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Extreme and moderate solutions of nonoscillatory second order half-linear differential equations

Volume 128 / 2022

Jaroslav Jaroš, Takaŝi Kusano, Tomoyuki Tanigawa Annales Polonici Mathematici 128 (2022), 49-85 MSC: Primary 34C11; Secondary 34D05. DOI: 10.4064/ap201216-12-8 Published online: 25 November 2021

Abstract

An existence and asymptotic theory is built for second order half-linear differential equations of the form $${\rm (A)}\qquad (p(t)|x^{\prime }|^\alpha {\rm sgn} \,x^{\prime })^{\prime } =q(t)|x|^\alpha {\rm sgn}\, x, \quad t \geq a \gt 0, $$ where $\alpha \gt 0$ is constant and $p(t)$ and $q(t)$ are positive continuous functions on $[a,\infty )$, in which a crucial role is played by a pair of the generalized Riccati differential equations $${\rm (R1)}\qquad u^{\prime }=q(t)-\alpha p(t)^{-{1}/{\alpha }}|u|^{1+{1}/{\alpha }},$$ $${\rm (R2)}\qquad v^{\prime }=p(t)^{-{1}/{\alpha }}-\frac {1}{\alpha }q(t)|v|^{1+\alpha }$$ associated with (A). An essential part of the theory is the construction of nonoscillatory solutions $x(t)$ of (A) enjoying explicit exponential-integral representations in terms of solutions $u(t)$ of (R1) or in terms of solutions $v(t)$ of (R2).

Authors

  • Jaroslav JarošDepartment of Mathematical Analysis
    and Numerical Mathematics
    Faculty of Mathematics, Physics and Informatics
    Comenius University
    842 48 Bratislava, Slovakia
    e-mail
  • Takaŝi KusanoDepartment of Mathematics
    Faculty of Science
    Hiroshima University
    739-8526 Higashi-Hiroshima, Japan
    e-mail
  • Tomoyuki TanigawaDepartment of Mathematical Sciences
    Osaka Prefecture University
    599-8531 Osaka, Japan
    e-mail

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