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Let α,β,γ,δ ≥ 0 and ϱ:= γβ + αγ + αδ > 0. Let ψ(t) = β + αt, ϕ(t) = γ + δ - γt, t ∈ [0,1]. We study the existence of positive solutions for the m-point boundary value problem ⎧u” + h(t)f(u) = 0, 0 < t < 1, ⎨$\alpha u\left(0\right)-\beta u^{\text{'}}\left(0\right)={\sum }_{i=1}^{m-2}{a}_{i}u\left({\xi }_i\right)$ ⎩$\gamma u\left(1\right)+\delta u^{\text{'}}\left(1\right)={\sum }_{i=1}^{m-2}{b}_{i}u\left({\xi }_i\right)$, where ${\xi }_i\in (0,1)$, $a_i,b_i\in (0,\infty )$ (for i ∈ 1,…,m-2) are given constants satisfying $\varrho -{\sum }_{i=1}^{m-2}{a}_i\varphi \left({\xi }_i\right)>0$, $\varrho -{\sum }_{i=1}^{m-2}{b}_i\psi \left({\xi }_i\right)>0$ and $\Delta :=\left|\begin{array}{cc}-{\sum }_{i=1}^{m-2}{a}_i\psi \left({\xi }_i\right)& \varrho -{\sum }_{i=1}^{m-2}{a}_i\varphi \left({\xi }_i\right)\\ \varrho -{\sum }_{i=1}^{m-2}{b}_i\psi \left({\xi }_i\right)& -{\sum }_{i=1}^{m-2}{b}_i\varphi \left({\xi }_i\right)\end{array}\right|<0$. We show the existence of positive solutions if f is either superlinear or sublinear by a simple application of a fixed point theorem in cones. Our result extends a result established by Erbe and Wang for two-point...

We study the existence of positive solutions of the nonlinear fourth order problem $u^{\left(4\right)}\left(x\right)=\lambda a\left(x\right)f\left(u\left(x\right)\right)$, u(0) = u’(0) = u”(1) = u”’(1) = 0, where a: [0,1] → ℝ may change sign, f(0) < 0, and λ < 0 is sufficiently small. Our approach is based on the Leray-Schauder fixed point theorem.

We study the existence of nodal solutions of the $m$-point boundary value problem $$u^{\text{'}\text{'}}+f\left(u\right)=0,0<t<1,u^{\text{'}}\left(0\right)=0,u\left(1\right)=\sum _{i=1}^{m-2}{\alpha }_{i}u\left({\eta }_i\right)$$ where ${\eta }_i\in \mathbb{Q}$ $(i=1,2,\cdots ,m-2)$ with $0<{\eta }_1<{\eta }_2<\cdots <{\eta }_{m-2}<1$, and ${\alpha }_i\in ℝ$ $(i=1,2,\cdots ,m-2)$ with ${\alpha }_i>0$ and $0<{\sum }_{i=1}^{m-2}{\alpha }_i<1$. We give conditions on the ratio $f\left(s\right)/s$ at infinity and zero that guarantee the existence of nodal solutions. The proofs of the main results are based on bifurcation techniques.

We consider boundary value problems for nonlinear $2m$th-order eigenvalue problem $$\begin{array}{cc}\hfill {(-1)}^m{u}^{\left(2m\right)}\left(t\right)& =\lambda a\left(t\right)f\left(u\right(t\left)\right),0<t<1,\hfill \\ \hfill u^{\left(2i\right)}\left(0\right)& =u^{\left(2i\right)}\left(1\right)=0,i=0,1,2,\cdots ,m-1.\hfill \end{array}$$ where $a\in C\left(\right[0,1],[0,\infty \left)\right)$ and $a\left(t_0\right)>0$ for some $t_0\in [0,1]$, $f\in C\left(\right[0,\infty ),[0,\infty \left)\right)$ and $f\left(s\right)>0$ for $s>0$, and $f_0=\infty$, where $f_0={lim}_{s\to 0^{+}}f\left(s\right)/s$. We investigate the global structure of positive solutions by using Rabinowitz’s global bifurcation theorem.

In this paper, we are concerned with the existence of one-signed solutions of four-point boundary value problems $$-u^{\text{'}\text{'}}+Mu=rg\left(t\right)f\left(u\right),u\left(0\right)=u\left(\epsilon \right),u\left(1\right)=u(1-\epsilon )$$ and $$u^{\text{'}\text{'}}+Mu=rg\left(t\right)f\left(u\right),u\left(0\right)=u\left(\epsilon \right),u\left(1\right)=u(1-\epsilon ),$$ where $\epsilon \in (0,1/2)$, $M\in (0,\infty )$ is a constant and $r>0$ is a parameter, $g\in C\left(\right[0,1],(0,+\infty \left)\right)$, $f\in C(ℝ,ℝ)$ with $sf\left(s\right)>0$ for $s\ne 0$. The proof of the main results is based upon bifurcation techniques.

Let $E=\{u\in C^1[0,1]:u\left(0\right)=u\left(1\right)=0\}$. Let $S_k^{\nu }$ with $\nu =\{+,-\}$ denote the set of functions $u\in E$ which have exactly $k-1$ interior nodal zeros in (0, 1) and $\nu u$ be positive near $0$. We show the existence of $S$-shaped connected component of $S_k^{\nu }$-solutions of the problem $$\left\{\begin{array}{cc}{\left({\displaystyle \frac{u^{\text{'}}}{\sqrt{1-u^{\text{'}2}}}}\right)}^{\text{'}}+\lambda a\left(x\right)f\left(u\right)=0,\hfill & x\in (0,1),\hfill \\ u\left(0\right)=u\left(1\right)=0,\hfill \end{array}\right.$$ where $\lambda >0$ is a parameter, $a\in C\left(\right[0,1],(0,\infty \left)\right)$. We determine the intervals of parameter $\lambda$ in which the above problem has one, two or three $S_k^{\nu }$-solutions. The proofs of the main results are based upon the bifurcation technique.