## Bifurcation from interval and positive solutions of a nonlinear fourth-order boundary value problem.(English)Zbl 1200.34023

Consider the fourth-order boundary value problem
$u^{(4)}(t)=f(t,u(t),u''(t)), t \in (0,1),$
$u(0) =u(1)=u''(0)=u''(1)=0,$
where $$f:[0,1] \times [0,+\infty) \times (-\infty,0] \to [0,+\infty)$$ is continuous, such that $$f(t,0,0)=0$$ and satisfies a technical condition ensuring that, roughly speaking, $$f$$ is not necessarily linearizable at $$(0,0)$$ and $$(+\infty,-\infty).$$ Moreover, it is assumed that there exist a non-negative function $$c_1$$ and a non-negative constant $$c_2$$ such that $$c_1(t)+c_2>0$$ and $$f(t,u,p) \geq c_1(t)u-c_2 p$$ for all $$t,u,p.$$ The authors give a sufficient condition, expressed in terms of the generalized eigenvalues of the associated BVP $$u^{(4)}=\lambda(A(t)u-B(t)u''),$$ for the existence of at least one positive solution to the given BVP. The proof is performed by applying the global P. H. Rabinowitz bifurcation theorem [Rocky Mountain J. Math. 3, 161–202 (1973; Zbl 0255.47069)].

### MSC:

 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34C23 Bifurcation theory for ordinary differential equations

### Keywords:

Fourth-order ODE; positive solution; eigenvalue; bifurcation

Zbl 0255.47069
Full Text:

### References:

 [1] Gupta, C.P., Existence and uniqueness theorems for a bending of an elastic beam equation, Appl. anal., 26, 289-304, (1988) · Zbl 0611.34015 [2] Gupta, C.P., Existence and uniqueness results for the bending of an elastic beam equation at resonance, J. math. anal. appl., 135, 208-225, (1988) · Zbl 0655.73001 [3] Agarwal, R.P., On fourth order boundary value problems arising in beam analysis, Differential integral equations, 2, 1, 91-110, (1989) · Zbl 0715.34032 [4] Del Pino, M.A.; Manásevich, R.F., Existence for a fourth-order boundary value problem under a two-parameter nonresonance condition, Proc. amer. math. soc., 112, 1, 81-86, (1991) · Zbl 0725.34020 [5] Ma, Ruyun; Wang, Haiyan, On the existence of positive solutions of fourth-order ordinary differential equations, Appl. anal., 59, 225-231, (1995) · Zbl 0841.34019 [6] Bai, Zhanbing; Wang, Haiyan, On positive solutions of some nonlinear fourth-order beam equations, J. math. anal. appl., 270, 2, 357-368, (2002) · Zbl 1006.34023 [7] Li, Yongxiang, Positive solutions of fourth-order boundary value problems with two parameters, J. math. anal. appl., 281, 477-484, (2003) · Zbl 1030.34016 [8] Chai, Guoqing, Existence of positive solutions for fourth-order boundary value problem with variable parameters, Nonlinear anal., 26, 289-304, (2007) · Zbl 1113.34008 [9] Ma, Ruyun, Existence of positive solutions of a fourth-order boundary value problem, Appl. math. comput., 168, 1219-1231, (2005) · Zbl 1082.34023 [10] Ma, Ruyun, Nodal solutions of boundary value problems of fourth-order ordinary differential equations, J. math. anal. appl., 319, 424-434, (2006) · Zbl 1098.34012 [11] Ma, Ruyun, Nodal solutions for a fourth-order two-point boundary value problem, J. math. anal. appl., 314, 254-265, (2006) · Zbl 1085.34015 [12] Bai, Z.; Wang, H., On positive solutions of some nonlinear fourth-order beam equations, J. math. anal. appl., 270, 357-368, (2002) · Zbl 1006.34023 [13] Chu, Jifeng; O’Regan, Donal, Positive solutions for regular and singular fourth-order boundary value problems, Commun. appl. anal., 10, 185-199, (2006) · Zbl 1123.34015 [14] Dalmasso, R., Uniqueness of positive solutions for some nonlinear fourth-order equations, J. math. anal. appl., 201, 152-168, (1996) · Zbl 0856.34024 [15] Korman, P., Uniqueness and exact multiplicity of solutions for a class of fourth-order semilinear problems, Proc. roy. soc. edinburghsect. A, 134, 179-190, (2004) · Zbl 1060.34014 [16] Rynne, P., Infinitely many solutions of superlinear fourth order boundary value problems, Topol. methods nonlinear anal., 19, 303-312, (2002) · Zbl 1017.34015 [17] Webb, J.R.L.; Infante, G.; Franco, D., Positive solutions of nonlinear fourth order boundary value problems with local and nonlocal boundary conditions, Proc. roy. soc. edinburghsect. A, 138, 427-446, (2008) · Zbl 1167.34004 [18] Yao, Q., Positive solutions for eigenvalue problems of fourth-order elastic beam equations, Appl. math. lett., 17, 237-243, (2004) · Zbl 1072.34022 [19] Yao, Q., On the positive solutions of a nonlinear fourth-order boundary value problem with two parameters, Appl. anal., 83, 97-107, (2004) · Zbl 1051.34018 [20] Rabinowitz, P.H., Some aspects of nonlinear eigenvalue problems, Rocky mountain consortium symposium on nonlinear eigenvalue problems (Santa Fe, N.M., 1971), Rocky mountain J. math., 3, 161-202, (1973) · Zbl 0255.47069 [21] Kim, Chan-Gyun; Lee, Yong-Hoon, Existence and multiplicity results for nonlinear boundary value problems, Comput. math. appl., 55, 12, 2870-2886, (2008) · Zbl 1142.34319 [22] Gulgowski, J., Bifurcation of solutions of nonlinear sturm – liouville problems, J. inequal. appl., 6, 483-506, (2001) · Zbl 1088.34518 [23] Gulgowski, J., Global bifurcation and multiplicity results for sturm – liouville problems, Nodea nonlinear differential equations appl., 14, 559-568, (2007) · Zbl 1137.34324 [24] Zeidler, E., Nonlinear functional analysis and its applications, I. fixed -point theorems, (1985), Springer-Verlag New York [25] Deimling, K., Nonlinear functional analysis, (1985), Springer-Verlag Berlin · Zbl 0559.47040 [26] Taylor, A.E., Introduction to functional analysis, (1963), John Wiley & Sons London
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