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Existence results for fractional order boundary value problem with nonlocal non-separated type multi-point integral boundary conditions

Department of Mathematics and Statistics, International Islamic University, H-10, Islamabad, Pakistan

Special Issues: Initial and Boundary Value Problems for Differential Equations

In this article, we discuss the existence of solutions of a fractional boundary value problem of order m ∈ (1, 2], with nonlocal non-separated type integral multipoint boundary conditions. Shaefer type and Krasnoselskii’s fixed point theorems are used to prove existence results for the given problem. To establish the uniqueness of solutions Banach contraction principle is used. The criteria for HyersUlam stability of the given boundary value problem is also discussed. Some examples are included for the illustration of our results.
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1. S. Abbas, M. Benchohra, G. M. N'Guérékata, Topics in Fractional Differential Equations, Springer Science & Business Media, 2012.

2. B. Ahmad, J. J. Nieto, Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions, Bound. value probl., 2009 (2009), 708576.

3. B. Ahmad, S. Sivasundaram, Existence of solutions for impulsive integral boundary value problems of fractional order, Nonlinear Anal. Hybri., 4 (2010), 134-141.    

4. B. Ahmad, S. K. Ntouyas, A. Alsaedi, On a coupled system of fractional differential equations with coupled nonlocal and integral boundary conditions, Chaos Soliton. Fract., 1 (2016), 234-241.

5. A. Alsaedi, M. Alsulami, R. P. Agarwal, et al. Some new nonlinear second-order boundary value problems on an arbitrary domain, Adv. Differ. Equ., 2018 (2018), 227.

6. B. Ahmad, S. K. Ntouyas, A. Alsaedi, Fractional order differential systems involving right Caputo and left Riemann-Liouville fractional derivatives with nonlocal coupled conditions, Bound. Value Probl., 2019 (2019), 109.

7. B. Ahmad, S. K. Ntouyas, A. Alsaedi, Existence theory for fractional differential equations with nonlocal integro-multipoint boundary conditions with applications, Appl. Eng. Life Soc. Sci., 1 (2019), 271.

8. B. Ahmad, S. K. Ntouyas, A. Alsaedi, et al. Existence theory for fractional differential equations with non-separated type nonlocal multi-point and multi-strip boundary conditions, Adv. Differ. Equ., 2018 (2018), 89.

9. R. P. Agarwal, A. Alsaedi, N, Alghamdi, et al. Existence results for multi-term fractional differential equations with nonlocal multi-point and multi-strip boundary conditions, Adv. Differ. Equ., 2018 (2018), 342.

10. K. Balachandran, J. Y. Park, Nonlocal Cauchy problem for abstract fractional semilinear evolution equations, Nonlinear Anal. Theor., 71 (2009), 4471-4475.    

11. L. Byszewski, Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. anal. Appl., 162 (1991), 494-505.    

12. M. Caputo, Distributed order differential equations modelling dielectric induction and diffusion, Fract. Calc. Appl. Anal., 4 (2001), 421-442.

13. R. J. DiPerna, P. L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability, Ann. Math., 1 (1989), 321-366.

14. J. Ginibre, G. Velo, The global Cauchy problem for the non linear Klein-Gordon equation, Math. Z., 189 (1985), 487-505.    

15. D. H. Hyers, On the stability of the linear functional equation, P. Natl. Acad. Sci. USA, 27 (1941), 222.

16. A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science Limited, 2006.

17. V. Lakshmikantham, A. S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal. Theor., 69 (2008), 2677-2682.    

18. V. Lakshmikantham, Theory of fractional functional differential equations, Nonlinear Anal. Theor., 69 (2008), 3337-3343.    

19. S. K. Ntouyas, A. Alsaedi, B. Ahmad, Existence theorems for mixed Riemann-Liouville and Caputo fractional differential equations and inclusions with nonlocal fractional integrodifferential boundary conditions, Fract. Fract., 3 (2019), 21.

20. K. Oldham, J. Spanier, The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order, Elsevier, 1974.

21. I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Elsevier, 1998.

22. T. M. Rassias, On the stability of the linear mapping in Banach spaces, P. Am. Math. Soc., 72 (1978), 297-300.    

23. I. A. Rus, Ulam stabilities of ordinary differential equations in a Banach space, Carpathian J. Math., 2010 (2010), 103-107.

24. G. S. Teodoro, J. T. Machado, E. C. De Oliveira, A review of definitions of fractional derivatives and other operators, J. Comput. Phys., 388 (2019), 195-208.    

25. S. M. Ulam, Problems in Modern mathematics, New York: John-Wiley & Sons Inc., 1964.

26. Y. Zhou, J. R. Wang, L. Zhang, Basic Theory of Fractional Differential Equations, World Scientific, 2016.

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

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