Ulam–Hyers–Mittag-Leffler stability of fractional-order delay differential equations

In this paper, we first prove two existence and uniqueness results for fractional-order delay differential equation with respect to Chebyshev and Bielecki norms. Secondly, we prove the above equation is Ulam–Hyers–Mittag-Leffler stable on a compact interval. Finally, two examples are also provided to illustrate our results.

[1]  S.-M. Jung,et al.  Hyers-Ulam stability of linear differential equations of first order , 2004, Appl. Math. Lett..

[2]  Andreas Thom,et al.  On Ulam stability , 2010, 1010.0565.

[3]  Dorian Popa,et al.  Hyers–Ulam stability of a first order partial differential equation , 2012 .

[4]  JinRong Wang,et al.  Eα-Ulam type stability of fractional order ordinary differential equations , 2014 .

[5]  Ioan A. Rus,et al.  ULAM STABILITY OF ORDINARY DIFFERENTIAL EQUATIONS , 2009 .

[6]  Ioan A. Rus,et al.  Ulam stabilities of ordinary differential equations in a Banach space , 2010 .

[7]  Dorian Popa,et al.  Hyers-Ulam stability of Euler's equation , 2011, Appl. Math. Lett..

[8]  JinRong Wang,et al.  Nonlinear impulsive problems for fractional differential equations and Ulam stability , 2012, Comput. Math. Appl..

[9]  Soon-Mo Jung,et al.  On the stability of Laplace's equation , 2013, Appl. Math. Lett..

[10]  Rabha W. Ibrahim,et al.  GENERALIZED ULAM-HYERS STABILITY FOR FRACTIONAL DIFFERENTIAL EQUATIONS , 2012 .

[11]  George Isac,et al.  Stability of Functional Equations in Several Variables , 1998 .

[12]  Guotao Wang,et al.  New Existence Results and Comparison Principles for Impulsive Integral Boundary Value Problem with Lower and Upper Solutions in Reversed Order , 2011 .

[13]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations, Volume 204 (North-Holland Mathematics Studies) , 2006 .

[14]  Yong Zhou,et al.  New concepts and results in stability of fractional differential equations , 2012 .

[15]  Szilárd András,et al.  Ulam-Hyers stability of dynamic equations on time scales via Picard operators , 2013, Appl. Math. Comput..

[16]  V. Ilea,et al.  Ulam stability for a delay differential equation , 2013 .

[17]  Yong Zhou,et al.  Ulam’s type stability of impulsive ordinary differential equations☆ , 2012 .

[18]  I. Rus.  Gronwall Lemma Approach to the Hyers–Ulam–Rassias Stability of an Integral Equation , 2010 .

[19]  Rabha W. Ibrahim,et al.  Ulam Stability for Fractional Differential Equation in Complex Domain , 2012 .

[20]  A. R. Ioan.  GRONWALL LEMMAS: TEN OPEN PROBLEMS , 2009 .

[21]  Yong Zhou,et al.  Ulam stability and data dependence for fractional differential equations with Caputo derivative , 2011 .

[22]  S. András,et al.  On the Ulam–Hyers stability of first order differential systems with nonlocal initial conditions , 2013 .

[23]  Rabha W. Ibrahim,et al.  Complex Transforms for Systems of Fractional Differential Equations , 2012 .

[24]  Rabha W. Ibrahim,et al.  Ulam-Hyers Stability for Cauchy Fractional Differential Equation in the Unit Disk , 2012 .

[25]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .