In the paper we prove an Ambrosetti-Prodi type result for solutions $u$ of the third-order nonlinear differential equation, satisfying $u^{\text{'}}\left(0\right)=u^{\text{'}}\left(1\right)=u\left(\eta \right)=0,0\le \eta \le 1$.

Let $X$ be the Banach space of $C^0$-functions on $\langle 0,1\rangle$ with the sup norm and $\alpha ,\beta \in X\to R$ be continuous increasing functionals, $\alpha \left(0\right)=\beta \left(0\right)=0$. This paper deals with the functional differential equation (1) $x^{\text{'}\text{'}\text{'}}\left(t\right)=Q[x,x^{\text{'}},x^{\text{'}\text{'}}\left(t\right)]\left(t\right)$, where $Q:X^2\times R\to X$ is locally bounded continuous operator. Some theorems about the existence of two different solutions of (1) satisfying the functional boundary conditions $\alpha \left(x\right)=0=\beta \left(x^{\text{'}}\right)$, $x^{\text{'}\text{'}}\left(1\right)-x^{\text{'}\text{'}}\left(0\right)=0$ are given. The method of proof makes use of Schauder linearizatin technique and the Schauder fixed point theorem. The results are modified for 2nd order functional...

The existence of nonzero nonnegative solutions are established for semilinear equations at resonance with the zero solution and possessing at most linear growth. Applications are given to nonlinear boundary value problems of ordinary differential equations.

In this paper, a class of damped vibration problems with impulsive effects is considered. An existence result is obtained by using the variational method and the critical point theorem due to Brezis and Nirenberg. The obtained result is also valid and new for the corresponding second-order impulsive Hamiltonian system. Finally, an example is presented to illustrate the feasibility and effectiveness of the result.

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.

We study the existence of periodic solutions for Liénard-type p-Laplacian systems with variable coefficients by means of the topological degree theory. We present sufficient conditions for the existence of periodic solutions, improving some known results.