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Boundary value problems for fractional differential equation with causal operators


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A.A. Kilbas, H.M. Srivastava, J.J. Trujillo.(2006), Theory and Applications of Fractional Differential Equations, in: North-Holland Mathematics Studies, vol. 204, Elsevier Science BV, Amsterdam.KilbasA.A.SrivastavaH.M.TrujilloJ.J.2006Theory and Applications of Fractional Differential Equations in:North-Holland Mathematics Studies204Elsevier Science BVAmsterdamSearch in Google Scholar

K.S.Miller, B.Ross.(1993), An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York.MillerK.S.RossB.1993An Introduction to the Fractional Calculus and Differential EquationsJohn WileyNew YorkSearch in Google Scholar

C. Corduneanu.(2002), Functional Equations with Causal Operators, in: Stability and Control: Theory, Methods and Applications, vol. 16, Taylor Francis, London.CorduneanuC.2002Functional Equations with Causal Operators in:Stability and Control: Theory, Methods and Applications16Taylor FrancisLondonSearch in Google Scholar

V. Lakshmikantham, S. Leela, Z. Drici, F.A. McRae.(2010), Theory of Causal Differential Equations, Atlantis Stud. Math. Eng. Sci. vol. 5, World Scientific, Singapore.LakshmikanthamV.LeelaS.DriciZ.McRaeF.A.2010Theory of Causal Differential EquationsAtlantis Stud. Math. Eng. Sci5World ScientificSingapore10.2991/978-94-91216-25-1Search in Google Scholar

T. Jankowski.(2008), Boundary value problems with causal operators, Nonlinear Anal.68, 3625-3632. 10.1016/j.na.2007.04.005JankowskiT.2008Boundary value problems with causal operatorsNonlinear Anal683625363210.1016/j.na.2007.04.005Open DOISearch in Google Scholar

V. Lupulescu.(2008), Causal functional differential equations in Banach spaces, Nonlinear Anal.69, 4787-4795. 10.1016/j.na.2007.11.028LupulescuV.2008Causal functional differential equations in Banach spacesNonlinear Anal694787479510.1016/j.na.2007.11.028Open DOISearch in Google Scholar

V.Lakshmikantham, S.Leela, J.V.Devi.(2009), Theory of fractional dynamic systems, Cambridge Scientific Publish-ers,Cambridge.LakshmikanthamV.LeelaS.DeviJ.V.2009Theory of fractional dynamic systemsCambridge Scientific PublishersCambridgeSearch in Google Scholar

Z.Drici, F.A. McRae, J. Vasundhara Devi.(2005), Set differential equations with causal operators, Math. Probl. Eng. 2, 185-194. 10.1155/MPE.2005.185DriciZ.McRaeF.A.Vasundhara DeviJ.2005Set differential equations with causal operatorsMath. Probl. Eng218519410.1155/MPE.2005.185Open DOISearch in Google Scholar

Z.Drici, F.A. McRae, J. Vasundhara Devi.(2005), Differential equations with causal operators in a Banach space, Nonlinear Anal. 62, 301-313. 10.1016/j.na.2005.02.117DriciZ.McRaeF.A.Vasundhara DeviJ.2005Differential equations with causal operators in a Banach spaceNonlinear Anal6230131310.1016/j.na.2005.02.117Open DOISearch in Google Scholar

Z. Drici, F.A. McRae, J. Vasundhara Devi.(2006), Monotone iterative technique for periodic value problems with causal operators, Nonlinear Anal. 64,1271-1277. 10.1016/j.na.2005.06.033DriciZ.McRaeF.A. Vasundhara DeviJ. 2006Monotone iterative technique for periodic value problems with causal operatorsNonlinear Anal641271127710.1016/j.na.2005.06.033Open DOISearch in Google Scholar

M.El-shahed, J.J.Nieto.(2010), Nontrivial solutions for a nonlinear multi-point boundary value problem of fractional order, Comput. Math. Appl. 59, 3438-3443. 10.1016/j.camwa.2010.03.031El-shahedMNietoJ.J.2010Nontrivial solutions for a nonlinear multi-point boundary value problem of fractional orderComput. Math. Appl5934383443doi 10.1016/j.camwa.2010.03.031Open DOISearch in Google Scholar

R. Agarwal, Y. Zhou, J. Wang, X. Luo.(2011), Fractional functional differential equations with causal operators in Banach spaces, Math. Comput. Model. 54, 1440-1452. 10.1016/j.mcm.2011.04.016AgarwalR.ZhouY.WangJ.LuoX.2011Fractional functional differential equations with causal operators in Banach spacesMath. Comput. Model5414401452doi 10.1016/j.mcm.2011.04.016Open DOISearch in Google Scholar

Z.LV, J.Liang, T.Xiao.(2011), Solutions to the cauchy problem for differential equations in Banach spaces with fractional order, Comput. Math. Appl. 62, 1303-1311. 10.1016/j.camwa.2011.04.027LvZ.LiangJ.XiaoT.2011Solutions to the cauchy problem for differential equations in Banach spaces with fractional orderComput. Math. Appl6213031311doi 10.1016/j.camwa.2011.04.027Open DOISearch in Google Scholar

G.S. Ladde, V.Lakshmikantham, A.S. Vatsala.(1985), Monotone iterative techniques for nonlinear differential equations, Pitman, Boston.LaddeG.S.LakshmikanthamV.VatsalaA.S.1985Monotone iterative techniques for nonlinear differential equations, PitmanBostonSearch in Google Scholar

J.J. Nieto.(1997), An abstract monotone iterative technique, Nonlinear Anal., 28, 1923-1933. 10.1016/S0362-546X(97)89710-6NietoJ.J.1997An abstract monotone iterative techniqueNonlinear Anal.2819231933doi 10.1016/S0362-546X(97)89710-6Open DOISearch in Google Scholar

D.Jiang, J.J. Nieto, W. Zuo.(2004), On monotone method for first and second order periodic boundary value problems and periodic solutions of functional differential equations, J. Math.Anal. Appl. 289, 691-699. 10.1016/j.jmaa.2003.09.020JiangD.NietoJ.J.ZuoW.2004On monotone method for first and second order periodic boundary value problems and periodic solutions of functional differential equationsJ. Math.Anal. Appl289691699doi10.1016/j.jmaa.2003.09.020Open DOISearch in Google Scholar

Z.Drici, F.A.McRae, J.Vasundhara Devi.(2006), Monotone iterative technique for periodic boundary value problems with causal operators, Nonlinear Anal. 64, 1271-1277. 10.1016/j.na.2005.06.033DriciZ.McRaeF.A.Vasundhara DeviJ.2006Monotone iterative technique for periodic boundary value problems with causal operatorsNonlinear Anal6412711277doi 10.1016/j.na.2005.06.033Open DOISearch in Google Scholar

Z.B.Bai, H.S. L ü.(2005), Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl. 311, 495-505. 10.1016/j.jmaa.2005.02.052BaiZ.B.H.S.2005Positive solutions for boundary value problem of nonlinear fractional differential equationJ. Math. Anal. Appl311495505doi 10.1016/j.jmaa.2005.02.052Open DOISearch in Google Scholar

J.Wei.(2010), The constant variation formulae for singular fractional differential systems with delay, Comput. Math. Appl. 59, 1184-1190. 10.1016/j.camwa.2009.07.010WeiJ.2010The constant variation formulae for singular fractional differential systems with delay, ComputMath. Appl5911841190doi 10.1016/j.camwa.2009.07.010Open DOISearch in Google Scholar

C.F.Li, X.N.Luo and Y.Zhou.(2010), Existence of positive solutions of the boundary problem for nonlinear fractional differential equations, Comput. Math. Appl. 59, 1363-1375. 10.1016/j.camwa.2009.06.029LiC.F.LuoX.N.ZhouY.2010Existence of positive solutions of the boundary problem for nonlinear fractional differential equations, Comput.Math. Appl5913631375doi 10.1016/j.camwa.2009.06.029Open DOISearch in Google Scholar

Zhang Shuqin.(2011), Existence of a solution for the fractional differential equation with nonlinear boundary conditions, Comput. Math. Appl. 61, 1202-1208. 10.1016/j.camwa.2010.12.071Zhang Shuqin2011Existence of a solution for the fractional differential equation with nonlinear boundary conditions, ComputMath. Appl6112021208doi 10.1016/j.camwa.2010.12.071Open DOISearch in Google Scholar

Zhang Shuqin.(2010), Positive solutions to singular boundary value problem for nonlinear fractional differential equation, Comput. Math. Appl. 59, 1300-1309. 10.1016/j.camwa.2009.06.034Zhang Shuqin2010Positive solutions to singular boundary value problem for nonlinear fractional differential equation, ComputMath. Appl5913001309doi 10.1016/j.camwa.2009.06.034Open DOISearch in Google Scholar

J.Wang, Y.Zhou.(2011), A class of fractional evolution equations and optimal controls, Nonlinear Anal. 12, 262-272. 10.1016/j.nonrwa.2010.06.013WangJ.ZhouY.2011A class of fractional evolution equations and optimal controlsNonlinear Anal12262272doi 10.1016/j.nonrwa.2010.06.013Open DOISearch in Google Scholar

A.Cabada, G.Wang.(2012), Positive solutions of nonlinear fractional differential equations with integral boundary value conditions, J. Math. Anal. Appl. 389, 403-411. 10.1016/j.jmaa.2011.11.065CabadaA.WangG.2012Positive solutions of nonlinear fractional differential equations with integral boundary value conditionsJ. Math. Anal. Appl389403411doi 10.1016/j.jmaa.2011.11.065Open DOISearch in Google Scholar

Y.Li.(2010), Solving a nonlinear fractional differential equation using Chebyshev wavelets. Commun.Nonlinear. Sci.Numer. Simul. 15, 2284-2292. 10.1016/j.cnsns.2009.09.020LiY.2010Solving a nonlinear fractional differential equation using Chebyshev wavelets. Commun.NonlinearSci.Numer. Simul1522842292doi 10.1016/j.cnsns.2009.09.020Open DOISearch in Google Scholar

Y.Zhao, S.Sun, Z.Han.(2011), The existence of multiple positive solutions for boundary value problems of nonlinear fractional differential equations, Commun.Nonlinear. Sci.Numer. Simul. 16, 2086-2097. 10.1016/j.cnsns.2010.08.017ZhaoY.SunS.HanZ.2011The existence of multiple positive solutions for boundary value problems of nonlinear fractional differential equations, Commun.Nonlinear.Sci.Numer. Simul.1620862097doi 10.1016/j.cnsns.2010.08.017Open DOISearch in Google Scholar

J.Jiang, C. F. Li, and H. Chen.(2013), Existence of solutions for set differential equations involving causal operator with memory in Banach space, J. Appl.Math.Comput. 41, 183-196. 10.1007/s12190-012-0604-6JiangJ.LiC. F.ChenH.2013Existence of solutions for set differential equations involving causal operator with memory in Banach space,J. Appl.Math.Comput41183196doi 10.1007/s12190-012-0604-6Open DOISearch in Google Scholar

J. Jiang J, C.F. Li , D. Cao D, et al. (2015), Existence and uniqueness of solution for fractional differential equation with causal operators in Banach spaces, Mediterr. J. Math. 12(3), 751-769. 10.1007/s00009-014-0435-9Jiang JJ.LiC.F.Cao DD.et al2015Existence and uniqueness of solution for fractional differential equation with causal operators in Banach spaces, MediterrJ. Math123751769doi 10.1007/s00009-014-0435-9Open DOISearch in Google Scholar

J. Jiang, D. Cao D, H. Chen.(2015), The fixed point approach to the stability of fractional differential equations with causal operators, Qualitative Theory of Dynamical Systems, 1-16. 10.1007/s12346-015-0136-1JiangJ.CaoD.ChenD. H.2015The fixed point approach to the stability of fractional differential equations with causal operatorsQualitative Theory of Dynamical Systems116doi 10.1007/s12346-015-0136-1Open DOISearch in Google Scholar

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