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Lemma 2. Suppose \( f\left( z\right) \) is holomorphic in a circle \( \left| {z - {z}_{0}}\right| \leqq R \), and has at least \( n \) zeros in the circle \( \left| {z - {z}_{0}}\right| \leqq r < R \) (counting multiplicities). Assume \( f\left( {z}_{0}\right) \neq 0 \) . Then\n\n\[{\left\lbrack \left( R - r\right) /r\... | Proof. We may assume \( {z}_{0} = 0 \) . Let\n\n\[f\left( z\right) = \mathop{\prod }\limits_{{i = 1}}^{n}\left( {z - {a}_{i}}\right) \varphi \left( z\right)\]\n\nwhere \( {a}_{i} \) are the zeros of \( f \) in the small circle. Then obviously on the large circle, we have\n\n\[\left| {\varphi \left( z\right) }\right| \l... | Yes |
Proposition 3. Let a be a number \( > 0 \) . Then \( {L}^{\prime }/L\left( s\right) \) is bounded for \( \operatorname{Re}\left( s\right) = 1 + a \), and\n\n\[{\Gamma }^{\prime }/\Gamma \left( s\right) = \log s + O\left( {1/{\left| s\right| }^{2}}\right)\]\n\n\[\\text{for}\\operatorname{Re}\left( s\right) = 1 + a\\text... | Proof. The first assertion follows at once from the product expansion of \( L \), and the second follows from Stirling’s formula (differentiating inside the integral giving the error term). | No |
Lemma 3. On the space of almost BCL functions, the preceding limit exists. | Proof. In a neighborhood of 0 , the denominator\n\n\[ \n\\left| {{e}^{x/2} - {e}^{-x/2}}\\right| \n\]\n\nbehaves like \( \\left| x\\right| \\left( {\\;\\operatorname{mod}\\{x}^{2}\\right) \) . It will then suffice to prove our assertion when we replace this denominator by \( \\left| x\\right| \) and \( \\beta \) by a f... | Yes |
Lemma 5. Let \( \\left\\{ {\\beta }_{n}\\right\\} \) be a sequence of \( {BCL} \) functions, converging to a \( {BCL} \) function \( \\beta \) . Assume also that the functions \( \\left\\{ {\\beta }_{n}\\right\\} \) are uniformly bounded, that the convergence is uniform on every compact set, and that the Lipshitz const... | Proof. We write for each \( n \) ,\n\n\[ \n{\\beta }_{n}\\left( x\\right) = {\\beta }_{n}\\left( x\\right) - {\\beta }_{n}\\left( 0\\right) + {\\beta }_{n}\\left( 0\\right) .\n\] \n\nThen \( {\\beta }_{n}\\left( 0\\right) \) converges to \( \\beta \\left( 0\\right) \) . This reduces the proof to considering the sum of ... | Yes |
Lemma 7. The functional \( W \) is a distribution. Let \( \beta \) be a BCL function. The convolution of \( W \) with \( {T}_{\beta } \) (the distribution represented by \( \beta \) ) is represented by the function whose value at \( x \) is \( W\left( {{\beta }^{ - }x}\right) \) . Symbolically, \[ \left( {W * {T}_{\bet... | Proof. If \( T \) is a distribution, which is represented outside some compact set by a function tending exponentially to 0 at infinity, and \( \alpha \) is a \( {C}^{\infty } \) - function which is bounded, then by the theory of distributions, one knows that \[ T * {T}_{\alpha } \] is represented by the function \( T\... | No |
Lemma 8. Let \( x \) be a characteristic function of an interval which does not hove 0 as its endpoints. Then the distribution \( W * {T}_{\chi } \) is represented by a \( {C}^{\infty } \) -function locally at every point other than the endpoints of the interval, and its value at such a point \( x \) is the value \( W\... | Proof. This follows from the general properties of convolutions of distributions, e.g. TD, Chapter VI, Theorem III of §3 and Theorem XI of §4. | No |
Lemma 9. The convergence of this limit is uniform on every compact set. | Proof. The first assertion is clear. As to the second, observe that the sum in the expression for \( {\mathrm{q}}_{M} \) is bounded from below by\n\n\[ \frac{n + \frac{1}{2}}{{\left( n + \frac{1}{2}\right) }^{2} + {t}^{2}} \leqq \frac{1}{n + \frac{1}{2}}. \]\n\nSay \( t > 0 \) . For \( M \leqq t \) the expression for \... | No |
Lemma 10. As a distribution, \[ {\widehat{T}}_{\mathrm{q}} = \widehat{\mathrm{q}} = - {\pi W} \] | Proof. The boundedness condition of Lemma 9 insures that the limit of the Fourier transforms is the Fourier transform of the limit. (Use TD, Example 3 of Chapter VII, §7.) | No |
Proposition 4. Let \( F \) be a function satisfying properties (A) and (B). Let\n\n\[ \n\psi = {\widehat{F}}^{ - }\text{. }\n\]\n\nThen \( \langle \psi ,\Psi \rangle \) exists, and\n\n\[ \n\langle \psi ,\mathrm{T}\rangle = \mathop{\lim }\limits_{{M \rightarrow \infty }}\left\langle {F,{\widehat{\mathrm{T}}}_{M}}\right\... | Proof. Writing \( F \) as a sum of an even function and an odd function, i.e.\n\n\[ \nF\left( x\right) = \frac{F\left( x\right) + F\left( {-x}\right) }{2} + \frac{F\left( x\right) - F\left( {-x}\right) }{2},\n\]\n\nhas the same effect on the Fourier transform. Thus it suffices to prove our proposition for even and odd ... | Yes |
Denoting as usual by \( {T}_{f} \) the distribution represented by a function \( f \), we have\n\n\[ {\widehat{T}}_{\psi \mathrm{q}} = {\widehat{T}}_{\mathrm{q}} * {T}_{F} \] | Proof. This follows from the theory of distributions, and hypotheses (i) and (ii), because \( W \) decreases rapidly and \( {T}_{f} \) is tempered \( ({TD} \) , Theorem XV of Chapter VII, §8). | No |
Lemma 12. Let \( f \) be the function \( \psi \mathrm{T} \) . Except at \( \pm {a}_{\nu },{\widehat{T}}_{f} \) is represented by the continuous function given by the integral\n\n\[{\int }_{-\infty }^{+\infty }f\left( t\right) {e}^{-{itx}}{dt}\] | Proof. Let \( A \) be a compact interval, \( - T \leqq t \leqq T \), and let \( {f}_{A} \) be the function \( f \) multiplied by the characteristic function of \( A \) . Our statement is true if we replace \( f \) by \( {f}_{A} \) . Hence \( {T}_{fA} \) approaches \( {T}_{f} \) as a tempered distribution, and consequen... | Yes |
Theorem 2.1 If \( {R}_{0} \leq 1 \), then \( {E}_{f} \) is globally asymptotically stable. | Proof Let us consider the Lyapunov function\n\n\[ \n{U}_{f}\left( {T, I, V}\right) = \frac{1}{1 + a{T}_{0}}\left( {T - {T}_{0} - {T}_{0}\ln \frac{T}{{T}_{0}}}\right) + I + \frac{p}{k}V + \frac{m}{h}C. \]\n\nIt is easily seen that \( {U}_{f}\left( {T, I, V, C}\right) \geq 0 \) and \( {U}_{f}\left( {T, I, V, C}\right) = ... | Yes |
Theorem 2.2 Assume that \( f : \left\lbrack {0, T}\right\rbrack \times R \rightarrow R \) is continuous and satisfies\n\n\[ \mathop{\lim }\limits_{{u \rightarrow 0}}\mathop{\sup }\limits_{{t \in \left\lbrack {0, T}\right\rbrack }}\left| \frac{f\left( {t, u}\right) }{u}\right| = m \]\n\nwhere \( m \) is a positive const... | In order to use Leray-Schauder nonlinear alternative, we set \( r < \delta \) . Using the same method in the proof of Theorem 2.1, we can prove \( A\overline{{K}_{r}} \subseteq \overline{{K}_{r}} \) is a completely continuous operator. Taking \( u \in \partial {K}_{r} \) such that \( u = {\mu Au},0 < \mu < 1 \) . By th... | Yes |
Consider the following anti-periodic boundary value problem for the third-order differential equation\n\[ \left\{ \begin{array}{l} {u}^{\prime \prime \prime } - {u}^{\prime } + \left( {t + 1}\right) {u}^{2} = 0,\;t \in J = \left\lbrack {0,2}\right\rbrack , \\ {u}^{\prime }\left( 0\right) + {u}^{\prime }\left( 2\right) ... | Here, \( \rho = \lambda = 1, T = 2, f\left( {t, u}\right) = \left( {t + 1}\right) {u}^{2} \) is continuous and\n\[ \mathop{\lim }\limits_{{u \rightarrow 0}}\mathop{\sup }\limits_{{t \in \left\lbrack {0,2}\right\rbrack }}\left| \frac{f\left( {t, u}\right) }{u}\right| = 0,\;\frac{\left( {{e}^{\varrho T} - 1}\right) \lamb... | Yes |
Consider the following anti-periodic boundary value problem for the third-order differential equation\n\[ \n\\left\\{ \\begin{array}{l} {u}^{\\prime \\prime \\prime } - 4{u}^{\\prime } + \\frac{1}{2}\\sin \\left( {u{e}^{ut}}\\right) = 0,\\;t \\in J = \\left\\lbrack {0,\\pi }\\right\\rbrack , \\\\ {u}^{\\prime }\\left( ... | Obviously. The anti-periodic boundary value problem for the third-order differential equation(3.2) satisfies all assumptions of Theorem 2.2. Hence, it has at least one solution. | No |
Theorem 1.1 \\( \\; \\) Let \\( \\left( {{u}_{1}\\left( {x, t}\\right) ,{u}_{2}\\left( {x, t}\\right) ,{u}_{3}\\left( {x, t}\\right) ,{u}_{4}\\left( {x, t}\\right) }\\right) \\in \\lbrack C\\left( {\\overline{\\Omega }\\times \\lbrack 0, T}\\right) )\\xrightarrow[]{\\cap {C}^{2,1}}\\left( {\\Omega \\times \\left( {0, T... | Proof Since the reaction functions of (0.1) are smooth in \\( {\\Re }_{ + }^{3} \\), from the standarded theory of PDE shows that has a unique local solution. By the strong maximum principle (see [6]), we know that if \\( {u}_{i0}\\left( x\\right) \\neq 0\\left( {i = 1,2,3,4}\\right) \\), then \\( {u}_{i}o\\left( {x, t... | Yes |
Theorem 1.1 For the model (1.3), if \( g \leq k \leq g + m,{v}_{1} \leq h \), then \( 0 \leq {c}_{i0}\left( t\right) \leq 1,0 \leq \) \( {c}_{e}\left( t\right) \leq 1 \), for any \( t \in \left\lbrack {0, T}\right\rbrack \) . | Proof The proof is completed by the standard argument [5], we omit it here. | No |
Theorem 2.2 If \( \left( {{u}^{ * },{v}^{ * }}\right) \) is an optimal control and \( \left( {{p}^{ * },{c}_{0}^{ * },{c}_{e}^{ * }}\right) \) is the corresponding optimal state, then\n\n\[ \n{u}_{i}^{ * }\left( {a, t}\right) = {\mathcal{L}}_{i}\left( \frac{\left\lbrack {{w}_{i}\left( {a, t}\right) - {q}_{i}\left( {a, ... | Proof The existence of a unique bound solution to the system (2.13) can be treated in a same manner as that for (1.3). For any given \( \left( {{\nu }_{1},{\nu }_{2}}\right) \in {\mathcal{F}}_{\mathcal{U}}\left( {{u}^{ * },{v}^{ * }}\right) \) (the tangent cone of at \( \left( {{u}^{ * },{v}^{ * }}\right) ,\;{u}^{ * } ... | Yes |
Consider the delay integral equation\n\n\[ \nx\left( t\right) = {\int }_{t - \sigma }^{t}\frac{a\left( s\right) }{1 + {x}^{2}\left( s\right) }{ds},\;t \in \mathbb{R}.\n\]\n\nwhich is a model arising in an epidemic problem. If \( a\left( t\right) \in {WPAA}\left( {\mathbb{R},{\mathbb{R}}^{ + },{\rho }_{1},{\rho }_{2}}\r... | Let \( \psi \left( \alpha \right) = \frac{1 + {\alpha }^{2}}{2},\alpha \in \left( {0,1}\right) \), it is easy to see that \( \psi \in \Lambda \) . For all \( t \in \mathbb{R},\alpha \in \left( {0,1}\right) \) and \( x \in \left\lbrack {0,\alpha }\right\rbrack , \) one has \( \left( {1\widetilde{ - }{\alpha }^{2}}\right... | Yes |
Consider the following Lasota-Wazewska model:\n\n\[ \n{x}^{\prime }\left( t\right) = - \lambda \left( t\right) x\left( t\right) + \mu \left( t\right) {e}^{-{\gamma x}\left( {t - \tau }\right) },\;t \in \mathbb{R}, \n\] \n\nwhich is a model for the survival of red blood cells in an animal. Assume that \( \gamma ,\tau > ... | Let \( \alpha \left( t\right) = h\left( t\right) = 0,\beta \left( t\right) = 1, f\left( {t, x}\right) = \mu \left( {t + \tau }\right) {e}^{-{\gamma x}} \) and \( a\left( {t, s}\right) = {e}^{-{\int }_{t - s + \tau }^{t}\lambda \left( u\right) {du}}{\mathbf{1}}_{\left\lbrack \tau , + \infty \right\rbrack }\left( s\right... | Yes |
In (4), let \( f\left( {t, x}\right) = \frac{v\left( t\right) }{\sqrt{1 + x}}, v\left( t\right) \mathrel{\text{:=}} m\sin \frac{1}{{\cos }^{2}t + {\cos }^{2}{\pi t} + 2} + m{\left( 1 + {t}^{2}\right) }^{-1} \) , where \( 0 < m < \frac{1}{4 + \pi } \), then \( v\left( t\right) \in {WPAA}\left( {\mathbb{R},{\mathbb{R}}^{... | Let \( \psi \left( \alpha \right) = \sqrt{\alpha },\alpha \in \left( {0,1}\right) \), then \( \psi \in \Lambda \) . For all \( t \in \mathbb{R},\alpha \in \left( {0,1}\right) \) and \( x \in \left\lbrack {0,\alpha }\right\rbrack \), one has\n\n\[ \frac{f\left( {t,{\alpha }^{-1}x}\right) }{f\left( {t, x}\right) } = \fra... | Yes |
Lemma 1.4 \( {}^{\left\lbrack 4\right\rbrack } \) Let \( \alpha > 0 \), and \( f;g : \left\lbrack {a;b}\right\rbrack \rightarrow R \) be \( {C}^{n} \) functions. Then,\n\n\[ \n{\int }_{a}^{b}g{\left( t\right) }^{C}{D}_{{a}^{ + }}^{\alpha }f\left( t\right) {dt} = {\int }_{a}^{b}f\left( t\right) {D}_{{b}^{ - }}^{\alpha }... | In the particular case when \( 0 < \alpha < 1 \), by Lemma1.4. we have \n\n\[ \n{\int }_{a}^{b}g{\left( t\right) }^{C}{D}_{{a}^{ + }}^{\alpha }f\left( t\right) {dt} = {\int }_{a}^{b}f\left( t\right) {D}_{{b}^{ - }}^{\alpha }g\left( t\right) {dt} + {\left. f\left( t\right) {I}_{{b}^{ - }}^{1 - \alpha }g\left( t\right) \... | Yes |
Theorem 3.2 Let \( L\left( {t;y;z;u}\right) \) be jointly convex (concave) in \( \left( {y, z, u}\right) \) . If \( {y}_{0} \) satisfies conditions (3), then \( {y}_{0} \) is a global minimizer (maximizer) to problems (1)-(2). | Proof We will give the proof for only the convex case (and similarly we can prove it for the concave case). Since \( L \) is jointly convex in \( \left( {y, z, u}\right) \) for any admissible function \( {y}_{0} + \eta \left( t\right) ,\eta \left( t\right) \) is an admissible function such that \( \eta \left( a\right) ... | Yes |
Theorem 2.1 Suppose that assumptions \( \left( {A}_{1}\right) \left( {A}_{2}\right) \) and \( \left( {A}_{3}\right) \) hold. Then the prey species \( \left( {{x}_{1}\left( t\right) ,{x}_{2}\left( t\right) }\right) \) of system (0.1) is permanent. | Here, we prove Theorem 2.1 under assumptions \( \left( {A}_{1}\right) \left( {A}_{2}\right) \) and \( \left( {A}_{3}\right) . We will use the Proposition 2.1 and Proposition 2.2 to complete the proof of Theorem 2.1. | No |
Theorem 2.2 Suppose that \( \left( {A}_{1}\right) \left( {A}_{2}\right) \) and \( \left( {A}_{3}\right) \) hold. If\n\n\[ \n{A}_{\omega }\left\lbrack {-g\left( t\right) + \frac{h\left( t\right) {x}_{1}^{ * }\left( t\right) }{k\left( t\right) + {\left( {x}_{1}^{ * }\left( t\right) \right) }^{2}}}\right\rbrack > 0.\n\]\n... | Here, we prove Theorem 2.2 under assumptions \( \left( {A}_{1}\right) \left( {A}_{2}\right) \) and \( \left( {A}_{3}\right) \) . We will use the Proposition 2.3 and Proposition 2.4 to complete the proof of Theorem 2.2. | No |
Lemma 1.2 Assume \( \left( {H}_{1}\right) - \left( {H}_{6}\right) \) hold. Then the solutions of Eq. (5) and (6) are defined on \( \lbrack - \bar{\tau },\infty ) \), positive and bounded on \( \lbrack 0,\infty ) \) . | Proof By Lemma 1.1, we only need to prove that the solutions of (7) and (8) are defined on \( \lbrack - \bar{\tau },\infty ) \) and are positive on \( \lbrack 0,\infty ) \) . From (7) and (8) we have that for any\n\n\[ \varphi \in L\left( {\left\lbrack {-\widetilde{\tau },0}\right\rbrack ,\lbrack 0,\infty }\right) ),\;... | Yes |
Example 2.1 Consider the impulsive delay differential equation\n\n\[ \left\{ \begin{array}{ll} {x}^{\prime }\left( t\right) = - \left( {\log 2}\right) \left| {\sin t}\right| x\left( t\right) + \frac{3}{{8\pi }\widetilde{b}}\left\lbrack {{\left( x\left( t - \tau \right) + 1\right) }^{2}{e}^{-x\left( {t - \tau }\right) }... | where \( \tau > 0 \) is a fixed constant, \( {b}_{k} \in \left( {-1,0}\right) \) . Set \( a\left( t\right) = - \left( {\log 2}\right) \left| {\sin t}\right| ,\omega = \pi, f\left( {t, u}\right) = \) \( \frac{3}{{8\pi }\bar{b}}\left\lbrack {{\left( u + 1\right) }^{2}{e}^{-u} + 1}\right\rbrack \) ; then we have \( \bar{b... | Yes |
Theorem 0.1 Suppose that \( \left( {H}_{1}\right) \) and \( \left( {H}_{2}\right) \) hold. Then, for any \( \tau > 0 \) sufficiently small, there exist speeds \( c \) such that the Eq. \( \left( {0.1}\right) \) with \( \left( {0.6}\right) \) has a traveling wave solution \( u\left( {x, t}\right) = U\left( z\right) \) c... | ## 1 The Proof of Theorem 0.1\n\nTo prove Theorem 0.1, we first introduce the following result on invariant manifolds which is due to Fenichel \( {}^{\left\lbrack 9\right\rbrack } \) . For convenience, we use a version of this theorem due to Jones \( {}^{\left\lbrack {10}\right\rbrack } \) .\n\nLemma 1.1 (Geometric Si | No |
Lemma 1.1 (Geometric Singular Perturbation Theorem). Given a \( {C}^{\infty } \) vector field of the form \( {x}^{\prime } = f\left( {x, y,\varepsilon }\right) ,\;{y}^{\prime } = {\varepsilon g}\left( {x, y,\varepsilon }\right) \) such that when \( \varepsilon = 0 \), the system has a compact, normally hyperbolic manif... | (I) which is locally invariant under the flow of the system;\n\n(II) which is \( {C}^{r} \) in \( x, y \) and \( \varepsilon \) ;\n\n(III) for which \( {\mathcal{M}}_{\varepsilon } = \left\{ {\left( {x, y}\right) \mid x = {h}^{\varepsilon }\left( y\right) }\right\} \) for some \( {C}^{r} \) function \( {h}^{\varepsilon... | Yes |
Lemma 1.3 System (1.1) has a positive periodic solution \( {S}^{ * }\left( t\right) \) and for every solution \( S\left( t\right) \) of (1.1), \( \left| {S\left( t\right) - {S}^{ * }\left( t\right) }\right| \rightarrow 0 \) as \( t \rightarrow \infty \), where \( {S}^{ * }\left( t\right) = \frac{q{e}^{-D\left( {t - {nT... | Therefore, system (0.1) has a microorganism-free periodic solution\n\n\[ \left( {{S}^{ * }\left( t\right) ,0,0}\right) = \left( {\frac{q{e}^{-D\left( {t - {nT}}\right) }}{1 - {e}^{-{DT}}},0,0}\right) ,\;t \in ({nT},\left( {n + 1}\right) T\rbrack . \] | Yes |
Theorem 3.1 For each positive solution \( \left( {S\left( t\right) ,{x}_{1}\left( t\right) ,{x}_{2}\left( t\right) }\right) \) of system (0.1), there exists a constant \( M > 0 \) such that \( S\left( t\right) \leq M,{x}_{i}\left( t\right) \leq M, i = 1,2 \) for \( t \) large enough. | Proof Define a function\n\n\[ V\left( t\right) = S\left( t\right) + \frac{{e}^{D{\tau }_{1}}{x}_{1}\left( {t + {\tau }_{1}}\right) }{\alpha } + \frac{{e}^{D{\tau }_{2}}{x}_{2}\left( {t + {\tau }_{2}}\right) }{\beta }, \]\n\nthen the upper right derivative of \( V\left( t\right) \) along a solution of (0.1) is described... | Yes |
Lemma 2.1 The system (2.1) is uniformly bounded. | The Proof is referenced in [10]. | No |
Theorem 2.1 Assume that \( {a}_{11}^{iL} > 0,{a}_{22}^{jL} > 0\left( {i = 0,1,\ldots ,{2m};j = 0,1,\ldots, m}\right) ,{b}_{k}^{L} > \) \( 0,{r}_{k}^{L} > 0,{d}_{k}^{L} > 0,{f}_{k}^{L} > 0,{e}_{k}^{L} > 0\left( {k = 1,2}\right) \), then system (0.3) is permanent. | Proof Combined with Lemma 1.6 and Lemma 1.7, the conclusion is obvious. | No |
Lemma 2.2 Assume that conditions \( {\left( H1\right) }^{\prime } \) and \( {\left( H2\right) }^{\prime } \) hold, then\n\n\[ \operatorname{Re}\left( {{d\lambda }\left( \mu \right) /{d\mu }\left| {}_{\mu = {\mu }_{1}^{ + }}\right. }\right) > 0,\operatorname{Re}\left( {{d\lambda }\left( \mu \right) /{\left. d\mu \right|... | Proof Substituting \( \lambda \left( \mu \right) \) into (2.3), and by derivative for both sides of it we have\n\n\[ {d\lambda }\left( \mu \right) /{d\mu } = 2{g}^{\prime }\left( 0\right) {e}^{-{\lambda \tau }}/\left( {{2\tau \mu }{g}^{\prime }\left( 0\right) {e}^{-{\lambda \tau }} - 1}\right) . \]\n\nBy \( \mu = {\mu ... | Yes |
Theorem 2.3\n\n(1) when \( \mu \in \left( {{\mu }_{1}^{ - },{\mu }_{1}^{ + }}\right) \), the trivial equilibrium \( \left( {0,0}\right) \) of equation (2.1) is asymptotic stable.\n\n(2) when \( \mu \in R \smallsetminus \left\lbrack {{\mu }_{1}^{ - },{\mu }_{1}^{ + }}\right\rbrack \), the trivial equilibrium \( \left( {... | The conclusion of \( \left( 1\right) \) and \( \left( 2\right) \) hold obviously. In fact, notice that from lemma 2.2, we can obtain that as \( \mu < {\mu }_{1}^{ - } \) and \( \mu > {\mu }_{1}^{ + } \), the equation (2.3) has at least a positive real root, and get the conclusion (2). | No |
Lemma 3.2 \( \; \) If the \( - {b}_{i}\left( t\right) + \mathop{\sum }\limits_{{j = 1}}^{n}\left( {{a}_{ij}\left( t\right) + {a}_{ij}^{T}\left( t\right) }\right) \neq 0,\; \) then the system (3.1) \( \; \) at most there exists a equilibrium solution on the same sub-domain. | By definition 3.1, similar lemma1 of Theorem in [17]. Lemma 3.2 can be proved. Omit these details. | No |
Theorem 3.5 Assume that (3.3), (3.6) hold, if there exists a positively definite diagram \( \begin{matrix} \text{matrix}\;D\left( t\right) = \text{diag}\left( {{d}_{1}\left( t\right) ,\cdots ,{d}_{n}\left( t\right) }\right) , \\ \min {d}_{i}\left( t\right) = {d}_{i}^{\prime } > 0, i = 1,\cdots, n,\;\text{such}\;\text{t... | Proof For the \( z, y \in {\Omega }_{0} \subset {R}^{n} \), select the Lyapunov function \( V\left( {z, t}\right) = {z}^{2} \), then by system (3.7) to yield the following that operator for \[ {LV}\left( {z, y}\right) = 2{z}^{T}\left( {-B\left( t\right) + A\left( t\right) \Phi \left( {{x}^{ * }\left( t\right) }\right) ... | Yes |
Lemma 1.3 The equation \(\Delta \left( {\lambda, c}\right) = 0\) has no such root with the form \(\lambda = {\lambda }_{1}\left( c\right) + {i\beta },\beta \neq 0.\) | Proof Assume, for a contradiction, that the equation \(\Delta \left( {\lambda, c}\right) = 0\) has such root \(\lambda = {\lambda }_{1}\left( c\right) + {i\beta }.\) Thus we have\n\n\[- c{\lambda }_{1} + D\left\lbrack {{e}^{{\lambda }_{1}}\cos \beta + {e}^{-{\lambda }_{1}}\cos \beta - 2}\right\rbrack - d +\n\]\n\[ \fra... | Yes |
Theorem 2.2 Assume \( \left( {H}_{1}\right) - \left( {H}_{4}\right) \) hold. \( {c}_{ * },{\lambda }_{ * } \) and \( {\lambda }_{1} \) are defined in Lemma 1.1. Let \( \widetilde{\phi }\left( {x + {ct}}\right) \) is a traveling wavefront of (0.2) with speed \( c > {c}_{ * } \) satisfying (1.9). Then \( \bar{\phi }\left... | Proof From Theorem 2.1, we have\n\n\[ \mathop{\lim }\limits_{{\xi \rightarrow - \infty }}\bar{\phi }\left( \xi \right) {e}^{-{\lambda }_{1}\xi } = \bar{\rho }\;\text{ and }\;\mathop{\lim }\limits_{{\xi \rightarrow - \infty }}\phi \left( \xi \right) {e}^{-{\lambda }_{1}\xi } = \rho . \]\n\nLet \( {\xi }_{0} = \frac{1}{{... | Yes |
Lemma 1 If \( S\left( 0\right) > 0, D\left( 0\right) > 0, R\left( 0\right) > 0, Q\left( 0\right) > 0 \), the solution \( S\left( t\right), D\left( t\right), R\left( t\right), Q\left( t\right) \) of system (1) is positive for all \( t > 0 \) . | Proof If the conclusion does not hold, then at least one of \( S\left( t\right), D\left( t\right), R\left( t\right), Q\left( t\right) \) is not positive. Thus, we have the following four cases.\n\n1) there exists a first time \( {t}_{1} \) such that\n\n\[ S\left( {t}_{1}\right) = 0,{S}^{\prime }\left( {t}_{1}\right) < ... | Yes |
Lemma 2 All feasible solutions of the system (1) are bounded and enter the region\n\n\\[ \n\\Omega = \\left\\{ {\\left( {S, D, R, Q}\\right) \\in {R}_{ + }^{4} : S + D + R + Q \\leq \\frac{\\Lambda }{\\mu }}\\right\\} .\n\\] | Proof Let \\( \\left( {S, D, R, Q}\\right) \\in {R}_{ + }^{4} \\) be any solution with positive initial condition, adding the first four equations of (1), we have\n\n\\[ \n\\frac{d}{dt}\\left( {S + D + R + Q}\\right) = \\Lambda - {\\mu S} - {\\mu D} - {\\mu R} - {\\mu Q} - {d}_{1}D - {d}_{2}R - {d}_{3}Q =\n\\]\n\n\\[ \... | Yes |
Theorem 2.1 The drinking model (1) has two steady states as follows:\n\n(1) The drinking-free equilibrium \( {E}_{0}\left( {\frac{\Lambda }{\mu },0,0,0}\right) \) always exists for all parameter values.\n\n\( \left( 2\right) \; \) If \( \;{R}_{0} > 1,\;{the}\;{system}\;\left( 1\right) \;{has}\;a\;{unique}\;{positive}\;... | Moveover, \( {E}^{ * }\left( {{S}^{ * },{D}^{ * },{R}^{ * },{Q}^{ * }}\right) \) satisfies the following equality:\n\n\[ \n{S}^{ * } = \frac{\left( {\mu + {d}_{1} + \phi - \frac{\rho \phi }{\rho + \mu + {d}_{2} + \omega }}\right) \left( {\frac{\phi }{\left( \rho + \mu + {d}_{2} + \omega \right) } + \frac{\phi \omega }{... | Yes |
Theorem 2.2 If \( {R}_{0} < 1 \), drinking-free equilibrium \( {E}_{0} \) is globally asymptotically stable. | Proof We introduce the following Lyapunov function:\n\n\[ V = \left( {\rho + \omega + {d}_{2} + \mu }\right) D + {\rho R}. \]\n\nThe derivative of \( V \) is given by\n\n\[ {V}^{\prime } = \left( {\rho + \omega + {d}_{2} + \mu }\right) {D}^{\prime } + \rho {R}^{\prime } = \]\n\n\[ \left( {\rho + \omega + {d}_{2} + \mu ... | Yes |
Lemma 1.3 If \( x\left( t\right) \in P \cap \left( {{\bar{\Omega }}_{2} \smallsetminus {\Omega }_{1}}\right) ,{H}_{2} \) hold, then \( {\left( \Phi x\right) }_{i}\left( t\right) > 0, i = 1,2,3 \) . | Proof Since \( x\left( t\right) \in P \cap \left( {{\bar{\Omega }}_{2} \smallsetminus {\Omega }_{1}}\right) \), from(1.9),(1.10) and \( {H}_{2} \) we have\n\n\[ \n{\left( \Phi x\right) }_{1}\left( t\right) = {\int }_{t}^{t + \omega }{G}_{1}\left( {t,\sigma }\right) {x}_{1}\left( \sigma \right) \left\lbrack {-{a}_{11}\l... | Yes |
Theorem 2.1 In addition to the existence of positive periodic solutions, assume that\n\n\[ \mathop{\inf }\limits_{{t \in \lbrack 0, + \infty )}}\left\lbrack {{a}_{ii}\left( t\right) - \mathop{\sum }\limits_{{j = 1, j \neq i}}^{3}{\int }_{-\infty }^{0}{K}_{ji}\left( s\right) {a}_{ji}\left( {t - s}\right) \mathrm{d}s}\ri... | Proof Let \( x\left( t\right) = {\left( {x}_{1}\left( t\right) ,{x}_{2}\left( t\right) ,{x}_{3}\left( t\right) \right) }^{T} \) be any positive solution of system (0.3). Consider a Lyapunov functional \( V\left( t\right) \) defined by\n\n\[ V\left( t\right) = {V}_{1}\left( t\right) + {V}_{2}\left( t\right) ,\;t \geq 0,... | Yes |
Lemma 1 The solution of equation (3) satisfies \( \mathop{\lim }\limits_{{t \rightarrow + \infty }}u\left( t\right) = {u}_{1} \) if \( r > p \) . | Proof In equation (3), let\n\n\[ F\left( u\right) = {ru}\left( {1 - \frac{u}{K}}\right) - \frac{pu}{1 + {qu}}. \]\n\nThen equation (3) has three equilibrium points:\n\n\[ {u}_{0} = 0,\;{u}_{1} = \frac{-\bar{a} + \sqrt{{\bar{a}}^{2} - 4\bar{c}}}{2},\;{u}_{2} = \frac{-\bar{a} - \sqrt{{\bar{a}}^{2} - 4\bar{c}}}{2}. \]\n\n... | Yes |
Lemma 1.3 Consider the equation\n\n\\[ \n\\left\\{ \\begin{array}{l} {x}^{\\prime }\\left( t\\right) = \\alpha \\left( t\\right) x\\left( t\\right), t \\in J, t \\neq {t}_{k}, \\\\ x\\left( {t}_{k}^{ + }\\right) = x\\left( {t}_{k}\\right) + p, \\end{array}\\right.\n\\]\n\n(1.9)\n\nwhere \\( \\alpha \\left( t\\right) \\... | Proof It is easy to show that the following function\n\n\\[ \n\\bar{x}\\left( t\\right) = \\left\\{ \\begin{array}{ll} {x}^{ * }\\left( t\\right) , & t \\in (0,\\omega \\rbrack , \\\\ x\\left( {t - {j\\omega }}\\right) , & t \\in ({j\\omega },\\left( {j + 1}\\right) \\omega \\rbrack, j \\in {Z}^{ + }, \\end{array}\\rig... | Yes |
Lemma 2.2 Let \( d\left( t\right) \) be a continuous \( \omega \) -periodic function. \( \omega > 0 \) and \( {\int }_{0}^{\omega }d\left( t\right) {dt} > 0 \), Then we have that\n\n\[ \n{e}^{c\left( {t - s}\right) } \leq {e}^{1 + {D\omega } + {\int }_{s}^{t}d\left( r\right) {dr}},\;\text{for}\;t \geq s, \n\] \n\n\( \t... | Proof Put \( t = s + {n\omega } + \mu \) where \( n \in \{ 0,1,2,\cdots \} \) and \( 0 \leq \mu < \omega \) . Then we have\n\n\[ \n{e}^{1 + {D\omega } + {\int }_{s}^{t}d\left( r\right) {dr}} = {e}^{1 + {D\omega } + {\int }_{s}^{s + {n\omega } + \mu }d\left( r\right) {dr}} \geq \n\] \n\n\[ \n{e}^{1 + n{\int }_{0}^{\omeg... | Yes |
Theorem 2.4 Suppose that \( {\int }_{0}^{\omega }d\left( t\right) {dt} > 0 \) and\n\n\[ \n{\int }_{0}^{\omega }a\left( s\right) {ds} + \ln \mathop{\prod }\limits_{{i = 1}}^{q}\left( {1 + {g}_{i}}\right) < 0 \n\]\n\n(2.8)\n\nhold, then the solution \( \left( {x\left( t\right), y\left( t\right) }\right) \) of system \( \... | Proof From initial value condition (0.2), one can easily see that \( x\left( t\right) > 0, y\left( t\right) > 0 \) for \( t \geq 0 \) and \( \mathop{\lim }\limits_{{t \rightarrow \infty }}y\left( t\right) \geq {2}^{-1}\mathop{\min }\limits_{{t \in R}}{y}^{ * }\left( t\right) > 0 \) . Hence,\n\n\[ \n{x}^{\prime }\left( ... | Yes |
Theorem 1.1 If \( {R}_{0} < 1 \), the system (3) only have a disease-free equilibrium point \( {E}_{0} = \) \( \left( {\frac{\lambda }{d},0,0,0}\right) \) ; If \( {R}_{0}^{ * } \leq 1 < {R}_{0} \), the immune response does not play an important role, so we can ignore the immune response, this implies \( \frac{dz}{dt} \... | Proof When the system does not exist the immune respond and \( {R}_{0} > 1 \), means that the disease breaks out, and the immune system dosen't play a role in defending the disease. In view of mathematics, means \( {R}_{0}^{ * } \leq 1 < {R}_{0} \), then model (3) reduces to model (2).\n\nSetting the right hand side of... | Yes |
Theorem 1.2 When \( {R}_{0} < 1 \), the disease-free equilibrium point \( {E}_{0} \) is globally asymptotically stable. | Proof When \( {R}_{0} < 1 \), model (3) reduces to model (2). Meanwhile the proof is given in the paper [9]. | No |
Lemma 1 (Positive invariant and bounded) \( \textit{System (1.1) is positively invariant and bounded} \) in \( {R}_{ + }^{3} \) . Moreover, we have\n\n\[ \mathop{\limsup }\limits_{{t \rightarrow + \infty }}x\left( t\right) \leq 1,\;\mathop{\limsup }\limits_{{t \rightarrow + \infty }}y\left( t\right) \leq \frac{{a}_{1} ... | Proof By the continuity argument, we can easily prove that system (1.1) is positively invariant in \( {R}_{ + }^{3} \) . Then from the first equation of (1.1) we have \( {x}^{\prime } \leq x\left( {1 - x}\right) \), thus we can conclude that\n\n\[ \mathop{\limsup }\limits_{{t \rightarrow + \infty }}x\left( t\right) \le... | Yes |
Proposition 1 (Subsystem of species \( x \) and \( y \) ) The system (1.2) is globally stable at \( \left( {0,\frac{{a}_{1}}{{a}_{2}}}\right) \) if and only if \( {a}_{2} \leq {a}_{1} \) while it is globally stable at \( \left( {\frac{{a}_{2} - {a}_{1}}{{a}_{4} + {a}_{2}},\frac{{a}_{4} + {a}_{1}}{{a}_{4} + {a}_{2}}}\ri... | Proof If \( {a}_{2} < {a}_{1} \), then (1.2) has only one locally asymptotically stable boundary equilibrium \( \left( {0,\frac{{a}_{1}}{{a}_{2}}}\right) \) and two unstable equilibria \( \left( {0,0}\right) \) and \( \left( {1,0}\right) \) . Since \( \mathop{\liminf }\limits_{{t \rightarrow \infty }}y\left( t\right) \... | Yes |
Theorem 1 Species \( y \) is persistent in \( {R}_{ + }^{3} \) for system (1.1) if \( \mu > \beta + \frac{{b}_{1}\left( {{a}_{1} + {a}_{4}}\right) }{{a}_{2}} \) . In addition, System (1.1) has global stability at \( \left( {0,\frac{{a}_{1}}{{a}_{2}},0}\right) \) if \( \mu > \beta + \frac{{b}_{1}\left( {{a}_{1} + {a}_{4... | Proof Assume that \( {a}_{1} > {a}_{2} \) for system (1.1). From Lemma 1, there is a \( T > 0 \) large enough such that for any \( t > T \), if \( \mu > \beta + \frac{{b}_{1}\left( {{a}_{1} + {a}_{4}}\right) }{{a}_{2}} \), we have\n\n\[ \n{z}^{\prime } = z\left( {{b}_{1}y + \frac{\beta x}{a + x} - \mu }\right) \leq z\l... | Yes |
Species \( x \) is persistent in \( {R}_{ + }^{3} \) for system (1.1) if \( \mu > \frac{{a}_{1}{b}_{1}}{{a}_{2}} \) and \( \frac{{a}_{1}}{{a}_{2}} < 1 \) or \( \min \left\{ {{a}_{1},{a}_{2}}\right\} > a{a}_{3} \) and \( \mu < \) \( \min \left\{ {\frac{a{a}_{3}{b}_{1} - {a}_{1}{b}_{1}}{a{a}_{3} - {a}_{2}},\frac{{a}_{1}{... | Proof According to Lemma 1, we can restrict the dynamics of system (1.1) on the compact set \( C = \left\lbrack {0, M}\right\rbrack \times \left\lbrack {0, M}\right\rbrack \times \left\lbrack {0, M}\right\rbrack \), where \( M \) is positive constant. The omega limit set of the \( y - z \) subsystem restricted on \( C ... | Yes |
Theorem 3 If \( {b}_{1} < \min \left( {\mu ,\frac{\beta {a}_{3}}{{a}_{4}}}\right) \), then equilibrium \( \left( {0,\frac{\mu }{{b}_{1}},\frac{{a}_{1}{b}_{1} - {a}_{2}\mu }{{a}_{3}{b}_{1}}}\right) = \left( {0,{y}^{ * },{z}^{ * }}\right) \) is globally asymptotically stable. | Proof Let\n\n\[ V = x + c\left( {y - {y}^{ * } - {y}^{ * }\ln \left( {y/{y}^{ * }}\right) }\right) + d\left( {z - {z}^{ * } - {z}^{ * }\ln \left( {z/{z}^{ * }}\right) }\right) ,\]\n\nwhere \( c \) and \( d \) are constants to be chosen later. \( V \) is continuous on the positive quadrant. It is also easy to verify tha... | Yes |
Lemma 1.1 \( \; \) If \( b = 0 \) , \( a{f}^{\prime }\left( 0\right) < 1 + c \), then the zero equilibrium \( {u}^{ * } \) of system (0.2) and (0.3) is local asymptotically stable. | Proof When \( b = 0 \), the characteristic equation (1.3) yields\n\n\[ \lambda + c{k}^{2} + 1 - a{f}^{\prime }\left( 0\right) = 0,\;k = 1,2,\ldots \]\n\n(1.4)\n\nIt is clear that all roots of (1.4) have negative real parts if \( a{f}^{\prime }\left( 0\right) < 1 + c \) . So the zero equilibrium \( {u}^{ * } \) of syste... | Yes |
Lemma 1.2 Eq. (1.3) have purely imaginary roots if and only if \( b = {b}_{j}^{\left( k\right) }, k = 1,2,\ldots ;j = \) \( 0,1,2,\ldots \) Moreover, when \( b = {b}_{0}^{\left( 1\right) } \), all the roots of Equations (1.3), except \( \pm i{\omega }_{0}^{\left( 1\right) } \), have negative real parts. | Proof From the definition of \( {b}_{j}^{\left( k\right) } \), Equation (1.3) have purely imaginary roots if and only if \( b = {b}_{j}^{\left( k\right) } \) . In addition, we know that \( {b}_{0}^{\left( 1\right) } \) is the first value of \( b > 0 \) such that Equation (1.3) have roots appearing on the imaginary axis... | Yes |
Lemma 1.3 If \( a{f}^{\prime }\left( 0\right) < 1 + c \), then\n\n\[{\left. \frac{d\left( {\operatorname{Re}\lambda }\right) }{db}\right| }_{b = {b}_{j}^{\left( k\right) }} > 0.\] | Proof Differentiating Equation (1.3) with respect to \( b \), we obtain\n\n\[ \frac{d\lambda }{db} - {ab\tau }{f}^{\prime }\left( 0\right) {e}^{-{\lambda \tau }}\frac{d\lambda }{db} + a{f}^{\prime }\left( 0\right) {e}^{-{\lambda \tau }} = 0.\]\n\nThis gives\n\n\[ \frac{d\lambda }{db} = \frac{a{f}^{\prime }\left( 0\righ... | Yes |
Example 4.1 Consider the following system\n\n\\[ \n\\begin{cases} {dx}\\left( t\\right) = & x\\left( t\\right) \\left\\lbrack {\\left( {{0.8} + {0.1}\\sin t}\\right) - \\left( {{0.6} + {0.1}\\sin t}\\right) x\\left( t\\right) - }\\right. \\\\ & \\left. \\frac{\\left( {{0.69} + {0.1}\\sin t}\\right) x\\left( t\\right) }... | Let \\( x\\left( 0\\right) = {1.0},{t}_{k} = k,{I}_{k} = {\\mathrm{e}}^{0.08} - 1 \\), then \\( {\\Phi }^{ * } = {0.94} > \\widehat{b}/2\\sqrt{\\widetilde{c}} = {0.7} \\) . It follows from Theorem 1.1 that species \\( x\\left( t\\right) \\) of system (4.1) is weakly persistent (see Figure 1). | Yes |
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