×

On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order. (English) Zbl 1207.45014

The authors consider the four-point nonlocal boundary value problem in a Banach space \(X\), i.e.
\[ ^cD^qx(t)=f(t,x(t),(\phi x)(t),(\psi x)(t)),\;\;0<t<1, \;\;1<q<2, \]
\[ x'(0)+ax(\eta_1)=0, \;\;bx'(1)+x(\eta_2)=0,\;\;0<\eta_1\leq \eta_2<1, \]
where \(^cD\) is the Caputo’s fractional derivative, \(f:[0,1]\times X\times X\times X\times X\to X\) is continuous, \(\phi\), \(\psi\) are Volterra integral operators and \(a,c\in (0,1)\). By using fixed point arguments they prove an existence and uniqueness result of solutions for the problem above.

MSC:

45N05 Abstract integral equations, integral equations in abstract spaces
45J05 Integro-ordinary differential equations
45G10 Other nonlinear integral equations
26A33 Fractional derivatives and integrals
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] N’Guerekata, G.M., A Cauchy problem for some fractional abstract differential equation with non local conditions, Nonlinear anal., 70, 1873-1876, (2009) · Zbl 1166.34320
[2] Ahmad, B.; Sivasundaram, S., Existence and uniqueness results for nonlinear boundary value problems of fractional differential equations with separated boundary conditions, Commun. appl. anal., 13, 121-228, (2009) · Zbl 1180.34003
[3] Lakshmikantham, V.; Leela, S.; Vasundhara Devi, J., Theory of fractional dynamic systems, (2009), Cambridge Academic Publishers Cambridge · Zbl 1188.37002
[4] Ahmad, B.; Sivasundaram, S., Existence of solutions for impulsive integral boundary value problems of fractional order, Nonlinear anal. hybrid syst., 4, 134-141, (2010) · Zbl 1187.34038
[5] Ahmad, B., Existence of solutions for irregular boundary value problems of nonlinear fractional differential equations, Appl. math. lett., 23, 390-394, (2010) · Zbl 1198.34007
[6] Ahmad, B.; Nieto, J.J., Existence of solutions for nonlocal boundary value problems of higher order nonlinear fractional differential equations, Abstr. appl. anal., 494720, 9, (2009) · Zbl 1186.34009
[7] Ahmad, B.; Nieto, J.J., Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions, Comput. math. appl., 58, 1838-1843, (2009) · Zbl 1205.34003
[8] Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J., Theory and applications of fractional differential equations, North-holland mathematical studies, vol. 204, (2006), Elsevier Science B.V. Amsterdam · Zbl 1092.45003
[9] Miller, K.S.; Ross, B., An introduction to the fractional calculus and differential equations, (1993), John Wiley New York · Zbl 0789.26002
[10] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional integrals and derivatives. Theory and applications. Edited and with a foreword by S.M. Nikol’skii. Translated from the 1987 Russian original. Revised by the authors. Gordon and Breach Science Publishers, Yverdon, 1993.
[11] Podlubny, I., Fractional differential equations, Mathematical science and engineering, vol. 198, (1999), Academic Press San Diego · Zbl 0918.34010
[12] Zhang, S., Positive solutions for boundary-value problems of nonlinear fractional differential equations, Electron. J. differ. equat., 36, 12, (2006)
[13] Lakshmikantham, V.; Vatsala, A.S., Basic theory of fractional differential equations, Nonlinear anal., 69, 8, 2677-2682, (2008) · Zbl 1161.34001
[14] Lakshmikantham, V.; Vatsala, A.S., Theory of fractional differential inequalities and applications, Commun. appl. anal., 11, 3-4, 395-402, (2007) · Zbl 1159.34006
[15] Lakshmikantham, V.; Vatsala, A.S., General uniqueness and monotone iterative technique for fractional differential equations, Appl. math. lett., 21, 8, 828-834, (2008) · Zbl 1161.34031
[16] Agarwal, R.P.; Benchora, M.; Hamani, S., Boundary value problems for fractional differential equations, Georgian math. J., 16, 3, 401-411, (2009) · Zbl 1179.26011
[17] Bai, Z.; Liu, H., Positive solutions for boundary value problem of nonlinear fractional differential equation, J. math. anal. appl., 311, 2, 495-505, (2005) · Zbl 1079.34048
[18] Fix, G.J.; Roop, J.P., Least squares finite element solution of a fractional order two-point boundary value problem, Comput. math. appl., 48, 1017-1033, (2004) · Zbl 1069.65094
[19] Jafari, H.; Daftardar-Gejji, V., Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method, Appl. math. comput., 180, 700-706, (2006) · Zbl 1102.65136
[20] Agrawal, O.P., Formulation of Euler-larange equations for fractional variational problems, J. math. anal. appl., 272, 368-379, (2002) · Zbl 1070.49013
[21] Ried, W.T., An integro differential boundary value problem, Amer. J. math., 60, 2, 257-292, (1938) · JFM 64.0439.02
[22] Appel, J.; S Kalitvin, A.; Zabrejko, P.P., Boundary value problems for integro differential equation of barbashin type, J. integral equat. appl., 6, 1, 1-30, (1994) · Zbl 0808.45012
[23] Momani, S.; Noor, M.A., Numerical methods for fourth-order fractional integro-differential equations, Appl. math. comput., 182, 754-760, (2006) · Zbl 1107.65120
[24] Momani, S.; Qaralleh, A., An efficient method for solving systems of fractional integro-differential equations, Comput. math. appl., 52, 459-470, (2006) · Zbl 1137.65072
[25] Rawashdeh, E.A., Numerical solution of fractional integro-differential equations by collocation method, Appl. math. comput., 176, 1-6, (2006) · Zbl 1100.65126
[26] Arikoglu, A.; Ozkol, I., Solution of fractional integro-differential equations by using fractional differential transform method, Chaos, solitons fractals, 40, 521-529, (2009) · Zbl 1197.45001
[27] Wu, J.; Liu, Y., Existence and uniqueness of solutions for the fractional integro-diferential equations in banch spaces, Electron. J. differ. equat., 2009, 129, 1-8, (2009)
[28] Agarwal, R.P.; O’Regan, D.; Wong, P.J.Y., Positive solutions of differential, difference, and integral equations, (1999), Kluwer Academic Publishers Boston · Zbl 0923.39002
[29] Guo, D.; Lakshmikantham, V., Nonlinear problems in abstract cones, (1988), Academic Press San Diego · Zbl 0661.47045
[30] Karaca, I.Y., Fourth-order four-point boundary value problem on time scales, Applied mathematics letters, 21, 1057-1063, (2008) · Zbl 1170.34309
[31] Bai, C.Z.; Yang, D.D.; Zhu, H.B., Existence of solutions for fourth-order differential equation with four-point boundary conditions, Appl. math. lett., 20, 4, 1131-1136, (2007) · Zbl 1140.34308
[32] Ma, D.X.; Yang, X.Z., Upper and lower solution method for fourth-order four-point boundary value problems, Journal of computational and applied mathematics, 223, 543-551, (2009) · Zbl 1181.65106
[33] Zhong, Y.L.; Chen, S.H.; Wang, C.P., Existence results for a fourth-order ordinary differential equation with a four-point boundary condition, Appl. math. lett., 21, 465-470, (2008) · Zbl 1141.34305
[34] Rachunkova, I., Multiplicity results for four-point boundary value problems, Nonlinear anal., 18, 495-505, (1992) · Zbl 0756.34026
[35] Gupta, C.P., A Dirichlet type multi-point boundary value problem for second-order ordinary differential equations, Nonlinear anal., 26, 925-931, (1996) · Zbl 0847.34018
[36] Graef, J.R.; Qian, C.; Yang, B., A three-point boundary value problem for nonlinear fourth-order differential equations, J. math. anal. appl., 287, 1, 217-233, (2003) · Zbl 1054.34038
[37] Hamani, S.; Benchora, M.; Graef, John R., Existence results for boundary value problems with nonlinear fractional inclusions and integral conditions, Electronic journal of differential equations, 2010, 20, 1-16, (2010) · Zbl 1185.26010
[38] Agarwal, R.P.; O’Regan, D.; Yan, B., Positive solutions for singular three-point boundary-value problems, Electronic journal of differential equations, 2008, 116, 1-20, (2008) · Zbl 1179.34019
[39] Geng, F.Z.; Cui, M.G., Solving singular nonlinear second-order periodic boundary value problems in the reproducing kernel space, Appl. math. comput., 192, 389-398, (2007) · Zbl 1193.34017
[40] Jackson, L.K., Uniqueness of solutions of boundary value problems for ordinary differential equations, SIAM J. appl. math., 24, 535-538, (1973) · Zbl 0237.34030
[41] Jackson, L.K., Existence and uniqueness of solutions of boundary value problems for third-order differential equations, J. diff. eqs., 13, 432-437, (1973) · Zbl 0256.34018
[42] Henderson, J., Existence of solutions of right focal point boundary value problems for ordinary differential equations, Nonlinear anal., 5, 9, 989-1002, (1981) · Zbl 0468.34010
[43] Clark, S.; Henderson, J., Uniqueness implies existence and uniqueness criterion for non local boundary value problems for third-order differential equations, Proc. amer. math. soc., 134, 3363-3372, (2006) · Zbl 1120.34010
[44] Ehme, J.; Hankerson, D., Existence of solutions for right focal boundary value problems, Nonlinear anal., 18, 2, 191-197, (1992) · Zbl 0755.34016
[45] Henderson, J.; McGwier, R.W., Uniqueness, existence, and optimality for fourth-order Lipschitz equations, Journal of differential eqs., 67, 3, 414-440, (1987) · Zbl 0642.34005
[46] Peterson, A.C., Focal green’s functions for fourth-order differential equations, Journal of mathematical analysis and applications, 75, 2, 602-610, (1980) · Zbl 0439.34026
[47] Peterson, A.C., Existence-uniqueness for focal-point boundary value problems, SIAM J. math. anal., 12, 173-185, (1982) · Zbl 0473.34009
[48] Podlubny, I., Geometric and physical interpretation of fractional integration and frac- tional differentiation, Dedicated to the 60th anniversary of prof. francesco mainardi. fract. calc. appl. anal., 5, 4, 367-386, (2002) · Zbl 1042.26003
[49] Smart, D.R., Fixed point theorems, (1980), Cambridge University Press Cambridge · Zbl 0427.47036
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.