References

  1. Abbas M.I. Controllability and Hyers-Ulam stability results of initial value problems for fractional differential equations via generalized proportional-Caputo fractional derivative. Miskolc Math. Notes 2021, 22 (2), 491–502. doi:10.18514/MMN.2021.3470
  2. Abbas M.I., Ragusa M.A. On the hybrid fractional differential equations with fractional proportional derivatives of a function with respect to a certain function. Symmetry 2021, 13 (2), 264. doi:10.3390/sym13020264
  3. Abbas M.I. Existence results and the Ulam stability for fractional differential equations with hybrid proportional-Caputo derivatives. J. Nonlinear Funct. Anal. 2020, 2020, 1–14. doi:10.23952/jnfa.2020.48
  4. Abbas M.I. Ulam stability of fractional impulsive differential equations with Riemann-Liouville integral boundary conditions. J. Contemp. Math. Anal. 2015, 50 (5), 209–219. doi:10.3103/S1068362315050015
  5. Abbas M.I. Existence and uniqueness of Mittag-Leffler-Ulam stable solution for fractional integro-differential equations with nonlocal initial conditions. Eur. J. Pure Appl. Math. 2015, 8 (4), 478–498.
  6. Alzabut J., Abdeljawad T., Jarad F., Sudsutad W. A Gronwall inequality via the generalized proportional fractional derivative with applications. J. Inequal. Appl. 2019, 2019, article number 101. doi:10.1186/s13660-019-2052-4
  7. Agarwal R., Hristova S., O’Regan D. Ulam type stability for non-instantaneous impulsive Caputo fractional differential equations with finite state dependent delay. Georgian Math. J. 2021, 28 (4), 499–517. doi:10.1515/gmj-2020-2061
  8. Baleanu D., Diethelm K., Scalas E., Trujillo J.J. Fractional Calculus Models and Numerical Methods. In: Luo A.C.J., Sanjuan M.A.F. (Eds.) Series on Complexity, Nonlinearity and Chaos, 5. World Scientific: Singapore, 2012.
  9. Benchohra M., Bouriah S., Nieto J.J. Existence and Ulam stability for nonlinear implicit differential equations with Riemann-Liouville fractional derivative. Demonstr. Math. 2019, 52, 437–450. doi:10.1515/dema-2019-0032
  10. Boucenna D., Baleanu D., Makhlouf A., Nagy A.M. Analysis and numerical solution of the generalized proportional fractional Cauchy problem. Appl. Numer. Math. 2021, 167, 173–186. doi:10.1016/j.apnum.2021.04.015
  11. Cuong D.X. On the Hyers-Ulam stability of Riemann-Liouville multi-order fractional differential equations. Afr. Mat. 2019, 30 (4), 1041–1047. doi:10.1007/s13370-019-00701-3
  12. Ferraoun S., Dahmani Z. Existence and stability of solutions of a class of hybrid fractional differential equations involving RL-operator. J. Interdisciplinary Math. 2020, 23 (4), 885–903. doi:10.1080/09720502.2020.1727617
  13. Geritz S.A.H., Kisdi E. Mathematical ecology: why mechanistic models? J. Math. Biol. 2012, 65 (6–7), 1411–1415. doi:10.1007/s00285-011-0496-3
  14. Hristova S., Ivanova K. Ulam type stability of non-instantaneous impulsive Riemann-Liouville fractional differential equations (changed lower bound of the fractional derivative). AIP Conf. Proc. 2019, 2159 (1), 030015. doi:10.1063/1.5127480
  15. Hristova S., Abbas M.I. Explicit solutions of initial value problems for fractional generalized proportional differential equations with and without impulses. Symmetry 2021, 13 (6), 996. doi:10.3390/sym13060996
  16. Hyers D.H., Isac G., Rassias Th.M. Stability of Functional Equations in Several Variables. Birkhäuser, Basel, 1998.
  17. Hyers D.H. On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA. 1941, 27 (4), 222–224. doi:10.1073/pnas.27.4.222
  18. Jarad F., Abdeljawad T., Alzabut J. Generalized fractional derivatives generated by a class of local proportional derivatives. Eur. Phys. J. Spec. Top. 2017, 226 (16), 3457–3471. doi:10.1140/epjst/e2018-00021-7
  19. Jarad F., Alqudah M.A., Abdeljawad T. On more general forms of proportional fractional operators. Open Math. 2020, 18 (1), 167–176. doi:10.1515/math-2020-0014
  20. Jung S.M. Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor, 2001.
  21. Ibrahim R.W. Generalized Ulam-Hyers stability for fractional differential equations. Int. J. Math. Anal. (N.S.) 2012, 23 (05), 1250056. doi:10.1142/S0129167X12500565
  22. Laadjal Z., Abdeljawad T., Jarad F. On existence-uniqueness results for proportional fractional differential equations and incomplete gamma functions. Adv. Difference Equ. 2020, 2020, 641. doi:10.1186/s13662-020-03043-8
  23. Kilbas A.A., Srivastava H., Trujillo J.J. Theory and Applications of Fractional Differential Equations. In: Jan van Mill (Ed.) North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.
  24. Mainardi F. Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. World Scientific Publishing Company: Singapore, Hackensack, NJ, USA, London, UK, Hong Kong, China, 2010. doi:10.1142/p614
  25. Miller K.S., Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley-Interscience, John-Wiley and Sons: New York, NY, USA, 1993.
  26. Podlubny I. Fractional Differential Equations. Academic Press, San Diego, 1999.
  27. Rassias Th.M. On the stability of linear mappings in Banach spaces. Proc. Amer. Math. Soc. 1978, 72, 297–300. doi:10.1090/S0002-9939-1978-0507327-1
  28. Rus I.A. Ulam stabilities of ordinary differential equations in a Banach space. Carpathian J. Math. 2010, 26 (1), 103–107.
  29. Samko S., Kilbas A., Marichev O. Fractional Integrals and Drivatives. Gordon and Breach Science Publishers, Longhorne, PA, 1993.
  30. Srivastava H.M., Saad K.M. Some new models of the time-fractional gas dynamics equation. Adv. Math. Models Appl. 2018, 3 (1), 5–17.
  31. Sudsutad W., Alzabut J., Nontasawatsri A., Thaiprayoon C. Stability analysis for a generalized proportional fractional Langevin equation with variable coefficient and mixed integro-differential boundary conditions. J. Nonlinear Funct. Anal. 2020, 2020, article ID 23, 1–24. doi:10.23952/jnfa.2020.23
  32. Ulam S.M. A Collection of Mathematical Problems. Interscience Publishers, New York, 1968.
  33. Wang J.R., Zhang Y. Ulam-Hyers-Mittag-Leffler stability of fractional-order delay differential equations. Optimization 2014, 63 (8), 1181–1190. doi:10.1080/02331934.2014.906597.
  34. Xu L., Dong Q., Li G. Existence and Hyers-Ulam stability for three-point boundary value problems with Riemann-Liouville fractional derivatives and integrals. Adv. Difference Equ. 2018, 2018, article number 458. doi:10.1186/s13662-018-1903-5