The paper establishes sufficient conditions for the existence of solutions of Neumann’s problem for the differential equation $\mu y"+ky=f(t,y)$ which tend to the solution of the reduced problem $ky=f(t,y)$ on $[0,1]$ as $\mu \to 0.$

n this paper we investigate systems of linear integral equations in the space ${𝔾}_L^n$ of $n$-vector valued functions which are regulated on the closed interval $[0,1]$ (i.e. such that can have only discontinuities of the first kind in $[0,1]$) and left-continuous in the corresponding open interval $(0,1).$ In particular, we are interested in systems of the form x(t) - A(t)x(0) - 01B(t,s)[d x(s)] = f(t), where $f\in {𝔾}_L^n$, the columns of the $n\times n$-matrix valued function $A$ belong to ${𝔾}_L^n$, the entries of $B(t,.)$ have a bounded variation...

Let $\Omega$ be a bounded $C^{\infty }$ domain in ${ℝ}^n$. The paper deals with inequalities of Hardy type related to the function spaces $B_{pq}^s\left(\Omega \right)$ and $F_{pq}^s\left(\Omega \right)$.

We consider the existence of positive solutions of -pu=g(x)|u|p-2u+h(x)|u|q-2u+f(x)|u|p*-2u(1) in ${ℝ}^N$, where $\lambda ,\alpha \in ℝ$, $1<p<N$, $p^{*}=Np/(N-p)$, the critical Sobolev exponent, and $1<q<p^{*}$, $q\ne p$. Let ${\lambda }_1^{+}>0$ be the principal eigenvalue of -pu=g(x)|u|p-2u    in ,        g(x)|u|p>0, (2) with $u_1^{+}>0$ the associated eigenfunction. We prove that, if ${\int }_{{ℝ}^N}f{\left|u_1^{+}\right|}^{p^{*}}<0$, ${\int }_{{ℝ}^N}h{\left|u_1^{+}\right|}^q>0$ if $1<q<p$ and ${\int }_{{ℝ}^N}h{\left|u_1^{+}\right|}^q<0$ if $p<q<p^{*}$, then there exist ${\lambda }^{*}>{\lambda }_1^{+}$ and ${\alpha }^{*}>0$, such that for $\lambda \in [{\lambda }_1^{+},{\lambda }^{*})$ and $\alpha \in [0,{\alpha }^{*})$, (1) has at least one positive solution.

Let $G\subset {ℝ}^m$ $(m\ge 2)$ be an open set with a compact boundary $B$ and let $\sigma \ge 0$ be a finite measure on $B$. Consider the space $L^1\left(\sigma \right)$ of all $\sigma$-integrable functions on $B$ and, for each $f\in L^1\left(\sigma \right)$, denote by $f\sigma$ the signed measure on $B$ arising by multiplying $\sigma$ by $f$ in the usual way. ${𝒩}_{\sigma }f$ denotes the weak normal derivative (w.r. to $G$) of the Newtonian (in case $m>2$) or the logarithmic (in case $n=2$) potential of $f\sigma$, correspondingly. Sharp geometric estimates are obtained for the essential norms of the operator ${𝒩}_{\sigma }-\alpha I$ (here $\alpha \in ℝ$...