In this paper we obtain existence conditions and a closed form of the general solution of higher order singular regular boundary value problems. The approach is based on the concept of co-solution of algebraic matrix equations of polynomial type that permits the treatment of the problem without considering an extended first order system as it has been done in the known literature.

A couple ($\sigma ,\tau$) of lower and upper slopes for the resonant second order boundary value problem $$x^{\text{'}\text{'}}=f(t,x,x^{\text{'}}),x\left(0\right)=0,x^{\text{'}}\left(1\right)={\int }_0^1{x}^{\text{'}}\left(s\right)\mathrm{d}g\left(s\right),$$ with $g$ increasing on $[0,1]$ such that ${\int }_0^{1}dg=1$, is a couple of functions $\sigma ,\tau \in C^1\left([0,1]\right)$ such that $\sigma \left(t\right)\le \tau \left(t\right)$ for all $t\in [0,1]$, $$\begin{array}{c}{\sigma }^{\text{'}}\left(t\right)\ge f(t,x,\sigma \left(t\right)),\sigma \left(1\right)\le {\int }_0^1\sigma \left(s\right)\mathrm{d}g\left(s\right),\\ {\tau }^{\text{'}}\left(t\right)\le f(t,x,\tau \left(t\right)),\tau \left(1\right)\ge {\int }_0^1\tau \left(s\right)\mathrm{d}g\left(s\right),\end{array}$$ in the stripe ${\int }_0^t\sigma \left(s\right)\mathrm{d}s\le x\le {\int }_0^t\tau \left(s\right)\mathrm{d}s$ and $t\in [0,1]$. It is proved that the existence of such a couple $(\sigma ,\tau )$ implies the existence and localization of a solution to the boundary value problem. Multiplicity results are also obtained.

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...

In the paper, we obtain the existence of symmetric or monotone positive solutions and establish a corresponding iterative scheme for the equation ${\left({\phi }_p\left(u^{\text{'}}\right)\right)}^{\text{'}}+q\left(t\right)f\left(u\right)=0$, $0<t<1$, where ${\phi }_p\left(s\right):={\left|s\right|}^{p-2}s$, $p>1$, subject to nonlinear boundary condition. The main tool is the monotone iterative technique. Here, the coefficient $q\left(t\right)$ may be singular at $t=0,1$.