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Uniqueness and asymptotic behavior of positive solutions for a fractional-order integral boundary value problem. (English) Zbl 1253.35201
Summary: We study a model arising from porous media, electromagnetic, and signal processing of wireless communication system \(-\mathcal D^\alpha_{\mathbf{t}} x(t) = f(t, x(t), x'(t), x''(t), \dots, x^{(n-2)}(t))\), \(0 < t < 1\), \(x(0) = x'(0) = \cdots = x^{(n-2)}(0) = 0\), \(x^{(n-2)}(1) = \int^1_0 x^{(n-2)}(s)dA(s)\) where \(n - 1 < \alpha \leq n\), \(n \in \mathbb N\), and \(n \geq 2, \mathcal D^\alpha_{\mathbf{t}}\) is the standard Riemann-Liouville derivative, \(\int^1_0 x(s)dA(s)\) is the linear functional given by the Riemann-Stieltjes integrals, \(A\) is a function of bounded variation, and \(dA\) can be a changing-sign measure. The existence, uniqueness, and asymptotic behavior of positive solutions to the singular nonlocal integral boundary value problem for fractional differential equation are obtained. Our analysis relies on Schauder’s fixed-point theorem and upper and lower solution method.

35R11 Fractional partial differential equations
35B40 Asymptotic behavior of solutions to PDEs
35B09 Positive solutions to PDEs
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
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[1] T. T. Hartley, C. F. Lorenzo, and H. K. Qammer, “Chaos in a fractional order Chua’s system,” IEEE Transactions on Circuits and Systems I, vol. 42, no. 8, pp. 485-490, 1995. · doi:10.1109/81.404062
[2] S. M. Caputo, “Free modes splitting and alterations of electrochemically polarizable media,” Rendiconti Lincei, vol. 4, no. 2, pp. 89-98, 1993. · doi:10.1007/BF03001421
[3] T. J. Anastasio, “The fractional-order dynamics of brainstem vestibulo-oculomotor neurons,” Biological Cybernetics, vol. 72, no. 1, pp. 69-79, 1994. · doi:10.1007/BF00206239
[4] R. A. Khan, “The generalized method of quasilinearization and nonlinear boundary value problems with integral boundary conditions,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 19, pp. 1-15, 2003. · Zbl 1055.34033 · emis:journals/EJQTDE/2003/200319.html · eudml:124169
[5] J. M. Gallardo, “Second-order differential operators with integral boundary conditions and generation of analytic semigroups,” The Rocky Mountain Journal of Mathematics, vol. 30, no. 4, pp. 1265-1291, 2000. · Zbl 0984.34014 · doi:10.1216/rmjm/1021477351 · math.la.asu.edu
[6] G. L. Karakostas and P. Ch. Tsamatos, “Multiple positive solutions of some Fredholm integral equations arisen from nonlocal boundary-value problems,” Electronic Journal of Differential Equations, vol. 30, pp. 1-17, 2002. · Zbl 0998.45004 · emis:journals/EJDE/Volumes/2002/30/abstr.html · eudml:122167
[7] B. Ahmad, S. K. Ntouyas, and A. Alsaedi, “New existence results for nonlinear fractional differential equations with three-point integral boundary conditions,” Advances in Difference Equations, vol. 2011, Article ID 107384, 11 pages, 2011. · Zbl 1204.34005 · doi:10.1155/2011/107384 · eudml:227693
[8] M. Feng, X. Liu, and H. Feng, “The existence of positive solution to a nonlinear fractional differential equation with integral boundary conditions,” Advances in Difference Equations, vol. 2011, Article ID 546038, 14 pages, 2011. · Zbl 1219.34009 · doi:10.1155/2011/546038 · eudml:229496
[9] C. Corduneanu, Integral Equations and Applications, Cambridge University Press, Cambridge, UK, 1991. · Zbl 0714.45002 · doi:10.1017/CBO9780511569395
[10] R. P. Agarwal and D. O’Regan, Infinite Interval Problems for Differential, Difference and Integral Equations, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001. · Zbl 0988.34002 · doi:10.1007/978-94-010-0718-4
[11] R. Ma and N. Castaneda, “Existence of solutions of nonlinear m-point boundary-value problems,” Journal of Mathematical Analysis and Applications, vol. 256, no. 2, pp. 556-567, 2001. · Zbl 0988.34009 · doi:10.1006/jmaa.2000.7320
[12] P. W. Eloe and B. Ahmad, “Positive solutions of a nonlinear nth order boundary value problem with nonlocal conditions,” Applied Mathematics Letters, vol. 18, no. 5, pp. 521-527, 2005. · Zbl 1074.34022 · doi:10.1016/j.aml.2004.05.009
[13] X. Hao, L. Liu, Y. Wu, and Q. Sun, “Positive solutions for nonlinear nth-order singular eigenvalue problem with nonlocal conditions,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 73, no. 6, pp. 1653-1662, 2010. · Zbl 1202.34038 · doi:10.1016/j.na.2010.04.074
[14] C. Yuan, “Multiple positive solutions for (n-1,1)-type semipositone conjugate boundary value problems of nonlinear fractional differential equations,” Electronic Journal of Qualitative Theory of Differential Equations, no. 36, pp. 1-12, 2010. · Zbl 1210.34008 · emis:journals/EJQTDE/2010/201036.html
[15] S. Zhang, “Positive solutions to singular boundary value problem for nonlinear fractional differential equation,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1300-1309, 2010. · Zbl 1189.34050 · doi:10.1016/j.camwa.2009.06.034
[16] X. Zhang and Y. Han, “Existence and uniqueness of positive solutions for higher order nonlocal fractional differential equations,” Applied Mathematics Letters, vol. 25, no. 3, pp. 555-560, 2012. · Zbl 1244.34009 · doi:10.1016/j.aml.2011.09.058
[17] X. Zhang, L. Liu, and Y. Wu, “The eigenvalue problem for a singular higher order fractional differential equation involving fractional derivatives,” Applied Mathematics and Computation, vol. 218, no. 17, pp. 8526-8536, 2012. · Zbl 1254.34016 · doi:10.1016/j.amc.2012.02.014
[18] X. Zhang, L. Liu, B. Wiwatanapataphee, and Y. Wu, “Positive solutions of eigenvalue problems for a class of fractional differential equations with derivatives,” Abstract and Applied Analysis, vol. 2012, Article ID 512127, 16 pages, 2012. · Zbl 1242.34015 · doi:10.1155/2012/512127
[19] X. Zhang, L. Liu, and Y. Wu, “Multiple positive solutions of a singular fractional differential equation with negatively perturbed term,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 1263-1274, 2012. · Zbl 1255.34010 · doi:10.1016/j.mcm.2011.10.006
[20] X. Zhang, L. Liu, and Y. Wu, “Existence results for multiple positive solutions of nonlinear higher order perturbed fractional differential equations with derivatives,” Applied Mathematics and Computation. In press. · Zbl 1296.34046 · doi:10.1016/j.amc.2012.07.046
[21] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, Ny, USA, 1993. · Zbl 0789.26002
[22] I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1999. · Zbl 0924.34008
[23] J. R. L. Webb and G. Infante, “Non-local boundary value problems of arbitrary order,” Journal of the London Mathematical Society, vol. 79, no. 1, pp. 238-258, 2009. · Zbl 1165.34010 · doi:10.1112/jlms/jdn066
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