Hilfer-Hadamard Fractional Differential Equations; Existence and Attractivity
Keywords:
Hilfer-Hadamard fractional derivative, Schauder fixed- point Theorem, uniformly locally attractingAbstract
This work deals with a class of Hilfer-Hadamard differential equations. Existence and stability of solutions are presented. We use an appropriate fixed point theorem.
References
S. Abbas, W. Albarakati and M. Benchohra, Existence and attractivity results for Volterra type nonlinear multi-delay Hadamard-Stieltjes fractional integral equations, PanAmer. Math. J. 26 (2016), 1-17.
S. Abbas and M. Benchohra, Existence and attractivity for fractional order integral equations in Frchet spaces, Discuss. Math. Differ. Incl. Control Optim. 33 (2013), 1-17.
S. Abbas and M. Benchohra, Existence and stability of nonlinear fractional order Riemann-Liouville, Volterra-Stieltjes multi-delay integral equations, J. Integ. Equat. Appl. 25 ( 2013 ), 143-158.
S. Abbas, M. Benchohra, and T. Diagana, Existence and attractivity results for some fractional order partial integrodifferential equations with delay, Afr. Diaspora J. Math. 15 (2013), 87-100.
S. Abbas, M. Benchohra and J. Henderson, Asymptotic attractive nonlinear frac- tional order Riemann-Liouville integral equations in Banach algebras, Nonlin. Stud. 20 (2013), 1-10.
S. Abbas, M. Benchohra and J. Henderson, Existence and attractivity results for Hilfer fractional differential equations, J. Math. Sci. 243 (2019), 347-357.
S. Abbas, M. Benchohra, J.-E. Lagreg, A. Alsaedi, Y. Zhou, Existence and Ulam stability for fractional differential equations of Hilfer-Hadamard type, Adv. Difference Equ. 180 (2017), 1-14.
S. Abbas, M. Benchohra and G. M. N’Gu´ er´ ekata, Advanced Fractional Differential and Integral Equations, Nova Sci. Publ., New York, 2014.
S. Abbas, M. Benchohra, and G. M.N’ Gu´ er´ ekata, Topics in Fractional Differential Equations, Dev. Math., 27, Springer, New York, 2015.
S. Abbas, M. Benchohra, and J. J. Nieto, Global attractivity of solutions for non-linear fractional order Riemann-Liouville Volterra-Stieltjes partial integral equations, Electron. J. Qual. Theory Differ. Equat 81 (2012), 1-15.
H. Afshari, E. Karapinar, A discussion on the existence of positive solutions of the boundary value problems via ψ-Hilfer fractional derivative on b-metric spaces. Adv. Difference Equ. 2020, 616.
M. Benchohra, J. Henderson, S. K. Ntouyas and A. Ouahab, Existence results for functional differential equations of fractional order, J. Math. Anal. Appl. 338 (2008), 1340-1350.
N. Bouteraa, S. Benaicha, The uniqueness of positive solution for higher-order nonlinear fractional differential equation with fractional multi-point boundary conditions, Adv. Theory Nonl. Anal. Appl. 2 (2) (2018), 74-84.
C. Corduneanu, Integral Equations and Stability of Feedback Systems, Acad. Press, New York, 1973.
A. Granas, J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003.
R. Hilfer, Applications of Fractional Calculus in Physics , World Scientific, Singa- pore, 2000.
R. Hilfer, Threefold introduction to fractional derivatives, R. Klages et al. (editors), Anomalous Transp.: Found. Appl., WileyVCH, Weinheim , pp. (2008), 17-73.
R. Kamocki and C. Obcznski, On fractional Cauchy-type problems containing Hilfer’s derivative, Electron. J. Qual. Theory Differ. Equ. (2016), No. 50, 1-12.
E. Karapinar, T. Abdeljawad, F. Jarad, Applying new fixed point theorems on fractional and ordinary differential equations. Adv. Difference Equ. 2019, Paper No. 421, 25 pp.
M.D. Kassim, K.M. Furati, N.-E. Tatar, On a differential equation involving Hilfer-Hadamard fractional derivative, Abstr. Appl. Anal. (2012), Article ID 391062.
M.D. Kassim, N.E. Tatar, Well-posedness and stability for a differential problem with Hilfer-Hadamard fractional derivative, Abstr. Appl. Anal. 1 (2013), 1-12.
A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
V. Lakshmikantham and J. Vasundhara Devi, Theory of fractional differential equations in a Banach space, Eur. J. Pure Appl. Math. 1 (2008), 38-45.
V. Lakshmikantham and A.S. Vatsala, Basic theory of fractional differential equations, Nonlin. Anal. 69 (2008), 2677-2682.
V. Lakshmikantham and A. S. Vatsala, General uniqueness and monotone iterative technique for fractional differential equations, Appl. Math. Lett. 21 (2008), 828-834.
K. Oldham, J. Spanier, The Fractional Calculus, Acad. Press, New York, 1974.
Podlubny, Fractional Differential Equations, in: Mathematics in Science and Engineering, 198, Acad. Press, New York, 1999.
S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications , Gordon Breach, Tokyo-Paris-Berlin, 1993.
S. Muthaiah, M. Murugesan, N. G. Thangaraj, Existence of solutions for nonlocal boundary value problem of Hadamard fractional differential equations, Adv. Theory Nonl. Anal. Appl. 3 (3) (2019), 162-173.
V. E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to the Dynamics of Particles, Fields and Media, Springer, Beijing-Heidelberg, 2010.
Z. Tomovski, R. Hilfer and H. M. Srivastava, Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler-type functions, Integral Transforms Spec. Funct. 21 (2010), No. 11, 797-814.
J.-R. Wang and Y. Zhang, Nonlocal initial value problems for differential equations with Hilfer fractional derivative, Appl. Math. Comput. 266 (2015), 850-859.
Y. Zhou, J.-R. Wang, L. Zhang, Basic Theory of Fractional Differential Equations, Second edition. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017.
