New existence results are presented for the two point singular “resonant” boundary value problem $\frac{1}{p}{\left(p{y}^{\text{'}}\right)}^{\text{'}}+ry+{\lambda }_{m}qy=f(t,y,p{y}^{\text{'}})$ a.eȯn $[0,1]$ with $y$ satisfying Sturm Liouville or Periodic boundary conditions. Here ${\lambda }_m$ is the ${(m+1)}^{st}$ eigenvalue of $\frac{1}{pq}[{\left(p{u}^{\text{'}}\right)}^{\text{'}}+rpu]+\lambda u=0$ a.eȯn $[0,1]$ with $u$ satisfying Sturm Liouville or Periodic boundary data.

In this paper we consider the nonlocal (nonstandard) Cauchy problem for differential inclusions in Banach spaces x'(t) ∈ F(t,x(t)), x(0)=g(x), t ∈ [0,T] = I. Investigation over some multivalued integrals allow us to prove the existence of solutions for considered problem. We concentrate on the problems for which the assumptions are expressed in terms of the weak topology in a Banach space. We recall and improve earlier papers of this type. The paper is complemented...

Sufficient conditions are given for the solvability of an impulsive Dirichlet boundary value problem to forced nonlinear differential equations involving the combination of viscous and dry frictions. Apart from the solvability, also the explicit estimates of solutions and their derivatives are obtained. As an application, an illustrative example is given, and the corresponding numerical solution is obtained by applying Matlab software.

We establish Vallée Poussin type disconjugacy and disfocality criteria for the half-linear second order differential equation $u^{\text{'}\text{'}}={p\left(t\right)\left|u\right|}^{\alpha }{\left|u^{\text{'}}\right|}^{1-\alpha }sgnu+g\left(t\right)u^{\text{'}}$, where α ∈ (0,1] and the functions $p,g\in L_{loc}(a,b)$ are allowed to have singularities at the end points t = a, t = b of the interval under consideration.

A class of linear elliptic operators has an important qualitative property, the so-called maximum principle. In this paper we investigate how this property can be preserved on the discrete level when an interior penalty discontinuous Galerkin method is applied for the discretization of a 1D elliptic operator. We give mesh conditions for the symmetric and for the incomplete method that establish some connection between the mesh size and the penalty parameter. We then investigate the sharpness of...

In this article, we consider the operator $L$ defined by the differential expression $$\ell \left(y\right)=-y^{\text{'}\text{'}}+q\left(x\right)y,-\infty <x<\infty$$ in $L_2(-\infty ,\infty )$, where $q$ is a complex valued function. Discussing the spectrum, we prove that $L$ has a finite number of eigenvalues and spectral singularities, if the condition $$\underset{-\infty <x<\infty }{sup}\left\{exp\left(ϵ\sqrt{\left|x\right|}\right)\left|q\left(x\right)\right|\right\}<\infty ,ϵ>0$$ holds. Later we investigate the properties of the principal functions corresponding to the eigenvalues and the spectral singularities.