Theory and Modern Applications
Table 3 Monotonicity of map \(\varphi (y_{k}^{+})\)
Case | Interval | \(\mathcal{A}(y_{k}^{+})\) | f(y) | \(\varphi (y_{k}^{+} )\) |
|---|---|---|---|---|
\(J_{21}\)(a) | In | \([0,{y_{{Q_{0}}}}]\) | \([0,{y_{{Q_{1}}}}]\) | \([0,{y_{{Q_{0}}}}]\) |
De | \(({y_{{Q_{0}}}}, + \infty )\) | – | \(({y_{{Q_{0}}}}, + \infty )\) | |
\(J_{21}\)(b) | In | \([0,{y_{{Q_{0}}}}]\) | – | \(({y_{{Q_{0}}}}, + \infty )\) |
De | \(({y_{{Q_{0}}}}, + \infty )\) | \([0,{y_{{Q_{1}}}}]\) | \([0,{y_{{Q_{0}}}}]\) | |
\(J_{21}\)(c) | In | \([0,{y_{{Q_{0}}}}]\) | \(( {\frac{{\sqrt {\tau \delta } - 1}}{\delta },{y_{{Q_{1}}}}} ]\) | \(({y_{d}},{y_{{Q_{0}}}})\), \(({y_{u}}, + \infty )\) |
De | \(({y_{{Q_{0}}}}, + \infty )\) | \([ {0,\frac{{\sqrt {\tau \delta } - 1}}{\delta }} ]\) | \([0,{y_{d}}]\), \([{y_{{Q_{0}}}},{y_{u}}]\) | |
\(J_{22}\)(d) | In | \([0,{y_{{P_{2}}}}]\) | \([0,{y_{{T}}}]\) | \([0,{y_{{P_{2}}}}]\) |
De | \([{y_{{P_{1}}}}, + \infty )\) | – | \([{y_{{P_{1}}}}, + \infty )\) | |
\(J_{22}\)(e) | In | \([0,{y_{{P_{2}}}}]\) | – | \([{y_{{P_{1}}}}, + \infty )\) |
De | \([{y_{{P_{1}}}}, + \infty )\) | \([0,{y_{{T}}}]\) | \([0,{y_{{P_{2}}}}]\) | |
\(J_{22}\)(f) | In | \([0,{y_{{P_{2}}}}]\) | \(( {\frac{{\sqrt {\tau \delta } - 1}}{\delta },{y_{{T}}}} ]\) | \(({y_{D}},{y_{{P_{2}}}}]\), \(({y_{U}}, + \infty )\) |
De | \([{y_{{P_{1}}}}, + \infty )\) | \([ {0,\frac{{\sqrt {\tau \delta } - 1}}{\delta }} ]\) | \([0,{y_{D}}]\), \([{y_{{P_{1}}}},{y_{U}}]\) |