Existence of solutions for nonlocal boundary value problem for Caputo nonlinear fractional differential inclusion

Bouteraa Noureddine; Slimane Benaicha

doi:10.33187/jmsm.414747

EN

Existence of solutions for nonlocal boundary value problem for Caputo nonlinear fractional differential inclusion

Abstract

This paper deals with the existence of solutions for nonlinear fractional differential inclusions supplemented with three-point boundary conditions. First, we investigate it for $ L^{1}$-Caratheodory convex-compact valued multifunction. Then, we investigate it for nonconvex-compact valued multifunction via some conditions. Two illustrative examples are presented at the end of the paper to illustrate the validity of our results.

Keywords

Existence,Fractional differential inclusion,Fixed point theorem,Nonlocal boundary

References

[1] A. Lasota and Z. Opial, An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations, Bull. Acad. Pol. Sci. Set. Sci. Math. Astronom. Phy. 13 (1965), 781–786.
[2] A. Ral, V. Gupta and R. P. Agarwal, Applications of q-calculus in operator theory., Springer, New York, (2013).
[3] A. Granas and J. Dugundji, Fixed Point Theory., Springer-Verlag, New York, 2005.
[4] A. A. Kilbas, H. M. Srivastava and J. J. Trijullo, Theory and applications of fractional differential equations, Elsevier Science b. V., Amsterdam, (2006).
[5] A. Ral, V. Gupta and R. P. Agarwal, Applications of q-calculus in operator theory., Springer, New York, (2013).
[6] A. Rezaigia and S. Kelaiaia, Existence results for third-order differential inclusion with three-point boundary value problems, Acta. Math. Univ. Comenianae, 2 (2016), 311–318.
[7] A. M. A. EL-Sayed, F. M. Nesreen and EL-Haddade, Existence of integrable solutions for a functional integral inclusion, Diff. Equa. Cont. Proce, 3 (2017).
[8] A. Cernia, Existence of solutions for a certain boundary value problem associated to a fourth-order differential inclusion, Inter. Jour. Anal. Appl. Vol 14, 1 (2017), 27–33.
[9] B. Ahmad, J. S. Ntouyas and I. Purnaras, Existence results for nonlocal boundary value problems of nonlinear fractional q-difference equations, Adv. Differ. Equ-NY, 2012 (2012), 140, 15 pages.
[10] C. R. Adamas, On the linear ordinary q-difference equation, Ann. Math, 30 (1928), 195–205.

[11] D. Baleanu, H. Mohammadi and S. H. Rezapour, The existence of solutions for a nonlinear mixed problem of singular fractional equations, Adv. Difference Equa. 2013 (2013), 12 pages.
[12] H. Covitz and S. B. Jr. Nadler, Multivalued contraction mappings in generalized metric spaces, Israel J. Math. 8 (1970), 5–11.
[13] H. H. Alsulami, S. K. Ntouyas, S. A. Al-Mezel, B. Ahmad and A. Alsaedi, A study of third-order single-valued and multi-valued problems with integral boundary conditions, Bound. Value. Prob. (2015), 20 pages.
[14] J. P. Aubin and A. Cellina, Differential Inclusions, Springer-Verlag, 2012.
[15] K. Demling, Multivalued Differential equations, Walter De Gryter, Berlin-New-York 1982.
[16] L. Gorniewcz, Topological Fixed Point Theory of Multivalued Map pings, Mathematics and Its Applications vol. 495, Kluwer Academic Publishers, Dordrecht 1999.
[17] M. Kisielewicz, Differential Inclusions and Optimal Control, Mathematics and Its Applications (East European Series), vol. 44, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1991.
[18] N. Bouteraa and S. Benaicha, Existence of solutions for three-point boundary value problem for nonlinear fractional differential equations, Analele Universitatii Oradea Fasc. Matematica, Tom XXIV (2017), Issue No. 2, 109–119.
[19] N. Bouteraa, S. Benaicha and H. Djourdem, Positive solutions for nonlinear fractional differential equation with nonlocal boundary conditions, Universal Journal of Mathematics and Applications. 1(1), (2018), 39–45.
[20] N. Bouteraa and S. Benaicha, The uniqueness of positive solution for nonlinear fractional differential equation with nonlocal boundary conditions, Analele Universitatii Oradea Fasc. Matematica, Volume 25 (2018), Issue No. 2, To appear.
[21] O. Agrawal, Some generalized fractional calculus operators and their applications in integral equations, Fract. Cal. Appl. Anal. 15 (2012), 700–711.
[22] P. M. Rajkovic, S. D. Marancovic and M. S. Stankovic, On q-analogues of Caputo derivative and Mettag-Leffter function, Frac. Calc. Appl. Anal. 10 (2007), 359–373.
[23] R. P. Agarwal, D. Baleanu, V. Hedayati and S. H. Rezapour, Two fractional derivative inclusion problems via integral boundary condition, Appl. Math. Comput. 257 (2015), 205–212.
[24] R. P. Agarwal, S. K. Ntouyas, B. Ahmad and M. S. Alhouthuali, Existence of solutions for integro-differential equations of fractional order with nonlocal tree-point fractional boundary condition, Adv. Difference Equ. 2013 (2013) .
[25] R. Ghorbanian, V. Hedayati, M. Postolache and S. H. Rezapour, On a fractional differential inclusion via a new integral boundary condition, J. Inequal. Appl. 2014 (2014), 20 pages.
[26] S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, John Wiley and Sons, Inc. New York, (1993).
[27] S. K. Ntouyas, Existence of solutions for fractional differencial inclusions with integral boundary conditions, Bound. Value Prob. 2015 (2015), 14 pages.
[28] S. K. Ntouyas, S. Etemad and J. Tariboon, Existence results for multi-term fractional diffrential inclusions, Adv. Difference Equ. 2015 (2015), 15 pages.
[29] S. K. Ntouyas, S. Etemad, J. Tariboon and W. Sutsutad, Boundary value problems for Riemann-Liouville nonlinear fractional diffrential inclusions with nonlocal Hadamard fractional integral conditions, Medittter. J. Math. 2015 (2015), 16 pages.
[30] S. Etemad, S. K. Ntouyas and J. Tariboon, Existence results for three-point boundary value problems for nonlinear fractional diffrential equations. J. Nonlinear Sci. Appl. 9 (2016).
[31] P. D. Phung and L. X. Truong, On a fractional differential inclusion with integral boundary conditions in Banach space, Fract. Calc. Appl. Anal. 16(3) (2013), 538-558.
[32] R. P. Agarwal and B. Ahmad, Existence theory for anti-periodic boundary value problems of fractional differential equations and inclusions, J. Appl. Math. Comput. 62 (2011).
[33] R. Ghorbanian, V. Hedayati, M. Postolache and S. H. Rezapour, On a fractional differential inclusion via a new integral boundary condition, J. Inequal. Appl. 2014 (2014), 20 pages.
[34] R. P. Agarwal, D. Baleanu, V. Hedayati and S. H. Rezapour, Two fractional derivative inclusion problems via integral boundary condition, Appl. Math. Comput. 257 (2015), 205–212.
[35] S. Hu and N. Papageorgeou, Handbook of Multivalued Analysis, Volume I
[36] S. Suganya and M. M. Arjunan, Existence of Mild Solutions for Impulsive Fractional Integro-Differential Inclusions with State-Dependent Delay, Mathematics. 9 (2017), 2–16.
[37] V. Kac and P. Cheung, Quantum Calculus, Springer, New York, (2002).
[38] V. Berinde and M. Pacurar, The role of the Pompeiu-Hausdorff metric in fixed point theory, Creat. Math. Inform. 22(2) (2013), 35–42.

Details

Primary Language

English

Subjects

Mathematical Sciences

Journal Section

Research Article

Authors

Bouteraa Noureddine
Algeria

Slimane Benaicha This is me
Algeria

Publication Date

May 27, 2018

Submission Date

April 12, 2018

Acceptance Date

May 22, 2018

Published in Issue

Year 2018 Volume: 1 Number: 1

DOI

https://doi.org/10.33187/jmsm.414747

IZ

https://izlik.org/JA24XA58TU

APA

Noureddine, B., & Benaicha, S. (2018). Existence of solutions for nonlocal boundary value problem for Caputo nonlinear fractional differential inclusion. Journal of Mathematical Sciences and Modelling, 1(1), 45-55. https://doi.org/10.33187/jmsm.414747

AMA

1.Noureddine B, Benaicha S. Existence of solutions for nonlocal boundary value problem for Caputo nonlinear fractional differential inclusion. Journal of Mathematical Sciences and Modelling. 2018;1(1):45-55. doi:10.33187/jmsm.414747

Chicago

Noureddine, Bouteraa, and Slimane Benaicha. 2018. “Existence of Solutions for Nonlocal Boundary Value Problem for Caputo Nonlinear Fractional Differential Inclusion”. Journal of Mathematical Sciences and Modelling 1 (1): 45-55. https://doi.org/10.33187/jmsm.414747.

EndNote

Noureddine B, Benaicha S (May 1, 2018) Existence of solutions for nonlocal boundary value problem for Caputo nonlinear fractional differential inclusion. Journal of Mathematical Sciences and Modelling 1 1 45–55.

IEEE

[1]B. Noureddine and S. Benaicha, “Existence of solutions for nonlocal boundary value problem for Caputo nonlinear fractional differential inclusion”, Journal of Mathematical Sciences and Modelling, vol. 1, no. 1, pp. 45–55, May 2018, doi: 10.33187/jmsm.414747.

ISNAD

Noureddine, Bouteraa - Benaicha, Slimane. “Existence of Solutions for Nonlocal Boundary Value Problem for Caputo Nonlinear Fractional Differential Inclusion”. Journal of Mathematical Sciences and Modelling 1/1 (May 1, 2018): 45-55. https://doi.org/10.33187/jmsm.414747.

JAMA

1.Noureddine B, Benaicha S. Existence of solutions for nonlocal boundary value problem for Caputo nonlinear fractional differential inclusion. Journal of Mathematical Sciences and Modelling. 2018;1:45–55.

MLA

Noureddine, Bouteraa, and Slimane Benaicha. “Existence of Solutions for Nonlocal Boundary Value Problem for Caputo Nonlinear Fractional Differential Inclusion”. Journal of Mathematical Sciences and Modelling, vol. 1, no. 1, May 2018, pp. 45-55, doi:10.33187/jmsm.414747.

Vancouver

1.Bouteraa Noureddine, Slimane Benaicha. Existence of solutions for nonlocal boundary value problem for Caputo nonlinear fractional differential inclusion. Journal of Mathematical Sciences and Modelling. 2018 May 1;1(1):45-5. doi:10.33187/jmsm.414747