# zbMATH — the first resource for mathematics

Non-negative solutions of systems of ODEs with coupled boundary conditions. (English) Zbl 1280.34026
In this very well written and interesting paper, the authors study a coupled system of perturbed Hammerstein integral equations. Due to the generality of the system, the authors use it to study the existence of positive solutions of nonlocal boundary value problems for ordinary differential equations. In particular, the boundary conditions can be quite general, being as they are realized as Stieltjes integrals with positive measures. The technique used to study the problem is the classical fixed point index. The paper concludes with an illustrative example, which greatly assists the reader in appreciating the application and strength of the results stated in the paper. Any researcher interested in boundary value problems will find this paper to be of significant interest to him or her.

##### MSC:
 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces 47H11 Degree theory for nonlinear operators 47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
##### Keywords:
fixed point index; cone; system; nonnegative solution
Full Text:
##### References:
 [1] Agarwal, R.P.; O’Regan, D.; Wong, P.J.Y., Constant-sign solutions of a system of integral equations with integrable singularities, J integr eq appl, 19, 117-142, (2007) · Zbl 1136.45007 [2] Agarwal, R.P.; O’Regan, D.; Wong, P.J.Y., Constant-sign solutions for singular systems of Fredholm integral equations, Math methods appl sci, 33, 1783-1793, (2010) · Zbl 1205.45006 [3] Agarwal, R.P.; O’Regan, D.; Wong, P.J.Y., Constant-sign solutions for systems of singular integral equations of Hammerstein type, Math comput model, 50, 999-1025, (2009) · Zbl 1193.45023 [4] Ahmad, B.; Graef, J.R., Coupled systems of nonlinear fractional differential equations with nonlocal boundary conditions, Panamer math J, 19, 29-39, (2009) · Zbl 1180.34002 [5] 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 [6] Amann, H., Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM rev, 18, 620-709, (1976) · Zbl 0345.47044 [7] Amann H. Parabolic evolution equations with nonlinear boundary conditions, Part 1 (Berkeley, CA, 1983). p. 17-27. Proc. Sympos. Pure Math., vol. 45, Part 1. Providence, RI: Amer. Math. Soc.; 1986. [8] Asif, N.A.; Khan, R.A., Positive solutions to singular system with four-point coupled boundary conditions, J math anal appl, 386, 848-861, (2012) · Zbl 1232.34034 [9] Cui, Y.; Sun, J., On existence of positive solutions of coupled integral boundary value problems for a nonlinear singular superlinear differential system, Electron J qual theory differ eq no, 41, 1-13, (2012) [10] Franco D, Infante G, O’Regan D. Nontrivial solutions in abstract cones for Hammerstein integral systems. Dyn Contin Discrete Impuls Syst Ser A: Math Anal 14 (2007) 837-50. · Zbl 1139.45004 [11] Goodrich, C.S., Nonlocal systems of BVPs with asymptotically superlinear boundary conditions, Comment math univ carolin, 53, 79-97, (2012) · Zbl 1249.34054 [12] Goodrich CS. Nonlocal systems of BVPs with asymptotically sublinear boundary conditions. Appl Anal Discrete Math, in press. http://dx.doi.org/10.2298/AADM120329010G. [13] Guo, D.; Lakshmikantham, V., Nonlinear problems in abstract cones, (1988), Academic Press · Zbl 0661.47045 [14] Henderson, J.; Ntouyas, S.K.; Purnaras, I.K., Positive solutions for systems of second order four-point nonlinear boundary value problems, Commun appl anal, 12, 29-40, (2008) · Zbl 1166.34006 [15] Infante, G.; Pietramala, P., Eigenvalues and non-negative solutions of a system with nonlocal bcs, Nonlinear stud, 16, 187-196, (2009) · Zbl 1184.34027 [16] 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 [17] Infante, G.; Webb, J.R.L., Nonlinear nonlocal boundary value problems and perturbed Hammerstein integral equations, Proc edinb math soc, 49, 637-656, (2006) · Zbl 1115.34026 [18] Kang, P.; Wei, Z., Three positive solutions of singular nonlocal boundary value problems for systems of nonlinear second-order ordinary differential equations, Nonlinear anal, 70, 444-451, (2009) · Zbl 1169.34014 [19] Karakostas, G.L.; Tsamatos, P.Ch., Existence of multiple positive solutions for a nonlocal boundary value problem, Topol methods nonlinear anal, 19, 109-121, (2002) · Zbl 1071.34023 [20] Lazer, A.C.; McKenna, P.J., Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM rev, 32, 537-578, (1990) · Zbl 0725.73057 [21] Lan, K.Q., Multiple positive solutions of Hammerstein integral equations with singularities, Diff eq dynam syst, 8, 175-195, (2000) · Zbl 0977.45001 [22] Lan, K.Q., Multiple positive solutions of semilinear differential equations with singularities, J lond math soc, 63, 690-704, (2001) · Zbl 1032.34019 [23] Lan, K.Q.; Lin, W., Multiple positive solutions of systems of Hammerstein integral equations with applications to fractional differential equations, J lond math soc, 83, 449-469, (2011) · Zbl 1215.45004 [24] Leung, A., A semilinear reaction – diffusion prey – predator system with nonlinear coupled boundary conditions: equilibrium and stability, Indiana univ math J, 31, 223-241, (1982) · Zbl 0519.92021 [25] Lü, H.; Yu, H.; Liu, Y., Positive solutions for singular boundary value problems of a coupled system of differential equations, J math anal appl, 302, 14-29, (2005) · Zbl 1076.34022 [26] Matas A. Mathematical models of suspension bridges. PhD thesis, University of West Bohemia in Pilsen, Pilsen; 2004. [27] Mehmeti, F.A.; Nicaise, S., Nonlinear interaction problems, Nonlinear anal, 20, 27-61, (1993) · Zbl 0817.35035 [28] Precup R. Componentwise compression – expansion conditions for systems of nonlinear operator equations and applications. Math Model Eng Biol Med. p. 284-93. AIP Conf. Proc., vol. 1124. Melville, NY: Amer. Inst. Phys.; 2009. [29] Precup, R., Existence, localization and multiplicity results for positive radial solutions of semilinear elliptic systems, J math anal appl, 352, 48-56, (2009) · Zbl 1178.35162 [30] Sun, Y., Necessary and sufficient condition for the existence of positive solutions of a coupled system for elastic beam equations, J math anal appl, 357, 77-88, (2009) · Zbl 1182.34033 [31] Webb, J.R.L., Solutions of nonlinear equations in cones and positive linear operators, J lond math soc, 82, 420-436, (2010) · Zbl 1209.47017 [32] 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 [33] Webb, J.R.L.; Infante, G., Nonlocal boundary value problems of arbitrary order, J lond math soc, 79, 238-258, (2009) · Zbl 1165.34010 [34] Webb, J.R.L.; Infante, G.; Franco, D., Positive solutions of nonlinear fourth-order boundary-value problems with local and non-local boundary conditions, Proc R soc edin sect A, 138, 427-446, (2008) · Zbl 1167.34004 [35] Yang, Z., Positive solutions to a system of second-order nonlocal boundary value problems, Nonlinear anal, 62, 1251-1265, (2005) · Zbl 1089.34022 [36] Yang, Z.; Kong, L., Positive solutions of a system of second order boundary value problems involving first order derivatives via $$R_+^n$$-monotone matrices, Nonlinear anal, 75, 2037-2046, (2012) · Zbl 1242.34041 [37] Yang Z, Zhang Z. Positive solutions for a system of nonlinear singular Hammerstein integral equations via nonnegative matrices and applications. Positivity, in press. http://dx.doi.org/10.1007/s11117-011-0146-4. [38] Yuan, C.; Jiang, D.; O’Regan, D.; Agarwal, R.P., Multiple positive solutions to systems of nonlinear semipositone fractional differential equations with coupled boundary conditions, Electron J qual theory differ eq no, 13, 1-13, (2012) · Zbl 1340.34041
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.