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

Summary: We are concerned with the existence and uniqueness of positive solutions for the following singular nonlinear $\left(n-1,1\right)$ conjugate-type fractional differential equation with a nonlocal term

$\left\{\begin{array}{c}{D}_{0+}^{\alpha }x\left(t\right)+f\left(t,x\left(t\right)\right)=0,\phantom{\rule{4pt}{0ex}}0

where $\alpha \ge 2$, ${D}_{0+}^{\alpha }$ is the standard Riemann-Liouville derivative, $A$ is a function of bounded variation and ${\int }_{0}^{1}u\left(s\right)dA\left(s\right)$ denotes the Riemann-Stieltjes integral of $u$ with respect to $A$, $dA$ can be a signed measure.

##### MSC:
 34A08 Fractional differential equations 47N20 Applications of operator theory to differential and integral equations 34B18 Positive solutions of nonlinear boundary value problems for ODE
##### 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 · doi:10.1112/S0024610706023179 [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 · doi:10.1016/j.amc.2005.11.077 [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 · doi:10.1016/j.jmaa.2005.09.064 [4] Wei, Z.: A class of fourth order singular boundary value problems, Appl. math. Comput. 153, 865-884 (2004) · Zbl 1057.34006 · doi:10.1016/S0096-3003(03)00683-0 [5] Wei, Z.: Positive solutions of some singular m-point boundary value problems at nonresonance, Appl. math. Comput. 171, 433-449 (2005) [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 · doi:10.1016/S0893-9659(03)80044-7 [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 · doi:10.1016/j.na.2007.03.006 [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., No. 16, 1-8 (2010) · Zbl 1202.34045 · doi:emis:journals/EJQTDE/2010/201016.pdf [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 · doi:10.1016/j.amc.2007.08.080 [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 · doi:10.1016/j.na.2008.12.010 [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 · doi:10.1016/j.aml.2010.04.035 [12] Goodrich, C. S.: Existence of a positive solution to systems of differential equations of fractional order, Comput. math. Appl. 62, 1251-1268 (2011) [13] Wang, Y.: Positive solutions for a nonlocal fractional differential equation, Nonlinear anal. 74, 3599-3605 (2011) · Zbl 1220.34006 · doi:10.1016/j.na.2011.02.043 [14] Podlubny, I.: Fractional differential equations, Mathematics in science and engineering (1999) [15] Kilbas, A.; Srivastava, H.; Nieto, J.: Theory and applicational differential equations, (2006) [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., No. 36 (2010) · Zbl 1210.34008 · doi:emis:journals/EJQTDE/2010/201036.html [17] Webb, J.: Nonlocal conjugate type boundary value problems of higher order, Nonlinear anal. 71, 1933-1940 (2009) · Zbl 1181.34025 · doi:10.1016/j.na.2009.01.033