Let $g$: $𝐑\to 𝐑$ be a continuous function, $e$: $[0,1]\to 𝐑$ a function in $L^2[0,1]$ and let $c\in 𝐑$, $c\ne 0$ be given. It is proved that Duffing’s equation $u^{\text{'}\text{'}}+c{u}^{\text{'}}+g\left(u\right)=e\left(x\right)$, $0<x<1$, $u\left(0\right)=u\left(1\right)$, $u^{\text{'}}\left(0\right)=u^{\text{'}}\left(1\right)$ in the presence of the damping term has at least one solution provided there exists an $𝐑>0$ such that $g\left(u\right)u\ge 0$ for $\left|u\right|\ge 𝐑$ and ${\int }_0^{1}e\left(x\right)dx=0$. It is further proved that if $g$ is strictly increasing on $𝐑$ with ${lim}_{u\to -\infty }g\left(u\right)=-\infty$, ${lim}_{u\to \infty }g\left(u\right)=\infty$ and it Lipschitz continuous with Lipschitz constant $\alpha <4{\pi }^2+c^2$, then Duffing’s equation given above has exactly one solution for every $e\in L^2[0,1]$.

In this paper, we consider the nonlinear fourth order eigenvalue problem. We show the existence of family of unbounded continua of nontrivial solutions bifurcating from the line of trivial solutions. These global continua have properties similar to those found in Rabinowitz and Berestycki well-known global bifurcation theorems.

We consider nonlinear Sturm-Liouville problems with spectral parameter in the boundary condition. We investigate the structure of the set of bifurcation points, and study the behavior of two families of continua of nontrivial solutions of this problem contained in the classes of functions having oscillation properties of the eigenfunctions of the corresponding linear problem, and bifurcating from the points and intervals of the line of trivial solutions.

Some new oscillation criteria are obtained for second order elliptic differential equations with damping ${\sum }_{i,j=1}^n{D}_i\left[A_{ij}\left(x\right)D_{j}y\right]+{\sum }_{i=1}^n{b}_i\left(x\right)D_{i}y+q\left(x\right)f\left(y\right)=0$, x ∈ Ω, where Ω is an exterior domain in ℝⁿ. These criteria are different from most known ones in the sense that they are based on the information only on a sequence of subdomains of Ω ⊂ ℝⁿ, rather than on the whole exterior domain Ω. Our results are more natural in view of the Sturm Separation Theorem.

Oscillatory properties of solutions to the system of first-order linear difference equations $$\begin{array}{cc}\hfill \Delta u_k& =q_k{v}_k\hfill \\ \hfill \Delta v_k& =-p_k{u}_{k+1},\hfill \end{array}$$ are studied. It can be regarded as a discrete analogy of the linear Hamiltonian system of differential equations. We establish some new conditions, which provide oscillation of the considered system. Obtained results extend and improve, in certain sense, results presented in Opluštil (2011).

An extension of a result of R. Conti is given from which some integro-differential inequalities of the Gronwall-Bellman-Bihari type and a criterion for the continuation of solutions of a system of ordinary differential equations are deduced.