Home Existence result for nonlinear fractional differential equations with nonlocal fractional integro-differential boundary conditions in Banach spaces
Article
Licensed
Unlicensed Requires Authentication

Existence result for nonlinear fractional differential equations with nonlocal fractional integro-differential boundary conditions in Banach spaces

  • Djamila Seba ORCID logo EMAIL logo , Hamza Rebai and Johnny Henderson ORCID logo
Published/Copyright: February 15, 2019
Become an author with De Gruyter Brill

Abstract

The nonlinear fractional differential equation with nonlocal fractional integro-differential boundary conditions in Banach spaces is studied, an existence result is obtained by using the method associated with the technique of measures of noncompactness and an appropriate fixed point theorem. An example is given to illustrate the theory.

MSC 2010: 34A08; 34B15

References

[1] R. P. Agarwal, M. Benchohra and D. Seba, On the application of measure of noncompactness to the existence of solutions for fractional differential equations, Results Math. 55 (2009), no. 3–4, 221–230. 10.1007/s00025-009-0434-5Search in Google Scholar

[2] R. P. Agarwal, M. Meehan and D. O’Regan, Fixed Point Theory and Applications, Cambridge Tracts in Math. 141, Cambridge University, Cambridge, 2001. 10.1017/CBO9780511543005Search in Google Scholar

[3] B. Ahmad and A. Alsaedi, Nonlinear fractional differential equations with nonlocal fractional integro-differential boundary conditions, Bound. Value Probl. 2012 (2012), Paper No. 124. 10.1186/1687-2770-2012-124Search in Google Scholar

[4] B. Ahmad and S. K. Ntouyas, Existence results for a coupled system of Caputo type sequential fractional differential equations with nonlocal integral boundary conditions, Appl. Math. Comput. 266 (2015), 615–622. 10.1016/j.amc.2015.05.116Search in Google Scholar

[5] R. R. Akhmerov, M. I. Kamenskiĭ, A. S. Potapov, A. E. Rodkina and B. N. Sadovskiĭ, Measures of Noncompactness and Condensing Operators, Oper. Theory Adv. Appl. 55, Birkhäuser, Basel, 1992. 10.1007/978-3-0348-5727-7Search in Google Scholar

[6] D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus. Models and Numerical Methods, Ser. Complex. Nonlinearity Chaos 3, World Scientific, Hackensack, 2012. 10.1142/8180Search in Google Scholar

[7] J. Banaś and K. Goebel, Measures of Noncompactness in Banach Spaces, Lect. Notes Pure Appl. Math. 60, Marcel Dekker, New York, 1980. Search in Google Scholar

[8] M. Benchohra, J. Henderson and D. Seba, Measure of noncompactness and fractional differential equations in Banach spaces, Commun. Appl. Anal. 12 (2008), no. 4, 419–427. Search in Google Scholar

[9] M. Benchohra, J. Henderson and D. Seba, Boundary value problems for fractional differential inclusions in Banach spaces, Fract. Differ. Calc. 2 (2012), no. 1, 99–108. 10.7153/fdc-02-07Search in Google Scholar

[10] M. Benchohra, G. M. N’Guérékata and D. Seba, Measure of noncompactness and nondensely defined semilinear functional differential equations with fractional order, Cubo 12 (2010), no. 3, 35–48. 10.4067/S0719-06462010000300003Search in Google Scholar

[11] Y. Cui, Uniqueness of solution for boundary value problems for fractional differential equations, Appl. Math. Lett. 51 (2016), 48–54. 10.1016/j.aml.2015.07.002Search in Google Scholar

[12] J. R. Graef, L. Kong and B. Yang, Positive solutions for a fractional boundary value problem, Appl. Math. Lett. 56 (2016), 49–55. 10.1016/j.aml.2015.12.006Search in Google Scholar

[13] D. Guo, V. Lakshmikantham and X. Liu, Nonlinear Integral Equations in Abstract Spaces, Math. Appl. 373, Kluwer Academic, Dordrecht, 1996. 10.1007/978-1-4613-1281-9Search in Google Scholar

[14] M. I. Ismailov and M. Çiçek, Inverse source problem for a time-fractional diffusion equation with nonlocal boundary conditions, Appl. Math. Model. 40 (2016), no. 7–8, 4891–4899. 10.1016/j.apm.2015.12.020Search in Google Scholar

[15] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud. 204, Elsevier Science, Amsterdam, 2006. Search in Google Scholar

[16] H. Mönch, Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces, Nonlinear Anal. 4 (1980), no. 5, 985–999. 10.1016/0362-546X(80)90010-3Search in Google Scholar

[17] S. K. Ntouyas, Boundary value problems for nonlinear fractional differential equations and inclusions with nonlocal and integral boundary conditions, Fract. Differ. Calc. 3 (2013), no. 1, 1–20. 10.7494/OpMath.2013.33.1.117Search in Google Scholar

[18] S. K. Ntouyas, J. Tariboon and W. Sudsutad, Boundary value problems for Riemann–Liouville fractional differential inclusions with nonlocal Hadamard fractional integral conditions, Mediterr. J. Math. 13 (2016), no. 3, 939–954. 10.11650/tjm.20.2016.5654Search in Google Scholar

[19] D. O’Regan and S. Staněk, Fractional boundary value problems with singularities in space variables, Nonlinear Dynam. 71 (2013), no. 4, 641–652. 10.1007/s11071-012-0443-xSearch in Google Scholar

[20] I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calc. Appl. Anal. 5 (2002), no. 4, 367–386. Search in Google Scholar

[21] J. Sabatier, O. P. Agrawal and J. A. T. Machado, Advances in Fractional Calculus, Springer, Dordrecht, 2007. 10.1007/978-1-4020-6042-7Search in Google Scholar

[22] S. A. Szufla, On the application of measure of noncompactness to existence theorems, Rend. Semin. Mat. Univ. Padova 75 (1986), 1–14. Search in Google Scholar

[23] W. Xie, J. Xiao and Z. Luo, Existence of extremal solutions for nonlinear fractional differential equation with nonlinear boundary conditions, Appl. Math. Lett. 41 (2015), 46–51. 10.1016/j.aml.2014.10.014Search in Google Scholar

Received: 2017-02-01
Accepted: 2017-09-12
Published Online: 2019-02-15
Published in Print: 2021-02-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 23.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/gmj-2019-2009/html
Scroll to top button