Opuscula Math. 32, no. 3 (2012), 559-577

Opuscula Mathematica

On the asymptotic behaviour of solutions to a linear functional equation

Dariusz Sokołowski

Abstract. We investigate the asymptotic behaviour at infinity of solutions of the equation \[\varphi (x) = \int_S \varphi (x+M(s))\sigma(d s).\] We show among others that, under some assumptions, any positive solution of the equation which is integrable on a vicinity of infinity or vanishes at \(+\infty\) tends on some sequence to zero faster than some exponential function, but it does not vanish faster than another such function.

Keywords: linear functional equations and inequalities, solutions with a constant sign, asymptotic behaviour of solutions.

Mathematics Subject Classification: 39B12, 39B22, 39B62, 26A12.

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  • Dariusz Sokołowski
  • Silesian Universit, Institute of Mathematics, ul. Bankowa 14, 40-007 Katowice, Poland
  • Received: 2011-03-15.
  • Revised: 2011-08-08.
  • Accepted: 2011-09-17.
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Cite this article as:
Dariusz Sokołowski, On the asymptotic behaviour of solutions to a linear functional equation, Opuscula Math. 32, no. 3 (2012), 559-577, http://dx.doi.org/10.7494/OpMath.2012.32.3.559

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