### Positive solutions for the $(n,p)$ boundary value problem.

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A new error correction method for the stationary Navier-Stokes equations based on two local Gauss integrations is presented. Applying the orthogonal projection technique, we introduce two local Gauss integrations as a stabilizing term in the error correction method, and derive a new error correction method. In both the coarse solution computation step and the error computation step, a locally stabilizing term based on two local Gauss integrations is introduced. The stability and convergence of the...

The authors consider the boundary value problem with a two-parameter nonhomogeneous multi-point boundary condition $$\begin{array}{c}{u}^{\text{'}\text{'}}+g\left(t\right)f(t,u)=0,\phantom{\rule{1.0em}{0ex}}t\in (0,1),\\ u\left(0\right)=\alpha u\left(\xi \right)+\lambda ,\phantom{\rule{1.0em}{0ex}}u\left(1\right)=\beta u\left(\eta \right)+\mu .Criteriafortheexistenceofnontrivialsolutionsoftheproblemareestablished.Thenonlineartermf(t,x)maytakenegativevaluesandmaybeunboundedfrombelow.Conditionsaredeterminedbytherelationshipbetweenthebehavioroff(t,x)/xforxnear0and\pm \infty ,andthesmallestpositivecharacteristicvalueofanassociatedlinearintegraloperator.Theanalysismainlyreliesontopologicaldegreetheory.Thisworkcomplementssomerecentresultsintheliterature.Theresultsareillustratedwithexamples.\end{array}$$

We study a third order singular boundary value problem with multi-point boundary conditions. Sufficient conditions are obtained for the existence of positive solutions of the problem. Recent results in the literature are significantly extended and improved. Our analysis is mainly based on a nonlinear alternative of Leray-Schauder.

In this paper, we study the existence of oscillatory and nonoscillatory solutions of neutral differential equations of the form $x\left(t\right)-cx(t-r)$’$P\left(t\right)x(t-\theta )-Q\left(t\right)x(t-\delta )$=0 where $c>0$, $r>0$, $\theta >\delta \ge 0$ are constants, and $P$, $Q\in C({\mathbb{R}}^{+}\phantom{\rule{-0.166667em}{0ex}},{\mathbb{R}}^{+})$. We obtain some sufficient and some necessary conditions for the existence of bounded and unbounded positive solutions, as well as some sufficient conditions for the existence of bounded and unbounded oscillatory solutions.

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