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The author considers the quasilinear differential equations $$\begin{array}{c}{\left(r\left(t\right)\varphi \left(x^{\text{'}}\right)\right)}^{\text{'}}+q\left(t\right)f\left(x\right)=0,t\ge a\\ \multicolumn{2}{c}{\text{and}}\\ {\left(r\left(t\right)\varphi \left(x^{\text{'}}\right)\right)}^{\text{'}}+F(t,x)=\pm g\left(t\right),t\ge a.\end{array}$$ By means of topological tools there are established conditions ensuring the existence of nonnegative asymptotic decaying solutions of these equations.

We present a new approach to solving boundary value problems on noncompact intervals for second order differential equations in case of nonlocal conditions. Then we apply it to some problems in which an initial condition, an asymptotic condition and a global condition is present. The abstract method is based on the solvability of two auxiliary boundary value problems on compact and on noncompact intervals, and uses some continuity arguments and analysis in the phase space. As shown in the applications,...

We investigate two boundary value problems for the second order differential equation with $p$-Laplacian $${\left(a\left(t\right){\Phi }_p\left(x^{\text{'}}\right)\right)}^{\text{'}}=b\left(t\right)F\left(x\right),t\in I=[0,\infty ),$$ where $a$, $b$ are continuous positive functions on $I$. We give necessary and sufficient conditions which guarantee the existence of a unique (or at least one) positive solution, satisfying one of the following two boundary conditions: $$\mathrm{i})x\left(0\right)=c>0,\underset{t\to \infty }{lim}x\left(t\right)=0;\mathrm{ii})x^{\text{'}}\left(0\right)=d<0,\underset{t\to \infty }{lim}x\left(t\right)=0.$$