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Gli Autori provano alcuni nuovi criteri sufficienti, indipendenti da altri criteri da loro ottenuti in precedenza, perché gli integrali dell'equazione ${(a(t)x^{\prime })}^{\prime }+q(t)f(x)g(x^{\prime })=r(t)$ siano tutti non oscillatori.

Si esamina il comportamento asintotico delle soluzioni nel caso che ${lim}_{t\to \mathrm{\infty }}r(t)/q(t)$ sia positivo e $r(t)/q(t)$ sia monotona.

The authors consider the nonlinear difference equation $$x_{n+1}=\alpha x_n+x_{n-k}f\left(x_{n-k}\right),n=0,1,\cdots .1\text{where}\alpha \in (0,1),k\in \{0,1,\cdots \}\text{and}f\in C^1[[0,\infty ),[0,\infty )]\left(0\right)$$ 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.

We study the existence of positive solutions to the fourth-order two-point boundary value problem $$\left\{\begin{array}{cc}{u}^{\text{'}\text{'}\text{'}\text{'}}\left(t\right)+f(t,u\left(t\right))=0,\hfill & 0<t<1,\hfill \\ u^{\text{'}}\left(0\right)=u^{\text{'}}\left(1\right)=u^{\text{'}\text{'}}\left(0\right)=0,\hfill & u\left(0\right)=\alpha \left[u\right],\hfill \end{array}\right.$$ where $\alpha \left[u\right]={\int }_0^{1}u\left(t\right)\mathrm{d}A\left(t\right)$ is a Riemann-Stieltjes integral with $A\ge 0$ being a nondecreasing function of bounded variation and $f\in 𝒞([0,1]\times {ℝ}_{+},{ℝ}_{+})$. The sufficient conditions obtained are new and easy to apply. Their approach is based on Krasnoselskii’s fixed point theorem and the Avery-Peterson fixed point theorem.

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.

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

In this paper, the authors establish sufficient conditions for the existence of solutions to implicit fractional differential inclusions with nonlocal conditions. Both of the cases of convex and nonconvex valued right hand sides are considered.

This paper presents several sufficient conditions for the existence of at least one classical solution to impulsive fractional differential equations with a $p$-Laplacian and Dirichlet boundary conditions. Our technical approach is based on variational methods. Some recent results are extended and improved. Moreover, a concrete example of an application is presented.

The authors establish some new sufficient conditions under which all solutions of a certain class of nonlinear neutral delay differential equations of the third order are stable, bounded, and square integrable. Illustrative examples are given to demonstrate the main results.

We describe the nonlinear limit-point/limit-circle problem for the $n$-th order differential equation $$y^{\left(n\right)}+r\left(t\right)f(y,y^{\text{'}},\cdots ,y^{(n-1)})=0.$$ The results are then applied to higher order linear and nonlinear equations. A discussion of fourth order equations is included, and some directions for further research are indicated.

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 $P$, $Q\in C({ℝ}^{+},{ℝ}^{+})$. 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.

The authors consider the difference equation $${\Delta }^m[y_n-p_n{y}_{n-k}]+\delta q_n{y}_{\sigma (n+m-1)}=0(*)$$ where $m\ge 2$, $\delta =\pm 1$, $k\in N_0=\{0,1,2,\cdots \}$, $\Delta y_n=y_{n+1}-y_n$, $q_n>0$, and $\left\{\sigma \right(n\left)\right\}$ is a sequence of integers with $\sigma \left(n\right)\le n$ and ${lim}_{n\to \infty }\sigma \left(n\right)=\infty$. They obtain results on the classification of the set of nonoscillatory solutions of ($*$) and use a fixed point method to show the existence of solutions having certain types of asymptotic behavior. Examples illustrating the results are included.