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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
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##### 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
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