We apply Max Müller's Theorem to second order equations u'' = f(t,u,u') to obtain solutions between given functions v,w.

We study asymptotic properties of solutions for a system of second differential equations with $p$-Laplacian. The main purpose is to investigate lower estimates of singular solutions of second order differential equations with $p$-Laplacian ${\left(A\left(t\right){\Phi }_p\left(y^{\text{'}}\right)\right)}^{\text{'}}+B\left(t\right)g\left(y^{\text{'}}\right)+R\left(t\right)f\left(y\right)=e\left(t\right)$. Furthermore, we obtain results for a scalar equation.

We consider the nonlinear Dirichlet problem $$-u^{\text{'}\text{'}}-r\left(x\right){\left|u\right|}^{\sigma }u=\lambda u\text{in}(0,\infty ),u\left(0\right)=0\text{and}\underset{x\to \infty }{lim}u\left(x\right)=0,$$ and develop conditions for the function $r$ such that the considered problem has a positive classical solution. Moreover, we present some results showing that $\lambda =0$ is a bifurcation point in $W^{1,2}(0,\infty )$ and in $L^p(0,\infty )(2\le p\le \infty )$.

We prove the existence of a positive solution to the BVP $${\left(\Phi \left(t\right)u^{\text{'}}\left(t\right)\right)}^{\text{'}}=f(t,u\left(t\right)),u^{\text{'}}\left(0\right)=u\left(1\right)=0,$$ imposing some conditions on Φ and f. In particular, we assume $\Phi \left(t\right)f(t,u)$ to be decreasing in t. Our method combines variational and topological arguments and can be applied to some elliptic problems in annular domains. An $L_{\infty }$ bound for the solution is provided by the $L_{\infty }$ norm of any test function with negative energy.

The system of nonlinear differential equations $$x^{\text{'}}+p_1\left(t\right)x^{{\alpha }_1}+q_1\left(t\right)y^{{\beta }_1}=0,y^{\text{'}}+p_2\left(t\right)x^{{\alpha }_2}+q_2\left(t\right)y^{{\beta }_2}=0,A$$ is under consideration, where ${\alpha }_i$ and ${\beta }_i$ are positive constants and $p_i\left(t\right)$ and $q_i\left(t\right)$ are positive continuous functions on $[a,\infty )$. There are three types of different asymptotic behavior at infinity of positive solutions $\left(x\right(t),y(t\left)\right)$ of (). The aim of this paper is to establish criteria for the existence of solutions of these three types by means of fixed point techniques. Special emphasis is placed on those solutions with both components decreasing to zero as $t\to \infty$, which can be...

The paper presents an existence result for global solutions to the finite dimensional differential inclusion $y^{\text{'}}\in F\left(y\right),$$F$ being defined on a closed set $K.$ A priori bounds for such solutions are provided.