Existence and stability analysis to a coupled system of implicit type impulsive boundary value problems of fractional-order differential equations

In this paper, we study a coupled system of implicit impulsive boundary value problems (IBVPs) of fractional differential equations (FODEs). We use the Schaefer fixed point and Banach contraction theorems to obtain conditions for the existence and uniqueness of positive solutions. We discuss Hyers–Ulam (HU) type stability of the concerned solutions and provide an example for illustration of the obtained results.

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

[2]  K. Shah,et al.  On Ulam's type stability for a class of impulsive fractional differential equations with nonlinear integral boundary conditions , 2017 .

[3]  K. Shah,et al.  On Ulam’s Stability for a Coupled Systems of Nonlinear Implicit Fractional Differential Equations , 2019 .

[4]  Soon-Mo Jung,et al.  Hyers-Ulam stability of linear differential equations of first order, II , 2006, Appl. Math. Lett..

[5]  A. Granas,et al.  Fixed Point Theory , 2003 .

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

[7]  D. Baleanu,et al.  Existence and uniqueness theorem for a class of delay differential equations with left and right Caputo fractional derivatives , 2008 .

[8]  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.

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

[10]  Bashir Ahmad,et al.  Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions , 2009, Comput. Math. Appl..

[11]  Norbert K. Semmer,et al.  Taking the chance: Core self-evaluations predict relative gain in job resources following turnover , 2016, SpringerPlus.

[12]  K. Deimling.  Fixed Point Theory , 2008 .

[13]  A Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel , 2017, Journal of inequalities and applications.

[14]  A. Cabada,et al.  Positive solutions of nonlinear fractional differential equations with integral boundary value conditions , 2012 .

[15]  Soon-Mo Jung.  On the Hyers–Ulam Stability of the Functional Equations That Have the Quadratic Property☆ , 1998 .

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

[17]  K. Shah,et al.  Ulam stability results to a class of nonlinear implicit boundary value problems of impulsive fractional differential equations , 2019, Advances in Difference Equations.

[18]  Tongxing Li,et al.  Hyers--Ulam stability of nth order linear differential equations , 2016 .

[19]  K. Shah,et al.  Ulam–Hyers stability analysis to a class of nonlinear implicit impulsive fractional differential equations with three point boundary conditions , 2019, Advances in Difference Equations.

[20]  Hammad Khalil,et al.  Investigation of positive solution to a coupled system of impulsive boundary value problems for nonlinear fractional order differential equations , 2015 .

[21]  K. Shah,et al.  Existence and numerical solutions of a coupled system of integral BVP for fractional differential equations , 2018, Advances in Difference Equations.

[22]  Dumitru Baleanu,et al.  On the existence and the uniqueness theorem for fractional differential equations with bounded delay within Caputo derivatives , 2008 .

[23]  T. Abdeljawad,et al.  On Riemann‐Liouville fractional q–difference equations and their application to retarded logistic type model , 2018 .

[24]  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 .

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

[26]  L. Chou,et al.  An empirical analysis of land property lawsuits and rainfalls , 2016, SpringerPlus.

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

[28]  Michal Fečkan,et al.  Stability Analysis for a General Class of Non-instantaneous Impulsive Differential Equations , 2017 .

[29]  T. Abdeljawad,et al.  A generalized discrete fractional Gronwall inequality and its application on the uniqueness of solutions for nonlinear delay fractional difference system , 2018 .

[30]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

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

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

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

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

[35]  D. Baleanu,et al.  A generalized q-fractional Gronwall inequality and its applications to nonlinear delay q-fractional difference systems , 2016, Journal of Inequalities and Applications.

[36]  Leonie Kohl,et al.  Dynamics Of Automatic Control Systems , 2016 .

[37]  Michal Fečkan,et al.  Fractional order differential switched systems with coupled nonlocal initial and impulsive conditions , 2017 .

[38]  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 .

[39]  Thabet Abdeljawad,et al.  Discrete Mittag-Leffler kernel type fractional difference initial value problems and Gronwall's inequality , 2017, J. Comput. Appl. Math..

[40]  D. O’Regan,et al.  Fractional semilinear equations with causal operators , 2017 .

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

[42]  M. Shitikova,et al.  Applications of Fractional Calculus to Dynamic Problems of Linear and Nonlinear Hereditary Mechanics of Solids , 1997 .

[43]  Shengjun Li,et al.  Multiplicity of Positive Periodic Solutions to Second Order Singular Dynamical Systems , 2017, Mediterranean Journal of Mathematics.

[44]  K. Shah,et al.  Multiple positive solutions to a coupled systems of nonlinear fractional differential equations , 2016, SpringerPlus.

[45]  Donal O'Regan,et al.  On the orbital Hausdorff dependence of differential equations with non-instantaneous impulses , 2018 .

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

[47]  Kamal Shah,et al.  Hyers‐Ulam stability analysis to implicit Cauchy problem of fractional differential equations with impulsive conditions , 2018, Mathematical Methods in the Applied Sciences.

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