Study of Hilfer fractional evolution equations by the properties of controllability and stability

Abstract This paper is devoted to discuss Hilfer fractional evolution equations through its controllability and stability in a Banach space. We achieve our claims and conclusions by first demonstrating the nature and uniqueness of the suggested set of equations with their mild solutions. The outcomes of existence and uniqueness are obtained with the aid of the propagation family theory, non-compactness calculation methods and the fixed point technique. An example is also provided for the description of our main results.

[1]  C. Tunç,et al.  STABILITY RESULTS AND EXISTENCE THEOREMS FOR NONLINEAR DELAY-FRACTIONAL DIFFERENTIAL EQUATIONS WITH $ \varphi^*_P $-OPERATOR , 2020 .

[2]  Hans-Peter Heinz,et al.  On the behaviour of measures of noncompactness with respect to differentiation and integration of vector-valued functions , 1983 .

[3]  M. I. Troparevsky,et al.  A Review on Fractional Differential Equations and a Numerical Method to Solve Some Boundary Value Problems , 2019, Nonlinear Systems -Theoretical Aspects and Recent Applications.

[4]  Rathinasamy Sakthivel,et al.  On the approximate controllability of semilinear fractional differential systems , 2011, Comput. Math. Appl..

[5]  D. S. Oliveira,et al.  Hilfer–Katugampola fractional derivatives , 2018 .

[6]  Apassara Suechoei,et al.  Existence uniqueness and stability of mild solutions for semilinear ψ-Caputo fractional evolution equations , 2020, Advances in Difference Equations.

[7]  M. Muslim,et al.  Controllability of Fractional Differential Equation of Order α∈(1,2] With Non‐Instantaneous Impulses , 2018 .

[8]  J. Rassias.  Solution of a problem of Ulam , 1989 .

[9]  F. Jarad,et al.  Existence and data dependence theorems for solutions of an ABC-fractional order impulsive system , 2020 .

[10]  J. F. Gómez‐Aguilar,et al.  Stability analysis and numerical solutions of fractional order HIV/AIDS model , 2019, Chaos, Solitons & Fractals.

[11]  J. Vanterler da C. Sousa,et al.  On the ψ-Hilfer fractional derivative , 2017, Commun. Nonlinear Sci. Numer. Simul..

[12]  Samuel M Rankin,et al.  A partial functional differential equation of Sobolev type , 1983 .

[13]  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.

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

[15]  T. Abdeljawad,et al.  Generalized fractional derivatives and Laplace transform , 2020, Discrete & Continuous Dynamical Systems - S.

[16]  D. Baleanu,et al.  A hybrid functions numerical scheme for fractional optimal control problems: Application to nonanalytic dynamic systems , 2017 .

[17]  D. Baleanu,et al.  Spline collocation methods for systems of fuzzy fractional differential equations , 2020 .

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

[19]  Dumitru Baleanu,et al.  Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems , 2011, Comput. Math. Appl..

[20]  D. Baleanu,et al.  Collocation methods for terminal value problems of tempered fractional differential equations , 2020 .

[21]  D. Baleanu,et al.  On the fractional optimal control problems with a general derivative operator , 2019, Asian Journal of Control.

[22]  V. Lakshmikantham,et al.  Nonlinear Integral Equations in Abstract Spaces , 1996 .

[23]  P. Kumam,et al.  Existence and uniqueness for ψ‐Hilfer fractional differential equation with nonlocal multi‐point condition , 2020, Mathematical Methods in the Applied Sciences.

[24]  Delfim F. M. Torres,et al.  A survey on fuzzy fractional differential and optimal control nonlocal evolution equations , 2017, J. Comput. Appl. Math..

[25]  D. Baleanu,et al.  A New Iterative Method for the Numerical Solution of High-Order Non-linear Fractional Boundary Value Problems , 2020, Frontiers in Physics.

[26]  M. Raizada,et al.  Shifts in the Gut Microbiota Composition Due to Depleted Bone Marrow Beta Adrenergic Signaling Are Associated with Suppressed Inflammatory Transcriptional Networks in the Mouse Colon , 2017, Front. Physiol..

[27]  T. Abdeljawad,et al.  S-asymptotically $ \omega $-periodic mild solutions and stability analysis of Hilfer fractional evolution equations , 2020 .

[28]  T. Abdeljawad,et al.  Existence of mild solutions for impulsive neutral Hilfer fractional evolution equations , 2020, Advances in Difference Equations.

[29]  Yongjin Li,et al.  Hyers–Ulam stability of linear fractional differential equations with variable coefficients , 2020 .

[30]  Ignace Tchangou Toudjeu,et al.  Analysis of recent fractional evolution equations and applications , 2019, Chaos, Solitons & Fractals.

[31]  Józef Banaś,et al.  Measures of Noncompactness in Banach Spaces , 1980 .

[32]  D. Baleanu,et al.  Planar System-Masses in an Equilateral Triangle: Numerical Study within Fractional Calculus , 2020, Computer Modeling in Engineering & Sciences.

[33]  D. Baleanu,et al.  On stability analysis and existence of positive solutions for a general non-linear fractional differential equations , 2020 .

[34]  Aziz Khan,et al.  Analytical solutions of time-fractional wave equation by double Laplace transform method , 2019, The European Physical Journal Plus.

[35]  D. Baleanu,et al.  System of fractional differential algebraic equations with applications , 2019, Chaos, Solitons & Fractals.

[36]  Michal Fečkan,et al.  Controllability of Sobolev type fractional evolution systems , 2014 .

[37]  Ricardo Almeida,et al.  A Caputo fractional derivative of a function with respect to another function , 2016, Commun. Nonlinear Sci. Numer. Simul..

[38]  Wen Chen,et al.  Stability analysis and a numerical scheme for fractional Klein‐Gordon equations , 2018, Mathematical Methods in the Applied Sciences.

[39]  Jin Liang,et al.  Abstract Degenerate Cauchy Problems in Locally Convex Spaces , 2001 .

[40]  Thabet Abdeljawad,et al.  Stability and numerical simulation of a fractional order plant-nectar-pollinator model , 2020 .