Nonoscillatory solutions of half-linear Euler-type equation with n terms

We consider the half-linear Euler-type equation with n terms (Phi(x '))'+(gamma(p)/t(p) + Sigma(n-1)(j=1) mu(p)/t(p)Log(j)(2)t +mu/t(p)Log(n)(2)t)Phi(x)=0, Phi(x)=|x|(p-1)sgnx in the subcritical case when 01. The solutions of this nonoscillatory equation cannot be found in an explicit form and can be studied only asymptotically. In this paper, with the use of the perturbation principle, modified Riccati technique, and the fixed point theorem, we establish an asymptotic formula for one of its solutions

Remarks on the existence of nonoscillatory solutions of half-linear ordinary differential equations, II

summary:We consider the half-linear differential equation of the form $$(p(t)|x^{\prime }|^{\alpha }\mathrm{sgn}\,x^{\prime })^{\prime } + q(t)|x|^{\alpha }\mathrm{sgn}\,x = 0\,, \quad t \ge t_{0} \,,$$ under the assumption that $p(t)^{-1/\alpha }$ is integrable on $[t_{0}, \infty )$. It is shown that if a certain condition is satisfied, then the above equation has a pair of nonoscillatory solutions with specific asymptotic behavior as $t \rightarrow \infty$

Existence and asymptotic behavior of nonoscillatory solutions of half-linear ordinary differential equations

We consider the half-linear differential equation $$(|x'|^{\alpha}\mathrm{sgn}\,x')' + q(t)|x|^{\alpha}\mathrm{sgn}\,x = 0, \quad t \geq t_{0},$$ under the condition $$\lim_{t\to\infty}t^{\alpha}\int_{t}^{\infty}q(s)ds = \frac{\alpha^{\alpha}}{(\alpha+1)^{\alpha+1}}.$$ It is shown that if certain additional conditions are satisfied, then the above equation has a pair of nonoscillatory solutions with specific asymptotic behavior as $t\to\infty$

Half-linear differential equations : regular variation, principal solutions, and asymptotic classes

We are interested in the structure of the solution space of second-order half-linear differential equations taking into account various classifications regarding asymptotics of solutions. We focus on an exhaustive analysis of the relations among several types of classes which include the classes constructed with respect to the values of the limits of solutions and their quasiderivatives, the classes of regularly varying solutions, the classes of principal and nonprincipal solutions, and the classes of the solutions that obey certain asymptotic formulae. Many of our observations are new even in the case of linear differential equations, and we provide also the revision of existing results

On increasing solutions of half-linear delay differential equations

We establish conditions guaranteeing that all eventually positive increas-ing solutions of a half-linear delay differential equation are regularly varying andderive precise asymptotic formulae for them. The results presented here are new alsoin the linear case and some of the observations are original also for non-functionalequations. A substantial difference is pointed out between the delayed and non-delayed case for eventually positive decreasing solutions

Hille-Nehari type criteria and conditionally oscillatory half-linear differential equations

We study perturbations of the generalized conditionally oscillatory half-linear equation of the Riemann-Weber type. We formulate new oscillation and nonoscillation criteria for this equation and find a perturbation such that the perturbed Riemann-Weber type equation is conditionally oscillatory

Half-linear differential equations: Regular variation, principal solutions, and asymptotic classes

We are interested in the structure of the solution space of second-order half-linear differential equations taking into account various classifications regarding asymptotics of solutions. We focus on an exhaustive analysis of the relations among several types of classes which include the classes constructed with respect to the values of the limits of solutions and their quasiderivatives, the classes of regularly varying solutions, the classes of principal and nonprincipal solutions, and the classes of the so-lutions that obey certain asymptotic formulae. Many of our observations are new even in the case of linear differential equations, and we provide also the revision of existing results

Nonlinear differential equations in the framework of the Karamata theory

Cílem této diplomové práce je sjednotit a zobecnit známé výsledky z literatury, studovat asymptotické chování kladných regulárně se měnících řešení jisté třídy nelineárních diferenciálních rovnic (tzv. skoro pololineárních diferenciálních rovnic) pomocí dostupných nástrojů. Tato práce zahrnuje popis teorie regulární variace, některé informace o nelineárních diferenciálních rovnicích různých typů, detailní odvození výsledků týkajících se asymptotického chování řešení a příklady aplikace získaných výsledků.The goal of the thesis is to unify and generalize known results from literature, to study asymptotic behaviour of positive regularly varying solutions to the certain type of non-linear differential equations (known as nearly-half-linear differential equations) using available tools. This work includes description of theory of regular variation, some information on non-linear differential equations of various types, detailed derivations of results related to asymptotic behaviour of the solutions and examples of application of obtained results.

q-Karamatine funkcije i asimptotska svojstva rešenja nelinearnih q-diferencnih jednačina

The purpose of the doctoral dissertation is to determine the conditions for the existence and to examine in detail the asymptotic properties of solutions of the second order nonlinear q-difference equations, with an application of the theory of q-regular variation. The half-linear q-difference equation was analyzed in the framework of q-regular variation. Necessary and sufficient conditions for the existence of q-regularly varying solutions of the half-linear q- difference equation were obtained. Moreover, sufficient conditions for all eventually positive solutions to be q-regularly varying were examined. In cases where this is possible, the application of q-Karamata’s integration theorem and properties of q-regularly varying functions have been used to determine the precise asymptotic formula of different types of solutions, which accurately describes the behavior of these solutions in long time intervals, which is of special importance from the point of view of application. The obtained results in the q-calculus were compared with the known results in the continuous and the discrete case, but also, they were used to obtain new results in the discrete asymptotic theory. The sublinear second order q-difference equation of Emden-Fowler type was also analyzed in the framework of q-regularly varying functions. Assuming that the coefficients of this equation are q-regularly varying functions, necessary and sufficient conditions for the existence of strongly increasing and strongly decreasing solutions, as well as their asymptotic representations at infinity, have been determined. Moreover, it was shown that all q-regularly varying solutions of the same regularity index have the same asymptotic representation at infinity. The obtained results enabled the complete structure of the set of q-regularly varying solutions to be presented