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Jiang, Min; Zhong, Shouming

Existence and multiple positive solutions for boundary value problem of fractional differential equation with \(p\)-Laplacian operator. (English) Zbl 1474.34035

Abstr. Appl. Anal. 2014, Article ID 512426, 18 p. (2014).
Summary: This paper investigates the existence, multiplicity, nonexistence, and uniqueness of positive solutions to a kind of two-point boundary value problem for nonlinear fractional differential equations with \(p\)-Laplacian operator. By using fixed point techniques combining with partially ordered structure of Banach space, we establish some criteria for existence and uniqueness of positive solution of fractional differential equations with \(p\)-Laplacian operator in terms of different value of parameter. In particular, the dependence of positive solution on the parameter was obtained. Finally, several illustrative examples are given to support the obtained new results. The study of illustrative examples shows that the obtained results are applicable.

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MSC:

34A08 Fractional ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
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References:

[1] Podlubny, I., Fractional Differential Equations, 198 (1999), San Diego, Calif, USA: Academic Press, San Diego, Calif, USA · Zbl 0918.34010
[2] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations. Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204 (2006), Amsterdam, The Netherlands: Elsevier Science B.V., Amsterdam, The Netherlands · Zbl 1092.45003
[3] Rossikhin, Y. A.; Shitikova, M. V., Application of fractional derivatives to the analysis of damped vibrations of viscoelastic single mass systems, Acta Mechanica, 120, 1-4, 109-125 (1997) · Zbl 0901.73030 · doi:10.1007/BF01174319
[4] Sabatier, J.; Agrawal, O.; Machado, J. T., Advances in Fractional Calculus. Advances in Fractional Calculus, Theoretical developments and applications in physics and engineering (2007), Berlin, Germany: Springer, Berlin, Germany · Zbl 1116.00014 · doi:10.1007/978-1-4020-6042-7
[5] Engheta, N., On fractional calculus and fractional multipoles in electromagnetism, IEEE Transactions on Antennas and Propagation, 44, 4, 554-566 (1996) · Zbl 0944.78506 · doi:10.1109/8.489308
[6] Hilfer, R., Applications of Fractional Calculus in Physics (2000), Singapore: World Scientific, Singapore · Zbl 0998.26002 · doi:10.1142/9789812817747
[7] Bagley, R. L.; Calico, R. A., Fractional order state equations for the control of viscoelastically damped structures, Journal of Guidance, Control, and Dynamics, 14, 2, 304-311 (1991)
[8] Zhou, W.-X.; Chu, Y.-D., Existence of solutions for fractional differential equations with multi-point boundary conditions, Communications in Nonlinear Science and Numerical Simulation, 17, 3, 1142-1148 (2012) · Zbl 1245.35153 · doi:10.1016/j.cnsns.2011.07.019
[9] Ahmad, B.; Wang, G., A study of an impulsive four-point nonlocal boundary value problem of nonlinear fractional differential equations, Computers & Mathematics with Applications, 62, 3, 1341-1349 (2011) · Zbl 1228.34012 · doi:10.1016/j.camwa.2011.04.033
[10] Ahmad, B.; Sivasundaram, S., On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order, Applied Mathematics and Computation, 217, 2, 480-487 (2010) · Zbl 1207.45014 · doi:10.1016/j.amc.2010.05.080
[11] Nyamoradi, N., Positive solutions for multi-point boundary value problems for nonlinear fractional differential equations, Journal of Contemporary Mathematical Analysis, 48, 4, 145-157 (2013) · Zbl 1287.34005
[12] Ahmad, B.; Nieto, J. J., Existence of solutions for nonlocal boundary value problems of higher-order nonlinear fractional differential equations, Abstract and Applied Analysis, 2009 (2009) · Zbl 1186.34009 · doi:10.1155/2009/494720
[13] Liu, X.; Jia, M.; Xiang, X., On the solvability of a fractional differential equation model involving the \(p\)-Laplacian operator, Computers & Mathematics with Applications, 64, 10, 3267-3275 (2012) · Zbl 1268.34020 · doi:10.1016/j.camwa.2012.03.001
[14] Wang, J.; Xiang, H.; Liu, Z., Existence of concave positive solutions for boundary value problem of nonlinear fractional differential equation with \(p\)-Laplacian operator, International Journal of Mathematics and Mathematical Sciences, 2010 (2010) · Zbl 1198.34008 · doi:10.1155/2010/495138
[15] Wang, J.; Xiang, H., Upper and lower solutions method for a class of singular fractional boundary value problems with \(p\)-Laplacian operator, Abstract and Applied Analysis, 2010 (2010) · Zbl 1209.34005 · doi:10.1155/2010/971824
[16] Lu, H.; Han, Z., Existence of positive solutions for boundary-value problem of fractional differential equation with \(p\)-laplacian operator, The American Jouranl of Engineering and Technology Research, 11, 3757-3764 (2011)
[17] Chai, G., Positive solutions for boundary value problem of fractional differential equation with \(p\)-Laplacian operator, Boundary Value Problems, 2012, article 18 (2012) · Zbl 1275.34008 · doi:10.1186/1687-2770-2012-18
[18] Chen, T.; Liu, W.; Hu, Z., A boundary value problem for fractional differential equation with \(p\)-Laplacian operator at resonance, Nonlinear Analysis: Theory, Methods & Applications, 75, 6, 3210-3217 (2012) · Zbl 1253.34010 · doi:10.1016/j.na.2011.12.020
[19] Chen, T.; Liu, W., An anti-periodic boundary value problem for the fractional differential equation with a \(p\)-Laplacian operator, Applied Mathematics Letters of Rapid Publication, 25, 11, 1671-1675 (2012) · Zbl 1248.35219 · doi:10.1016/j.aml.2012.01.035
[20] Li, S. J.; Zhang, X. G.; Wu, Y. H.; Caccetta, L., Extremal solutions for \(p\)-Laplacian differential systems via iterative computation, Applied Mathematics and Computation, 26, 1151-1158 (2013) · Zbl 1316.34006
[21] Liu, Z.; Lu, L., A class of BVPs for nonlinear fractional differential equations with \(p\)-Laplacian operator, Electronic Journal of Qualitative Theory of Differential Equations, 70, 1-16 (2012) · Zbl 1340.34299
[22] Liu, Z.; Lu, L.; Szántó, I., Existence of solutions for fractional impulsive differential equations with \(p\)-Laplacian operator, Acta Mathematica Hungarica, 141, 3, 203-219 (2013) · Zbl 1313.34242 · doi:10.1007/s10474-013-0305-0
[23] Bai, Z., Eigenvalue intervals for a class of fractional boundary value problem, Computers & Mathematics with Applications, 64, 10, 3253-3257 (2012) · Zbl 1272.34006 · doi:10.1016/j.camwa.2012.01.004
[24] Zhang, X.; Wang, L.; Sun, Q., Existence of positive solutions for a class of nonlinear fractional differential equations with integral boundary conditions and a parameter, Applied Mathematics and Computation, 226, 708-718 (2014) · Zbl 1354.34049 · doi:10.1016/j.amc.2013.10.089
[25] Guo, D. J.; Lakshmikantham, V., Nonlinear Problems in Abstract Cones, 5 (1988), New York, NY, USA: Academic Press, New York, NY, USA · Zbl 0661.47045
[26] Krasnosel’skii, M. A., Topological Methods in the Theory of Nonlinear Integral Equations. Topological Methods in the Theory of Nonlinear Integral Equations, Translated by A. H. Armstrong (1964), Pergamon, Turkey: Elmsford, Pergamon, Turkey · Zbl 0111.30303
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