The Fixed Point Approach to the Stability of Fractional Differential Equations with Causal Operators

In this paper, the Hyers–Ulam (HU) stability and Hyers–Ulam–Rassias (HUR) stability of fractional differential equations with causal operators (FDEwCO) are investigated. The techniques rely on a fixed point theorem which is employed to study the HUR stability for FDEwCO on both bounded and unbounded time intervals as well as HU stability on bounded time interval. Finally, two typical examples are given to demonstrate the applications of theoretical results proposed.

[1]  Viorel Radu,et al.  THE FIXED POINT ALTERNATIVE AND THE STABILITY OF FUNCTIONAL EQUATIONS , 2003 .

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

[3]  V. Lakshmikantham,et al.  Theory of Causal Differential Equations , 2010 .

[4]  Vasile Lupulescu,et al.  Causal functional differential equations in Banach spaces , 2008 .

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

[6]  Xia Li,et al.  New stability results for fractional integral equation , 2012, Comput. Math. Appl..

[7]  G. Forti,et al.  Comments on the core of the direct method for proving Hyers–Ulam stability of functional equations , 2004 .

[8]  Ping Gong,et al.  Existence of Positive Solutions for a Class of Higher-Order Caputo Fractional Differential Equation , 2014, Qualitative Theory of Dynamical Systems.

[9]  Takeshi Miura,et al.  A characterization of Hyers–Ulam stability of first order linear differential operators , 2003 .

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

[11]  D. H. Hyers.  On the Stability of the Linear Functional Equation. , 1941, Proceedings of the National Academy of Sciences of the United States of America.

[12]  J. Vasundhara Devi,et al.  Differential equations with causal operators in a Banach space , 2005 .

[13]  Changpin Li,et al.  A survey on the stability of fractional differential equations , 2011 .

[14]  Constantin Corduneanu,et al.  Functional Equations with Causal Operators , 2002 .

[15]  C. F. Li,et al.  Existence and Uniqueness of Solution for Fractional Differential Equation with Causal Operators in Banach Spaces , 2014, Mediterranean Journal of Mathematics.

[16]  JinRong Wang,et al.  Mittag-Leffler-Ulam stabilities of fractional evolution equations , 2012, Appl. Math. Lett..

[17]  C. F. Li,et al.  Existence of solutions for set differential equations involving causal operator with memory in Banach space , 2013 .

[18]  Ioan A. Rus,et al.  REMARKS ON ULAM STABILITY OF THE OPERATORIAL EQUATIONS , 2009 .

[19]  T. Rassias.  On the stability of the linear mapping in Banach spaces , 1978 .

[20]  Zsolt Páles,et al.  A fixed point approach to stability of functional equations , 2011 .

[21]  Ravi P. Agarwal,et al.  Fractional functional differential equations with causal operators in Banach spaces , 2011, Math. Comput. Model..

[22]  JinRong Wang,et al.  Ulam–Hyers–Mittag-Leffler stability of fractional-order delay differential equations , 2014 .

[23]  I. Podlubny.  Fractional differential equations , 1998 .

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

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

[26]  Soon-Mo Jung,et al.  HYERS-ULAM-RASSIAS STABILITY OF THE BANACH SPACE VALUED LINEAR DIFFERENTIAL EQUATIONS y′ = λy , 2004 .

[27]  Yangquan Chen,et al.  Computers and Mathematics with Applications Stability of Fractional-order Nonlinear Dynamic Systems: Lyapunov Direct Method and Generalized Mittag–leffler Stability , 2022 .

[28]  JinRong Wang,et al.  A class of nonlinear differential equations with fractional integrable impulses , 2014, Commun. Nonlinear Sci. Numer. Simul..

[29]  Soon-Mo Jung,et al.  Hyers-Ulam stability of linear differential equations of first order, III , 2005 .

[30]  Soon-Mo Jung,et al.  Laplace transform and Hyers–Ulam stability of linear differential equations , 2013 .

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

[32]  P. Gǎvruţa,et al.  Fixed Points and Generalized Hyers-Ulam Stability , 2012 .

[33]  Krzysztof Ciepliński,et al.  A fixed point approach to the stability of functional equations in non-Archimedean metric spaces , 2011 .

[34]  Bo Zhang,et al.  Fixed points and stability in differential equations with variable delays , 2005 .