### Ambrosetti-Prodi type results in a system of second and fourth-order ordinary differential equations.

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We use the method of quasilinearization to boundary value problems of ordinary differential equations showing that the corresponding monotone iterations converge to the unique solution of our problem and this convergence is quadratic.

We consider the boundary value problem involving the one dimensional $p$-Laplacian, and establish the precise intervals of the parameter for the existence and non-existence of solutions with prescribed numbers of zeros. Our argument is based on the shooting method together with the qualitative theory for half-linear differential equations.

Applying two three critical points theorems, we prove the existence of at least three anti-periodic solutions for a second-order impulsive differential inclusion with a perturbed nonlinearity and two parameters.

We consider boundary value problems for nonlinear $2m$th-order eigenvalue problem $$\begin{array}{cc}\hfill {(-1)}^{m}{u}^{\left(2m\right)}\left(t\right)& =\lambda a\left(t\right)f\left(u\right(t\left)\right),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}0<t<1,\hfill \\ \hfill {u}^{\left(2i\right)}\left(0\right)& ={u}^{\left(2i\right)}\left(1\right)=0,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}i=0,1,2,\cdots ,m-1.\hfill \end{array}$$ 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.

We study the existence and multiplicity of positive nonsymmetric and sign-changing nonantisymmetric solutions of a nonlinear second order ordinary differential equation with symmetric nonlinear boundary conditions, where both of the nonlinearities are of power type. The given problem has already been studied by other authors, but the number of its positive nonsymmetric and sign-changing nonantisymmetric solutions has been determined only under some technical conditions. It was a long-standing open...

We consider a Lidstone boundary value problem in ${\mathbb{R}}^{k}$ at resonance. We prove the existence of a solution under the assumption that the nonlinear part is a Carathéodory map and conditions similar to those of Landesman-Lazer are satisfied.

We consider the classical nonlinear fourth-order two-point boundary value problem $$\left\{\begin{array}{c}{u}^{\left(4\right)}\left(t\right)=\lambda h\left(t\right)f(t,u\left(t\right),{u}^{\text{'}}\left(t\right),{u}^{\text{'}\text{'}}\left(t\right)),\phantom{\rule{1.0em}{0ex}}0<t<1,\hfill \\ u\left(0\right)={u}^{\text{'}}\left(1\right)={u}^{\text{'}\text{'}}\left(0\right)={u}^{\text{'}\text{'}\text{'}}\left(1\right)=0.\hfill \end{array}\right.$$ In this problem, the nonlinear term $h\left(t\right)f(t,u\left(t\right),{u}^{\text{'}}\left(t\right),{u}^{\text{'}\text{'}}\left(t\right))$ contains the first and second derivatives of the unknown function, and the function $h\left(t\right)f(t,x,y,z)$ may be singular at $t=0$, $t=1$ and at $x=0$, $y=0$, $z=0$. By introducing suitable height functions and applying the fixed point theorem on the cone, we establish several local existence theorems on positive solutions and obtain the corresponding eigenvalue intervals.

The authors consider the boundary value problem with a two-parameter nonhomogeneous multi-point boundary condition $$\begin{array}{c}{u}^{\text{'}\text{'}}+g\left(t\right)f(t,u)=0,\phantom{\rule{1.0em}{0ex}}t\in (0,1),\\ u\left(0\right)=\alpha u\left(\xi \right)+\lambda ,\phantom{\rule{1.0em}{0ex}}u\left(1\right)=\beta u\left(\eta \right)+\mu .Criteriafortheexistenceofnontrivialsolutionsoftheproblemareestablished.Thenonlineartermf(t,x)maytakenegativevaluesandmaybeunboundedfrombelow.Conditionsaredeterminedbytherelationshipbetweenthebehavioroff(t,x)/xforxnear0and\pm \infty ,andthesmallestpositivecharacteristicvalueofanassociatedlinearintegraloperator.Theanalysismainlyreliesontopologicaldegreetheory.Thisworkcomplementssomerecentresultsintheliterature.Theresultsareillustratedwithexamples.\end{array}$$

In this paper, we are interested in the study of bifurcation solutions of nonlinear wave equation of elastic beams located on elastic foundations with small perturbation by using local method of Lyapunov-Schmidt.We showed that the bifurcation equation corresponding to the elastic beams equation is given by the nonlinear system of two equations. Also, we found the parameters equation of the Discriminant set of the specified problem as well as the bifurcation diagram.