Upper and lower solutions method for higher order fractional initial value problems

Guezane-Lakoud, Assia; Khaldi, Rabah

doi:10.1080/1726037X.2017.1327162

Open Access Article

Upper and lower solutions method for higher order fractional initial value problems

^*Assia Guezane-LakoudCorresponding authorA_GUEZANE@YAHOO.FRLABORATORY OF ADVANCED MATERIALSBADJI MOKHTAR-ANNABA UNIVERSITYDEPARTMENT OF MATHEMATICS, ANNABA, P.O. BOX 12, 23000, ALGERIAView full profile → , Rabah KhaldiLABORATORY OF ADVANCED MATERIALSBADJI MOKHTAR-ANNABA UNIVERSITYDEPARTMENT OF MATHEMATICS, ANNABA, P.O. BOX 12, 23000, ALGERIAView full profile →

* Corresponding author · click or hover a name for details

pp. 29–35Vol. 15Issue 1January 2017DOI: 10.1080/1726037X.2017.1327162XML

Received:: 24 May 2016
Accepted:: 26 Oct 2016
Published Online:: 12 Jun 2017
Article type:: Article
Language:: EN
Article no.:: 1726037X.2017.1327162
Pages:: 29–35

Abstract

This paper concerns the existence of solution to an initial fractional problem of arbitrary order. The main tools for this study are the lower ad upper solutions method and Schauder’s fixed point Theorem.

Keywords

Initial value problemFractional derivativeUpper and lower solutions methodExistence of solution

Subject Classifications

34B1534A0826A3334A12

References

C.De Coster, P.Habets, Two-Point Boundary Value Problems: Lower and Upper Solutions, Mathematics in Science and Engineering, Vol 205, Series Editor: C.K. Chui, Elsevier, 2006 .
A.Elhaffaf, M.Naceri, Existence theorems for a fifth-order boundary value problem , Journal of Mathematics and System Science 4 , 1–5 , ( 2014 ).
D.Franco, Juan J.Nieto, D.O’Regan, Upper and lower solutions for first order problems with nonlinear boundary conditions , Extracta mathematicae 18 , 153 – 160 , ( 2003 ).
A.Guezane-Lakoud, Initial value problem of fractional order, Cogent Math 2, Art.ID 1004797, (2015) .
A.Guezane-Lakoud, R.Khaldi, Existence results for a fractional boundary value problem with fractional Lidstone conditions , J.Appl.Math.Comput 49 , 261 – 268 , ( 2015 ). doi: 10.1007/s12190-014-0837-7
D.Jiang, Y.Yang, J.Chu, J, D.O’Regan, The monotone method for Neumann functional differential equations with upper and lower solutions in the reverse order. Nonlinear Anal ., Theory Methods Appl 67 , 2815 – 2828 ( 2007 ). doi: 10.1016/j.na.2006.09.042
M.Jia, X.Liu, Multiplicity of solutions for integral boundary value problems of fractional differential equations with upper and lower solutions , Applied Mathematics and Computation , Vol. 232 , 313 – 323 , ( 2014 ). doi: 10.1016/j.amc.2014.01.073
A. A.Kilbas, H. M.Srivastava, and J. J.Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies , Elsevier Science , Amsterdam, The Netherlands, 2006 .
F.Li, M.Jia, X.Liu, C.Li, and G.Li, Existence and uniqueness of solutions of second-order three-point boundary value problems with upper and lower solutions in the reversed order , Nonlinear Anal , 68 , 2381 – 2388 , ( 2008 ). doi: 10.1016/j.na.2007.01.065
I.Podlubny, Fractional Differential Equation , Academic Press , Sain Diego1999 .
S. G.Samko, A. A.Kilbas, and O. I.Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, Switzerland, 1993 .
A.Shi, S.Zhang, Upper and lower solutions method and a fractional differential equation boundary value problem , Electronic Journal of Qualitative Theory of Differential Equations , 30 , 1 – 13 , ( 2009 ). doi: 10.14232/ejqtde.2009.1.30

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[1] C.De Coster, P.Habets, Two-Point Boundary Value Problems: Lower and Upper Solutions, Mathematics in Science and Engineering, Vol 205, Series Editor: C.K. Chui, Elsevier, 2006 .

[2] A.Elhaffaf, M.Naceri, Existence theorems for a fifth-order boundary value problem , Journal of Mathematics and System Science 4 , 1–5 , ( 2014 ).

[3] D.Franco, Juan J.Nieto, D.O’Regan, Upper and lower solutions for first order problems with nonlinear boundary conditions , Extracta mathematicae 18 , 153 – 160 , ( 2003 ).

[4] A.Guezane-Lakoud, Initial value problem of fractional order, Cogent Math 2, Art.ID 1004797, (2015) .

[5] A.Guezane-Lakoud, R.Khaldi, Existence results for a fractional boundary value problem with fractional Lidstone conditions , J.Appl.Math.Comput 49 , 261 – 268 , ( 2015 ). doi: 10.1007/s12190-014-0837-7

[6] D.Jiang, Y.Yang, J.Chu, J, D.O’Regan, The monotone method for Neumann functional differential equations with upper and lower solutions in the reverse order. Nonlinear Anal ., Theory Methods Appl 67 , 2815 – 2828 ( 2007 ). doi: 10.1016/j.na.2006.09.042

[7] M.Jia, X.Liu, Multiplicity of solutions for integral boundary value problems of fractional differential equations with upper and lower solutions , Applied Mathematics and Computation , Vol. 232 , 313 – 323 , ( 2014 ). doi: 10.1016/j.amc.2014.01.073

[8] A. A.Kilbas, H. M.Srivastava, and J. J.Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies , Elsevier Science , Amsterdam, The Netherlands, 2006 .

[9] F.Li, M.Jia, X.Liu, C.Li, and G.Li, Existence and uniqueness of solutions of second-order three-point boundary value problems with upper and lower solutions in the reversed order , Nonlinear Anal , 68 , 2381 – 2388 , ( 2008 ). doi: 10.1016/j.na.2007.01.065

[10] I.Podlubny, Fractional Differential Equation , Academic Press , Sain Diego1999 .

[11] S. G.Samko, A. A.Kilbas, and O. I.Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, Switzerland, 1993 .

[12] A.Shi, S.Zhang, Upper and lower solutions method and a fractional differential equation boundary value problem , Electronic Journal of Qualitative Theory of Differential Equations , 30 , 1 – 13 , ( 2009 ). doi: 10.14232/ejqtde.2009.1.30

Journal of Dynamical Systems and Geometric Theories

WoS JIF 2026 : 0.5 (Q3)

Upper and lower solutions method for higher order fractional initial value problems

Abstract

Keywords

Subject Classifications

References