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Periodic solutions for indefinite singular equations with singularities in the spatial variable and non-monotone nonlinearity

  • * Corresponding author: Manuel Zamora

    * Corresponding author: Manuel Zamora
The first author is supported by projects UBACyT 20020120100029BA and CONICET PIP11220130100006CO, and the second author was supported by FONDECYT, project no. 11140203.
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  • TWe prove the existence of $T$-periodic solutions for the second order non-linear equation

    $ {\left( {\frac{{u'}}{{\sqrt {1 - {{u'}^2}} }}} \right)^\prime } = h(t)g(u), $

    where the non-linear term $g$ has two singularities and the weight function $h$ changes sign. We find a relation between the degeneracy of the zeroes of the weight function and the order of one of the singularities of the non-linear term. The proof is based on the classical Leray-Schauder continuation theorem. Some applications to important mathematical models are presented.

    Mathematics Subject Classification: Primary: 34C25, 34B18; Secondary: 34B30.


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  • Figure 1.  The figure illustrates a possible behaviour of any $T-$periodic solution $u$ of (20) when $t_*$ is included on $[\bar{\ell}_i, \bar{\ell}_i')$ (Case Ⅰ.) or if $t_*\in (\bar{\ell_i}, \bar{\ell}_i']$ (Case Ⅱ.).

    Figure 2.  The figure illustrates a possible behaviour of the $T-$periodic solution $u$ of (16) when $t_*$ is included on $[a_i, a_i']$, distinguishing if $t_*\in [a_i, \bar{\ell}_i)$ (Case Ⅰ. a)) or $t_*\in (\bar{\ell}_i', a_i']$ (Case Ⅰ. b)).

    Figure 3.  The figure illustrates a possible behaviour of the $T-$periodic solution $u$ of (16) on the interval $[\bar{\ell}_i', \ell_i']$, assuming that $u(a_i')<\alpha+2\delta$.

    Figure 4.  The figure illustrates a possible behaviour of the $T-$periodic solution $u$ of (16) on the interval $[\bar{\ell}_i', \ell_i']$, assuming that $u(a_i')\geq \alpha+2\delta$.

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