We establish not only sufficient but also necessary conditions for existence of solutions to a singular multi-point third-order boundary value problem posed on the half-line. Our existence results are based on the Krasnosel’skii fixed point theorem on cone compression and expansion. Nonexistence results are proved under suitable a priori estimates. The nonlinearity $f=f(t,x,y)$ which satisfies upper and lower-homogeneity conditions in the space variables $x,y$ may be also singular at time $t=0$. Two examples of applications...

The problem on the existence of a positive in the interval $]a,b[$ solution of the boundary value problem $$u^{\text{'}\text{'}}=f(t,u)+g(t,u)u^{\text{'}};u(a+)=0,u(b-)=0$$ is considered, where the functions $f$ and $g]a,b[\times ]0,+\infty [\to ℝ$ satisfy the local Carathéodory conditions. The possibility for the functions $f$ and $g$ to have singularities in the first argument (for $t=a$ and $t=b$) and in the phase variable (for $u=0$) is not excluded. Sufficient and, in some cases, necessary and sufficient conditions for the solvability of that problem are established.

Bounds on the spectrum of the Schur complements of subdomain stiffness matrices with respect to the interior variables are key ingredients in the analysis of many domain decomposition methods. Here we are interested in the analysis of floating clusters, i.e. subdomains without prescribed Dirichlet conditions that are decomposed into still smaller subdomains glued on primal level in some nodes and/or by some averages. We give the estimates of the regular condition number of the Schur complements...

Existence results for semilinear operator equations without the assumption of normal cones are obtained by the properties of a fixed point index for A-proper semilinear operators established by Cremins. As an application, the existence of positive solutions for a second order m-point boundary value problem at resonance is considered.

We present necessary and sufficient conditions for the existence of unbounded increasing solutions to ordinary differential equations with mean curvature operator. The results illustrate the asymptotic proximity of such solutions with those of an auxiliary linear equation on the threshold of oscillation. A new oscillation criterion for equations with mean curvature operator, extending Leighton criterion for linear Sturm-Liouville equation, is also derived.

We study the singular periodic boundary value problem of the form $${\left(\phi \left(u^{\text{'}}\right)\right)}^{\text{'}}+h\left(u\right)u^{\text{'}}=g\left(u\right)+e\left(t\right),u\left(0\right)=u\left(T\right),u^{\text{'}}\left(0\right)=u^{\text{'}}\left(T\right),$$ where $\phi ℝ\to ℝ$ is an increasing and odd homeomorphism such that $\phi \left(ℝ\right)=ℝ,$$h\in C[0,\infty ...$

The paper is concerned with a stochastic delay predator-prey model under regime switching. Sufficient conditions for extinction and non-persistence in the mean of the system are established. The threshold between persistence and extinction is also obtained for each population. Some numerical simulations are introduced to support our main results.

The paper is concerned with existence results for positive solutions and maximal positive solutions of singular mixed boundary value problems. Nonlinearities h(t;x;y) in differential equations admit a time singularity at t=0 and/or at t=T and a strong singularity at x=0.

We consider the classical nonlinear fourth-order two-point boundary value problem $$\left\{\begin{array}{c}{u}^{\left(4\right)}\left(t\right)=\lambda h\left(t\right)f(t,u\left(t\right),u^{\text{'}}\left(t\right),u^{\text{'}\text{'}}\left(t\right)),0<t<1,\hfill \\ u\left(0\right)=u^{\text{'}}\left(1\right)=u^{\text{'}\text{'}}\left(0\right)=u^{\text{'}\text{'}\text{'}}\left(1\right)=0.\hfill \end{array}\right.$$ In this problem, the nonlinear term $h\left(t\right)f(t,u\left(t\right),u^{\text{'}}\left(t\right),u^{\text{'}\text{'}}\left(t\right))$ contains the first and second derivatives of the unknown function, and the function $h\left(t\right)f(t,x,y,z)$ may be singular at $t=0$, $t=1$ and at $x=0$, $y=0$, $z=0$. By introducing suitable height functions and applying the fixed point theorem on the cone, we establish several local existence theorems on positive solutions and obtain the corresponding eigenvalue intervals.

This paper studies positive solutions and eigenvalue intervals of a nonlinear third-order two-point boundary value problem. The nonlinear term is allowed to be singular with respect to both the time and space variables. By constructing a proper cone and applying the Guo-Krasnosel'skii fixed point theorem, the eigenvalue intervals for which there exist one, two, three or infinitely many positive solutions are obtained.