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Asymptotic integration of nonlinear $$\phi$$-Laplacian differential equations. (English) Zbl 1192.34059
The aim is to study the existence of solutions to initial values problems for a $$\phi$$-Laplace-like operator of the form:
$\begin{cases} (\phi(u'))'+f(t,u,u')=0, t\geq t_0\geq 1; \\ u(t_0)=u_0, u'(t_0)=u_1, \end{cases}$
where $$\phi:{\mathbb{R}}\to{\mathbb{R}}$$ is an increasing homeomorphism with a locally Lipchitzian’s invers satisfying $$\phi(0)=0$$ and the non-linearity $$f$$ verifies the inequality
$|f(t,u,v)|\leq h(t)\left[p_1\left(\frac{|u|}{t}\right)+p_2(|v|)\right], \quad t\geq t_0,u,v\in{\mathbb{R}}.$
Here, the function $$h:[t_0,+\infty)\to[0,+\infty)$$ is continuous and the functions $$p_1,p_2:[0,+\infty)\to(0,+\infty)$$ are continuous and nondecreasing.
This type of homeomorphism encompasses the usual $$p$$-Laplacian operator with $$0<p\leq 2$$.
The authors derive sufficient conditions under which all global solutions are asymptotic to $$at+b$$, as $$t\to +\infty$$, where $$a,b$$ are real numbers.
The paper generalizes the results of R. P. Agarwal, S. Djebali, T. Moussaoui and O. G. Mustafa [J. Comput. Appl. Math. 202, No. 2, 352–376 (2007; Zbl 1123.34038)] and the result of Ch. G. Philos, I. K. Purnaras and P. Ch. Tsamatos [Nonlinear Anal., Theory Methods Appl. 59, No. 7 (A), 1157–1179 (2004; Zbl 1094.34032)] to the $$\phi$$-Laplacian-like problems.

MSC:
 34D05 Asymptotic properties of solutions to ordinary differential equations 34E10 Perturbations, asymptotics of solutions to ordinary differential equations 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
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References:
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