Global Behavior of an Arbitrary-Order Nonlinear Difference Equation with a Nonnegative Function
Let <inline-formula> <math display="inline"> <semantics> <mrow> <mi>k</mi> <mo>,</mo> <mi>l</mi> </mrow> </semantics> </math> </inline-formula> be two integers with <inline-formula> <math display=&...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-05-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/5/825 |
| _version_ | 1797567568915988480 |
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| author | Wen-Xiu Ma |
| author_facet | Wen-Xiu Ma |
| author_sort | Wen-Xiu Ma |
| collection | DOAJ |
| description | Let <inline-formula> <math display="inline"> <semantics> <mrow> <mi>k</mi> <mo>,</mo> <mi>l</mi> </mrow> </semantics> </math> </inline-formula> be two integers with <inline-formula> <math display="inline"> <semantics> <mrow> <mi>k</mi> <mo>≥</mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>l</mi> <mo>≥</mo> <mn>2</mn> </mrow> </semantics> </math> </inline-formula>, <i>c</i> a real number greater than or equal to 1, and <i>f</i> a multivariable function satisfying <inline-formula> <math display="inline"> <semantics> <mrow> <mi>f</mi> <mo>(</mo> <msub> <mi>w</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>w</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>w</mi> <mn>3</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>w</mi> <mi>l</mi> </msub> <mo>)</mo> <mo>≥</mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula> when <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>w</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>w</mi> <mn>2</mn> </msub> <mo>≥</mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula>. We consider an arbitrary order nonlinear difference equation with the indicated function <i>f</i>: <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>z</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mi>c</mi> <mrow> <mo>(</mo> <msub> <mi>z</mi> <mi>n</mi> </msub> <mo>+</mo> <msub> <mi>z</mi> <mrow> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mi>c</mi> <mo>−</mo> <mn>1</mn> <mo>)</mo> </mrow> <msub> <mi>z</mi> <mi>n</mi> </msub> <msub> <mi>z</mi> <mrow> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msub> <mo>+</mo> <mi>c</mi> <mi>f</mi> <mrow> <mo>(</mo> <msub> <mi>z</mi> <mi>n</mi> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>w</mi> <mn>3</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>w</mi> <mi>l</mi> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <msub> <mi>z</mi> <mi>n</mi> </msub> <msub> <mi>z</mi> <mrow> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msub> <mo>+</mo> <mi>f</mi> <mrow> <mo>(</mo> <msub> <mi>z</mi> <mi>n</mi> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>w</mi> <mn>3</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>w</mi> <mi>l</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>c</mi> </mrow> </mfrac> <mo>,</mo> <mspace width="4pt"></mspace> <mi>n</mi> <mo>≥</mo> <mn>0</mn> <mo>,</mo> </mrow> </semantics> </math> </inline-formula> where initial values <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>z</mi> <mrow> <mo>−</mo> <mi>k</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow> <mo>−</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>z</mi> <mn>0</mn> </msub> </mrow> </semantics> </math> </inline-formula> are positive and <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>w</mi> <mi>i</mi> </msub> <mo>,</mo> <mspace width="4pt"></mspace> <mi>i</mi> <mo>≥</mo> <mn>3</mn> </mrow> </semantics> </math> </inline-formula>, are arbitrary functions of <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>z</mi> <mi>j</mi> </msub> <mo>,</mo> <mspace width="4pt"></mspace> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>≤</mo> <mi>j</mi> <mo>≤</mo> <mi>n</mi> </mrow> </semantics> </math> </inline-formula>. We classify its solutions into three types with different asymptotic behaviors, and verify the global asymptotic stability of its positive equilibrium solution <inline-formula> <math display="inline"> <semantics> <mrow> <mover accent="true"> <mi>z</mi> <mo>¯</mo> </mover> <mo>=</mo> <mi>c</mi> </mrow> </semantics> </math> </inline-formula>. |
| first_indexed | 2024-03-10T19:44:47Z |
| format | Article |
| id | doaj.art-6d05fcaa77a34404a44bfd9ae2c20ef8 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T19:44:47Z |
| publishDate | 2020-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-6d05fcaa77a34404a44bfd9ae2c20ef82023-11-20T00:57:13ZengMDPI AGMathematics2227-73902020-05-018582510.3390/math8050825Global Behavior of an Arbitrary-Order Nonlinear Difference Equation with a Nonnegative FunctionWen-Xiu Ma0Department of Mathematics, Zhejiang Normal University, Jinhua 321004, ChinaLet <inline-formula> <math display="inline"> <semantics> <mrow> <mi>k</mi> <mo>,</mo> <mi>l</mi> </mrow> </semantics> </math> </inline-formula> be two integers with <inline-formula> <math display="inline"> <semantics> <mrow> <mi>k</mi> <mo>≥</mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>l</mi> <mo>≥</mo> <mn>2</mn> </mrow> </semantics> </math> </inline-formula>, <i>c</i> a real number greater than or equal to 1, and <i>f</i> a multivariable function satisfying <inline-formula> <math display="inline"> <semantics> <mrow> <mi>f</mi> <mo>(</mo> <msub> <mi>w</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>w</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>w</mi> <mn>3</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>w</mi> <mi>l</mi> </msub> <mo>)</mo> <mo>≥</mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula> when <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>w</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>w</mi> <mn>2</mn> </msub> <mo>≥</mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula>. We consider an arbitrary order nonlinear difference equation with the indicated function <i>f</i>: <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>z</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mi>c</mi> <mrow> <mo>(</mo> <msub> <mi>z</mi> <mi>n</mi> </msub> <mo>+</mo> <msub> <mi>z</mi> <mrow> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mi>c</mi> <mo>−</mo> <mn>1</mn> <mo>)</mo> </mrow> <msub> <mi>z</mi> <mi>n</mi> </msub> <msub> <mi>z</mi> <mrow> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msub> <mo>+</mo> <mi>c</mi> <mi>f</mi> <mrow> <mo>(</mo> <msub> <mi>z</mi> <mi>n</mi> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>w</mi> <mn>3</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>w</mi> <mi>l</mi> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <msub> <mi>z</mi> <mi>n</mi> </msub> <msub> <mi>z</mi> <mrow> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msub> <mo>+</mo> <mi>f</mi> <mrow> <mo>(</mo> <msub> <mi>z</mi> <mi>n</mi> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>w</mi> <mn>3</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>w</mi> <mi>l</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>c</mi> </mrow> </mfrac> <mo>,</mo> <mspace width="4pt"></mspace> <mi>n</mi> <mo>≥</mo> <mn>0</mn> <mo>,</mo> </mrow> </semantics> </math> </inline-formula> where initial values <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>z</mi> <mrow> <mo>−</mo> <mi>k</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>z</mi> <mrow> <mo>−</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>z</mi> <mn>0</mn> </msub> </mrow> </semantics> </math> </inline-formula> are positive and <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>w</mi> <mi>i</mi> </msub> <mo>,</mo> <mspace width="4pt"></mspace> <mi>i</mi> <mo>≥</mo> <mn>3</mn> </mrow> </semantics> </math> </inline-formula>, are arbitrary functions of <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>z</mi> <mi>j</mi> </msub> <mo>,</mo> <mspace width="4pt"></mspace> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>≤</mo> <mi>j</mi> <mo>≤</mo> <mi>n</mi> </mrow> </semantics> </math> </inline-formula>. We classify its solutions into three types with different asymptotic behaviors, and verify the global asymptotic stability of its positive equilibrium solution <inline-formula> <math display="inline"> <semantics> <mrow> <mover accent="true"> <mi>z</mi> <mo>¯</mo> </mover> <mo>=</mo> <mi>c</mi> </mrow> </semantics> </math> </inline-formula>.https://www.mdpi.com/2227-7390/8/5/825difference equationpositive equilibriumoscillatory solutionstrong negative feedbackglobal asymptotic stability |
| spellingShingle | Wen-Xiu Ma Global Behavior of an Arbitrary-Order Nonlinear Difference Equation with a Nonnegative Function Mathematics difference equation positive equilibrium oscillatory solution strong negative feedback global asymptotic stability |
| title | Global Behavior of an Arbitrary-Order Nonlinear Difference Equation with a Nonnegative Function |
| title_full | Global Behavior of an Arbitrary-Order Nonlinear Difference Equation with a Nonnegative Function |
| title_fullStr | Global Behavior of an Arbitrary-Order Nonlinear Difference Equation with a Nonnegative Function |
| title_full_unstemmed | Global Behavior of an Arbitrary-Order Nonlinear Difference Equation with a Nonnegative Function |
| title_short | Global Behavior of an Arbitrary-Order Nonlinear Difference Equation with a Nonnegative Function |
| title_sort | global behavior of an arbitrary order nonlinear difference equation with a nonnegative function |
| topic | difference equation positive equilibrium oscillatory solution strong negative feedback global asymptotic stability |
| url | https://www.mdpi.com/2227-7390/8/5/825 |
| work_keys_str_mv | AT wenxiuma globalbehaviorofanarbitraryordernonlineardifferenceequationwithanonnegativefunction |