Existence and uniqueness of positive solutions for higher order nonlocal fractional differential equations. (English) Zbl 1244.34009

Zhang, Xinguang; Han, Yuefeng

Appl. Math. Lett. 25, No. 3, 555-560 (2012).

Summary: We are concerned with the existence and uniqueness of positive solutions for the following singular nonlinear \((n-1,1)\) conjugate-type fractional differential equation with a nonlocal term \[ \begin{cases} D^\alpha_{0+}x(t)+f(t,x(t))=0,\;0<t<1,\;n-1<\alpha\leq n,\\ x^{(k)}(0) =0,\;0\leq k\leq n-2,\;x(1)=\int^1_0x(s)dA(s),\end{cases} \] where \(\alpha \geq 2\), \(D^\alpha_{0+}\) is the standard Riemann-Liouville derivative, \(A\) is a function of bounded variation and \(\int^1_0u(s)dA(s)\) denotes the Riemann-Stieltjes integral of \(u\) with respect to \(A\), \(dA\) can be a signed measure.

MSC:

34A08	Fractional ordinary differential equations
47N20	Applications of operator theory to differential and integral equations
34B18	Positive solutions to nonlinear boundary value problems for ordinary differential equations

Keywords:

monotone iterative technique; fractional differential equation; existence and uniqueness; positive solution

References:

[1]	Webb, J.; Infante, G., Positive solutions of nonlocal boundary value problems: a unified approach, J. Lond. Math. Soc., 74, 673-693 (2006) · Zbl 1115.34028
[2]	Wei, Z., A necessary and sufficient condition for the existence of positive solutions of singular super-linear \(m\)-point boundary value problems, Appl. Math. Comput., 179, 67-78 (2006) · Zbl 1166.34305
[3]	Wei, Z.; Pang, C., The method of lower and upper solutions for fourth order singular \(m\)-point boundary value problems, J. Math. Anal. Appl., 322, 675-692 (2006) · Zbl 1112.34010
[4]	Wei, Z., A class of fourth order singular boundary value problems, Appl. Math. Comput., 153, 865-884 (2004) · Zbl 1057.34006
[5]	Wei, Z., Positive solutions of some singular \(m\)-point boundary value problems at nonresonance, Appl. Math. Comput., 171, 433-449 (2005) · Zbl 1085.34017
[6]	Hao, Z.; Liu, L.; Debnath, L., A necessary and sufficient condition for the existence of positive solutions of fourth-order singular boundary value problems, Appl. Math. Lett., 16, 279-285 (2003) · Zbl 1055.34047
[7]	Zhang, X.; Liu, L., A necessary and sufficient condition for positive solutions for fourth-order multi-point boundary value problems with \(p\)-Laplacian, Nonlinear Anal., 68, 3127-3137 (2008) · Zbl 1143.34016
[8]	Mao, J.; Zhao, Z.; Xu, N., On existence and uniqueness of positive solutions for integral boundary value problems, Electron. J. Qual. Theory Differ. Equ., 16, 1-8 (2010) · Zbl 1202.34045
[9]	Du, X.; Zhao, Z., Existence and uniqueness of positive solutions to a class of singular \(m\)-point boundary value problems, Appl. Math. Comput., 198, 487-493 (2008) · Zbl 1158.34315
[10]	Webb, J.; Zima, M., Multiple positive solutions of resonant and non-resonant nonlocal boundary value problems, Nonlinear Anal., 71, 1369-1378 (2009) · Zbl 1179.34023
[11]	Goodrich, C. S., Existence of a positive solution to a class of fractional differential equations, Appl. Math. Lett., 23, 1050-1055 (2010) · Zbl 1204.34007
[12]	Goodrich, C. S., Existence of a positive solution to systems of differential equations of fractional order, Comput. Math. Appl., 62, 1251-1268 (2011) · Zbl 1253.34012
[13]	Wang, Y., Positive solutions for a nonlocal fractional differential equation, Nonlinear Anal., 74, 3599-3605 (2011) · Zbl 1220.34006
[14]	Podlubny, I., (Fractional Differential Equations. Fractional Differential Equations, Mathematics in Science and Engineering (1999), Academic Press: Academic Press New York, London, Toronto) · Zbl 0924.34008
[15]	Kilbas, A.; Srivastava, H.; Nieto, J., Theory and Applicational Differential Equations (2006), Elsevier: Elsevier Amsterdam · Zbl 1092.45003
[16]	Yuan, C., Multiple positive solutions for \((n - 1, 1)\)-type semipositone conjugate boundary value problems of nonlinear fractional differential equations, Electron. J. Qual. Theory Differ. Equ., 36 (2010), 12 p. · Zbl 1210.34008
[17]	Webb, J., Nonlocal conjugate type boundary value problems of higher order, Nonlinear Anal., 71, 1933-1940 (2009) · Zbl 1181.34025