### Nonoscillatory solutions for nonlinear discrete systems

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A characterization of oscillation and nonoscillation of the Emden-Fowler difference equation $$\Delta \left({a}_{n}{\left|\Delta {x}_{n}\right|}^{\alpha}sgn\Delta {x}_{n}\right)+{b}_{n}{\left|{x}_{n+1}\right|}^{\beta}sgn{x}_{n+1}=0$$ is given, jointly with some asymptotic properties. The problem of the coexistence of all possible types of nonoscillatory solutions is also considered and a comparison with recent analogous results, stated in the half-linear case, is made.

Consider the third order differential operator $L$ given by $$L(\xb7)\equiv \phantom{\rule{0.166667em}{0ex}}\frac{1}{{a}_{3}\left(t\right)}\frac{\text{d}}{\text{d}t}\frac{1}{{a}_{2}\left(t\right)}\frac{\text{d}}{\text{d}t}\frac{1}{{a}_{1}\left(t\right)}\frac{\text{d}}{\text{d}t}\phantom{\rule{0.166667em}{0ex}}(\xb7)$$ and the related linear differential equation $L\left(x\right)\left(t\right)+x\left(t\right)=0$. We study the relations between $L$, its adjoint operator, the canonical representation of $L$, the operator obtained by a cyclic permutation of coefficients ${a}_{i}$, $i=1,2,3$, in $L$ and the relations between the corresponding equations. We give the commutative diagrams for such equations and show some applications (oscillation, property A).

We investigate two boundary value problems for the second order differential equation with $p$-Laplacian $${\left(a\left(t\right){\Phi}_{p}\left({x}^{\text{'}}\right)\right)}^{\text{'}}=b\left(t\right)F\left(x\right),\phantom{\rule{1.0em}{0ex}}t\in I=[0,\infty ),$$ 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: $$\mathrm{i})\phantom{\rule{4pt}{0ex}}x\left(0\right)=c>0,\phantom{\rule{4pt}{0ex}}\underset{t\to \infty}{lim}x\left(t\right)=0;\phantom{\rule{1.0em}{0ex}}\mathrm{ii})\phantom{\rule{4pt}{0ex}}{x}^{\text{'}}\left(0\right)=d<0,\phantom{\rule{4pt}{0ex}}\underset{t\to \infty}{lim}x\left(t\right)=0.$$

Some asymptotic properties of principal solutions of the half-linear differential equation $${\left(a\left(t\right)\Phi \left({x}^{\text{'}}\right)\right)}^{\text{'}}+b\left(t\right)\Phi \left(x\right)=0\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{2.0em}{0ex}}(*)$$ $\Phi \left(u\right)={\left|u\right|}^{p-2}u$, $p>1$, is the $p$-Laplacian operator, are considered. It is shown that principal solutions of (*) are, roughly speaking, the smallest solutions in a neighborhood of infinity, like in the linear case. Some integral characterizations of principal solutions of (), which completes previous results, are presented as well.

Globally positive solutions for the third order differential equation with the damping term and delay, $${x}^{\text{'}\text{'}\text{'}}+q\left(t\right){x}^{\text{'}}\left(t\right)-r\left(t\right)f\left(x\left(\phi \left(t\right)\right)\right)=0,$$ are studied in the case where the corresponding second order differential equation $${y}^{\text{'}\text{'}}+q\left(t\right)y=0$$ is oscillatory. Necessary and sufficient conditions for all nonoscillatory solutions of (*) to be unbounded are given. Furthermore, oscillation criteria ensuring that any solution is either oscillatory or unbounded together with its first and second derivatives are presented. The comparison of results with those...

Oscillatory properties of the second order nonlinear equation $${\left(r\left(t\right){x}^{\text{'}}\right)}^{\text{'}}+q\left(t\right)f\left(x\right)=0$$ are investigated. In particular, criteria for the existence of at least one oscillatory solution and for the global monotonicity properties of nonoscillatory solutions are established. The possible coexistence of oscillatory and nonoscillatory solutions is studied too.

We study solutions tending to nonzero constants for the third order differential equation with the damping term $${\left({a}_{1}\left(t\right){\left({a}_{2}\left(t\right){x}^{\text{'}}\left(t\right)\right)}^{\text{'}}\right)}^{\text{'}}+q\left(t\right){x}^{\text{'}}\left(t\right)+r\left(t\right)f\left(x\left(\varphi \left(t\right)\right)\right)=0$$ in the case when the corresponding second order differential equation is oscillatory.

Asymptotic properties of the half-linear difference equation $$\Delta ({a}_{n}|\Delta {x}_{n}{|}^{\alpha}\mathrm{s}gn\Delta {x}_{n})={b}_{n}{\left|{x}_{n+1}\right|}^{\alpha}\mathrm{s}gn{x}_{n+1}\phantom{\rule{2.0em}{0ex}}(*)$$ are investigated by means of some summation criteria. Recessive solutions and the Riccati difference equation associated to $(*)$ are considered too. Our approach is based on a classification of solutions of $(*)$ and on some summation inequalities for double series, which can be used also in other different contexts.

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