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Picone identity for a class of nonlinear differential equations is established and various qualitative results (such as Wirtinger-type inequality and the existence of zeros of first components of solutions) are obtained with the help of this new formula.

We present an integral comparison theorem which guarantees the global existence of a solution of the generalized Riccati equation on the given interval $[a,b)$ when it is known that certain majorant Riccati equation has a global solution on $[a,b)$.

In the paper we present an identity of the Picone type for a class of nonlinear differential operators of the second order involving an arbitrary norm $H$ in ${ℝ}^n$ which is continuously differentiable for $x\ne 0$ and such that $H^p$ is strictly convex for some $p>1$. Two important special cases are the $p$-Laplacian and the so-called pseudo $p$-Laplacian. The identity is then used to establish a variety of comparison results concerning nonlinear degenerate elliptic equations which involve such operators. We also get criteria...

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...

Two identities of the Picone type for a class of half-linear differential systems in the plane are established and the Sturmian comparison theory for such systems is developed with the help of these new formulas.

We extend the classical Leighton comparison theorem to a class of quasilinear forced second order differential equations $$\left(r\left(t\right)\right|x^{\text{'}}{|}^{\alpha -2}{x}^{\text{'}}{)}^{\text{'}}+c\left(t\right){\left|x\right|}^{\beta -2}x=f\left(t\right),1<\alpha \le \beta ,t\in I=(a,b),(*)$$ where the endpoints $a$, $b$ of the interval $I$ are allowed to be singular. Some applications of this statement in the oscillation theory of (*) are suggested.

Positive solutions of the nonlinear second-order differential equation $\left(p\left(t\right)\right|x^{\text{'}}{|}^{\alpha -1}{x}^{\text{'}}{)}^{\text{'}}+q\left(t\right){\left|x\right|}^{\beta -1}x=0,\alpha >\beta >0,$ are studied under the assumption that p, q are generalized regularly varying functions. An application of the theory of regular variation gives the possibility of obtaining necessary and sufficient conditions for existence of three possible types of intermediate solutions, together with the precise information about asymptotic behavior at infinity of all solutions belonging to each type of solution classes.

Criteria for oscillatory behavior of solutions of fourth order half-linear differential equations of the form $$\left(\right|y^{\text{'}\text{'}}{|}^{\alpha }\mathrm{sgn}{y}^{\text{'}\text{'}}{)}^{\text{'}\text{'}}+q\left(t\right){\left|y\right|}^{\alpha }\mathrm{sgn}y=0,t\ge a>0,A$$ where $\alpha >0$ is a constant and $q\left(t\right)$ is positive continuous function on $[a,\infty )$, are given in terms of an increasing continuously differentiable function $\omega \left(t\right)$ from $[a,\infty )$ to $(0,\infty )$ which satisfies ${\int }_a^{\infty }1/\left(t\omega \left(t\right)\right)dt<\infty$.

In the paper a comparison theory of Sturm-Picone type is developed for the pair of nonlinear second-order ordinary differential equations first of which is the quasilinear differential equation with an oscillatory forcing term and the second is the so-called half-linear differential equation. Use is made of a new nonlinear version of the Picone’s formula.