×

Existence of solutions of nonlinear and non-local fractional boundary value problems. (English) Zbl 1440.34007

The following fractional analog of the nonlinear equation is condidered \[ ^CD^\alpha u(t) + f(t,u(t))=0, \ \ \ 0 <t <1, \tag{1} \] with non-local boundary conditions \[ u'(0)=-\delta[u], \ u'(1)=\beta[u] - \frac{1}{b} u(\eta), \ b>0, \ \eta \in [0,1],\tag{2} \] where \(^CD^\alpha\) denotes the Caputo fractional derivative of order \(\alpha \in (1,2)\); \(\delta[.],\beta[.]\) are suitable linear and continuous functionals on \(C[0,1]\) and \(f: [0,1] \times \mathbb{R} \to [0,\infty)\) is a continuous function.
By Krasnoselskii’s fixed point theorem and fixed point index theory, the authors establish some new results on the existence of non-zero solutions of the boundary value problem (1)–(2) depending on the linear functionals and positive parameters.
Sufficient conditions for the existence of positive solutions in a particular case of the boundary conditions are obtained.
Some examples illustrating the obtained results are given, too.

MSC:

34A08 Fractional ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ahmad, B., Agarwal, R.: On nonlocal fractional boundary value problems. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 18, 535-544 (2011) · Zbl 1230.26003
[2] Ahmad, B., Nieto, J.J.: Anti-periodic fractional boundary value problems with nonlinear term depending on lower order derivative. Fract. Calc. Appl. Anal. 15, 451-462 (2012) · Zbl 1281.34005
[3] Ahmad, B., Nieto, J.J.: Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions. Comput. Math. Appl. 58, 1838-1843 (2009) · Zbl 1205.34003 · doi:10.1016/j.camwa.2009.07.091
[4] Ahmad, B., Nieto, J.J.: Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions. Bound. Value Probl. 36, 9 (2011) · Zbl 1275.45004
[5] Ahmad, B., Nieto, J.J., Alsaedi, A., El-Shahed, M.: A study of nonlinear Langevin equation involving two fractional orders in different intervals. Nonlinear Anal. Real World Appl. 13(2), 599-606 (2012) · Zbl 1238.34008 · doi:10.1016/j.nonrwa.2011.07.052
[6] Baleanu, D., Agarwal, R.P., Khan, H., Khan, R.A., Jafari, H.: On the existence of solution for fractional differential equations of order \[3 <\delta_1\leqslant 43\]<δ1⩽4. Adv. Differ. Equ. 362, 9 (2015) · Zbl 1422.34020
[7] Baleanu, D., Nazemi, S.Z., Rezapour, S.: A \[k\] k-dimensional system of fractional neutral functional differential equations with bounded delay. Abstr. Appl. Anal. (2014). https://doi.org/10.1155/2014/524761 · Zbl 1475.34051
[8] Benchohra, M., Cabada, A., Seba, D.: An existence result for nonlinear fractional differential equations on Banach spaces. Bound. Value Probl. (2009). https://doi.org/10.1155/2009/628916 · Zbl 1181.34007 · doi:10.1155/2009/628916
[9] Butzer, PL; Westphal, U.; Hilfer, R. (ed.), An introduction to fractional calculus (2000), New Jersey · Zbl 0987.26005
[10] Cabada, A., Cid, J.A., Infante, G.: New criteria for the existence of non-trivial fixed points in cones. Fixed Point Theory Appl. 125, 12 (2013) · Zbl 1428.47017
[11] Cabada, A., Hamdi, A.: Nonlinear fractional differential equations with integral boundary value conditions. Appl. Math. Comput. 228, 251-257 (2014) · Zbl 1364.34010
[12] Cabada, A., Infante, G., Tojo, F.A.F.: Nonzero solutions of perturbed Hammerstein integral equations with deviated arguments and applications. Topol. Methods Nonlinear Anal. 47, 265-287 (2016) · Zbl 1366.45005
[13] Cabada, A., Infante, G.: Positive solutions of a nonlocal Caputo fractional BVP. Dyn. Syst. Appl. 23(4), 715-722 (2014) · Zbl 1310.34005
[14] Cabada, A., Saavedra, L.: Existence of solutions for \[n^{th}\] nth-order nonlinear differential boundary value problems by means of fixed point theorems. Nonlinear Anal. Real World Appl. 12, 180-206 (2018) · Zbl 1414.34018 · doi:10.1016/j.nonrwa.2017.12.008
[15] Cabada, A., Wang, G.: Positive solutions of nonlinear fractional differential equations with integral boundary value conditions. J. Math. Anal. Appl. 389, 403-411 (2012) · Zbl 1232.34010 · doi:10.1016/j.jmaa.2011.11.065
[16] Fan, H., Ma, R.: Loss of positivity in a nonlinear second order ordinary differential equations. Nonlinear Anal. 71(1), 437-444 (2009) · Zbl 1172.34301 · doi:10.1016/j.na.2008.10.117
[17] Gorenflo, R.; Mainardi, F.; Carpinteri, A. (ed.); Mainardi, F. (ed.), Fractional calculus: integral and differential equations of fractional order, 223-276 (1997), New York · Zbl 1438.26010 · doi:10.1007/978-3-7091-2664-6_5
[18] Guidotti, P., Merino, S.: Gradual loss of positivity and hidden invariant cones in a scalar heat equation. Differ. Integral Equ. 13, 1551-1568 (2000) · Zbl 0983.35013
[19] Infante, G.: Nonlocal boundary value problems with two nonlinear boundary conditions. Commun. Appl. Anal. 12(3), 279-288 (2008) · Zbl 1198.34025
[20] Infante, G.: Positive solutions of some nonlinear BVPs involving singularities and integral BCs. Discrete Contin. Dyn. Syst. 1, 99-106 (2008) · Zbl 1160.34018 · doi:10.3934/dcdss.2008.1.99
[21] Infante, G., Webb, J. R. L.: Loss of positivity in a nonlinear scalar heat equation. NoDEA Nonlinear Differ. Equ. Appl. https://doi.org/10.1007/s00030-005-0039-y · Zbl 1112.34017 · doi:10.1007/s00030-005-0039-y
[22] Infante, G., Webb, J.R.L.: Nonlinear non-local boundary-value problems and perturbed Hammerstein integral equations. Proc. Edinb. Math. Soc. 49(3), 637-656 (2006) · Zbl 1115.34026 · doi:10.1017/S0013091505000532
[23] Infante, G., Webb, J.R.L.: Three point boundary value problems with solutions that change sign. J. Integral Equ. Appl. 15, 37-57 (2003) · Zbl 1055.34023 · doi:10.1216/jiea/1181074944
[24] Infante, G., Pietramala, P.: Perturbed Hammerstein integral inclusions with solutions that change sign. Comment. Math. Univ. Carolin. 50(4), 591-605 (2009) · Zbl 1212.45009
[25] Karatsompanis, I., Palamides, P.K.: Polynomial approximation to a non-local boundary value problem. Comput. Math. Appl. 60, 3058-3071 (2010) · Zbl 1207.65099 · doi:10.1016/j.camwa.2010.10.006
[26] Kilbas, A., Srivastava, H.M., Trujillo, J.: Theory and Applications of Fractional Differential Equations, North Holland Mathematics Studies 204. Elsevier, Amsterdam (2006) · Zbl 1092.45003
[27] Lloyd, N.G.: Degree Theory, Cambridge Tracts in Mathematics 73. Cambridge University Press, Cambridge (1978) · Zbl 0367.47001
[28] Mainardi, F.; Carpinteri, A. (ed.); Mainardi, F. (ed.), Fractional calculus: some basic problems in continuum and statistical mechanics, 291-348 (1997), New York · Zbl 0917.73004 · doi:10.1007/978-3-7091-2664-6_7
[29] Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993) · Zbl 0789.26002
[30] Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974) · Zbl 0292.26011
[31] Palamides, P.K., Infante, G., Pietramala, P.: Nontrivial solutions of a nonlinear heat flow problem via Sperner lemma. Appl. Math. Lett. 22, 1444-1450 (2009) · Zbl 1173.34311 · doi:10.1016/j.aml.2009.03.014
[32] Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Academic Press, San Diego (1999) · Zbl 0924.34008
[33] Samko, S., Kilbas, A., Maricev, O.: Fractional Integrals and Derivatives. Gordon & Breach, New York (1993) · Zbl 0818.26003
[34] Webb, J.R.L.: Multiple positive solutions of some nonlinear heat flow problems. Discrete Contin. Dyn. Syst. (2005). https://doi.org/10.3934/proc.2005.2005.895 · Zbl 1161.34007 · doi:10.3934/proc.2005.2005.895
[35] Webb, J.R.L.: Existence of positive solutions for a thermostat model. Nonlinear Anal. Real World Appl. 13, 923-938 (2012) · Zbl 1238.34033 · doi:10.1016/j.nonrwa.2011.08.027
[36] Webb, J.R.L.: Optimal constants in a nonlocal boundary value problem. Nonlinear Anal. 63, 672-685 (2005) · Zbl 1153.34320 · doi:10.1016/j.na.2005.02.055
[37] Zhang, S.: Positive solutions for boundary-value problems of nonlinear fractional differential equations. Electron. J. Differ. Equ. 2006, 1-12 (2006) · Zbl 1096.34016 · doi:10.1155/ADE/2006/90479
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.