We investigate the global asymptotic behavior of solutions of the difference equation <mml:math id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>x</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mstyle displaystyle="true"><mml:msubsup><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mrow><mml:msub><mml:mi>x</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mstyle></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mi>A</mml:mi><mml:msub><mml:mi>x</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>, <mml:math id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>ℕ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math>, where <mml:math id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>A</mml:mi><mml:mo>∈</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>∞</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>, <mml:math id="E4" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>k</mml:mi><mml:mo>∈</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:mrow></mml:math>, and the initial values <mml:math id="E5" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>x</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:msub><mml:mi>x</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math> are arbitrary negative numbers. Asymptotics of some positive solutions of the equation are also found.

Stevo Stevic

On a Discrete Epidemic Model

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Discrete Dynamics in Nature and Society publishes research that links basic and applied research relating to discrete dynamics of complex systems encountered in the natural and social sciences.

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Discrete Dynamics in Nature and Society

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Table of Contents: 2007

On a Discrete Epidemic Model