Existence of positive solution for a sixth-order differential system with variable parameters. (English) Zbl 1301.34030
Let \(A, B, C, D\in C[0,1]\) with \(D>0\), and let \(\mu>0\) be a parameter. The authors investigate the existence of positive solutions for the sixth-order differential system
\[
-u^{(6)}+A(t)u^{(4)}+B(t)u^{(2)}+C(t)u=(D(t)+u)\varphi+f(t,u),\quad -\varphi''+\lambda\varphi=\mu u,
\]
\(u(0)=u(1)=u''(0)=u''(1)=u^{(4)}(0)=u^{(4)}(1)=0\), \(\varphi(0)=\varphi(1)=0\).
Using a fixed point theorem and an operator spectral theorem they give some new existence results.
Using a fixed point theorem and an operator spectral theorem they give some new existence results.
Reviewer: Ruyun Ma (Lanzhou)
MSC:
| 34B18 | Positive solutions to nonlinear boundary value problems for ordinary differential equations |
| 34B15 | Nonlinear boundary value problems for ordinary differential equations |
| 47N20 | Applications of operator theory to differential and integral equations |
References:
| [1] | Yao, Q.: Successive iteration and positive solution for nonlinear second-order boundary-value problems. Comput. Math. Appl. 50, 433–444 (2005) · Zbl 1096.34015 · doi:10.1016/j.camwa.2005.03.006 |
| [2] | Bai, Z., Ge, W.: Existence of three positive solutions for some second-order boundary-value problems. Comput. Math. Appl. 48, 699–707 (2004) · Zbl 1066.34019 · doi:10.1016/j.camwa.2004.03.002 |
| [3] | Feng, M.: Existence of symmetric positive solutions for a boundary-value problem with integral boundary conditions. Appl. Math. Lett. 24, 1419–1427 (2011) · Zbl 1221.34062 · doi:10.1016/j.aml.2011.03.023 |
| [4] | Karakostas, G.L., Tsamatos, P.Ch.: Positive solutions of a boundary-value problem for second order ordinary differential equations. Electron. J. Differ. Equ. 49, 1–9 (2000) · Zbl 0957.34022 |
| [5] | Wang, Y.-M.: On 2nth-order nonlinear multi-point boundary-value problems. Math. Comput. Model. 51, 1251–1259 (2010) · Zbl 1206.34036 · doi:10.1016/j.mcm.2010.01.008 |
| [6] | Cao, D., Ma, R.: Positive solutions to a second order multi-point boundary-value problem. Electron. J. Differ. Equ. 65, 1–8 (2000) · Zbl 0964.34022 |
| [7] | Bai, D., Feng, H.: Three positive solutions for m-point boundary-value problems with one-dimensional p-Laplacian. Electron. J. Differ. Equ. 75, 1–10 (2011) · Zbl 1230.34026 |
| [8] | Webb, J.R.L.: Positive solutions of a boundary-value problem with integral boundary conditions. Electron. J. Differ. Equ. 55, 1–10 (2011) · Zbl 1229.34039 |
| [9] | Chasreechai, S., Tariboon, J.: Positive solutions to generalized second-order three-point integral boundary-value problems. Electron. J. Differ. Equ. 14, 1–14 (2011) · Zbl 1211.34025 |
| [10] | Yang, A., Sun, B., Ge, W.: Existence of positive solutions for self-adjoint boundary value problems with integral boundary conditions at resonance. Electron. J. Differ. Equ. 11, 1–8 (2011) · Zbl 1211.34031 · doi:10.1155/2011/404917 |
| [11] | Xia, J., Liu, Y.: Monotone positive solutions for p-Laplacian equations with sign-changing coefficients and multi-point boundary conditions. Electron. J. Differ. Equ. 22, 1–20 (2010) · Zbl 1188.34033 · doi:10.1007/s10884-010-9163-4 |
| [12] | Ji, C., Yan, B.: Positive solution for second-order singular three-point boundary-value problems with sign-changing nonlinearities. Electron. J. Differ. Equ. 38, 1–9 (2010) · Zbl 1201.34031 |
| [13] | Agarval, R.P., O’Regan, D.: Upper and lower solutions for singular problems with nonlinear data. Nonlinear Differ. Equ. Appl. 9, 419–440 (2002) · Zbl 1025.34019 · doi:10.1007/PL00012607 |
| [14] | Erbe, L.H., Wang, H.: On the existence of positive solutions of ordinary differential equations. Proc. Am. Math. Soc. 120, 743–748 (1994) · Zbl 0802.34018 · doi:10.1090/S0002-9939-1994-1204373-9 |
| [15] | Gupta, C.P.: Existence and uniqueness theorems for a bending of an elastic beam equation. Appl. Anal. 26, 289–304 (1988) · Zbl 0611.34015 · doi:10.1080/00036818808839715 |
| [16] | Gupta, C.P.: Existence and uniqueness results for some fourth order fully quasilinear boundary value problem. Appl. Anal. 36, 169–175 (1990) · Zbl 0713.34025 · doi:10.1080/00036819008839930 |
| [17] | Liu, B.: Positive solutions of fourth-order two point boundary value problem. Appl. Math. Comput. 148, 407–420 (2004) · Zbl 1039.34018 · doi:10.1016/S0096-3003(02)00857-3 |
| [18] | Ma, R.: Positive solutions of fourth-order two point boundary value problems. Ann. Differ. Equ. 15, 305–313 (1999) · Zbl 0964.34021 |
| [19] | Ma, R., Zhang, J., Fu, S.: The method of lower and upper solutions for fourth-order two point boundary value problems. J. Math. Anal. Appl. 215, 415–422 (1997) · Zbl 0892.34009 · doi:10.1006/jmaa.1997.5639 |
| [20] | Ma, R., Wang, H.: On the existence of positive solutions of fourth-order ordinary differential equations. Appl. Anal. 59, 225–231 (1995) · Zbl 0841.34019 · doi:10.1080/00036819508840401 |
| [21] | Aftabizadeh, A.R.: Existence and uniqueness theorems for fourth-order boundary problems. J. Math. Anal. Appl. 116, 415–426 (1986) · Zbl 0634.34009 · doi:10.1016/S0022-247X(86)80006-3 |
| [22] | Yang, Y.: Fourt-order two-point boundary value problems. Proc. Am. Math. Soc. 104, 175–180 (1988) · Zbl 0671.34016 · doi:10.1090/S0002-9939-1988-0958062-3 |
| [23] | Del Pino, M.A., Manasevich, R.F.: Existence for fourth-order boundary problem under a two-parameter nonresonance condition. Proc. Am. Math. Soc. 112, 81–86 (1991) · Zbl 0725.34020 · doi:10.1090/S0002-9939-1991-1043407-9 |
| [24] | Li, Y.: Positive solution of fourth-order boundary value problems with two parameters. J. Math. Anal. Appl. 281, 477–484 (2003) · Zbl 1030.34016 · doi:10.1016/S0022-247X(03)00131-8 |
| [25] | Chai, G.: Existence of positive solutions for fourth-order boundary value problem with variable parameters. Nonlinear Anal. 66, 870–880 (2007) · Zbl 1113.34008 · doi:10.1016/j.na.2005.12.028 |
| [26] | Wang, F., An, Y.: Positive solutions for a second-order differential systems. J. Math. Anal. Appl. 373, 370–375 (2011) · Zbl 1217.34036 · doi:10.1016/j.jmaa.2010.07.031 |
| [27] | Li, W.: The existence and multiplicity of positive solutions of nonlinear sixth-order boundary value problem with three variable coefficients. Bound. Value Probl. 2012 22 (2012) · Zbl 1277.34025 · doi:10.1186/1687-2770-2012-22 |
| [28] | Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones. Academic Press, New York (1988) · Zbl 0661.47045 |
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