International Journal of Nonlinear Analysis and Applications

Existence and stability analysis for nonlinear ψ-Hilfer fractional differential equations with nonlocal integral boundary conditions

Document Type : Research Paper

Authors

1 Department of Mathematics, Faculty of Sciences, University of Annaba, Annaba, Algeria

2 Department of Mathematics and Informatics, University of Souk Ahras, Souk Ahras, Algeria

3 Laboratory of Analysis and Control of Differential Equations,University of 8 May 1945 Guelma, Algeria

Abstract

In this paper, we study the existence and uniqueness of mild solutions for nonlinear fractional differential equations subject to nonlocal integral boundary conditions in the frame of a ψ-Hilfer fractional derivative. Further, we discuss different kinds of stability of Ulam-Hyers for mild solutions to the given problem. Using the fixed point theorems together with generalized Gronwall inequality the desired outcomes are proven. The obtained results generalize many previous works that contain special cases of function ψ. At the end, some pertinent examples demonstrating the effectiveness of the theoretical results are presented.

Keywords

[1] MS. Abdo and S. Panchal, Fractional integro-differential equations involving ψ-Hilfer fractional derivative, Adv. Appl. Math. Mech. 11 (2019) 1–22.
[2] MS. Abdo, SK. Panchal and HS. Hussien, Fractional integro-differential equations with nonlocal conditions and ψ-Hilfer fractional derivative, Mathematical Modelling and Analysis 24(4) (2019) 564–584.
[3] MS. Abdo, SK. Panchal and HA. Wahash, Ulam-Hyers-Mittag-Leffler stability for a ψ-Hilfer problem with fractional order and infinite delay, Results in Applied Mathematics 7 (2020) 100115.
[4] MS. Abdo, STM. Thabet and B. Ahmad, The existence and Ulam–Hyers stability results for ψ-Hilfer fractional integro-differential equations, J. Pseudo-Differ. Oper. Appl. 11 (2020) 1757–1780.
[5] MA. Almalahi, MS. Abdo, SK. Panchal, On the theory of fractional terminal value problem with ψ-Hilfer fractional derivative, AIMS Mathematics 5(5) (2020) 4889–4.
[6] MA. Almalahi, MS. Abdo and SK. Panchal, Existence and Ulam-Hyers Mittag-Leffler stability results of ψ-Hilfer nonlocal Cauchy problem, Rendiconti del Circolo Matematico di Palermo Series 2 70(1) (2021) 57–77.
[7] MA. Almalahi, MS. Abdo and SK. Panchal, Existence and Ulam–Hyers stability results of a coupled system of ψ-Hilfer sequential fractional differential equations, Results in Applied Mathematics 10 (2021) 100142.
[8] MA. Almalahi and SK. Panchal, On the theory of ψ-Hilfer nonlocal Cauchy problem, Journal of Siberian Federal University, Mathematics & Physics 14(2) (2021) 159–175.
[9] S. Asawasamrit, A. Kijjathanakorn, SK. Ntouyas and J. Tariboon, Nonlocal boundary value problems for Hilfer fractional differential equations, Bull Korean Math Soc. 55(6) (2018) 1639–1657.
[10] A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, Therm. Sci. 20(2) (2016) 763–69.
[11] A. Atangana, Fractal-fractional differentiation and integration: connecting fractal calculus and fractional calculus to predict complex system, Chaos Solitons Fractals 102 (2017), 396–406.
[12] D. Baleanu, JAT. Machado and ACJ. Luo, Fractional dynamics and control, Springer, New York, 2002.
[13] L. Debnath, Recent applications of fractional calculus to science and engineering, Int. J. Math. Math. Sci. 54 (2003) 3413–3442.
[14] K. Diethelm, The analysis of fractional differential equations, Lecture Notes in Mathematics, Springer-Verlag, Berlin, Heidelberg, 2010.
[15] KM. Furati and MD. Kassim, Existence and uniqueness for a problem involving Hilfer fractional derivative, Computers Mathematics with Applications 64(6) (2012) 1616–1626.
[16] Z. Gao and X. Yu, Existence results for BVP of a class of Hilfer fractional differential equations, J. Appl. Math. Comput. 56 (2018) 217–233.
[17] R. Hilfer, Applications of fractional calculus in physics, World Scientific, Singapore, 2000.
[18] AA. Kilbas, HM. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier Science B. V., Amsterdam, 2006.
[19] V. Lakshmikantham, S. Leela and JV. Devi, Theory of fractional dynamic systems, Cambridge Scientific Publishers, Cambridge, 2009.
[20] R. Magin, Fractional calculus in bioengineering, Critical Rev. Biomed. Eng. 32 (2004) 1–104.
[21] DA. Mali and KD. Kucche, Nonlocal boundary value problem for generalized Hilfer implicit fractional differential equations, Math. Methods Appl. Sci. 43(15) (2020) 8608–8631.
[22] A. Lachouri, MS. Abdo, A. Ardjouni, B. Abdalla and T. Abdeljawad, Hilfer fractional differential inclusions with Erd´elyi-Kober fractional integral boundary condition, Advances in Difference Equations 2021(1) (2021) 1–17.
[23] A. Lachouri, A. Ardjouni and A. Djoudi, Existence and Ulam stability results for nonlinear hybrid implicit Caputo fractional differential equations, Mathematica Moravica 24(1) (2020) 109–122.
[24] A. Lachouri, A. Ardjouni and A. Djoudi, Existence and ulam stability results for fractional differential equations with mixed nonlocal conditions, Azerbaijan Journal of Mathematics 11(2) (2021) 78–97.
[25] I. Podlubny, Fractional differential equations, Academic Press, San Diego, 1999.
[26] TM. Rassians, On the stability of linear mappings in Banach spaces, Proc. Amer. Math. Soc. 72 (1978) 297–300.
[27] IA. Rus, Ulam stability of ordinary differential equations, Stud. Univ. Babes-Bolyai Math. 54(4) (2009) 125–133.
[28] IA. Rus, Ulam stabilities of ordinary differential equations in a Banach space, Carpathian J. Math. 26 (2010) 103–107.
[29] W. Shatanawi, A. Boutiara, MS. Abdo, MB. Jeelani and K. Abodayeh, Nonlocal and multiple-point fractional boundary value problem in the frame of a generalized Hilfer derivative, Adv. Differ. Equ. 2021 (2021) 294.
[30] DR. Smart, Fixed point theorems, Cambridge Tracts in Mathematics, no. 66, Cambridge University Press, LondonNew York, 1974.
[31] JVC. Sousa and ECD. Oliveira, On the Ψ-Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simula. 60 (2018) 72–91.
[32] JVC. Sousa and ECD. Oliveira, On the Ulam-Hyers-Rassias stability for nonlinear fractional differential equations using the Ψ -Hilfer operator, J. Fixed Point Theory Appl. 20 (2018) 96.
[33] JVC. Sousa and ECD. Oliveira, Ulam-Hyers stability of a nonlinear fractional Volterra integro-differential equation, Appl. Math. Lett. 81 (2018) 50–56.
[34] VE. Tarasov, Fractional dynamics: Application of fractional calculus to dynamics of particles, Fields and Media. Springer, New York, 2011.
[35] STM. Thabet, B. Ahmad and RP. Agarwal, On abstract Hilfer fractional integrodifferential equations with boundary conditions, Arab Journal of Mathematical Sciences 26(1/2) (2019) 107–125.
[36] H. Ye, J. Gao and Y. Ding, A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl. 328(2) (2007) 1075–1081.
[37] HA. Wahash, MS. Abdo and SK. Panchal, Fractional integro-differential equations with nonlocal conditions and generalized ψ-Hilfer fractional derivative, Ufa Mathematical Journal 11(4) (2019) 114–133.
[38] J. Wang, L. Lv and Y. Zhou, Ulam stability and data dependence for fractional differential equations with Caputo derivative, Electron. J. Qual. Theory Differ. Equ. 2011(63) (2011) 1–10.
International Journal of Nonlinear Analysis and Applications
Volume 13, Issue 1
March 2022
Pages 2617-2633
  • Receive Date: 02 August 2021
  • Revise Date: 20 October 2021
  • Accept Date: 12 December 2021