### Qualitative Analysis of Nonlinear Hilfer Fractional Implicit Differential Equations in a Banach space

#### Keywords:

Fixed point theorems, Hilfer fractional derivative, implicit differential equations, Ulam’s stability#### Abstract

This article focuses on the class of nonlinear implicit Hilfer-type fractional differential equations. By using the non-linear growth condition, we have derived the existence of at least one solution by applying Schauder’s fixed point theorem and using Lipschitz conditions, we have derived the uniqueness of the solution with the help of the Banach contraction principle. In addition, we have discussed the stability analysis by using Ulam-Hyers and Ulam-Hyers-Rassias stabilities. All results of this paper are established in a Banach space instead of R. We illustrate our results with the help of two examples.

## References

S. Abbas, M. Benchohra, J.E. Lazreg, Y. Zhou, A survey on Hadamard and Hilfer fractional differential equations analysis and stability, Chaos Solitons Fractals 102 (2017) 47–71.

W. Albarakati, M. Benchohra, S. Bouriah, Existence and stability results for nonlinear implicit fractional differential equations with delay and impulses, Differ. Equ. Appl. 2 (2016) 273–293.

. Andras, J.J Kolumban, On the Ulam-Hyers stability of first order differential systems with nonlocal initial conditions, Nonlinear Anal. Theory Methods Appl. 82 (2013) 1–11.

M. Benchohra, E. Karapınar, J.E. Lazreg, A. Salim, Hybrid Fractional Differential Equations. In Fractional Differential Equations: New Advancements for Generalized Fractional Derivatives Cham: Springer Nature Switzerland (2023) 31-76.

M. Benchohra, E. Karapınar, J.E. Lazreg, A. Salim, Implicit Fractional Differential Equations. In Advanced Topics in Fractional Differential Equations: A Fixed Point Approach Cham: Springer Nature Switzerland (2023) 35–76.

S. Bouriah, M. Benchohra, J.R. Graef, Nonlinear implicit differential equations of fractional order at resonanc, Electron. J. Differ. Equ. 324 (2016) 1–10.

C. Derbazi, Z. Baitich, M. Feckan, Some new uniqueness and Ulam stability results for a class of Multi-Terms fractional differential equations in the framework of Generalized Caputo Fractional Derivative using the ϕ-fractional Bielecki-type norm, Turk. J. Math. (2021) DOI: 10.3906/mat-2011-92.

K. Dhawan, R.K. Vats, A.K. Nain, A. Shukla, Well-posedness and Ulam-Hyers stability of Hilfer Fractional Differential Equations of order (1, 2] with Nonlocal Boundary Conditions, Bulletin des Sciences Math´ematiques 191 (2024) 103401.

K. Dhawan, R.K. Vats, R.P. Agarwal, Qualitative analysis of couple fractional differential equations involving Hilfer Derivative, An. St. Univ. Ovidius Constanta. 30(1) (2022) 191–217.

K. Dhawan, R.K. Vats, S. Kumar, A. Kumar, Existence and Stability analysis for nonlinear boundary value problem involving Caputo fractional derivative, Dyn. Contin. Discrete Impuls. Syst. 30 (2023) 107–121.

K. Dhawan, R.K. Vats, V. Vijaykumar, Analysis of Neutral Fractional Differential Equation via the Method of Upper and Lower Solution, Qual. Theory Dyn. Syst. 22(93) (2023) 1–15.

K.M. Furati, M.D. Kassim, N.E. Tatar, Existence and uniqueness for a problem involving Hilfer fractional derivative, Comput. Math. Appl. 64, (2012) 1616–1626.

H. Gu, J.J. Trujillo, Existence of mild solution for evolution equation with Hilfer fractional derivative, Appl. Math. Comput. 257, (2015) 344–354.

R. Hilfer, Applications of fractional calculus in Physics, World Scientific, Singapore (2000).

R. Hilfer, Y. Luchko, Z. Tomovski, Operational method for the solution of fractional differential equations with generalized Riemann-Lioville fractional derivative, Fract. Calc. Appl. Anal. 12 (2009) 289–318.

D.H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. 27 (1941) 222–224.

R.W. Ibrahim, Generalized Ulam-Hyers stability for fractional differential equations, Int. J. Math. 23(5) (2012) 1–9.

S.M. Jung, Hyers-Ulam stability of linear differential equations of first order, Appl. Math. Lett. 17 (2004) 1135–1140 .

E. Karapınar, N. Benkhettou, J.E. Lazreg, M. Benchohra, Fractional differential equations with maxima on time scale via Picard operators, Filomat. 37 (2) (2023) 393–402.

E. Karapınar, A Survey On The Fixed Point Theorems Via Admissible Mapping, Cuadernos de desarrollo aplicados a las. 11 (2022) 1–25.

A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V, Amsterdam (2006).

R. Magin, Fractional calculus in bioengineering, Crit. Rev. Biom. Eng. 32 (2004) 1–104.

P. Muniyappan, S. Rajan, Hyers-Ulam-Rassias stability of fractional differential equation, Int. J. Pure Appl. Math. 102 (2015) 631-642.

I. Podlubny, Fractional Differential Equations, Academic Press, San Diego (1999).

A. Nain R. Vats A. Kumar Coupled fractional differential equations involving Caputo–Hadamard derivative with nonlocal boundary conditions, Math Meth Appl Sci. (2020) 1–13.

I.A. Rus, Ulam stabilities of ordinary differential equations in a Banach space, Carpath. J. Math. 26 (2010) 103–107.

H.L. Tidke, R.P. Mahajan, Existence and uniqueness of nonlinear implicit fractional differential equation with Rie mann–Liouville derivative, Am. J. Comput. Appl. Math. 7 (2) (2017) 46–50.

H.M. Srivastava, K. Dhawan, R.K. Vats, A.K. Nain, Well-posedness of a nonlinear Hilfer fractional derivative model for the Antarctic circumpolar current, Zeitschrift f¨ur angewandte Mathematik und Physik. 75 (2024) 1-19.

S.M. Ulam, A Collection of the Mathematical Problems Interscience, New York (1960).

S.K. Verma, R.K Vats, A. Kumar, V. Vijayakumar, A. Shukla, A discussion on the existence and uniqueness analysis for the coupled two-term fractional differential equations. Turk. J. Math. (2021) DOI: 10.3906/mat-2107-30.

D. Vivek, K. Kanagarajan, E.M. Elsayed, Some existence and stability results for Hilfer-fractional implicit differential equations with nonlocal conditions. Mediterr. J. Math. (2018) https://doi.org/10.1007/s00009-017-1061-0 .

M. Yang, Q. Wang, Existence of mild solutions for a class of Hilfer fractional evolution eqautions with nonlocal condition, Fract. Calc. Appl. Anal. 20 (2017) 679–705.

A. Zada, S. Ali, Y. Li, Ulam-type stability for a class of implicit fractional differential equations with non-instantaneous integral impulses and boundary condition, Adv. Differ. Equ. 317 (2017) 1–26.

R. Hilfer, Y. Luchko, Z. Tomovski, Operational method for the solution of fractional differential equations with generalized Riemann-Liouville fractional derivatives, Fract. Calc. Appl. Anal. 12 (2009) 299–318.

## Downloads

## Published

## Issue

## Section

## How to Cite

*Advances in the Theory of Nonlinear Analysis and Its Application*,

*7*(5), 141-154. https://doi.org/10.17762/atnaa.v7.i5.331