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More new results on integral inequalities for generalized $\mathcal{K}$-fractional conformable Integral operators
July  2021, 14(7): 2137-2150. doi: 10.3934/dcdss.2020177

## Bounded perturbation for evolution equations with a parameter & application to population dynamics

 Department of Mathematical Sciences, University of South Africa, Florida, 0003, South Africa

* Corresponding author: franckemile2006@yahoo.ca

Received  April 2019 Revised  September 2020 Published  July 2021 Early access  May 2021

Fund Project: This work was partially supported by the grant No: 105932 from the National Research Foundation (NRF) of South Africa

Evolution equations using derivatives of fractional order like Caputo's derivative or Riemann-Liouville's derivative have been intensively analyzed in numerous works. But the classical bounded perturbation theorem has been proven not to be in general true for these models, especially for solution operators of evolution equations with fractional order derivative $\alpha$ less than $1$ ($0<\alpha<1$), as shown by the example in the next section. This paper proposes an alternative way of dealing with this issue. We make use of the conventional time derivative with a new parameter to show the perturbations by bounded linear operators for linear evolution equations when the derivative order is less than one. The new parameter which happens to be fractional, characterizes the so-called $\beta$-derivative. Its fractional order parameter allows the use of concepts like revamped time to provide a relation between both strongly continuous two-parameter solution operators involved in the perturbation process. To validate the theory, we use an application to population dynamics and perform some numerical simulations that reveal some consistency with the expected results.

Citation: Emile Franc Doungmo Goufo. Bounded perturbation for evolution equations with a parameter & application to population dynamics. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2137-2150. doi: 10.3934/dcdss.2020177
##### References:
 [1] A. Atangana, Derivative with a New Parameter: Theory, Methods and Applications, Academic Press, 2016.  doi: 10.1016/B978-0-08-100644-3.00001-5. [2] A. Atangana and E. F. Doungmo Goufo, Extension of matched asymptotic method to fractional boundary layers problems, Math. Probl. Eng., 2014 (2014), 1-7.  doi: 10.1155/2014/107535. [3] E. G. Bazhlekova, Subordination principle for fractional evolution equations, Fract. Calc. Appl. Anal., 3 (2000), 213-230. [4] E. G. Bazhlekova, Perturbation and Approximation Properties for Abstract Evolution Equations of Fractional Order, Research Report RANA 00-05, Eindhoven University of Technology, Eindhoven 2000. [5] D. Brockmann and L. Hufnagel, Front propagation in reaction-superdiffusion dynamics: Taming Lévy flights with fluctuations, Phys. Review Lett., 98 (2007), 178301. doi: 10.1103/PhysRevLett.98.178301. [6] M. Caputo, Linear models of dissipation whose Q is almost frequency independent II, Geophys. J. R. Ast. Soc., 13 (1967), 529-539.  doi: 10.1111/j.1365-246X.1967.tb02303.x. [7] M. Caputo and M. Fabrizio, A New Definition of Fractional Derivative without Singular Kernel, Progr. Fract. Differ. Appl., 1 (2015), 1-13. [8] G. Greiner, A typical Perron-Frobenius theorem with applications to an age-dependent population equation, Infinite-Dimensional Systems, Springer, Berlin, Heidelberg, 1076 (1984), 86–100. doi: 10.1007/BFb0072769. [9] W. Desch and W. Schappacher, On relatively bounded perturbations of linear $C_0$-semigroups, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 11 (1984), 327-341. [10] K. Diethelm, N. J. Ford, A. D. Freed and Yu. Luchko, Algorithms for the fractional calculus: A selection of numerical methods, Comput. Methods Appl. Mech. Engrg., 194 (2005), 743-773.  doi: 10.1016/j.cma.2004.06.006. [11] E. F. Doungmo Goufo, Solvability of chaotic fractional systems with 3D four-scroll attractors, Chaos Solitons Fractals, 104 (2017), 443-451.  doi: 10.1016/j.chaos.2017.08.038. [12] E. F. Doungmo Goufo, Evolution equations with a parameter and application to transport-convection differential equations, Turkish J. Math., 41 (2017), 636-654.  doi: 10.3906/mat-1603-107. [13] E. F. Doungmo Goufo, Application of the Caputo-Fabrizio fractional derivative without singular kernel to Korteweg–de Vries–Burgers equation, Math. Model. Anal., 21 (2016), 188-198.  doi: 10.3846/13926292.2016.1145607. [14] E. F. Doungmo Goufo and A. Atangana, Dynamics of traveling waves of variable order hyperbolic Liouville equation: Regulation and control, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 645-662.  doi: 10.3934/dcdss.2020035. [15] E. F. Doungmo Goufo and J. J. Nieto, Attractors for fractional differential problems of transition to turbulent flows, J. Comput. Appl. Math., 339 (2018), 329-342.  doi: 10.1016/j.cam.2017.08.026. [16] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics (Book 194), New York, NY, USA: Springer-Verlag, 2000. [17] R. Hilfer, Application of Fractional Calculus in Physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812817747. [18] J. Kestin and L. N. Persen, The transfer of heat across a turbulent boundary layer at very high prandtl numbers, Int. J. Heat Mass Transfer, 5 (1962), 355-371.  doi: 10.1016/0017-9310(62)90026-1. [19] R. Khalil, M. Al Horani, A. Yousef and M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70.  doi: 10.1016/j.cam.2014.01.002. [20] D. Lutz, On bounded time-dependent perturbations of operator cosine functions, Aequationes Math., 23 (1981), 197-203.  doi: 10.1007/BF02188032. [21] B. Nagy, On cosine operator functions in Banach spaces, Acta Sci. Math. Szeged, 36 (1974), 281-289. [22] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Springer-Verlag, vol. 44, 1983. doi: 10.1007/978-1-4612-5561-1. [23] J. Prüss, Evolutionary Integral Equations and Applications, Birkhäuser, Basel–Boston–Berlin, 1993. doi: 10.1007/978-3-0348-8570-6. [24] H. Schlichting, Boundary-Layer Theory, (7 ed.). New York (USA): McGraw-Hill, 1979. [25] F. Verhulst, Methods and Applications of Singular Perturbations: Boundary Layers and Multiple Timescale Dynamics, Springer, Dordrecht, 2005. doi: 10.1007/0-387-28313-7.

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##### References:
 [1] A. Atangana, Derivative with a New Parameter: Theory, Methods and Applications, Academic Press, 2016.  doi: 10.1016/B978-0-08-100644-3.00001-5. [2] A. Atangana and E. F. Doungmo Goufo, Extension of matched asymptotic method to fractional boundary layers problems, Math. Probl. Eng., 2014 (2014), 1-7.  doi: 10.1155/2014/107535. [3] E. G. Bazhlekova, Subordination principle for fractional evolution equations, Fract. Calc. Appl. Anal., 3 (2000), 213-230. [4] E. G. Bazhlekova, Perturbation and Approximation Properties for Abstract Evolution Equations of Fractional Order, Research Report RANA 00-05, Eindhoven University of Technology, Eindhoven 2000. [5] D. Brockmann and L. Hufnagel, Front propagation in reaction-superdiffusion dynamics: Taming Lévy flights with fluctuations, Phys. Review Lett., 98 (2007), 178301. doi: 10.1103/PhysRevLett.98.178301. [6] M. Caputo, Linear models of dissipation whose Q is almost frequency independent II, Geophys. J. R. Ast. Soc., 13 (1967), 529-539.  doi: 10.1111/j.1365-246X.1967.tb02303.x. [7] M. Caputo and M. Fabrizio, A New Definition of Fractional Derivative without Singular Kernel, Progr. Fract. Differ. Appl., 1 (2015), 1-13. [8] G. Greiner, A typical Perron-Frobenius theorem with applications to an age-dependent population equation, Infinite-Dimensional Systems, Springer, Berlin, Heidelberg, 1076 (1984), 86–100. doi: 10.1007/BFb0072769. [9] W. Desch and W. Schappacher, On relatively bounded perturbations of linear $C_0$-semigroups, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 11 (1984), 327-341. [10] K. Diethelm, N. J. Ford, A. D. Freed and Yu. Luchko, Algorithms for the fractional calculus: A selection of numerical methods, Comput. Methods Appl. Mech. Engrg., 194 (2005), 743-773.  doi: 10.1016/j.cma.2004.06.006. [11] E. F. Doungmo Goufo, Solvability of chaotic fractional systems with 3D four-scroll attractors, Chaos Solitons Fractals, 104 (2017), 443-451.  doi: 10.1016/j.chaos.2017.08.038. [12] E. F. Doungmo Goufo, Evolution equations with a parameter and application to transport-convection differential equations, Turkish J. Math., 41 (2017), 636-654.  doi: 10.3906/mat-1603-107. [13] E. F. Doungmo Goufo, Application of the Caputo-Fabrizio fractional derivative without singular kernel to Korteweg–de Vries–Burgers equation, Math. Model. Anal., 21 (2016), 188-198.  doi: 10.3846/13926292.2016.1145607. [14] E. F. Doungmo Goufo and A. Atangana, Dynamics of traveling waves of variable order hyperbolic Liouville equation: Regulation and control, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 645-662.  doi: 10.3934/dcdss.2020035. [15] E. F. Doungmo Goufo and J. J. Nieto, Attractors for fractional differential problems of transition to turbulent flows, J. Comput. Appl. Math., 339 (2018), 329-342.  doi: 10.1016/j.cam.2017.08.026. [16] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics (Book 194), New York, NY, USA: Springer-Verlag, 2000. [17] R. Hilfer, Application of Fractional Calculus in Physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812817747. [18] J. Kestin and L. N. Persen, The transfer of heat across a turbulent boundary layer at very high prandtl numbers, Int. J. Heat Mass Transfer, 5 (1962), 355-371.  doi: 10.1016/0017-9310(62)90026-1. [19] R. Khalil, M. Al Horani, A. Yousef and M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70.  doi: 10.1016/j.cam.2014.01.002. [20] D. Lutz, On bounded time-dependent perturbations of operator cosine functions, Aequationes Math., 23 (1981), 197-203.  doi: 10.1007/BF02188032. [21] B. Nagy, On cosine operator functions in Banach spaces, Acta Sci. Math. Szeged, 36 (1974), 281-289. [22] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Springer-Verlag, vol. 44, 1983. doi: 10.1007/978-1-4612-5561-1. [23] J. Prüss, Evolutionary Integral Equations and Applications, Birkhäuser, Basel–Boston–Berlin, 1993. doi: 10.1007/978-3-0348-8570-6. [24] H. Schlichting, Boundary-Layer Theory, (7 ed.). New York (USA): McGraw-Hill, 1979. [25] F. Verhulst, Methods and Applications of Singular Perturbations: Boundary Layers and Multiple Timescale Dynamics, Springer, Dordrecht, 2005. doi: 10.1007/0-387-28313-7.
Numerical solutions showing the time evolution of the history intervals $[-\tau, 0], \ \ \ \tau>0,$
Numerical solutions showing evolution dynamics in the time interval [0, 25] for solutions to (33)-(34). Different delays are considered and similar trajectories are shown in both figures (integer and pure fractional order)
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