We study the vector $p$-Laplacian $$\left\}\begin{array}{c}-\left(\right|u^{\text{'}}{{|}^{p-2}{u}^{\text{'}})}^{\text{'}}=\nabla F(t,u)\text{a.e.}t\in [0,T],u\left(0\right)=u\left(T\right),u^{\text{'}}\left(0\right)=u^{\text{'}}\left(T\right),1<p<\infty .\hfill \end{array}\right.(*)$$ We prove that there exists a sequence $\left(u_n\right)$ of solutions of ($*$) such that $u_n$ is a critical point of $\varphi$ and another sequence $\left(u_n^{*}\right)$ of solutions of $(*)$ such that $u_n^{*}$ is a local minimum point of $\varphi$, where $\varphi$ is a functional defined below.

The existence of positive solutions for a nonlocal boundary-value problem with vector-valued response is investigated. We develop duality and variational principles for this problem. Our variational approach enables us to approximate solutions and give a measure of a duality gap between the primal and dual functional for minimizing sequences.

The existence of nonnegative radial solutions for some systems of m (m ≥ 1) quasilinear elliptic equations is proved by a simple application of a fixed point theorem in cones.

We study the existence of one-signed periodic solutions of the equations $$\begin{array}{ccc}& x^{\text{'}\text{'}}\left(t\right)-a^2\left(t\right)x\left(t\right)+\mu f(t,x\left(t\right),x^{\text{'}}\left(t\right))=0,\hfill & \hfill x^{\text{'}\text{'}}\left(t\right)+a^2\left(t\right)x\left(t\right)=\mu f(t,x\left(t\right),x^{\text{'}}\left(t\right)),\end{array}$$ where $\mu >0$, $a:(-\infty ,+\infty )\to (0,\infty )$ is continuous and 1-periodic, $f$ is a continuous and 1-periodic in the first variable and may take values of different signs. The Krasnosielski fixed point theorem on cone is used.

In this paper we prove the existence of periodic solutions for nonlinear impulsive viable problems monitored by differential inclusions of the type x′(t)∈F(t,x(t))+G(t,x(t)). Our existence theorems extend, in a broad sense, some propositions proved in [10] and improve a result due to Hristova-Bainov in [13].

A general theorem on the GBDT version of the Bäcklund-Darboux transformation for systems depending rationally on the spectral parameter is treated and its applications to nonlinear equations are given. Explicit solutions of direct and inverse problems for Dirac-type systems, including systems with singularities, and for the system auxiliary to the N-wave equation are reviewed. New results on explicit construction of the wave functions for radial...

In the paper it is proved that the generalized linear boundary value problem generates a Fredholm operator. Its index depends on the number of boundary conditions. The existence results of Landesman-Lazer type are given as an application to nonlinear problems by using dual generalized boundary value problems.

In this paper explicit expressions for solutions of Cauchy problems and two-point boundary value problems concerned with the generalized Riccati matrix differential equation are given. These explicit expressions are computable in terms of the data and solutions of certain algebraic Riccati equations related to the problem. The interplay between the algebraic and the differential problems is used in order to obtain approximate solutions of the differential problem in terms of those of the algebraic...