# Existence of Solutions for a Fractional Boundary Value Problem at Resonance

## DOI:

https://doi.org/10.52737/18291163-2022.14.15-1-16## Keywords:

Fractional Differential Equations, Boundary Value Problem, Caputo Derivative, Coincidence Degree Theory, Resonance## Abstract

In this paper, we focus on the existence of solutions to a fractional boundary value problem at resonance. By constructing suitable operators, we establish an existence theorem upon the coincidence degree theory of Mawhin.

## References

Z. Bai, W. Li and W. Ge, Existence and multiplicity of solutions for four-point boundary value problems at resonance. Nonlinear Anal., 60 (2005), no. 6, pp. 1151-1162. https://doi.org/10.1016/j.na.2004.10.013

M. Benchohra, S. Bouriah and J. Graef, Nonlinear implicit differential equations of fractional order at resonance. Electron. J. Differ. Equ., 324 (2016), pp. 1-10.

A. Guezane-Lakoud, S. Kouachi and F. Ellaggoune, Two point fractional boundary value problem at resonance. Int. J. Appl. Math., 33 (2015), no. 3-4, pp. 425-434. https://doi.org/10.14317/jami.2015.425

B.B. He, Existence of solutions to fractional differential equations with three-point boundary conditions at resonance with general conditions. Fract. Calc. Appl., 9 (2018), pp. 120-136.

L. Hu and S. Zhang, Existence of positive solutions to a periodic boundary value problems for nonlinear fractional differential equations at resonance. J. Fract. Calc. Appl., 8 (2017), no. 2, pp. 19-31.

Z. Hu, W. Liu and T. Chen, Two-point boundary value problems for fractional differential equations at resonance. Bull. Malaysian Math. Sci. Soc. 2, 36 (2013), pp. 747-755.

W. Jiang, X. Huang and B. Wang, Boundary value problems of fractional differential equations at resonance. Phys. Procedia, 25 (2012), pp. 965-972. https://doi.org/10.1016/j.phpro.2012.03.185

A.A. Kilbas, H.M. Srivastava and J.J Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2016.

J. Mawhin, Equivalence theorems for nonlinear operator equations and coincidence degree theory for some mappings in locally convex topological vector spaces. J. Diff. Equations, 12 (1972), no. 3, pp. 610-636. https://doi.org/10.1016/0022-0396(72)90028-9

J. Mawhin, Topological degree methods in nonlinear boundary value problems. In NSF-CBMS Regional Conference Series in Mathematics, 40, Amer. Math. Soc., Providence, RI, 1979. https://doi.org/10.1090/cbms/040

J. Mawhin, Topological degree and boundary value problems for nonlinear differential equations. In M. Furi, P. Zecca (eds) Topological Methods for Ordinary Differential Equations, Lecture Notes in Mathematics, 1537, Springer, Berlin, Heidelberg, 1993. https://doi.org/10.1007/bfb0085076

P.D. Phung and H.B. Minh, Existence of solutions to fractional boundary value problems at resonance in Hilbert spaces. Bound. Value Probl., 105 (2017), pp. 1-20. https://doi.org/10.1186/s13661-017-0836-3

P.D. Phung and L.X. Truong, On the existence of a three point boundary value problem at resonance in Rn. J. Math. Anal. Appl., 416 (2014), pp. 522-533.

P.D. Phung and L.X. Truong, Existence of solutions to three-point boundary-value problems at resonance. Electron. J. Diff. Eqns., 115 (2016), pp. 1-13.

I. Podlubny, Fractional Differential Equations, Mathematics in Sciences and Applications, Academic Press, New York, 1999.

S.G. Samko, A.A. Kilbas and O.I. Mirichev, Fractional Integral and Derivatives (Theory and Applications). Gordon and Breach, Switzerland, 1993.

S. Song and Y. Cui, Existence of solutions for integral boundary value problems of mixed fractional differential equations under resonance. Bound. Value Probl., 23 (2020), pp. 1-12. https://doi.org/10.1186/s13661-020-01332-5

Y. Wang and Y. Wu, Positive solutions of fractional differential equation boundary value problems at resonance, J. Appl. Anal. Comput., 10 (2020), pp. 2459-2475.

C. Yang and C. Zhai, Uniqueness of positive solutions for a fractional differential equation via a fixed point theorem of a sum operator. Electron J. Differ. Equ., 70 (2021), pp. 1-8.

C. Zhai and M. Hao, Fixed point theorems for mixed monotone operators with perturbation and applications to fractional differential equation boundary value problems. Nonlinear Anal., 75 (2012), no. 4, pp. 2542-2551. https://doi.org/10.1016/j.na.2011.10.048

C. Zhai and L. Xu, Properties of positive solutions to a class of four-point boundary value problem of Caputo fractional differential equations with a parameter. Commun. Nonlinear Sci. Numer. Simul., 19 (2014), no. 8, pp. 2820-2827. https://doi.org/10.1016/j.cnsns.2014.01.003

C. Zhai, W. Yan and C. Yang, A sum operator method for the existence and uniqueness of positive solutions to Riemann-Liouville fractional differential equation boundary value problems. Commun. Nonlinear Sci. Numer. Simul., 18 (2013), no. 4, pp. 858-866. https://doi.org/10.1016/j.cnsns.2012.08.037

Y.H. Zhang, Z.B. Bai and T.T. Feng, Existence results for a coupled system of nonlinear fractional three-point boundary value problems at resonance. Comput. Math. Appl., 61 (2011), no. 4, pp. 1032-1047. https://doi.org/10.1016/j.camwa.2010.12.053

H.C. Zhou, F.D. Ge and C.H. Kou, Existence of solutions to fractional differential equations with multi-point boundary conditions at resonance in Hilbert spaces. Electron. J. Differ. Equ., 61 (2016), pp. 1-16.

Y. Zhou, F. Jiao and J. Li, Existence and uniqueness for fractional neutral differential equations with infinite delay. Nonlinear Anal., 71 (2009), no. 7-8, pp. 3249-3256. https://doi.org/10.1016/j.na.2009.01.202

## Downloads

## Published

## Issue

## Section

## License

Copyright (c) 2022 Armenian Journal of Mathematics

This work is licensed under a Creative Commons Attribution 4.0 International License.

## How to Cite

*Armenian Journal of Mathematics*,

*14*(15), 1-16. https://doi.org/10.52737/18291163-2022.14.15-1-16