# Article

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Keywords:
positive solutions; boundary value problems; cone; fixed point theorem
Summary:
We study the existence of one-signed periodic solutions of the equations \begin{align} & x^{\prime \prime } (t) - a^2(t) x(t) + \mu f(t, x(t), x^{\prime }(t)) = 0, & x^{\prime \prime }(t) + a^2(t) x(t) = \mu f(t, x(t), x^{\prime }(t)), \end{align} where $\mu > 0$, $a: (-\infty , +\infty ) \rightarrow (0, \infty )$ is continuous and 1-periodic, $f$ is a continuous and 1-periodic in the first variable and may take values of different signs. The Krasnosielski fixed point theorem on cone is used.
References:
[1] Agarwal R. P., Grace S. R., O’Regan D.: Existence of positive solutions of semipositone Fredholm integral equation. Funkciałaj Equaciaj 45 (2002), 223–235. MR 1948600
[2] Agarwal R. P., O’Regan D., Wang J. Y.: Positive Solutions of Differential, Difference, Integral Equations. : Kluwer Academic Publishers, Dordrecht, Boston, London. 1999. MR 1680024
[3] Agarwal R. P., O’Regan D.: Infinite Interval Problems For Differential, Difference, Integral Equations. : Kluwer Acad. Publishers, Dordrecht, Boston, London. 2001. MR 1845855
[4] Deimling K.: Nonlinear Functional Analysis. : Springer, New York. 1985. MR 0787404
[5] Guo D., Lakshmikannthan V.: Nonlinear Problems in Abstract Cones. : Academic Press, San Diego. 1988. MR 0959889
[6] Santanilla J.: Nonnegative solutions to boundary value problems for nonlinear first and second order ordinary differential equations. J. Math. Annal. Appl. 126 (1987), 397–408. DOI 10.1016/0022-247X(87)90049-7 | MR 0900756 | Zbl 0629.34017
[7] Torres P. J.: Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem. J. Diff. Eq. 190 (2003), 643–662. DOI 10.1016/S0022-0396(02)00152-3 | MR 1970045
[8] Zima M.: Positive Operators in Banach Spaces, Their Applications. : Wydawnictwo Uniwersytetu Rzeszowskiego, Rzeszów. 2005. MR 2493071

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