Existence results for Riemann-Liouville fractional integro-differential inclusions with fractional nonlocal integral boundary conditions

Bashir Ahmad; Badrah Alghamdi; Ahmed Alsaedi; Sotiris K. Ntouyas; Bashir Ahmad; Badrah Alghamdi; Ahmed Alsaedi; Sotiris K. Ntouyas

doi:10.3934/math.2021416

AIMS Mathematics

2021, Volume 6, Issue 7: 7093-7110. doi: 10.3934/math.2021416

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Existence results for Riemann-Liouville fractional integro-differential inclusions with fractional nonlocal integral boundary conditions

1.
Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
2.
Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece

Received: 20 February 2021 Accepted: 20 April 2021 Published: 27 April 2021
MSC : 34A08, 34A60, 34B15

We introduce a new class of problems consisting of Riemann-Liouville fractional integro-differential inclusions supplemented with fractional nonlocal multi-point boundary conditions. The existence results for the given problem are derived in the weighted space with the aid of appropriate fixed point theorems for multi-valued maps. Numerical examples are constructed for the illustration of the obtained results.
- Riemann-Liouville fractional derivative,
- integro-differential inclusions,
- nonlocal multi-point boundary conditions,
- existence,
- fixed point
Citation: Bashir Ahmad, Badrah Alghamdi, Ahmed Alsaedi, Sotiris K. Ntouyas. Existence results for Riemann-Liouville fractional integro-differential inclusions with fractional nonlocal integral boundary conditions[J]. AIMS Mathematics, 2021, 6(7): 7093-7110. doi: 10.3934/math.2021416

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Abstract

We introduce a new class of problems consisting of Riemann-Liouville fractional integro-differential inclusions supplemented with fractional nonlocal multi-point boundary conditions. The existence results for the given problem are derived in the weighted space with the aid of appropriate fixed point theorems for multi-valued maps. Numerical examples are constructed for the illustration of the obtained results.

References

[1]	R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
[2]	F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical mechanis, In: Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi, Eds, 291–348, Springer-Verlag, Wien, 1997.
[3]	S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Yverdon, 1993.
[4]	K. Diethelm, The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics, Springer, New York, 2010.
[5]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.
[6]	I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
[7]	J. Sabatier, O. P. Agrawal, J. A. T. Machado, Eds., Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, 2007.
[8]	V. Lakshmikantham, S. Leela, J. Vasundhara Devi, Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, 2009.
[9]	Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014.
[10]	B. Ahmad, A. Alsaedi, S. K. Ntouyas, J. Tariboon, Hadamard-Type Fractional Differential Equations, Inclusions and Inequalities, Springer, Cham, Switzerland, 2017.
[11]	M. Kisielewicz, Stochastic Differential Inclusions and Applications, Springer: New York, NY, USA, 2013.
[12]	B. Ahmad, M. Alghanmi, S. K. Ntouyas, A. Alsaedi, A study of fractional differential equations and inclusions involving generalized Caputo-type derivative equipped with generalized fractional integral boundary conditions, AIMS Math., 4 (2019), 26–42. doi: 10.3934/Math.2019.1.26
[13]	B. Ahmad, A. Alsaedi, S. K. Ntouyas, Nonlinear Langevin equations and inclusions involving mixed fractional order derivatives and variable coefficient with fractional nonlocal-terminal conditions, AIMS Math.. 4 (2019), 626–647.
[14]	J. Tariboon, S. K. Ntouyas, B. Ahmad, A. Alsaedi, Existence results for sequential Riemann-Liouville and Caputo fractional differential inclusions with generalized fractional integral conditions, Mathematics, 2020, 8, 1044. doi: 10.3390/math8061044
[15]	S. K. Ntouyas, B. Ahmad, A. Alsaedi, Fractional order single-valued and multi-valued problems with integro-multistrip-multipoint boundary conditions, Fractal Fract., 2019, 4, 31.
[16]	A. Wongcharoen, S. K. Ntouyas, J. Tariboon, Nonlocal boundary value problems for Hilfer type pantograph fractional differential equations and inclusions, Adv. Differ. Equ., 2020 (2020), 279. doi: 10.1186/s13662-020-02747-1
[17]	S. K. Ntouyas, S. Etemad, On the existence of solutions for fractional differential inclusions with sum and integral boundary conditions, Appl. Math. Comput., 266 (2015), 235–243.
[18]	S. Belmor, F. Jarad, T. Abdeljawad, G. Kilinc, A study of boundary value problem for generalized fractional differential inclusion via endpoint theory for weak contractions, Adv. Differer. Equ, 2020 (2020), 348. doi: 10.1186/s13662-020-02811-w
[19]	R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
[20]	R. L. Magin, Fractional Calculus in Bioengineering, Begell House Publishers, Danbury, CT, 2006.
[21]	D. Baleanu, J. A. T. Machado, A. C. J. Luo, Fractional Dynamics and Control, Springer, New York, 2012.
[22]	N. Heymans, I. Podlubny, Physical interpretation of initial conditions for fractional differential equations with Riemann–Liouville fractional derivatives, Rheol. Acta., 45 (2006) 765-771.
[23]	T. F. Nonnenmacher, R. Metzler, On the Riemann-Liouville fractional calculus and some recent applications, Symposium in Honor of Benoit Mandelbrot, (Curaçao, 1995), Fractals 3 (1995), 557–566.
[24]	P. J. Torvik, R. L. Bagley, On the appearance of the fractional derivative in the behavior of real materials, J. Appl. Mech., 51 (1984), 294-298. doi: 10.1115/1.3167615
[25]	B. Ahmad, H. Batarfi, J. J. Nieto, O. Otero-Zarraquinos, W. Shammakh, Projectile motion via Riemann-Liouville calculus, Adv. Difference Equ., 2015:63 (2015), 14.
[26]	A. Alsaedi, J. J. Nieto, V. Venktesh, Fractional electrical circuits, Adv. Mech. Eng., 7 (2015), 1-7.
[27]	N. H. Tuan, N. H. Tuan, D. Baleanu, T. N. Thach, On a backward problem for fractional diffusion equation with Riemann-Liouville derivative, Math. Methods Appl. Sci., 43 (2020), 1292–1312. doi: 10.1002/mma.5943
[28]	M. Feng, X. Zhang, W. Ge, Existence theorems for a second order nonlinear differential equation with nonlocal boundary conditions and their applications, J. Appl. Math. Comput., 33 (2010), 137–153. doi: 10.1007/s12190-009-0278-x
[29]	L. Zheng, X. Zhang, Modeling and Analysis of Modern Fluid Problems. Mathematics in Science and Engineering, Elsevier/Academic Press: London, UK, 2017.
[30]	F. Nicoud, T. Schfonfeld, Integral boundary conditions for unsteady biomedical CFD applications, Int. J. Numer. Meth. Fluids, 40 (2002), 457–465. doi: 10.1002/fld.299
[31]	R. Čiegis, A. Bugajev, Numerical approximation of one model of the bacterial self-organization, Nonlinear Anal. Model. Control, 17 (2012), 253–270. doi: 10.15388/NA.17.3.14054
[32]	E. Yusufoglu, I. Turhan, A mixed boundary value problem in orthotropic strip containing a crack. J. Franklin Inst., 349 (2012), 2750–2769.
[33]	T. V. Renterghem, D. Botteldooren, K. Verheyen, Road traffic noise shielding by vegetation belts of limited depth, J. Sound Vib., 331 (2012), 2404–2425. doi: 10.1016/j.jsv.2012.01.006
[34]	H. F. Bohnenblust, S. Karlin, On a theorem of Ville, in: Contributions to the theory of games, vol Ⅰ, Princeton Univ. Press, 1950, 155–160.
[35]	A. Lasota, Z. Opial, An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys., 13 (1965), 781–786.
[36]	K. Deimling, Multivalued Differential Equations, Walter De Gruyter, Berlin-New York, 1992.
[37]	A. Granas, J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003.
[38]	H. Covitz, S. B. Nadler Jr., Multivalued contraction mappings in generalized metric spaces, Israel J. Math., 8 (1970), 5–11. doi: 10.1007/BF02771543
[39]	C. Castaing, M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics 580, Springer-Verlag, Berlin-Heidelberg-New York, 1977.

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