### On properties of nonlinear second-order systems under nonlinear impulse perturbations.

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The authors consider the nonlinear difference equation $${x}_{n+1}=\alpha {x}_{n}+{x}_{n-k}f\left({x}_{n-k}\right),\phantom{\rule{1.0em}{0ex}}n=0,1,\cdots .1\text{where}\alpha \in (0,1),\phantom{\rule{5.0pt}{0ex}}k\in \{0,1,\cdots \}\phantom{\rule{5.0pt}{0ex}}\text{and}\phantom{\rule{5.0pt}{0ex}}f\in {C}^{1}[[0,\infty ),[0,\infty )]\phantom{\rule{2.0em}{0ex}}\left(0\right)$$ with ${f}^{\text{'}}\left(x\right)<0$. They give sufficient conditions for the unique positive equilibrium of (0.1) to be a global attractor of all positive solutions. The results here are somewhat easier to apply than those of other authors. An application to a model of blood cell production is given.

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.

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