Analysis of Implicit Type Nonlinear Dynamical Problem of Impulsive Fractional Differential Equations

We study the existence, uniqueness, and various kinds of Ulam–Hyers stability of the solutions to a nonlinear implicit type dynamical problem of impulsive fractional differential equations with nonlocal boundary conditions involving Caputo derivative. We develop conditions for uniqueness and existence by using the classical fixed point theorems such as Banach fixed point theorem and Krasnoselskii’s fixed point theorem. For stability, we utilized classical functional analysis. Also, an example is given to demonstrate our main theoretical results.

[1]  Rabha W. Ibrahim,et al.  On the existence and uniqueness of solutions of a class of fractional differential equations , 2007 .

[2]  R. Lewandowski,et al.  Identification of parameters of the fractional rheological model of viscoelastic dampers , 2010 .

[3]  Rabha W. Ibrahim,et al.  On a fractional integral equation of periodic functions involving Weyl–Riesz operator in Banach algebras , 2008 .

[4]  N. Ford,et al.  Analysis of Fractional Differential Equations , 2002 .

[5]  A. El-Sayed,et al.  POSITIVE SOLUTIONS FOR A NONLOCAL MULTI-POINT BOUNDARY-VALUE PROBLEM OF FRACTIONAL AND SECOND ORDER , 2013 .

[6]  KAMAL SHAH,et al.  EXISTENCE AND UNIQUENESS OF SOLUTIONS TO FRACTIONAL ORDER MULTI-POINT BOUNDARY VALUE PROBLEMS , 2015 .

[7]  Zeeshan Ali,et al.  Existence and Stability Analysis of Three Point Boundary Value Problem , 2017 .

[8]  H. Akça,et al.  Continuous-time additive Hopfield-type neural networks with impulses , 2004 .

[9]  Margarita Rivero,et al.  On systems of linear fractional differential equations with constant coefficients , 2007, Appl. Math. Comput..

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

[11]  S. Abbas.  On The Generalized Ulam-Hyers-Rassias Stability For Darboux Problem For Partial Fractional Implicit Dierential Equations , 2014 .

[12]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.

[13]  Soon-Mo Jung.  Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis , 2011 .

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

[15]  Tongxing Li,et al.  Connections between Hyers-Ulam stability and uniform exponential stability of discrete evolution families of bounded linear operators over Banach spaces , 2016 .

[16]  Leon O. Chua,et al.  Cellular neural networks: applications , 1988 .

[17]  Fajun Yu,et al.  Integrable coupling system of fractional soliton equation hierarchy , 2009 .

[18]  Yongjin Li,et al.  ON HYERS-ULAM STABILITY OF NONLINEAR DIFFERENTIAL EQUATIONS , 2015 .

[19]  Akbar Zada,et al.  Hyers-Ulam stability of non-autonomous systems in terms of boundedness of Cauchy problems , 2015, Appl. Math. Comput..

[20]  S. Ulam.  A collection of mathematical problems , 1960 .

[21]  Yongjin Li,et al.  On the Hyers-Ulam Stability of First-Order Impulsive Delay Differential Equations , 2016 .

[22]  Drumi D. Bainov,et al.  Oscillation of the bounded solutions of impulsive differential-difference equations of second order , 2000, Appl. Math. Comput..

[23]  E. P. Popov,et al.  The Dynamics of Automatic Control Systems , 1964, Mathematical Gazette.

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

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

[26]  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.

[27]  K. Shah,et al.  Existence and uniqueness of positive solutions to a coupled system of nonlinear fractional order differential equations with anti periodic boundary conditions , 2015 .

[28]  K. Miller,et al.  An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993 .

[29]  K. Shah,et al.  Existence of Positive Solution to a Class of Fractional Differential Equations with Three Point Boundary Conditions , 2016 .

[30]  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 .

[31]  M. Benchohra,et al.  Existence and Stability Results for Nonlinear Boundary Value Problem for Implicit Differential Equations of Fractional Order , 2015 .

[32]  K. Shah,et al.  Upper and Lower Solutions to a Coupled System of Nonlinear Fractional Differential Equations , 2016 .

[33]  Tongxing Li,et al.  Stability of higher-order nonlinear impulsive differential equations , 2016 .

[34]  Zhanbing Bai,et al.  Impulsive Boundary Value Problem for Differential Equations with Fractional Order , 2013 .

[35]  Xinwei Su,et al.  Boundary value problem for a coupled system of nonlinear fractional differential equations , 2009, Appl. Math. Lett..

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

[37]  R. Hilfer.  Applications Of Fractional Calculus In Physics , 2000 .

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

[39]  Rahmat Ali Khan,et al.  A note on boundary value problems for a coupled system of fractional differential equations , 2011, Comput. Math. Appl..