Ulam–Hyers type stability for \(\psi \)-Hilfer fractional differential equations with impulses and delay

• Abdo MS, Panchal SK, Wahash HA (2020) Ulam–Hyers–Mittag-Leffler stability for a \(\psi \)-Hilfer problem with fractional order and infinite delay. Results Appl Math 7(100):115

  MATH  Google Scholar 

• Almeida R (2017) A Caputo fractional derivative of a function with respect to another function. Commun Nonlinear Sci Numer Simulat 44:460–481

  Article  MathSciNet  Google Scholar 

• Antonov A, Nenov S, Tsvetkov T (2019) Impulsive controllability of tumor growth. Dyn Syst Appl 28(1):93–109

  Google Scholar 

• Bainov DD, Dishliev AB (1990) Population dynamics control in regard to minimizing the time necessary for the regeneration of a biomass taken away from the population. Esaim Math Model Numer Anal 24(6):681–691

  Article  MathSciNet  Google Scholar 

• Ballinger GH (1999) Qualitative theory of impulsive delay differential equations. PhD thesis, Univesity of Waterloo

• Foryś U, Poleszczuk J, Liu T (2014) Logistic tumor growth with delay and impulsive treatment. Math Popul Stud 21:146–158

  Article  MathSciNet  Google Scholar 

• Gopalsamy K (1992) Stability and oscillations in delay differential equations of population dynamics, mathematics and its applications, vol 74, 1st edn. Springer, Dordrecht

  Book  Google Scholar 

• Hutchinson GE (1948) Circular causal systems in ecology. Ann NY Acad Sci 50(4):221–246

  Article  Google Scholar 

• Keane A, Krauskopf B, Postlethwaite CM (2017) Climate models with delay differential equations. Chaos 27(11):114309

• Khader MM, Babatin MM (2013) On approximate solutions for fractional logistic differential equation. Math Probl Eng 2013:391901

• Kharade JP, Kucche KD (2020) On the impulsive implicit \(\psi \)-Hilfer fractional differential equations with delay. Math Meth Appl Sci 43(4):1938–1952

  Article  MathSciNet  Google Scholar 

• Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations, North-Holland mathematics studies, vol 204. Elsevier, Amsterdam

  Google Scholar 

• Kolmanovskii V, Myshkis A (1999) Introduction to the theory and applications of functional differential equations. Springer, Heidelberg

  Book  Google Scholar 

• Kuang Y (1993) Delay differential equations: with applications in population dynamics. Mathematics in science and engineering. Academic Press, San Diego

  Google Scholar 

• Lima KB (2020) Existence, uniqueness and Ulam-Hyers type stability for \(\psi \)-Hilfer fractional-order delay impulsive equations (Existência, unicidade e estabilidade do tipo Ulam-Hyers de solução de equações diferenciais fracionárias impulsivas \(\psi \)-Hilfer). PhD thesis, Universidade Estadual de Campinas

• Lima KB, Sousa JVdC, Capelas de Oliveira E (2019) Existence and uniqueness for \(\psi \)-Hilfer impulsive fractional differential equations (Existência e unicidade de solução de equações diferenciais fracionárias impulsivas \(\psi \)-Hilfer). Proc Ser Braz Soc Comput Appl Math 7(1):010381

• Liu Z, Wu J, Chen Y et al (2010) Impulsive perturbations in a periodic delay differential equation model of plankton allelopathy. Nonlinear Anal Real World Appl 11:432–445

  Article  MathSciNet  Google Scholar 

• Liu K, Wang J, O’Regan D (2019) Ulam–Hyers–Mittag-Leffler stability for \(\psi \)-Hilfer fractional-order delay differential equations. Adv Differ Equ 209:1337–1350

  MathSciNet  MATH  Google Scholar 

• Ma C (2019) A novel computational technique for impulsive fractional differential equations. Symmetry 11(216)

• Murray JD (2002) Mathematical biology, interdisciplinary applied mathematics, vol 17, 3rd edn. Springer, New York

  Google Scholar 

• Noupoue YYY, Tandoğdu Y, Awadalla M (2019) On numerical techniques for solving the fractional logistic differential equation. Adv Differ Equ 2019(108):1687–1847

• Pender J, Rand RH, Wesson E (2017) Queues with choice via delay differential equations. Int J Bifurc Chaos 27(4):1730016

• Rus IA (2009) Remarks on Ulam stability of the operatorial equations. Fixed Point Theory 10(2):305–320

  MathSciNet  MATH  Google Scholar 

• Ruschel S, Pereira T, Yanchuk S et al (2019) An SIQ delay differential equations model for disease control via isolation. J Math Biol 79:249–279

  Article  MathSciNet  Google Scholar 

• Saker SH, Alzabut JO (2009) On the impulsive delay hematopoiesis model with periodic coefficients. Rocky Mt J Math 39(5):1657–1688

  Article  MathSciNet  Google Scholar 

• Sales Teodoro G, Tenreiro Machado JA, Capelas de Oliveira E (2019) A review of definitions of fractional derivatives and other operators. J Comput Phys 388:195–208

  Article  MathSciNet  Google Scholar 

• Shi Z, Li Y, Cheng H (2019) Dynamic analysis of a pest management smith model with impulsive state feedback control and continuous delay. Mathematics 7:591

  Article  Google Scholar 

• Sousa JVdC, Capelas de Oliveira E (2018) On the \(\psi \)-Hilfer fractional derivative. Commun Nonlinear Sci Numer Simulat 60:72–91

  Article  MathSciNet  Google Scholar 

• Sousa JVdC, Capelas de Oliveira E (2019) Leibniz type rule: \(\psi \)-Hilfer fractional operator. Commun Nonlinear Sci Numer Simulat 77:305–311

  Article  MathSciNet  Google Scholar 

• Sousa JVdC, Oliveira DS, Capelas de Oliveira E (2021) A note on the mild solutions of Hilfer impulsive fractional differential equations. Chaos Solitons Fract 147:110944 (preprint)

  Article  MathSciNet  Google Scholar 

• Stamova I, Stamov G (2016) Applied impulsive mathematical models. CMS books in mathematics. Springer, Switzerland

  MATH  Google Scholar 

• Sweilam N, Khader M, Mahdy A (2012) Numerical studies for solving fractional-order logistic equation. Int J Pure Appl Math 78(8):1199–1210

  MATH  Google Scholar 

• Tamen AT, Dumont Y, Tewa JJ et al (2017) A minimalistic model of tree-grass interactions using impulsive differential equations and non-linear feedback functions of grass biomass onto fire-induced tree mortality. Math Comput Simul 133:265–297

  Article  MathSciNet  Google Scholar 

• Tang S, Xiao Y, Cheke RA (2010) Dynamical analysis of plant disease models with cultural control strategies and economic thresholds. Math Comput Simul 80(5):894–921

  Article  MathSciNet  Google Scholar 

• Wang J, Fečkan M, Zhou Y (2012) Ulam’s type stability of impulsive ordinary differential equations. J Math Anal Appl 365(1):258–264

  Article  MathSciNet  Google Scholar 

• Wang J, Zhou Y, Feckan M (2012) Nonlinear impulsive problems for fractional differential equations and Ulam stability. Comput Math Appl 64(10):3389–3405

  Article  MathSciNet  Google Scholar 

• Xiangsen L, Binxiang D (2019) Dynamics of a generalized predator-prey model with delay and impulse via the basic reproduction number. Math Methods Appl Sci 42:6878–6895

  Article  MathSciNet  Google Scholar 

• Yu J (2018) Modeling mosquito population suppression based on delay differential equations. SIAM J Appl Math 78(6):3168–3187

  Article  MathSciNet  Google Scholar 

• Zada A, Faisal S, Li Y (2016) On the Hyers–Ulam stability of first-order impulsive delay differential equations. J Funct Space 2016:1–6

  Article  MathSciNet  Google Scholar 

• Zhang H, Chen L, Nieto JJ (2008) A delayed epidemic model with stage-structure and pulses for pest management strategy. Nonlinear Anal Real World Appl 9:1714–1726

  Article  MathSciNet  Google Scholar 

• Zhang H, Georgescu P, Chen L (2008) On the impulsive controllability and bifurcation of a predator-pest model of IPM. Biosystems 93(3):151–171

  Article  Google Scholar 

• Zhang X, Tang S, Cheke RA, et al (2016) Modeling the effects of augmentation strategies on the control of dengue fever with an impulsive differential equation. Bull Math Biol 78(10):1968–2010

Download references