Global structure of one-sign solutions for a simply supported beam equation

In this paper, we consider the nonlinear eigenvalue problem u′′′′=λh(t)f(u),0<t<1,u(0)=u(1)=u″(0)=u″(1)=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{gathered} u''''= \lambda h(t)f(u),\quad 0< t< 1, \\ u(0)=u(1)=u''(0)=u''(1)=0, \\ \end{gathered} $$\end{document} where h∈C([0,1],(0,∞))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$h\in C([0,1], (0,\infty))$\end{document}; f∈C(R,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f\in C(\mathbb{R},\mathbb{R})$\end{document} and sf(s)>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$sf(s)>0$\end{document} for s≠0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$s\neq0$\end{document}, and f0=f∞=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f_{0}=f_{\infty}=0$\end{document}, f0=lim|s|→0f(s)/s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f_{0}=\lim_{|s|\rightarrow0}f(s)/s$\end{document}, f∞=lim|s|→∞f(s)/s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f_{\infty}=\lim_{|s|\rightarrow\infty}f(s)/s$\end{document}. We investigate the global structure of one-sign solutions by using bifurcation techniques.

Existence and multiplicity of positive solutions of (1.1) have been extensively studied by several authors, see [1,2,[5][6][7][8][9][10]13]. Cabada and Enguiça [2] developed the method of lower and upper solutions to show the existence and multiplicity of solutions, Jiang [6] and Li [7] proved the existence and multiplicity of solutions via the fixed point theorem in cone.
Bonanno and Di Bella [1] used variational method to obtain the following. In the present work, we attempt to give a direct and complete description of the global structure of one-sign solutions of (1.1) under the assumptions: 1] u(t) .
The rest of the paper is arranged as follows: In Sect. 2, we prove some properties of superior limit of certain infinity collection of connected sets. In Sect. 3, we state and prove some properties for the one-sign solutions (λ, u) of (1.1). Finally, in Sect. 4, we state and prove our main results.

Lemma 2.2 ([11]
) Let X be a Banach space and let {C n } be a family of closed connected subsets of X. Assume that (i) there exist z n ∈ C n , n = 1, 2, . . . , and z * ∈ X such that z n → z * ; Then there exists an unbounded component C in D and z * ∈ C.

Some preliminary results
Let us define an operator T λ : Y → Y by Proof Assume on the contrary that { y k ∞ } is bounded. Then for some constant M that is independent of k. Thus, it follows from the relation

Proof of the main results
We only deal with the global behavior of positive solutions of (1.1). The global behavior of negative solutions of (1.1) can be treated by a similar method. Let Σ + be the closure of the set of positive solutions of (1.1) in E. To prove Theorem 1.1, we will develop a bifurcation approach to treat the case f 0 = 0. Crucial to this approach is to construct a sequence of functions {f [n] } that is asymptotic linear at 0 and satisfies By means of the corresponding auxiliary equations, we obtain a sequence of unbounded components {C [n] + } via nonlinear Krein-Rutman bifurcation theorem, see Dancer [3] and Zeidler [15], and this enables us to find unbounded componentsζ satisfyinĝ ζ ⊂ lim sup  Let us consider as a bifurcation problem from the trivial solution u ≡ 0. Equation (4.7) can be converted to the equivalent equation By the fact (g [n] ) 0 > 0, the results of nonlinear Krein-Rutman theorem (see Dancer [3] and Zeidler [15,Corollary 15.
Proof We divide the proof into two steps.