We study the singular fractional-order boundary-value problem with a sign-changing nonlinear term $$, ${\mathcal{D}}^{{\mu}_{n-1}}x(1)={\sum}_{j=1}^{p-2}{a}_{j}{\mathcal{D}}^{{\mu}_{n-1}}x({\xi}_{j})$, where $$, $n\in \mathbb{N}$ and $n\ge 3$ with $$ and $$, $$ satisfying $$, ${\mathcal{D}}^{\alpha}$ is the standard Riemann-Liouville derivative, $f:[\mathrm{0,1}]\times {\mathbb{R}}^{n}\to \mathbb{R}$ is a sign-changing continuous function and may be unbounded from below with respect to ${x}_{i}$, and $p:(\mathrm{0,1})\to [0,\infty )$ is continuous. Some new results on the existence of nontrivial solutions for the above problem are obtained by computing the topological degree of a completely continuous field.

## References

K. Diethelm and A. D. Freed, “On the solutions of nonlinear fractional order differential equations used in the modelling of viscoplasticity,” in

*Scientific Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering and Molecular Properties*, F. Keil, W. Mackens, H. Voss, and J. Werthers, Eds., Springer, Heidelberg, Germany, 1999. K. Diethelm and A. D. Freed, “On the solutions of nonlinear fractional order differential equations used in the modelling of viscoplasticity,” in*Scientific Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering and Molecular Properties*, F. Keil, W. Mackens, H. Voss, and J. Werthers, Eds., Springer, Heidelberg, Germany, 1999. L. Gaul, P. Klein, and S. Kemple, “Damping description involving fractional operators,”

*Mechanical Systems and Signal Processing*, vol. 5, no. 2, pp. 81–88, 1991. L. Gaul, P. Klein, and S. Kemple, “Damping description involving fractional operators,”*Mechanical Systems and Signal Processing*, vol. 5, no. 2, pp. 81–88, 1991. W. G. Glockle and T. F. Nonnenmacher, “A fractional calculus approach to self-similar protein dynamics,”

*Biophysical Journal*, vol. 68, no. 1, pp. 46–53, 1995. W. G. Glockle and T. F. Nonnenmacher, “A fractional calculus approach to self-similar protein dynamics,”*Biophysical Journal*, vol. 68, no. 1, pp. 46–53, 1995. F. Mainardi, “Fractional calculus: some basic problems in continuum and statistical mechanics,” in

*Fractals and Fractional Calculus in Continuum Mechanics*, C. A. Carpinteri and F. Mainardi, Eds., Springer, Vienna, Austria, 1997. 0917.73004 MR1611587 F. Mainardi, “Fractional calculus: some basic problems in continuum and statistical mechanics,” in*Fractals and Fractional Calculus in Continuum Mechanics*, C. A. Carpinteri and F. Mainardi, Eds., Springer, Vienna, Austria, 1997. 0917.73004 MR1611587 R. Metzler, W. Schick, H. G. Kilian, and T. F. Nonnenmacher, “Relaxation in filled polymers: a fractional calculus approach,”

*The Journal of Chemical Physics*, vol. 103, no. 16, pp. 7180–7186, 1995. R. Metzler, W. Schick, H. G. Kilian, and T. F. Nonnenmacher, “Relaxation in filled polymers: a fractional calculus approach,”*The Journal of Chemical Physics*, vol. 103, no. 16, pp. 7180–7186, 1995. K. B. Oldham and J. Spanier,

*The Fractional Calculus*, Academic Press, New York, NY, USA, 1974. MR361633 0292.26011 K. B. Oldham and J. Spanier,*The Fractional Calculus*, Academic Press, New York, NY, USA, 1974. MR361633 0292.26011 X. Zhang, L. Liu, and Y. Wu, “Multiple positive solutions of a singular fractional differential equationwith negatively perturbed term,”

*Mathematical and Computer Modelling*, vol. 55, no. 3-4, pp. 1263–1274, 2012. MR2887513 1255.34010 10.1016/j.mcm.2011.10.006 X. Zhang, L. Liu, and Y. Wu, “Multiple positive solutions of a singular fractional differential equationwith negatively perturbed term,”*Mathematical and Computer Modelling*, vol. 55, no. 3-4, pp. 1263–1274, 2012. MR2887513 1255.34010 10.1016/j.mcm.2011.10.006 B. Ahmad and J. J. Nieto, “Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions,”

*Boundary Value Problems*, vol. 2011, article 36, 2011. MR2851530 10.1186/1687-2770-2011-36 1275.45004 B. Ahmad and J. J. Nieto, “Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions,”*Boundary Value Problems*, vol. 2011, article 36, 2011. MR2851530 10.1186/1687-2770-2011-36 1275.45004 C. S. Goodrich, “Existence of a positive solution to systems of differential equations of fractional order,”

*Computers & Mathematics with Applications*, vol. 62, no. 3, pp. 1251–1268, 2011. MR2824712 1253.34012 C. S. Goodrich, “Existence of a positive solution to systems of differential equations of fractional order,”*Computers & Mathematics with Applications*, vol. 62, no. 3, pp. 1251–1268, 2011. MR2824712 1253.34012 C. S. Goodrich, “Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions,”

*Computers & Mathematics with Applications*, vol. 61, no. 2, pp. 191–202, 2011. 1211.39002 MR2754129 C. S. Goodrich, “Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions,”*Computers & Mathematics with Applications*, vol. 61, no. 2, pp. 191–202, 2011. 1211.39002 MR2754129 C. S. Goodrich, “Positive solutions to boundary value problems with nonlinear boundary conditions,”

*Nonlinear Analysis*, vol. 75, no. 1, pp. 417–432, 2012. 1237.34153 MR2846811 C. S. Goodrich, “Positive solutions to boundary value problems with nonlinear boundary conditions,”*Nonlinear Analysis*, vol. 75, no. 1, pp. 417–432, 2012. 1237.34153 MR2846811 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. MR2856032 10.1016/j.aml.2011.09.058 1244.34009 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. MR2856032 10.1016/j.aml.2011.09.058 1244.34009 Y. Wang, L. Liu, and Y. Wu, “Positive solutions for a nonlocal fractional differential equation,”

*Nonlinear Analysis*, vol. 74, no. 11, pp. 3599–3605, 2011. 1220.34006 MR2803087 Y. Wang, L. Liu, and Y. Wu, “Positive solutions for a nonlocal fractional differential equation,”*Nonlinear Analysis*, vol. 74, no. 11, pp. 3599–3605, 2011. 1220.34006 MR2803087 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. MR2921344 1254.34016 10.1016/j.amc.2012.02.014 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. MR2921344 1254.34016 10.1016/j.amc.2012.02.014 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. MR2922960 1242.34015 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. MR2922960 1242.34015 J. Wu, X. Zhang, L. Liu, and Y. Wu, “Positive solutions of higher-order nonlinear fractional differential equations with changing-sign measure,”

*Advances in Difference Equations*, vol. 2012, article 71, 2012. MR2948729 1294.34028 J. Wu, X. Zhang, L. Liu, and Y. Wu, “Positive solutions of higher-order nonlinear fractional differential equations with changing-sign measure,”*Advances in Difference Equations*, vol. 2012, article 71, 2012. MR2948729 1294.34028 A. Castro, C. Maya, and R. Shivaji, “Nonlinear eigenvalue problems with semipositone,”

*Electronic Journal of Differential Equations*, no. 5, pp. 33–49, 2000. MR1799043 0959.35045 A. Castro, C. Maya, and R. Shivaji, “Nonlinear eigenvalue problems with semipositone,”*Electronic Journal of Differential Equations*, no. 5, pp. 33–49, 2000. MR1799043 0959.35045 V. Anuradha, D. D. Hai, and R. Shivaji, “Existence results for superlinear semipositone BVP's,”

*Proceedings of the American Mathematical Society*, vol. 124, no. 3, pp. 757–763, 1996. 0857.34032 MR1317029 10.1090/S0002-9939-96-03256-X V. Anuradha, D. D. Hai, and R. Shivaji, “Existence results for superlinear semipositone BVP's,”*Proceedings of the American Mathematical Society*, vol. 124, no. 3, pp. 757–763, 1996. 0857.34032 MR1317029 10.1090/S0002-9939-96-03256-X J. X. Sun, “Non-zero solutions to superlinear Hammerstein integral equations and applications,”

*Chinese Annals of Mathematics A*, vol. 7, no. 5, pp. 528–535, 1986. MR886319 0633.45006 J. X. Sun, “Non-zero solutions to superlinear Hammerstein integral equations and applications,”*Chinese Annals of Mathematics A*, vol. 7, no. 5, pp. 528–535, 1986. MR886319 0633.45006 F. Li and G. Han, “Existence of non-zero solutions to nonlinear Hammerstein integral equation,”

*Journal of Shanxi University (Natural Science Edition)*, vol. 26, pp. 283–286, 2003. F. Li and G. Han, “Existence of non-zero solutions to nonlinear Hammerstein integral equation,”*Journal of Shanxi University (Natural Science Edition)*, vol. 26, pp. 283–286, 2003. G. Han and Y. Wu, “Nontrivial solutions of singular two-point boundary value problems with sign-changing nonlinear terms,”

*Journal of Mathematical Analysis and Applications*, vol. 325, no. 2, pp. 1327–1338, 2007. 1111.34019 MR2270087 10.1016/j.jmaa.2006.02.076 G. Han and Y. Wu, “Nontrivial solutions of singular two-point boundary value problems with sign-changing nonlinear terms,”*Journal of Mathematical Analysis and Applications*, vol. 325, no. 2, pp. 1327–1338, 2007. 1111.34019 MR2270087 10.1016/j.jmaa.2006.02.076 J. Sun and G. Zhang, “Nontrivial solutions of singular superlinear Sturm-Liouville problems,”

*Journal of Mathematical Analysis and Applications*, vol. 313, no. 2, pp. 518–536, 2006. 1100.34019 MR2182515 10.1016/j.jmaa.2005.06.087 J. Sun and G. Zhang, “Nontrivial solutions of singular superlinear Sturm-Liouville problems,”*Journal of Mathematical Analysis and Applications*, vol. 313, no. 2, pp. 518–536, 2006. 1100.34019 MR2182515 10.1016/j.jmaa.2005.06.087 L. Liu, B. Liu, and Y. Wu, “Nontrivial solutions of $m$-point boundary value problems for singular second-order differential equations with a sign-changing nonlinear term,”

*Journal of Computational and Applied Mathematics*, vol. 224, no. 1, pp. 373–382, 2009. MR2474239 1163.34011 10.1016/j.cam.2008.05.007 L. Liu, B. Liu, and Y. Wu, “Nontrivial solutions of $m$-point boundary value problems for singular second-order differential equations with a sign-changing nonlinear term,”*Journal of Computational and Applied Mathematics*, vol. 224, no. 1, pp. 373–382, 2009. MR2474239 1163.34011 10.1016/j.cam.2008.05.007 K. Deimling,

*Nonlinear Functional Analysis*, Springer, Berlin, Germany, 1985. MR787404 0559.47040 K. Deimling,*Nonlinear Functional Analysis*, Springer, Berlin, Germany, 1985. MR787404 0559.47040 I. Podlubny,

*Fractional Differential Equations, Mathematics in Science and Engineering*, Academic Press, New York, NY, USA, 1999. MR1658022 0924.34008 I. Podlubny,*Fractional Differential Equations, Mathematics in Science and Engineering*, Academic Press, New York, NY, USA, 1999. MR1658022 0924.34008 A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo,

*Theory and Applications of Fractional Differential Equations*, Elsevier, Amsterdam, The Netherlands, 2006. MR2218073 1092.45003 A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo,*Theory and Applications of Fractional Differential Equations*, Elsevier, Amsterdam, The Netherlands, 2006. MR2218073 1092.45003 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*, vol. 36, pp. 1–12, 2010. MR2652066 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*, vol. 36, pp. 1–12, 2010. MR2652066 M. H. Protter and H. F. Weinberger,

*Maximum Principles in Differential Equations*, Prentice Hall, New York, NY, USA, 1967. MR219861 0549.35002 M. H. Protter and H. F. Weinberger,*Maximum Principles in Differential Equations*, Prentice Hall, New York, NY, USA, 1967. MR219861 0549.35002