## Abstract

In this work, we study the existence of solutions to a new class of boundary value problems consisting of a system of nonlinear differential equations with generalized fractional derivative operators of different orders and nonlocal boundary conditions containing Riemann-Stieltjes and generalized fractional integral operators. We emphasize that the nonlinearities in the given system are of general form as they depend on the unknown functions as well as their lower order generalized fractional derivatives. The uniqueness result for the given problem is proved by applying the Banach contraction mapping principle, while the existence of solutions for the given system is shown with the aid of Leray-Schauder alternative. Two concrete examples are given for illustrating the obtained results. The paper concludes with some interesting observations.

### Similar content being viewed by others

## References

Abbas, M.I., Fečkan, M.: Michal Feckan, Investigation of an implicit Hadamard fractional differential equation with Riemann-Stieltjes integral boundary condition. Math. Slovaca

**72**(4), 925–934 (2022)Ahmad, B., Alghanmi, M., Ntouyas, S.K., Alsaedi, A.: Fractional differential equations involving generalized derivative with Stieltjes and fractional integral boundary conditions. Appl. Math. Lett.

**84**, 111–117 (2018)Ahmad, B., Ntouyas, S.K.: Existence results for a coupled system of Caputo type sequential fractional differential equations with nonlocal integral boundary conditions. Appl. Math. Comput.

**266**, 615–622 (2015)Ahmad, B., Ntouyas, S., Alsaedi, A.: On a coupled system of fractional differential equations with coupled nonlocal and integral boundary conditions. Chaos Solitons Fractals

**83**, 234–241 (2016)Ahmad, B., Luca, R.: Existence of solutions for a system of fractional differential equations with coupled nonlocal boundary conditions. Frac. Calc. Appl. Anal.

**21**(2), 423–441 (2018)Ahmad, B., Alghanmi, M., Alsaedi, A.: Existence results for a nonlinear coupled system involving both Caputo and Riemann-Liouville generalized fractional derivatives and coupled integral boundary conditions. Rocky Mountain J. Math.

**50**(6), 1901–1922 (2020)Alruwaily, Y., Ahmad, B., Ntouyas, S.K., Alzaidi, A.S.M.: Existence Results for Coupled Nonlinear Sequential Fractional Differential Equations with Coupled Riemann-Stieltjes Integro-Multipoint Boundary Conditions. Fractal Fract.

**6**, 123 (2022). https://doi.org/10.3390/fractalfract6020123Arena, P., Caponetto, R., Fortuna, L., Porto, D.: Chaos in a fractional order Duffing system, In:Proceedings of the 1997 European conference on circuit theory and design(ECCTD97), Budapest, Hungary, 30 August-3. Budapest: Hungary: Technical University of Budapest (1997) 1259-1262

Asawasamrit, S., Thadang, Y., Ntouyas, S.K., Tariboon, J.: Non-Instantaneous Impulsive Boundary Value Problems Containing Caputo Fractional Derivative of a Function with Respect to Another Function and Riemann-Stieltjes Fractional Integral Boundary Conditions. Axioms

**10**(3), 130 (2021)Belmor, S., Ravichandran, C., Jarad, F.: Nonlinear generalized fractional differential equations with generalized fractional integral conditions. J. Taibah University Sci

**14**(1), 114–123 (2020)Belmor, S., Jarad, F., Abdeljawad, T., Kilinç, G.: A study of boundary value problem for generalized fractional differential in- clusion via endpoint theory for weak contractions. Advances in Difference Equations

**2020**(1), 1–11 (2020)Belmor, S., Jarad, F., Abdeljawad, T.: On Caputo-, Hadamard type coupled systems of nonconvex fractional differential inclusions. Adv. Diff. Equ.

**2021**(1), 1–12 (2021)Belmor, S., Jarad, F., Abdeljawad, T., Alqudah, M.A.: On fractional differential inclusion problems involving fractional order derivative with respect to another function. Fractals

**28**(08), 2040002 (2020)Faieghi, M., Kuntanapreeda, S., Delavari, H., Baleanu, D.: LMI-based stabilization of a class of fractional-order chaotic systems. Nonlinear Dyn.

**72**, 301–309 (2013)Ge, Z.M., Jhuang, W.R.: Chaos, control and synchronization of a fractional order rotational mechanical system with a centrifugal governor. Chaos Solitons Fractals

**33**, 270–289 (2007)Ge, Z.M., Ou, C.Y.: Chaos synchronization of fractional order modified Duffing systems with parameters excited by a chaotic signal. Chaos Solitons Fractals

**35**, 705–717 (2008)Granas, A., Dugundji, J.: Fixed Point Theory. Springer-Verlag, New York (2003)

Grigorenko, I., Grigorenko, E.: Chaotic dynamics of the fractional Lorenz system. Phys. Rev. Lett.

**91**, 034101 (2003)Hartley, T.T., Lorenzo, C.F., Killory, Q.H.: Chaos in a fractional order Chua’s system. IEEE Trans. CAS-I

**42**, 485–490 (1995)Henderson, J., Luca, R., Tudorache, A.: On a system of fractional differential equations with coupled integral boundary conditions. Fract. Calc. Appl. Anal.

**18**(2), 361–386 (2015). https://doi.org/10.1515/fca-2015-0024Hilfer, R. (ed.): Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)

Jiang, J., Liu, L., Wu, Y.: Positive solutions to singular fractional differential system with coupled boundary conditions. Comm. Nonlinear Sc. Num. Sim.

**18**(11), 3061–3074 (2013)Jiao, Z., Chen, Y.Q., Podlubny, I.: Distributed-order Dyn. Syst. Springer, New York (2012)

Katugampola, U.N.: New approach to a generalized fractional integral. Appl. Math. Comput.

**218**, 860–865 (2015)Katugampola, U.N.: A new approach to generalized fractional derivatives. Bull. Math. Anal. Appl.

**6**, 1–15 (2014)Kilbas, A.A., Trujillo, J.J.: Differential equations of fractional order: Methods, results and problems I. Appl. Anal.

**78**, 153–192 (2001)Kilbas, A.A., Trujillo, J.J.: Differential equations of fractional order: Methods, results and problems II. Appl. Anal.

**81**, 435–493 (2002)Kusnezov, D., Bulgac, A., Dang, G.D.: Quantum Levy processes and fractional kinetics. Phys. Rev. Lett.

**82**, 1136–11399 (1999)Lu, H., Sun, S., Yang, D., Teng, H.: Theory of fractional hybrid differential equations with linear perturbations of second type. Bound. Value Probl.

**2013**, 23 (2013)Lupinska, B., Odzijewicz, T.: A Lyapunov-type inequality with the Katugampola fractional derivative. Math. Methods Appl. Sci.

**41**, 8985–8996 (2018)Metzler, R., Klafter, J.: The random walks guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep.

**339**, 1–77 (2000)Ostoja-Starzewski, M.: Towards thermoelasticity of fractal media. J. Therm. Stress

**30**, 889–896 (2007)Povstenko, Y.Z.: Fract thermoelast. Springer, New York (2015)

Redhwan, S.S., Shaikh, S.L., Abdo, M.S.: Implicit fractional differential equation with anti-periodic boundary condition involving Caputo-Katugampola type. AIMS Math.

**5**(4), 3714–3730 (2020)Redhwan, S.S., Shaikh, S.L., Abdo, M.S., Shatanawi, W., Abodayeh, K., Almalahi, M.A., Aljaaidi, T.: Investigating a generalized Hilfer-type fractional differential equation with two-point and integral boundary conditions. AIMS Math.

**7**(2), 1856–1872 (2021)S.S. Redhwan, S.L. Shaikh, M.S. Abdo, Theory of Nonlinear Caputo-Katugampola Fractional Differential Equations, arXiv:1911.08884 13 (2019)

Redhwan, S.S., Shaikh, S.L., Abdo, M.S.: Caputo-Katugampola type implicit fractional differential equation with two-point anti-periodic boundary conditions. Results Nonlinear Anal.

**5**(1), 12–28 (2022)Sokolov, I.M., Klafter, J., Blumen, A.: Fractional kinetics. Phys. Today.

**55**, 48–54 (2002)Su, X.: Boundary value problem for a coupled system of nonlinear fractional differential equations. Appl. Math. Lett.

**22**, 64–69 (2009)Sabatier, J., Agarwal, O.P., Ttenreiro Machado, J.A.: Advances in Fractional Calculus. Theoretical Developments and Applications in Physics and Engineering, New York, Springer (2007)

Ttenreiro Machado, J.A.: Discrete time fractional-order controllers. Frac. Cal. App. Anal.

**4**, 47–66 (2001)Waheed, H., Zada, A., Rizwan, R., Popa, I.L.: Hyers-Ulam Stability for a Coupled System of Fractional Differential Equation With \(p\)-Laplacian Operator Having Integral Boundary Conditions. Qual. Theory Dyn. Syst.

**21**(3), Paper No. 92 (2022)Zada, A., Alam, M., Riaz, U.: Analysis of \(q\)-fractional implicit boundary value problems having stieltjes integral conditions. Math. Meth. Appl. Sci.

**44**(6), 4381–4413 (2020)Zhang, F., Chen, G., Li, C., Kurths, J.: Chaos synchronization in fractional differential systems. Phil. Trans. R. Soc. A

**371**, 20120155, 26 pages (2013)

## Acknowledgements

The authors would like to thank the referees for their suggestions and helpful comments which improved the presentation of the original manuscript.

## Author information

### Authors and Affiliations

### Corresponding author

## Ethics declarations

### Conflict of interest

The authors declare that they have no conflict of interest.

## Additional information

### Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

## About this article

### Cite this article

Nyamoradi, N., Ahmad, B. Generalized Fractional Differential Systems with Stieltjes Boundary Conditions.
*Qual. Theory Dyn. Syst.* **22**, 6 (2023). https://doi.org/10.1007/s12346-022-00703-w

Received:

Accepted:

Published:

DOI: https://doi.org/10.1007/s12346-022-00703-w