## 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:

  Gupta, C. P., Existence and uniqueness theorems for a bending of an elastic beam equation, Appl. Anal., 26, 289-304 (1988) · Zbl 0611.34015  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  Agarwal, R. P., On fourth order boundary value problems arising in beam analysis, Differential Integral Equations, 2, 1, 91-110 (1989) · Zbl 0715.34032  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  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  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  Li, Yongxiang, Positive solutions of fourth-order boundary value problems with two parameters, J. Math. Anal. Appl., 281, 477-484 (2003) · Zbl 1030.34016  Chai, Guoqing, Existence of positive solutions for fourth-order boundary value problem with variable parameters, Nonlinear Anal., 26, 289-304 (2007) · Zbl 1113.34008  Ma, Ruyun, Existence of positive solutions of a fourth-order boundary value problem, Appl. Math. Comput., 168, 1219-1231 (2005) · Zbl 1082.34023  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  Ma, Ruyun, Nodal solutions for a fourth-order two-point boundary value problem, J. Math. Anal. Appl., 314, 254-265 (2006) · Zbl 1085.34015  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  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  Dalmasso, R., Uniqueness of positive solutions for some nonlinear fourth-order equations, J. Math. Anal. Appl., 201, 152-168 (1996) · Zbl 0856.34024  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  Rynne, P., Infinitely many solutions of superlinear fourth order boundary value problems, Topol. Methods Nonlinear Anal., 19, 303-312 (2002) · Zbl 1017.34015  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  Yao, Q., Positive solutions for eigenvalue problems of fourth-order elastic beam equations, Appl. Math. Lett., 17, 237-243 (2004) · Zbl 1072.34022  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  Rabinowitz, P. H., Some aspects of nonlinear eigenvalue problems, Rocky Mountain Consortium Symposium on Nonlinear Eigenvalue Problems (Santa Fe, N.M., 1971). Rocky Mountain Consortium Symposium on Nonlinear Eigenvalue Problems (Santa Fe, N.M., 1971), Rocky Mountain J. Math., 3, 161-202 (1973) · Zbl 0255.47069  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  Gulgowski, J., Bifurcation of solutions of nonlinear Sturm-Liouville problems, J. Inequal. Appl., 6, 483-506 (2001) · Zbl 1088.34518  Gulgowski, J., Global bifurcation and multiplicity results for Sturm-Liouville problems, NoDEA Nonlinear Differential Equations Appl., 14, 559-568 (2007) · Zbl 1137.34324  Zeidler, E., Nonlinear Functional Analysis and its Applications, I. Fixed -Point Theorems (1985), Springer-Verlag: Springer-Verlag New York  Deimling, K., Nonlinear Functional Analysis (1985), Springer-Verlag: Springer-Verlag Berlin · Zbl 0559.47040  Taylor, A. E., Introduction to Functional Analysis (1963), John Wiley & Sons: John Wiley & Sons London
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