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Positive solutions to fractional differential equations involving Stieltjes integral conditions. (English) Zbl 1334.34058

Summary: In this paper, we investigate nonlocal boundary value problems for fractional differential equations with dependence on the first-order derivatives and deviating arguments. Sufficient conditions which guarantee the existence of at least three positive solutions are new and obtained by using the Avery-Peterson theorem. We discuss problems (1) and (2) when argument \(\beta\) can change the character on \([0,1]\), so in some subinterval \(I\) of \([0,1]\), it can be delayed in \(I\) and advanced in \([0, 1] \setminus I\). Moreover in our discussion problem (2) depends on delayed argument \(\alpha\). Examples illustrate the results.

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34A08 Fractional ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
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[1] Avery, R. I.; Peterson, A. C., Three positive fixed points of nonlinear operators on ordered Banach spaces, Comput. Math. Appl., 42, 313-322 (2001) · Zbl 1005.47051
[2] Bai, Z.; Lü, H., Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl., 311, 495-505 (2005) · Zbl 1079.34048
[3] Bai, Z., On positive solutions of a nonlocal fractional boundary value problem, Nonlinear Anal., 72, 916-924 (2010) · Zbl 1187.34026
[4] Feng, M.; Liu, X.; Feng, H., The existence of positive solutions to a nonlinear fractional differential equation with integral boundary conditions, Adv. Diff. Equ., 14 (2011), Art. ID 546038 · Zbl 1219.34009
[5] Goodrich, Ch., Existence of a positive solution to a class of fractional difference equations, Appl. Math. Lett., 23, 1050-1055 (2010) · Zbl 1204.34007
[6] Graef, J. R.; Webb, J. R.L., Third order boundary value problems with nonlocal boundary conditions, Nonlinear Anal., 71, 1542-1551 (2009) · Zbl 1189.34034
[7] Graef, J. R.; Kong, L.; Kong, Q.; Wang, M., Uniqueness of positive solutions of fractional boundary value problems with non-homogeneous integral boundary conditions, Fract. Calc. Appl. Anal., 15, 509-528 (2012) · Zbl 1279.34007
[9] Infante, G.; Pietramala, P., Existence and multiplicity of non-negative solutions for systems of perturbed Hammerstein integral equations, Nonlinear Anal., 71, 1301-1310 (2009) · Zbl 1169.45001
[10] Infante, G.; Pietramala, P.; Zima, M., Positive solutions for a class of nonlocal impulsive BVPs via fixed point index, Topol. Methods Nonlinear Anal., 36, 263-284 (2010) · Zbl 1237.34032
[11] Jankowski, T., Fractional differential equations with deviating arguments, Dyn. Syst. Appl., 17, 677-684 (2008) · Zbl 1202.34017
[12] Jankowski, T., Positive solutions for second order impulsive differential equations involving Stieltjes integral conditions, Nonlinear Anal., 74, 3775-3785 (2011) · Zbl 1221.34071
[13] Jankowski, T., Existence of positive solutions to third order differential equations with advanced arguments and nonlocal boundary conditions, Nonlinear Anal., 75, 913-923 (2012) · Zbl 1235.34179
[14] Jankowski, T., Positive solutions to second-order differential equations with dependence on the first-order derivative and nonlocal boundary conditions, Bound. Value Probl., 2013, 8, 21 (2013) · Zbl 1342.34036
[15] Jankowski, T., Nonnegative solutions to nonlocal boundary value problems for systems of second-order differential equations dependent on the first-order derivatives, Nonlinear Anal., 87, 83-101 (2013) · Zbl 1286.34096
[16] Jankowski, T., Fractional equations of Volterra type involving a Riemann-Liouville derivative, Appl. Math. Lett., 26, 344-350 (2013) · Zbl 1259.45007
[17] Jankowski, T., Initial value problems for neutral fractional differential equations involving a Riemann Liouville derivative, Appl. Math. Comput., 219, 7772-7776 (2013) · Zbl 1293.34101
[18] Jankowski, T., Existence results to delay fractional differential equations with nonlinear boundary conditions, Appl. Math. Comput., 219, 9155-9164 (2013) · Zbl 1294.34074
[19] Jankowski, T., Boundary problems for fractional differential equations, Appl. Math. Lett., 28, 14-19 (2014) · Zbl 1311.34014
[20] Jankowski, T., Positive solutions to Sturm-Liouville problems with non-local boundary conditions, Proc. R. Soc. Edinburgh Sect. A, 144, 119-138 (2014) · Zbl 1303.34049
[21] Jankowski, T., Fractional problems with advanced arguments, Appl. Math. Comput., 230, 371-381 (2014) · Zbl 1410.34186
[22] Jankowski, T., Monotone iterative method for first-order differential equations at resonance, Appl. Math. Comput., 233, 20-28 (2014) · Zbl 1334.34049
[23] Jiang, J.; Liu, L.; Wu, Y., Positive solutions for nonlinear fractional differential equations with boundary conditions involving Riemann-Stieltjes integrals, Abstr. Appl. Anal., 21 (2012), Art. ID 708192 · Zbl 1253.34015
[24] Kilbas, A. A.; Bonilla, B.; Trukhillo, Kh., Fractional integrals and derivatives, and differential equations of fractional order in weighted spaces of continuous functions (Russian), Dokl. Nats. Akad. Nauk Belarusi, 44, 18-22 (2000) · Zbl 1177.26011
[25] Kilbas, A. A.; Srivastava, H. R.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations. Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204 (2006), Elsevier: Elsevier Amsterdam · Zbl 1092.45003
[26] Lakshmikantham, V.; Leela, S.; Vasundhara, J., Theory of Fractional Dynamic Systems (2009), Cambridge Academic Publishers: Cambridge Academic Publishers Cambridge · Zbl 1188.37002
[27] Lin, L.; Liu, X.; Fang, H., Method of upper and lower solutions for fractional differential equations, Elektron. J. Diff. Equ., 100, 1-13 (2012) · Zbl 1260.34026
[29] Liu, Z.; Sun, J.; Szántó, I., Monotone iterative technique for Riemann-Liouville fractional integro-differential equations with advanced arguments, Results Math., 63, 1277-1287 (2013) · Zbl 1276.26017
[30] McRae, F. A., Monotone iterative technique and existence results for fractional differential equations, Nonlinear Anal., 71, 6093-6096 (2009) · Zbl 1260.34014
[31] Ntouyas, S. N.; Wang, G.; Zhang, L., Positive solutions of arbitrary order nonlinear fractional differential equations with advanced arguments, Opuscula Math., 31, 433-442 (2011) · Zbl 1235.34209
[32] Ramirez, J. D.; Vatsala, A. S., Monotone iterative technique for fractional differential equations with periodic boundary conditions, Opuscula Math., 29, 289-304 (2009) · Zbl 1197.26007
[33] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives: Theory and Applications (1993), Gordon and Breach Science Publishers: Gordon and Breach Science Publishers Switzerland · Zbl 0818.26003
[34] Wang, G., Monotone iterative technique for boundary value problems of a nonlinear fractional differential equations with deviating arguments, J. Comput. Appl. Math., 236, 2425-2430 (2012) · Zbl 1238.65077
[35] Wang, G.; Liu, S.; Baleanu, D.; Zhang, L., Existence results for nonlinear fractional differential equations involving different Riemann-Liouville fractional derivatives, Adv. Diff. Equ., 2013, 280 (2013) · Zbl 1375.34015
[36] Wanhg, Y.; Liu, L.; Wu, Y., Positive solutions for a nonlocal fractional differential equation, Nonlinear Anal., 74, 3599-3605 (2011) · Zbl 1220.34006
[37] Webb, J. R.L.; Infante, G., Positive solutions of nonlocal boundary value problems: a unified approach, J. Lond. Math. Soc., 74, 673-693 (2006) · Zbl 1115.34028
[38] Webb, J. R.L.; Infante, G., Positive solutions of nonlocal boundary value problems involving integral conditions, NoDEA Nonlinear Diff. Equ. Appl., 15, 45-67 (2008) · Zbl 1148.34021
[39] Webb, J. R.L.; Infante, G., Non-local boundary value problems of arbitrary order, J. Lond. Math. Soc., 79, 238-259 (2009) · Zbl 1165.34010
[40] Wei, Z.; Li, G.; Che, J., Initial value problems for fractional differential equations involving Riemann-Liouville sequential fractional derivative, J. Math. Anal. Appl., 367, 260-272 (2010) · Zbl 1191.34008
[41] Yuan, Ch., Multiple positive solutions for \((n - 1, 1)\)-type semipositone conjugate boundary value problems of nonlinear fractional differential equations, Elect. J. Qual. Theory Diff. Equ., 36, 12 (2010) · Zbl 1210.34008
[42] Zhang, S., Monotone iterative method for initial value problem involving Riemann-Liouville fractional derivatives, Nonlinear Anal., 71, 2087-2093 (2009) · Zbl 1172.26307
[43] Zhang, S., Positive solutions to singular boundary value problem for nonlinear fractional differential equations, Comput. Math. Appl., 59, 1300-1309 (2010) · Zbl 1189.34050
[44] Zhang, S.; Su, X., The existence of a solution for a fractional differential equation with nonlinear boundary conditions considered using upper and lower solutions in reversed order, Comput. Math. Appl., 62, 1269-1274 (2011) · Zbl 1228.34022
[45] Zhang, X.; Han, Y., Existence and uniqueness of positive solutions for higher order nonlocal fractional differential equations, Appl. Math. Lett., 25, 555-560 (2012) · Zbl 1244.34009
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