaram, Dynamics and stability results for Hilfer fractional type thermistor problem, Fractal Fract, 1(1) (2017), 5.
• [30] J. Wang, L. Lv, Y. Zhou, Ulam stability and data dependence for fractional differential equations with Caputo derivative,Electron J. Qual. Theory Differ. Equ.,63 (2011), 1-10.
• [31] J. Wang, Y. Zhou, New concepts and results in stability of fractional differential equations,Commun. Nonlinear Sci. Numer. Simul.,17 (2012),2530-2538.
• [32] J. Wang, Yong Zhou, Milan Medved, Existence and stability of fractional differential equations with Hadamard derivative, Topol. Methods Nonlinear Anal., 41(1) (2013), 113-133.