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Benchohra, Mouffak; Graef, John R.; Hamani, Samira

Existence results for boundary value problems with non-linear fractional differential equations. (English) Zbl 1198.26008

Appl. Anal. 87, No. 7, 851-863 (2008).
The authors study the boundary value problem for the nonlinear fractional differential equation
\[ ^cD^\alpha y(t)= f(t,y), \quad t\in [0,T] \]
with integral boundary conditions
\[ y(T)-y'(T)=\int_0^T g(s,y)\,ds, \quad y(T)+y'(T)=\int_0^T h(s,y)\,ds, \]
where \(1<\alpha \leq 2\) and \(^cD^\alpha y(t)\) is the Caputo fractional derivative, problems of such a kind arise in applications.
Under various assumptions on \(f(t,y)\), \(g(t,y)\) and \(h(t,y)\) they prove four theorems on the existence of the solution of the problem. In one of them, the uniqueness is also proved.
Reviewer: Stefan G. Samko (Faro)

Cited in 119 Documents

MSC:

26A33 Fractional derivatives and integrals

Keywords:

boundary value problem; Caputo fractional derivative; fractional integral; existence; uniqueness; fixed point; integral conditions
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\textit{M. Benchohra} et al., Appl. Anal. 87, No. 7, 851--863 (2008; Zbl 1198.26008)
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References:

[1] Diethelm K, Scientific Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering and Molecular Properties pp 217– (1999)
[2] DOI: 10.1016/0888-3270(91)90016-X
[3] DOI: 10.1016/S0006-3495(95)80157-8
[4] DOI: 10.1142/9789812817747
[5] Mainardi F, Fractals and Fractional Calculus in Continuum Mechanics pp 291– (1997)
[6] DOI: 10.1063/1.470346
[7] Oldham KB, The Fractional Calculus (1974)
[8] Kilbas AA, North-Holland Mathematics Studies 204 (2006)
[9] Miller KS, An Introduction to the Fractional Calculus and Differential Equations (1993)
[10] Samko SG, Theory and Applications (1993)
[11] Agarwal RP, Adv. Stud. Contemp. Math 16 pp 181– (2008)
[12] Benchohra M, Topol. Meth. Nonlinear Anal
[13] DOI: 10.1006/jmaa.1996.0456 · Zbl 0881.34005
[14] DOI: 10.1006/jmaa.2000.7194
[15] DOI: 10.1023/A:1019147432240 · Zbl 0926.65070
[16] El-Sayed AMA, J. Fract. Calc. 7 pp 89– (1995)
[17] DOI: 10.1007/BF02083817 · Zbl 0846.35001
[18] DOI: 10.1016/S0362-546X(97)00525-7 · Zbl 0934.34055
[19] Kaufmann ER, Electron. J. Qual. Theory Differ. Equ. 3 pp 11– (2007)
[20] DOI: 10.1007/s10625-005-0137-y · Zbl 1160.34301
[21] Momani SM, J. Fract. Calc. 24 pp 37– (2003)
[22] DOI: 10.1155/S0161171204302231 · Zbl 1069.34002
[23] DOI: 10.1023/A:1016556604320 · Zbl 1041.93022
[24] DOI: 10.1016/j.jmaa.2004.12.015 · Zbl 1088.34501
[25] DOI: 10.1016/j.na.2007.08.042 · Zbl 1161.34001
[26] Lakshmikantham V, Commun. Appl. Anal. 11 pp 395– (2007)
[27] DOI: 10.1016/j.aml.2007.09.006 · Zbl 1161.34031
[28] DOI: 10.1080/00036810601066350 · Zbl 1175.34080
[29] DOI: 10.1016/j.jmaa.2007.06.021 · Zbl 1209.34096
[30] Belarbi A, Commun. Appl. Anal. 11 pp 429– (2007)
[31] Benchohra M, Surv. Math. Appl. 3 pp 1– (2008)
[32] Belarbi A, Georgian Math. J. 13 pp 215– (2006)
[33] Benchohra M, Electron. J. Qual. Theory Differ. Equ. 15 pp 1– (2007) · Zbl 1182.34006
[34] Benchohra M, Rocky Mountain J. Math
[35] DOI: 10.1007/s00397-005-0043-5
[36] Podlubny I, Fract. Calculus Appl. Anal. 5 pp 367– (2002)
[37] Blayneh KW, Far East J. Dyn. Syst. 4 pp 125– (2002)
[38] DOI: 10.1016/0898-1221(85)90153-1 · Zbl 0565.92017
[39] Podlubny I, Fractional Differential Equation (1999)
[40] DOI: 10.1155/ADE/2006/90479 · Zbl 1134.39008
[41] Granas A, Fixed Point Theory (2003)
[42] DOI: 10.1002/mana.19981890103 · Zbl 0896.47042
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.