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A sum operator equation and applications to nonlinear elastic beam equations and Lane-Emden-Fowler equations. (English) Zbl 1207.47064
The paper studies the nonlinear operator equation Ax+Bx+Cx=x on ordered Banach spaces, where A is an α-concave operator, B an increasing sub-homogeneous operator, and C a homogeneous operator. By using the properties of cones and a fixed point theorem for increasing general β-concave operators, some new results on the existence and uniqueness of positive solutions are obtained. Applications are made to two classes of nonlinear problems; they include fourth-order two-point boundary value problems for elastic beam equations and elliptic boundary value problems for Lane-Emden-Fowler equations.
MSC:
47J05Equations involving nonlinear operators (general)
47H07Monotone and positive operators on ordered topological linear spaces
47N20Applications of operator theory to differential and integral equations
74K10Rods (beams, columns, shafts, arches, rings, etc.) in solid mechanics
34B10Nonlocal and multipoint boundary value problems for ODE
35J66Nonlinear boundary value problems for nonlinear elliptic equations
References:
[1]Adams, R. A.: Sobolev spaces, (1975)
[2]Agarwal, R. P.; O’regan, D.: A multiplicity result for second order impulsive differential equations via the Leggett Williams fixed point theorem, Appl. math. Comput. 161, 433-439 (2005) · Zbl 1070.34042 · doi:10.1016/j.amc.2003.12.096
[3]Alves, E.; Ma, T. F.; Pelicer, M. L.: Monotone positive solutions for a fourth order equation with nonlinear boundary conditions, Nonlinear anal. 71, 3834-3841 (2009) · Zbl 1177.34030 · doi:10.1016/j.na.2009.02.051
[4]Amann, H.: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM rev. 18, No. 4, 620-709 (1976) · Zbl 0345.47044 · doi:10.1137/1018114
[5]Avery, R. I.; Henderson, J.: Three symmetric positive solutions for a second order boundary value problem, Appl. math. Lett. 13, No. 3, 1-7 (2000) · Zbl 0961.34014 · doi:10.1016/S0893-9659(99)00177-9
[6]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 · doi:10.1016/S0898-1221(01)00156-0
[7]Bai, Z.: The upper and lower solution method for some fourth-order boundary value problems, Nonlinear anal. 67, 1704-1709 (2007) · Zbl 1122.34010 · doi:10.1016/j.na.2006.08.009
[8]Brezis, H.; Nirenberg, L.: Minima locaux relatifs a C1 et H1, C. R. Acad. sci. Paris 317, 465-472 (1993)
[9]Chen, Y. Z.: Stability of positive fixed points of nonlinear operators, Positivity 6, 47-57 (2002) · Zbl 1008.47048 · doi:10.1023/A:1012079817987
[10]Coclite, M. M.; Palmieri, G.: On a singular nonlinear Dirichlet problem, Comm. partial differential equations 14, 1315-1327 (1989) · Zbl 0692.35047 · doi:10.1080/03605308908820656
[11]Crandall, M. G.; Rabinowitz, P. H.; Tartar, L.: On a Dirichlet problem with a singular nonlinearity, Comm. partial differential equations 2, No. 2, 193-222 (1977) · Zbl 0362.35031 · doi:10.1080/03605307708820029
[12]Deimling, K.: Nonlinear functional analysis, (1985) · Zbl 0559.47040
[13]Dupaigne, L.; Ghergu, M.; Rădulescu, V.: Lane-Emden-Fowler equations with convection and singular potential, J. math. Pures appl. 87, 563-581 (2007) · Zbl 1122.35034 · doi:10.1016/j.matpur.2007.03.002
[14]Fulks, W.; Maybee, J. S.: A singular nonlinear elliptic equation, Osaka J. Math. 12, 1-19 (1960) · Zbl 0097.30202
[15]Ghergu, M.; Rădulescu, V.: Sublinear singular elliptic problems with two parameters, J. differential equations 195, 520-536 (2003) · Zbl 1039.35042 · doi:10.1016/S0022-0396(03)00105-0
[16]Ghergu, M.; Rădulescu, V.: Ground state solutions for the singular Lane-Emden-Fowler equation with sublinear convection term, J. math. Anal. appl. 333, 265-273 (2007) · Zbl 1211.35128 · doi:10.1016/j.jmaa.2006.09.074
[17]Ghergu, M.; Rădulescu, V.: Singular elliptic problems. Bifurcation and asymptotic analysis, Oxford lecture ser. Math. appl. 37 (2008)
[18]Graef, J. R.; Yang, B.: Positive solutions of a nonlinear fourth order boundary value problem, Comm. appl. Nonlinear anal. 14, No. 1, 61-73 (2007) · Zbl 1136.34024
[19]Guo, D. J.; Lakshmikantham, V.: Nonlinear problems in abstract cones, (1988)
[20]Hernández, J.; Mancebo, F.; Vega, J. M.: Positive solutions for singular nonlinear elliptic equations, Proc. roy. Soc. Edinburgh sect. A 137, 41-62 (2007) · Zbl 1180.35231 · doi:10.1017/S030821050500065X
[21]Ilic, D.; Rakocevic, V.: Common fixed points for maps on cone metric space, J. math. Anal. appl. 341, 876-882 (2008) · Zbl 1156.54023 · doi:10.1016/j.jmaa.2007.10.065
[22]Jachymski, J.: The contraction principle for mappings on a metric space with a graph, Proc. amer. Math. soc. 136, 1359-1373 (2008) · Zbl 1139.47040 · doi:10.1090/S0002-9939-07-09110-1
[23]Karaca, I. Y.: Nonlinear triple-point problems with change of sign, Comput. math. Appl. 55, No. 14, 691-703 (2008) · Zbl 1143.34013 · doi:10.1016/j.camwa.2007.04.032
[24]Kristăly, A.; Radulescu, V.; Varga, Cs.: Variational principles in mathematical physics, geometry, and economics: qualitative analysis of nonlinear equations and unilateral problems, Encyclopedia math. Appl. 136 (2010)
[25]Ladyzhenskaya, O. A.; Ural’ceva, N. N.: Linear and quasilinear elliptic equations, (1968) · Zbl 0164.13002
[26]Lazer, A. C.; Mckenna, P. J.: On a singular nonlinear elliptic boundary value problem, Proc. amer. Math. soc. 111, 721-730 (1991) · Zbl 0727.35057 · doi:10.2307/2048410
[27]Li, K.; Liang, J.; Xiao, T. J.: A fixed point theorem for convex and decreasing operators, Nonlinear anal. 63, e209-e216 (2005) · Zbl 1159.47306 · doi:10.1016/j.na.2004.12.014
[28]Li, Y.: Two-parameter nonresonance condition for the existence of fourth-order boundary value problems, J. math. Anal. appl. 308, 121-128 (2005) · Zbl 1071.34016 · doi:10.1016/j.jmaa.2004.11.021
[29]Liu, B.: Positive solutions of fourth-order two-point boundary value problems, Appl. math. Comput. 148, 407-420 (2004) · Zbl 1039.34018 · doi:10.1016/S0096-3003(02)00857-3
[30]Nieto, J. J.; Pouso, R. L.; Rodriguez-Lopez, R.: Fixed point theorems in ordered abstract spaces, Proc. amer. Math. soc. 135, 2505-2517 (2007) · Zbl 1126.47045 · doi:10.1090/S0002-9939-07-08729-1
[31]Nussbaum, R. D.: Iterated nonlinear maps and Hilbert’s projective metric, Mem. amer. Math. soc. 75, No. 391 (1988)
[32]Shi, J.; Yao, M.: On a singular nonlinear semilinear elliptic problem, Proc. roy. Soc. Edinburgh sect. A 128, 1389-1401 (1998) · Zbl 0919.35044 · doi:10.1017/S0308210500027384
[33]Stuart, C. A.: Existence and approximation of solutions of nonlinear elliptic equations, Math. Z. 147, 53-63 (1976) · Zbl 0324.35037 · doi:10.1007/BF01214274
[34]Wiegner, M.: A degenerate diffusion equation with a nonlinear source term, Nonlinear anal. 28, No. 12, 1977-1995 (1997) · Zbl 0874.35061 · doi:10.1016/S0362-546X(96)00027-2
[35]Yang, B.: Positive solutions for the beam equation under certain boundary value problems, Electron. J. Differential equations 2005, No. 78, 1-8 (2005) · Zbl 1075.34025 · doi:emis:journals/EJDE/Volumes/2005/78/abstr.html
[36]Yang, C.; Zhai, C. B.; Yan, J. R.: Positive solutions of three-point boundary value problem for second order differential equations with an advanced argument, Nonlinear anal. 65, 2013-2023 (2006) · Zbl 1113.34048 · doi:10.1016/j.na.2005.11.003
[37]Yao, Q.: Positive solutions for eigenvalue problems of fourth-order elastic beam equations, Appl. math. Lett. 17, 237-243 (2004) · Zbl 1072.34022 · doi:10.1016/S0893-9659(04)90037-7
[38]Zhai, C. B.; Guo, C. M.: On α-convex operators, J. math. Anal. appl. 316, 556-565 (2006) · Zbl 1094.47047 · doi:10.1016/j.jmaa.2005.04.064
[39]Zhai, C. B.; Yang, C.; Guo, C. M.: Positive solutions of operator equation on ordered Banach spaces and applications, Comput. math. Appl. 56, 3150-3156 (2008) · Zbl 1165.47308 · doi:10.1016/j.camwa.2008.09.005
[40]Zhang, X. P.: Existence and iteration of monotone positive solutions for an elastic beam with a corner, Nonlinear anal. Real world appl. 10, 2097-2103 (2009) · Zbl 1163.74478 · doi:10.1016/j.nonrwa.2008.03.017
[41]Zhang, Z. J.: The asymptotic behaviour of the unique solution for the singular Lane-Emden-Fowler equation, J. math. Anal. appl. 312, 33-43 (2005) · Zbl 1165.35377 · doi:10.1016/j.jmaa.2005.03.023
[42]Zhao, Z.: Multiple fixed points of a sum operator and applications, J. math. Anal. appl. 360, 1-6 (2009) · Zbl 1173.47038 · doi:10.1016/j.jmaa.2009.06.016