×

Existence of solutions for fractional differential equations of order \(q \in (2,3]\) with anti-periodic boundary conditions. (English) Zbl 1216.34003

The authors consider a fractional order differential equation of order \(q\), which lies between 2 and 3, with anti-periodic boundary conditions. They establish existence and uniqueness of solutions to fractional order differential equations with anti-periodic boundary conditions and for the two-point boundary value problem by using contraction mapping principal. They also prove the existence of at least one positive solution for the two-point fractional boundary value problem by using Krasnosel’skii’s fixed point theorem. Finally, the existence and uniqueness of solutions of the anti periodic boundary value problem is explained by an example.

MSC:

34A08 Fractional ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ahmad, B., Nieto, J.J.: Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions. Bound. Value Probl. (2009). Art. ID 708576, pages 11. doi: 10.1155/2009/708576 · Zbl 1167.45003
[2] Ahmad, B., Nieto, J.J.: Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray-Schauder degree theory. Topol. Methods Nonlinear Anal. (2009, to appear) · Zbl 1245.34008
[3] Ahmad, B., Otero-Espinar, V.: Existence of solutions for fractional differential inclusions with anti-periodic boundary conditions. Bound. Value Probl. (2009) Art. ID 625347, pages 11. doi: 10.1155/2009/625347 · Zbl 1172.34004
[4] Ahmad, B., Nieto, J.J.: Existence and approximation of solutions for a class of nonlinear impulsive functional differential equations with anti-periodic boundary conditions. Nonlinear Anal. 69, 3291–3298 (2008) · Zbl 1158.34049
[5] Ahmad, B.: Existence of solutions for second order nonlinear impulsive boundary value problems. Electron. J. Differ. Equ. 68, 1–7 (2009) · Zbl 1176.34031
[6] Chang, Y.-K., Nieto, J.J.: Some new existence results for fractional differential inclusions with boundary conditions. Math. Comput. Model. 49, 605–609 (2009) · Zbl 1165.34313
[7] Chen, Y., Nieto, J.J., O’Regan, D.: Antiperiodic solutions for fully nonlinear first-order differential equations. Math. Comput. Model. 46, 1183–1190 (2007) · Zbl 1142.34313
[8] Daftardar-Gejji, V., Bhalekar, S.: Boundary value problems for multi-term fractional differential equations. J. Math. Anal. Appl. 345, 754–765 (2008) · Zbl 1151.26004
[9] Gafiychuk, V., Datsko, B., Meleshko, V.: Mathematical modeling of time fractional reaction-diffusion systems. J. Comput. Appl. Math. 220, 215–225 (2008) · Zbl 1152.45008
[10] Hilfer, R. (ed.): Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000) · Zbl 0998.26002
[11] Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006) · Zbl 1092.45003
[12] Lakshmikantham, V., Vatsala, A.S.: Basic theory of fractional differential equations. Nonlinear Anal. 69, 2677–2682 (2008) · Zbl 1161.34001
[13] Lakshmikantham, V.: Theory of fractional functional differential equations. Nonlinear Anal. 69, 3337–3343 (2008) · Zbl 1162.34344
[14] Lakshmikantham, V., Vatsala, A.S.: General uniqueness and monotone iterative technique for fractional differential equations. Appl. Math. Lett. 21, 828–834 (2008) · Zbl 1161.34031
[15] Lakshmikantham, V., Leela, S., Vasundhara Devi, J.: Theory of Fractional Dynamic Systems. Cambridge Academic, Cambridge (2009) · Zbl 1188.37002
[16] Lim, S.C., Li, M., Teo, L.P.: Langevin equation with two fractional orders. Phys. Lett. A 372, 6309–6320 (2008) · Zbl 1225.82049
[17] 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
[18] Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999) · Zbl 0924.34008
[19] Rida, S.Z., El-Sherbiny, H.M., Arafa, A.A.M.: On the solution of the fractional nonlinear Schrödinger equation. Phys. Lett. A 372, 553–558 (2008) · Zbl 1217.81068
[20] Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives, Theory and Applications. Gordon and Breach, Yverdon (1993) · Zbl 0818.26003
[21] Smart, D.R.: Fixed Point Theorems. Cambridge University Press, Cambridge (1980) · Zbl 0427.47036
[22] Vasundhara Devi, J., Lakshmikantham, V.: Nonsmooth analysis and fractional differential equations. Nonlinear Anal. 70, 4151–4157 (2009) · Zbl 1237.49022
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.