## Solutions for an operator equation under the conditions of pairs of paralleled lower and upper solutions.(English)Zbl 1165.47044

Let $$X$$ and $$Z$$ be ordered Banach spaces, $$L: D(L)\to Z$$ be a linear operator and $$F$$ a continuous and bounded operator. The paper discusses the multiplicity of solutions of the operator equation
$L x=F(x)\qquad\tag $$*$$$
by the method of lower and upper solutions and fixed point index theory. When $$(*)$$ has two pairs of strict lower and upper solutions, the authors obtain a three solutions theorem, which is different to Amann’s three solutions theorem. Under three pairs of strict lower and upper solutions, they obtain existence results of five solutions and nine solutions. The abstract results are applied to a Sturm–Liouville boundary value problem.

### MSC:

 47J05 Equations involving nonlinear operators (general) 47H11 Degree theory for nonlinear operators 34B15 Nonlinear boundary value problems for ordinary differential equations 47H10 Fixed-point theorems 47N20 Applications of operator theory to differential and integral equations
Full Text:

### References:

 [1] C. De Coster, P. Habets, An overview of the method of lower and upper solutions for ODEs, in: Nonlinear Analysis and its Applications to Differential Equations, vol. 43, Birkhăuser, Boston, Basel, Berlin · Zbl 1132.34309 [2] Amann, H., Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM rev., 18, 620-709, (1976) · Zbl 0345.47044 [3] Jingxian, Sun; Xian, Xu, Three solutions theorems for nonlinear operator equations and applications, J. syst. sci. complex., 18, 1, 119-125, (2005) · Zbl 1097.47054 [4] Xian, Xu; O’Regan, Donal; Jingxian, Sun, Multiplicity results for three-point boundary value problems with a non-well-ordered upper and lower solution condition, Math. comput. modelling, 45, 189-200, (2007) · Zbl 1140.34009 [5] De Coster, C.; Henrard, M., Existence and localization of solution for elliptic problem in presence of lower and upper solutions without any order, J. differential equations, 145, 420-452, (1998) · Zbl 0908.35042 [6] Xian, Xu, Three solutions for three-point boundary value problems, Nonlinear anal., 62, 1053-1066, (2005) · Zbl 1076.34011 [7] Rachůnková, I.; Tvrdy, M., Non-ordered lower and upper functions in second order impulsive periodic problems, Dyn. contin. discrete impuls. syst., 12, 397-415, (2005) · Zbl 1086.34026 [8] Rachůnková, I., Upper and lower solutions and topological degree, J. math. anal. appl., 234, 311-327, (1999) · Zbl 1086.34017 [9] Rachůnková, I., Upper and lower solutions and multiplicity results, J. math. anal. appl., 246, 446-464, (2000) · Zbl 0961.34004 [10] Habets, P.; Omari, P., Existence and localization of second order elliptic problems using lower and upper solutions in the reversed order, Topol. methods nonlinear anal., 8, 25-56, (1996) · Zbl 0897.35030 [11] Omari, P., Non-ordered lower and upper solution and solvability of the periodic problem for Liénard and Rayleigh equations, Rend. istit. mat. univ. treiste, 20, 54-64, (1988) [12] Dajun, Guo; Lakshmikantham, V., Nonlinear problems in abstract cones, (1988), Academic Press Inc, New York · Zbl 0661.47045 [13] Zhongli Wei, Some problems in the theory of differential equations and their applications, Doctoral Thesis, Shandong University, Jinan, 1996 · Zbl 0897.45011 [14] Fuyi Li, Solutions of nonlinear operator equations and applications, Doctoral Thesis, Shandong University, Jinan, 1996 [15] Dajun, Guo, Nonlinear functional analysis and applications, (1994), Beijing Sci. & Tec. Press Beijing · Zbl 0807.34076 [16] Jingxian, Sun; Yujun, Cui, Multiple solutions for nonlinear operators and applications, Nonlinear anal., 66, 1999-2015, (2007) · Zbl 1120.47050 [17] Jingxian, Sun; Kemei, Zhang, On the number of fixed points of nonlinear operators and applications, J. syst. sci. complex., 16, 2, (2003) · Zbl 1131.47308 [18] Mohamed, El-Gebeily; Donal, O’Regan, Upper and lower solutions and quasilinearization for a class of second order singular nonlinear differential equations with nonlinear boundary conditions, Nonlinear anal. real world appl., 8, 2, 636-645, (2007) · Zbl 1152.34317 [19] Nieto, J.J.; Rodríguez-Lóez, Rosana, Periodic boundary value problem for non-Lipschitzian impulsive functional differential equations, J. math. anal. appl., 318, 2, 593-610, (2006) · Zbl 1101.34051 [20] Yongxiang, Li, Maximum principles and the method of upper and lower solutions for time-periodic problems of the telegraph equations, J. math. anal. appl., 327, 2, 997-1009, (2007) · Zbl 1108.35021 [21] Ali Khan, Rahmat; Nieto, J.J.; Rodríguez-Lóez, Rosana, Upper and lower solutions method for second order nonlinear four point boundary value problems, J. Korean math. soc., 43, 6, 1253-1268, (2006) · Zbl 1118.34009 [22] Jiang, Daqing; Nieto, J.J.; Zuo, Wenjie, On monotone method for first and second order periodic boundary value problems and periodic solutions of functional differential equations, J. math. anal. appl., 289, 2, 691-699, (2004) · Zbl 1134.34322 [23] Goncharov Vladimir, V.; Ornelas, Antnio, On minima of a functional of the gradient: upper and lower solutions, Nonlinear anal., 64, 7, 1437-1459, (2006) · Zbl 1102.49011 [24] Agarwal, R.P.; O’Regan, D.; Lakshmikantham, V.; Leela, S.A., Generalized upper and lower solution method for singular initial value problems, Comput. math. appl., 47, 4-5, 739-750, (2004) · Zbl 1098.34004 [25] Ladde, G.S.; Lakshmikantham, V.; Vatsala, A.S., Monotone iterative techniques for nonlinear differential equations, (1985), Pitman Boston, MA, USA · Zbl 0658.35003 [26] Nieto, J.J.; Jiang, Y.; Jurang, Y., Monotone iterative method for functional-differential equations, Nonlinear anal., 32, 741-747, (1998) · Zbl 0937.34053 [27] Drici, Z.; McRae, F.A.; Vasundhara Devi, J., Monotone iterative technique for periodic boundary value problems with causal operators, Nonlinear anal., 64, 1271-1277, (2006) · Zbl 1208.34103 [28] West, I.H.; Vatsala, A.S., Generalizedmonotone iterative method for initial value problems, Appl. math. lett., 17, 1231-1237, (2004) · Zbl 1112.34304 [29] Ahmad, B.; Sivasundaram, S., The monotone iterative technique for impulsive hybrid set valued integro-differential equations, Nonlinear anal., 65, 2260-2276, (2006) · Zbl 1111.45006
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.