Boundary value problems for fractional differential inclusions with Hadamard type derivatives in Banach spaces

John R. Graef, Nassim Guerraiche, Samira Hamani

Abstract


The authors establish sufficient conditions for the existence of solutions to boundary value problems for fractional differential inclusions involving the Hadamard type fractional derivative of order $\alpha \in (1,2]$ in Banach spaces. Their approach uses M\"onch's fixed point theorem and the Kuratowski measure of noncompacteness.

Keywords


Fractional differential inclusion; Hadamard-type fractional derivative; fractional integral; M\"onch's fixed point theorem; Kuratowski measure of noncompacteness

Full Text:

PDF

References


bibitem{AgBeHa2}R. P Agarwal, M. Benchohra, and S. Hamani,

A survey on existence results for boundary value problems

for nonlinear fractional differential equations and inclusions, {em Acta Applicandae Math.} {bf 109} (2010), 973--1033.

bibitem{AgBeSe} R. P. Agarwal, M. Benchohra, and D. Seba, An the application of measure of noncompactness to the existence of solutions for fractional differential equations, {em Results Math.} {bf 55 } (2009), 221--230.

bibitem{AgMeOr} R. P. Agarwal, M. Meehan, and D. O'Regan, {it Fixed Point Theory and Applications}, Cambridge Tracts in Mathematics {bf 141}, Cambridge University Press, Cambridge, 2001.

bibitem{AhNt} B. Ahmed and S. K. Ntouyas, Initial value problems for hybrid Hadamard

fractional equations, {em Electron. J. Differential Equations} {bf 2014} (2014), No. 161, pp. 1--8.

bibitem{AkKaPaRoSa} R. R. Akhmerov, M. I. Kamenski, A. S. Patapov, A. E. Rodkina, and B. N. Sadovski, {it Measures of Noncompactness and Condensing Operators} (Translated from the 1986 Russian original by A. Iacop), Operator theory: Advances and Applications, {bf 55}, Birkh"auser Verlag, Bassel, 1992.

bibitem{AuCe} J. P. Aubin and A. Cellina, {em Differential

Inclusions}, Springer-Verlag, Berlin-Heidelberg, New York, 1984.

bibitem{AuFr} J. P. Aubin and H. Frankowska, {em Set-Valued Analysis},

Birkhauser, Boston, 1990.

bibitem{BaGo} J. Banas and K. Goebel, {it Measure of Noncompactness in Banach Spaces}, Lecture Notes in Pure and Applied Mathematics,

Vol. {bf 60}, Dekker, New York.

bibitem{BaSa} J. Banas and K. Sadarangani, On some measures of noncompactness in the space of continuous functions, {em Nonlinear Anal.} {bf 60} (2008), 377--383.

bibitem{BeHaNt} M. Benchohra, S. Hamani, and S. K. Ntouyas,

Boundary value problems for differential equations with fractional

order, {em Surv. Math. Appl.} {bf 3} (2008), 1--12.

bibitem{BeHeSe1} M. Benchohra, J. Henderson, and D. Seba,

Measure of noncompactness and fractional differential equations

in Banach spaces, {em Commun. Appl. Anal.} {bf 12} (2008), 419--428.

bibitem{BeHeSe} M. Benchohra, J. Henderson, and D. Seba,

Boundary value problems for fractional differential inclusions

in Banach Space, {em Fract. Differ. Calc.} {bf 2} (2012), 99--108.

bibitem{BeNiSe} M. Benchohra, J. J. Nieto, and D. Seba,

Measure of noncompactness and fractional and hyperbolic partial fractional

differential equations in Banach space, {em Panamer. Math. J } {bf 20} (2010), 27--37.

bibitem{HG1} W. Benhamida, J. R. Graef, and S. Hamani,

Boundary value problems for fractional differential equations with integral and anti-periodic conditions in a Banach space, to appear.

bibitem{BuKiTr} P. L. Butzer, A. A. Kilbas, and J. J. Trujillo, Composition

of Hadamard-type fractional integration operators and the semigroup

property, {em J. Math. Anal. Appl.} {bf 269} (2002), 387--400.

bibitem{BuKiTr1} P. L. Butzer, A. A. Kilbas, and J. J. Trujillo, Fractional calculus in the Mellin setting and Hadamard-type fractional integrals, {em J. Math. Anal. Appl.} {bf 269} (2002), 1--27.

bibitem{BuKiTr2} P. L. Butzer, A. A. Kilbas, and J. J. Trujillo, Mellin transform analysis and integration by parts for Hadamard-type fractional integrals, {em J. Math. Anal. Appl.} {bf 270} (2002), 1--15.

bibitem{By} L. Byszewski, Theorems about existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, {em J. Math. Anal. Appl.} {bf 162} (1991), 494--505.

bibitem{By1} L. Byszewski, Existence and uniqueness of mild and classical solutions of semilinear functional-differential evolution nonlocal

Cauchy problem, in: {em Selected Problems of Mathematics,} 25-30, 50th Anniv. Cracow Univ. Technol. Anniv. Issue, 6, Cracow Univ. Technol., Krakw, 1995.

bibitem{CaVa} C. Castaing and M. Valadier, {em Convex Analysis

and Measurable Multifunctions}, Lecture Notes in Mathematics {bf

}, Springer-Verlag, Berlin-Heidelberg-New York, 1977.

bibitem{De} K. Deimling, {em Multivalued Differential Equations},

De Gruyter, Berlin-New York, 1992.

bibitem{GuLaLi} D. Guo, V. Lakshmikantham, and X. Liu, Nonlinear integral equations in abstract spaces, {em Mathematics and its Applications}, {bf 373}, Kluwer, Dordrecht, 1996.

bibitem{Ha} J. Hadamard, Essai sur l'etude des fonctions donnees par leur development de Taylor, {em J. Mat. Pure Appl.} Ser. 8 (1892), 101--186.

bibitem {HaBeGr} S. Hamani, M. Benchohra, and J. R. Graef,

Existence results for boundary-value problems with nonlinear

fractional differential inclusions and integral conditions,

{em Electron. J. Differential Equations} {bf 10} (2010), No. 20, pp. 1--16.

bibitem{He} H. P. Heinz, On the behavior of measure of

noncompactness with respect of differentiation and integration

of vector-valued function, {em Nonlinear. Anal} {bf 7} (1983),

--1371.

bibitem{Hil} R. Hilfer, {em Applications of Fractional Calculus in

Physics}, World Scientific, Singapore, 2000.

bibitem{KiMa} A. A. Kilbas and S. A. Marzan, Nonlinear

differential equations with the Caputo fractional derivative in the

space of continuously differentiable functions, {em Differential

Equations} {bf 41} (2005), 84--89.

bibitem{KST} A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo,

{em Theory and Applications of Fractional Differential Equations}.

North-Holland Mathematics Studies, {bf 204}, Elsevier,

Amsterdam, 2006.

bibitem {LaLe} V. Lakshmikantham and S. Leela, {it Nonlinear Differential Equations in Abstract Spaces}, International Series in Mathematics: Theory, Methods and Applications, {bf 2}, Pergamon Press, Oxford, 1981.

bibitem{LaOp} A. Lasota and Z. Opial, An application of the

Kakutani-Ky Fan theorem in the theory of ordinary

differential equation, {em Bull. Accd. Pol. Sci. Ser. Sci. Math.

Astronom. Phys.} {bf 13} (1965), 781--786.

bibitem{MoHaAl} S. M. Momani, S. B. Hadid, and Z. M.

Alawenh, Some analytical properties of solutions of diifferential

equations of noninteger order, {em Int. J. Math. Math. Sci.} {bf

} (2004), 697--701.

bibitem{Mo} H. M"{o}nch, Boundary value problem for nonlinear

ordinary differential equations of second order in Banach spaces,

{em Nonlinear Anal.} {bf 75} (1980), 985--999.

bibitem{OrPr} D. O'Regan and R. Precup, Fixed point theorems for

set-valued maps and existence principles for integral inclusions,

{em J. Math. Anal. Appl.} {bf 245} (2000), 594--612.

bibitem{Pod} I. Podlubny, {em Fractional Differential Equation}, Academic Press, San Diego, 1999.

bibitem{Sz} S. Szufla, On the application of measure of

noncompactness to existence theorems, {em Rend. Semin. Mat. Univ. Padova} {bf 75} (1986), 1--14.

bibitem{ThNtTa} P. Thiramanus, S. K. Ntouyas, and J. Tariboon, Existence and uniqueness results for Hadamard-type fractional differential equations with nonlocal fractional integral boundary conditions,

{em Abstr. Appl. Anal.} (2014), Art. ID 902054, 9 pp.

end{thebibliography}




DOI: http://dx.doi.org/10.24193/subbmath.2017.4.02

Refbacks

  • There are currently no refbacks.