## Sequential fractional differential equations with three-point boundary conditions.(English)Zbl 1268.34006

Summary: We study a nonlinear three-point boundary value problem of sequential fractional differential equations of order $$\alpha +1$$ with $$1<\alpha \leq 2$$. The expression for Green’s function of the associated problem involving the classical gamma function and the generalized incomplete gamma function is obtained. Some existence results are obtained by means of Banach’s contraction mapping principle and Krasnoselskii’s fixed point theorem. An illustrative example is also presented. Existence results for a three-point third-order nonlocal boundary value problem of nonlinear ordinary differential equations follow as a special case of our results.

### MSC:

 34A08 Fractional ordinary differential equations 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
Full Text:

### References:

 [1] Samko, S.G.; Kilbas, A.A.; Marichev, O.I., Fractional integrals and derivatives. theory and applications, (1993), Gordon and Breach Yverdon · Zbl 0818.26003 [2] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego · Zbl 0918.34010 [3] Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J., () [4] () [5] Ahmad, B.; Nieto, J.J., Existence of solutions for nonlocal boundary value problems of higher order nonlinear fractional differential equations, Abstr. appl. anal., (2009), 9. Art. ID 494720 · Zbl 1186.34009 [6] 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 [7] 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 [8] Nieto, J.J., Maximum principles for fractional differential equations derived from Mittag-Leffler functions, Appl. math. lett., 23, 1248-1251, (2010) · Zbl 1202.34019 [9] Baleanu, D.; Mustafa, O.G.; Agarwal, R.P., An existence result for a superlinear fractional differential equation, Appl. math. lett., 23, 1129-1132, (2010) · Zbl 1200.34004 [10] Zhang, S., Positive solutions to singular boundary value problem for nonlinear fractional differential equation, Comput. math. appl., 59, 1300-1309, (2010) · Zbl 1189.34050 [11] Bai, Z., On positive solutions of a nonlocal fractional boundary value problem, Nonlinear anal., 72, 916-924, (2010) · Zbl 1187.34026 [12] Sztonyk, P., Regularity of harmonic functions for anisotropic fractional Laplacians, Math. nachr., 283, 2, 89-311, (2010) · Zbl 1194.47044 [13] Wang, G.; Ahmad, B.; Zhang, L., Impulsive anti-periodic boundary value problem for nonlinear differential equations of fractional order, Nonlinear anal., 74, 792-804, (2011) · Zbl 1214.34009 [14] Bhalekar, S.; Daftardar-Gejji, V.; Baleanu, D., Fractional Bloch equation with delay, Comput. math. appl., 61, 1355-1365, (2011) · Zbl 1217.34123 [15] Ahmad, B.; Agarwal, Ravi P., On nonlocal fractional boundary value problems, Dyn. contin. discrete impuls. syst. ser. A math. anal., 18, 535-544, (2011) · Zbl 1230.26003 [16] Ahmad, B.; Nieto, J.J.; Pimentel, J., Some boundary value problems of fractional differential equations and inclusions, Comput. math. appl., 62, 1238-1250, (2011) · Zbl 1228.34011 [17] Tomovski, Z.; Hilfer, R.; Srivastava, H.M., Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler type functions, Integral transforms spec. funct., 21, 797-814, (2010) · Zbl 1213.26011 [18] Konjik, S.; Oparnica, L.; Zorica, D., Waves in viscoelastic media described by a linear fractional model, Integral transforms spec. funct., 22, 283-291, (2011) · Zbl 1225.26009 [19] Keyantuo, V.; Lizama, C., A characterization of periodic solutions for time-fractional differential equations in UMD spaces and applications, Math. nachr., 284, 494-506, (2011) · Zbl 1221.34012 [20] M.D. Riva, S. Yakubovich, On a Riemann-Liouville fractional analog of the Laplace operator with positive energy, Integral Transforms Spec. Funct. (in press) doi:10.1080/10652469.2011.576832. [21] Miller, K.S.; Ross, B., An introduction to the fractional calculus and fractional differential equations, (1993), Wiley and Sons New York · Zbl 0789.26002 [22] Wei, Z.; Dong, W., Periodic boundary value problems for riemann – liouville sequential fractional differential equations, Electron. J. qual. theory differ. equ., 87, 1-13, (2011) · Zbl 1340.34038 [23] Wei, Z.; Li, Q.; Che, J., Initial value problems for fractional differential equations involving riemann – liouville sequential fractional derivative, J. math. anal. appl., 367, 260-272, (2010) · Zbl 1191.34008 [24] Klimek, M., Sequential fractional differential equations with Hadamard derivative, Commun. nonlinear sci. numer. simul., 16, 4689-4697, (2011) · Zbl 1242.34009 [25] Baleanu, D.; Mustafa, O.G.; Agarwal, R.P., On L$${}^p$$-solutions for a class of sequential fractional differential equations, Appl. math. comput., 218, 2074-2081, (2011) · Zbl 1235.34008 [26] Bai, C., Impulsive periodic boundary value problems for fractional differential equation involving riemann – liouville sequential fractional derivative, J. math. anal. appl., 384, 211-231, (2011) · Zbl 1234.34005 [27] Bressan, A., Hyperbolic systems of conservation laws. the one-dimensional Cauchy problem, (2000), Oxford University Press · Zbl 0997.35002 [28] Akyildiz, F.T.; Bellout, H.; Vajravelu, K.; Van Gorder, R.A., Existence results for third order nonlinear boundary value problems arising in nano boundary layer fluid flows over stretching surfaces, Nonlinear anal. RWA, 12, 2919-2930, (2011) · Zbl 1231.35155 [29] Ahmad, B.; Nieto, J.J.; Alsaedi, A.; El-Shahed, M., A study of nonlinear Langevin equation involving two fractional orders in different intervals, Nonlinear anal. RWA, 13, 599-606, (2012) · Zbl 1238.34008 [30] Polyanin, A.D.; Zaitsev, V.F., Handbook of nonlinear partial differential equations, (2004), Chapman & Hall, CRC Boca Raton · Zbl 1024.35001 [31] I. Thompson, A note on the real zeros of the incomplete gamma function, Integral Transforms Spec. Funct. (in press) doi:10.1080/10652469.2011.597391. [32] Sebastian, N., A generalized gamma model associated with a Bessel function, Integral transforms spec. funct., 22, 631-645, (2011) · Zbl 1230.33007 [33] Krasnoselskii, M.A., Two remarks on the method of successive approximations, Uspekhi mat. nauk, 10, 123-127, (1955)
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.