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Eigenvalue problems for systems of nonlinear boundary value problems on time scales. (English) Zbl 1154.39020

The authors investigate two-dimensional systems of dynamic equation
\[ u^{\Delta\Delta}(t)+\lambda a(t)f(v(\sigma(t))=0, \quad v^{\Delta\Delta}(t)+\lambda b(t)g(u(\sigma(t))=0 \]
on a time scale \({\mathbb T}\subseteq[0,1]\), which are equipped with the boundary conditions
\[ u(0)=0=u(\sigma^2(1)), \quad v(0)=0=v(\sigma^2(1)). \]
Using a Guo-Krasnosel’skii fixed point theorem in cones, the authors determine values for the real parameter \(\lambda\) (the eigenvalue) such that there exist solutions \(u,v\) with positive components on the time scale interval \((0,\sigma^1(1))_{\mathbb T}\). The corresponding results are formulated in two theorems.

MSC:

39A12 Discrete version of topics in analysis
39A10 Additive difference equations
34L05 General spectral theory of ordinary differential operators
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References:

[1] Agarwal RP, O’Regan D: Triple solutions to boundary value problems on time scales.Applied Mathematics Letters 2000,13(4):7-11. 10.1016/S0893-9659(99)00200-1 · Zbl 0958.34021 · doi:10.1016/S0893-9659(99)00200-1
[2] Anderson DR: Eigenvalue intervals for a second-order mixed-conditions problem on time scale.International Journal of Nonlinear Differential Equations 2002, 7: 97-104.
[3] Anderson DR: Eigenvalue intervals for a two-point boundary value problem on a measure chain.Journal of Computational and Applied Mathematics 2002,141(1-2):57-64. 10.1016/S0377-0427(01)00435-6 · Zbl 1134.34310 · doi:10.1016/S0377-0427(01)00435-6
[4] Bohner M, Peterson A: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston, Mass, USA; 2001:x+358. · Zbl 0978.39001 · doi:10.1007/978-1-4612-0201-1
[5] Chyan CJ, Davis JM, Henderson J, Yin WKC: Eigenvalue comparisons for differential equations on a measure chain.Electronic Journal of Differential Equations 1998, (35):7 pp.. · Zbl 0912.34067
[6] Erbe LH, Peterson A: Positive solutions for a nonlinear differential equation on a measure chain.Mathematical and Computer Modelling 2000,32(5-6):571-585. 10.1016/S0895-7177(00)00154-0 · Zbl 0963.34020 · doi:10.1016/S0895-7177(00)00154-0
[7] He Z: Double positive solutions of boundary value problems forp-Laplacian dynamic equations on time scales.Applicable Analysis 2005,84(4):377-390. 10.1080/00036810500047956 · Zbl 1082.39002 · doi:10.1080/00036810500047956
[8] Chyan CJ, Henderson J: Eigenvalue problems for nonlinear differential equations on a measure chain.Journal of Mathematical Analysis and Applications 2000,245(2):547-559. 10.1006/jmaa.2000.6781 · Zbl 0953.34068 · doi:10.1006/jmaa.2000.6781
[9] Li W-T, Sun H-R: Multiple positive solutions for nonlinear dynamical systems on a measure chain.Journal of Computational and Applied Mathematics 2004,162(2):421-430. 10.1016/j.cam.2003.08.032 · Zbl 1045.39007 · doi:10.1016/j.cam.2003.08.032
[10] Sun H-R: Existence of positive solutions to second-order time scale systems.Computers & Mathematics with Applications 2005,49(1):131-145. 10.1016/j.camwa.2005.01.011 · Zbl 1075.34019 · doi:10.1016/j.camwa.2005.01.011
[11] Erbe LH, Wang H: On the existence of positive solutions of ordinary differential equations.Proceedings of the American Mathematical Society 1994,120(3):743-748. 10.1090/S0002-9939-1994-1204373-9 · Zbl 0802.34018 · doi:10.1090/S0002-9939-1994-1204373-9
[12] Guo DJ, Lakshmikantham V: Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering. Volume 5. Academic Press, Boston, Mass, USA; 1988:viii+275. · Zbl 0661.47045
[13] Henderson J, Wang H: Positive solutions for nonlinear eigenvalue problems.Journal of Mathematical Analysis and Applications 1997,208(1):252-259. 10.1006/jmaa.1997.5334 · Zbl 0876.34023 · doi:10.1006/jmaa.1997.5334
[14] Hu L, Wang L: Multiple positive solutions of boundary value problems for systems of nonlinear second-order differential equations.Journal of Mathematical Analysis and Applications 2007,335(2):1052-1060. 10.1016/j.jmaa.2006.11.031 · Zbl 1127.34010 · doi:10.1016/j.jmaa.2006.11.031
[15] Webb JRL: Positive solutions of some three point boundary value problems via fixed point index theory.Nonlinear Analysis: Theory, Methods & Applications 2001,47(7):4319-4332. 10.1016/S0362-546X(01)00547-8 · Zbl 1042.34527 · doi:10.1016/S0362-546X(01)00547-8
[16] Agarwal RP, O’Regan D, Wong PJY: Positive Solutions of Differential, Difference and Integral Equations. Kluwer Academic, Dordrecht, The Netherlands; 1999:xii+417. · Zbl 1157.34301 · doi:10.1007/978-94-015-9171-3
[17] Graef JR, Yang B: Boundary value problems for second order nonlinear ordinary differential equations.Communications in Applied Analysis 2002,6(2):273-288. · Zbl 1085.34514
[18] Infante G: Eigenvalues of some non-local boundary-value problems.Proceedings of the Edinburgh Mathematical Society 2003,46(1):75-86. · Zbl 1049.34015 · doi:10.1017/S0013091501001079
[19] Infante G, Webb JRL: Loss of positivity in a nonlinear scalar heat equation.Nonlinear Differential Equations and Applications 2006,13(2):249-261. 10.1007/s00030-005-0039-y · Zbl 1112.34017 · doi:10.1007/s00030-005-0039-y
[20] Henderson J, Wang H: Nonlinear eigenvalue problems for quasilinear systems.Computers & Mathematics with Applications 2005,49(11-12):1941-1949. 10.1016/j.camwa.2003.08.015 · Zbl 1092.34517 · doi:10.1016/j.camwa.2003.08.015
[21] Henderson J, Wang H: An eigenvalue problem for quasilinear systems.The Rocky Mountain Journal of Mathematics 2007,37(1):215-228. 10.1216/rmjm/1181069327 · Zbl 1149.34013 · doi:10.1216/rmjm/1181069327
[22] Ma R: Multiple nonnegative solutions of second-order systems of boundary value problems.Nonlinear Analysis: Theory, Methods & Applications 2000,42(6):1003-1010. 10.1016/S0362-546X(99)00152-2 · Zbl 0973.34014 · doi:10.1016/S0362-546X(99)00152-2
[23] Wang H: On the number of positive solutions of nonlinear systems.Journal of Mathematical Analysis and Applications 2003,281(1):287-306. · Zbl 1036.34032 · doi:10.1016/S0022-247X(03)00100-8
[24] Zhou Y, Xu Y: Positive solutions of three-point boundary value problems for systems of nonlinear second order ordinary differential equations.Journal of Mathematical Analysis and Applications 2006,320(2):578-590. 10.1016/j.jmaa.2005.07.014 · Zbl 1101.34008 · doi:10.1016/j.jmaa.2005.07.014
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