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Lemma 1 \( {}^{\left\lbrack {14}\right\rbrack } \) If \( u \in {H}^{3}\left( \Omega \right) ,{\overrightarrow{w}}^{h} \in {\overrightarrow{W}}^{h} \), then for all \( {v}^{h} \in {V}^{h} \) and \( \overrightarrow{p} \in {\left( {H}^{2}\left( \Omega \right) \right) }^{2} \), there hold \[ \left( {\nabla \left( {u - {I}_... | Using Lemma 1.1 in [14], we get that for all \( {\overrightarrow{w}}^{h} \in {\overrightarrow{W}}^{h}, u \in {H}^{3}\left( \Omega \right) \), there holds \[ \left( {\nabla \left( {u - {I}_{h}u}\right) ,{\overrightarrow{w}}^{h}}\right) \leq c{h}^{2}{\left| u\right| }_{3}{\begin{Vmatrix}{\overrightarrow{w}}^{h}\end{Vmatr... | Yes |
Lemma 2.4 Consider system (2.1). If there exist matrix \( P > 0 \) and positive scalar \( \mu \) such that the following matrix inequality holds:\n\n\[ \n- {\mu P} + {C}^{\mathrm{T}}{PC} \leq 0, \n\]\n\n(2.6)\n\nthen \( V\left( {t}_{k}\right) \leq {\mu V}\left( {t}_{k}^{ - }\right) \), where \( V\left( t\right) = {x}^{... | Proof By the second equation of system (2.1), we have\n\n\[ \nV\left( {t}_{k}\right) = {x}^{\mathrm{T}}\left( {t}_{k}\right) {Px}\left( {t}_{k}\right) = {x}^{\mathrm{T}}\left( {t}_{k}^{ - }\right) {C}^{\mathrm{T}}{PCx}\left( {t}_{k}^{ - }\right) \n\]\n\n\[ \n\leq \mu {x}^{\mathrm{T}}\left( {t}_{k}^{ - }\right) {Px}\lef... | Yes |
Lemma 2.5 Consider system (2.1) satisfying Assumption 2.1. If for given positive scalar \( \beta \), there exist matrix \( P > 0 \) and positive scalars \( 0 < {\beta }_{1} \leq 1,0 < {\beta }_{2} \leq 1,0 < \mu < 1,\alpha \) such that the following matrix inequality holds:\n\n\[ \left( \begin{matrix} \Phi & P & P & {B... | Proof By (2.8), for sufficiently small positive scalars \( \gamma \) and \( h \in \left( {0,1 - \mu }\right) \), we have\n\n\[ \widetilde{\Psi } = \left( \begin{matrix} {\Phi }_{1} & P & P & {B}^{\mathrm{T}}{PD} & {L}^{\mathrm{T}} & {M}^{\mathrm{T}} \\ * & - {\beta }_{1}I & 0 & 0 & 0 & 0 \\ * & * & - {\beta }_{2}I & 0 ... | Yes |
Theorem 3.2 Under Assumption 2.1, if there exist \( n \times n \) matrix \( P > 0 \), constants \( \lambda > 0,{q}_{1} > 0,\gamma \geq 1, q \geq \gamma {\mathrm{e}}^{\lambda \tau } \) and \( {\varepsilon }_{i} > 0\left( {i = 1,2,3}\right) \) such that the following conditions hold:\n\n(a) \( {PA} + {A}^{\mathrm{T}}P + ... | Proof Define the Lyapunov function \( V\left( {t, x\left( t\right) }\right) \in \left( {{\mathbb{R}}^{ + } \times {\mathbb{R}}^{n};{\mathbb{R}}^{ + }}\right) \) as \( V\left( {t, x\left( t\right) }\right) = \) \( {x}^{\mathrm{T}}\left( t\right) {Px}\left( t\right) \) . Set \( {c}_{1} = {\lambda }_{\min }\left( P\right)... | Yes |
Example 4.1 Consider the following impulsive nonlinear stochastic delay differential systems.\n\n\[ \n\begin{cases} \mathrm{d}x\left( t\right) = & \left\lbrack {{Ax}\left( t\right) + f\left( {x\left( t\right) }\right) + g\left( {x\left( {t - \tau \left( t\right) }\right) }\right) }\right\rbrack \mathrm{d}t \\ & + \left... | By Theorem 3.1, we can apply the impulsive control law \( x\left( {t}_{k}\right) = {Cx}\left( {t}_{k}^{ - }\right) \) to exponentially stabilize system (4.1), where \( C = \left\lbrack \begin{matrix} {0.4252} & 0 \\ 0 & {0.4252} \end{matrix}\right\rbrack ,{t}_{k} - {t}_{k - 1} = {0.2841} \) . Choosing \( \mu = {0.1808}... | Yes |
Consider the following first-order impulsive nonlinear stochastic delay differential systems. | By Theorem 3.2, we can apply the impulsive control law \( x\left( {t}_{k}\right) = {0.1228x}\left( {t}_{k}^{ - }\right) \) to exponentially stabilize system (4.2), where \( \mathop{\sup }\limits_{{k \in \mathbb{N}}}\left\{ {{t}_{k} - {t}_{k - 1}}\right\} \leq {0.03} \) . It is easy to see that \( L = M = 1 \) . In Theo... | Yes |
Theorem 2.1 The octant \( {\mathbb{R}}_{ + }^{5} = \left\{ {\left( {x, w, y, v, z}\right) \in {\mathbb{R}}^{5} : x, w, y, v, z \geq 0}\right\} \) is positively invariant with respect to (1.2). Moreover, all solutions of system (1.2) are uniformly bounded in the compact subset \( \Omega = \left\{ {\left( {x, w, y, v, z}... | Proof The positive invariance of the positive orthant is trivial. It remains to show that system (1.2) is uniformly bounded. Let \( \left( {x\left( t\right), w\left( t\right), y\left( t\right), v\left( t\right), z\left( t\right) }\right) \) be any solution of (1.2) with positive initial conditions (1.3).\n\nDefine\n\n\... | Yes |
If the componentwise strict complementarity condition (A4) holds, then the critical directions cone \( \mathcal{C}\left( {x}^{ * }\right) \) in (A2) is given by\n\n\[ \mathcal{C}\left( {x}^{ * }\right) = \left\{ {d \mid \mathcal{J}{g}^{j}\left( {x}^{ * }\right) d = 0, j \in \alpha ;\left\langle {{\widetilde{u}}_{j}^{ *... | where \( {u}^{ * } \in \Lambda \left( {x}^{ * }\right) ,{\widetilde{u}}_{j}^{ * } = \sqrt{\frac{2}{{\left( {u}_{j}^{ * }\right) }_{0}}}\left( {{\left( {u}_{j}^{ * }\right) }_{0};{\bar{u}}_{j}^{ * }}\right) \) . In this case, the critical directions cone \( \mathcal{C}\left( {x}^{ * }\right) \) is a linear subspace of \... | Yes |
Lemma 3.3 Let \( {x}^{ * } \) be a stationary point of (NSOP) and \( \Lambda \left( {x}^{ * }\right) \) be the set of its Lagrange multipliers. Then \[{\nabla }_{x}F\left( {{x}^{ * },{u}^{ * }, t}\right) = 0,\;\forall {u}^{ * } \in \Lambda \left( {x}^{ * }\right) ,\forall t > 0.\] | Proof It follows from Lemma 3.2 that we have \[ {\nabla }_{x}F\left( {{x}^{ * },{u}^{ * }, t}\right) = \nabla f\left( {x}^{ * }\right) - \mathop{\sum }\limits_{{j \in \alpha \cup {\gamma }_{1}}}\nabla {g}^{j}\left( {x}^{ * }\right) {\left( {t}^{-1}{g}_{0}^{j}\left( {x}^{ * }\right) + 1\right) }^{-2}{u}_{j}^{ * } - 2\ma... | Yes |
If Assumption (A1) holds, for any \( {u}^{ * } \in \Lambda \left( {x}^{ * }\right) \) such that the componentwise strict complementarity condition (A4), then for any \( t > 0, F\left( {x, u, t}\right) \) is twice continuously differentiable with respect to \( x \) at \( \left( {{x}^{ * },{u}^{ * }, t}\right) \) | \[ {\nabla }_{xx}^{2}F\left( {{x}^{ * },{u}^{ * }, t}\right) = {\nabla }_{xx}^{2}L\left( {{x}^{ * },{u}^{ * }}\right) + 2{t}^{-1}\mathop{\sum }\limits_{{j \in \alpha }}\nabla {g}^{j}\left( {x}^{ * }\right) \mathcal{L}\left( {u}_{j}^{ * }\right) \mathcal{J}{g}^{j}\left( {x}^{ * }\right) \] \[ + 2{t}^{-1}\mathop{\sum }\l... | Yes |
Theorem 2.1 Assume that \( \beta > 0 \) and \( \left| \alpha \right| < \sqrt{2\nu \beta },{u}_{0} \in {\mathbb{L}}^{2}\left( {\mathbb{R}}^{3}\right) \) with free divergence, and \( f \in {L}^{1}\left( {0, T;{\mathbb{L}}^{2}\left( {\mathbb{R}}^{3}\right) }\right) \) for any \( T > 0 \) . Then problem (1.1) admits a uniq... | Proof of Theorem 2.1 The existence result in Theorem 2.1 follows from Theorem 1.1 in [4]. Therefore, we only need to prove (2.2) and (2.3). Indeed, since \( A\left( u\right) \) is symmetric, we have \[ {\left| A\left( u\right) \right| }^{2}A\left( u\right) \cdot \nabla u = \frac{1}{2}{\left| A\left( u\right) \right| }^... | Yes |
Assume that \( \beta > 0,\left| \alpha \right| < \sqrt{2\nu \beta },{u}_{0},{v}_{0} \in {\mathbb{L}}^{2}\left( {\mathbb{R}}^{3}\right) \) are two vector fields with free divergences, and \( f, g \in {L}^{1}\left( {0, T;{\mathbb{L}}^{2}\left( {\mathbb{R}}^{3}\right) }\right) \) for any \( T > 0 \) . Let \( u, v \) be th... | \[ \parallel w\left( t\right) {\parallel }^{2} + \nu {\varepsilon }_{0}{\int }_{0}^{t}\parallel \nabla w\left( s\right) {\parallel }^{2}\mathrm{\;d}s + \beta {\varepsilon }_{0}{\int }_{0}^{t}\parallel A\left( w\right) \left( s\right) {\parallel }_{{\mathbb{L}}^{4}}^{4}\mathrm{\;d}s \]\n\[ \leq {C}_{2}\left( t\right) \l... | Yes |
Lemma 2.2 \( {}^{\left\lbrack {11}\right\rbrack } \) Assume that \( a > 0, b > 0 \) . Then\n\n\[ \n{y}^{\Delta }\left( t\right) \leq \left( \geq \right) y\left( t\right) \left( {b - {ay}\left( {\sigma \left( t\right) }\right) }\right), y\left( t\right) > 0, t \in {\left\lbrack {t}_{0}, + \infty \right) }_{\mathbb{T}} \... | implies\n\n\[ \ny\left( t\right) \leq \left( \geq \right) \frac{b}{a}\left\lbrack {1 + \left( {\frac{b}{{ay}\left( {t}_{0}\right) } - 1}\right) {e}_{\ominus b}\left( {t,{t}_{0}}\right) }\right\rbrack, t \in {\left\lbrack {t}_{0}, + \infty \right) }_{\mathbb{T}}. \n\] | Yes |
Lemma 2.4 Assume that \( a > 0, b > 0, - b \in {\mathcal{R}}^{ + } \), and \( y\left( t\right) > 0, t \in {\left\lbrack {t}_{0}, + \infty \right) }_{\mathbb{T}} \) . (i) If \( {y}^{\Delta }\left( t\right) \geq y\left( t\right) \left( {b - {ay}\left( t\right) }\right) \), then \( \mathop{\liminf }\limits_{{t \rightarrow... | Proof We only need to prove (i), since the proof of (ii) is similar. If \( y\left( {\sigma \left( t\right) }\right) \geq y\left( t\right), t \in {\left\lbrack {t}_{0}, + \infty \right) }_{\mathbb{T}} \), then \[ {y}^{\Delta }\left( t\right) \geq y\left( t\right) \left( {b - {ay}\left( t\right) }\right) \geq y\left( t\r... | Yes |
Lemma 2.6 Let \( \mathbb{T} \) be an almost periodic time scale. If \( f\left( t\right), g\left( t\right) \) are almost periodic functions, then, for any \( \varepsilon > 0, E\{ \varepsilon, f\} \cap E\{ \varepsilon, g\} \) is a nonempty relatively dense set in \( \mathbb{T} \) ; that is, for any given \( \varepsilon >... | \[ \left| {f\left( {t + \tau }\right) - f\left( t\right) }\right| < \varepsilon ,\left| {g\left( {t + \tau }\right) - g\left( t\right) }\right| < \varepsilon ,\forall t \in \mathbb{T}. \] | Yes |
Lemma 2.7 \( {}^{\left\lbrack {14}\right\rbrack } \) Suppose that there exists a Lyapunov function \( V\left( {t, x, y}\right) \in C\left( {\left\lbrack 0, + \infty \right) }_{\mathbb{T}}\right. \times \) \( \mathbb{D} \times \mathbb{D},\mathbb{R}) \) satisfying the following conditions:\n\n1) \( a\left( {\parallel x -... | Moreover, if there exists a solution \( x\left( t\right) \) of (2.3) such that \( x\left( t\right) \in S \), where \( S \subset \mathbb{D} \) is a compact set. Then there exists a unique uniformly asymptotically stable almost periodic solution \( p\left( t\right) \) of (2.3) in \( S \) . Furthermore, if \( f\left( {t, ... | Yes |
Theorem 3.1 Let \( \left( {x\left( t\right), y\left( t\right) }\right) \) be any positive solution of system (1.1) with initial condition (1.2). If \( \left( {\mathrm{H}}_{1}\right) - \left( {\mathrm{H}}_{2}\right) \) hold, then system (1.1) is permanent, that is, any positive solution \( \left( {x\left( t\right), y\le... | Proof Assume that \( \left( {{x}_{1}\left( t\right) ,{x}_{2}\left( t\right) ,{y}_{1}\left( t\right) ,{y}_{2}\left( t\right) }\right) \) be any positive solution of system (1.1) with initial condition (1.2). It follows from the first equation of system (1.1) and the inequality \( {\mathrm{e}}^{x} \geq 1 + x \) for \( x ... | Yes |
Theorem 4.1 Suppose the conditions \( \left( {\mathrm{H}}_{1}\right) - \left( {\mathrm{H}}_{2}\right) \) hold, and assume further that\n\n\( \left( {\mathrm{H}}_{3}\right) \lambda > 0 \) and \( - \lambda \in {\mathcal{R}}^{ + } \), where\n\n\[ \lambda = \min \left\{ {{r}_{1}^{l} - \frac{{r}_{2}^{u}\mathop{\max }\limits... | Proof From Lemma 4.1, system (1.1) has a bounded solution satisfying\n\n\( {m}_{1} \leq {x}_{1}\left( t\right) \leq {M}_{1},{m}_{2} \leq {x}_{2}\left( t\right) \leq {M}_{2},{m}_{3} \leq {y}_{1}\left( t\right) \leq {M}_{3},{m}_{4} \leq {y}_{2}\left( t\right) \leq {M}_{4},\forall t \in \mathbb{T}. \)\n\nThen\n\n\[ \left|... | Yes |
Assume that H1-H3 hold and \( \gamma \in \left( {{\gamma }_{2},{\gamma }_{1}}\right) \) . Let \( \left( {X\left( {t,\omega }\right), Y\left( {t,\omega }\right) }\right) \) be the solution of system (3.2) with initial date \( \left( {{X}_{0},{Y}_{0}}\right) \), then \( \left( {{X}_{0},{Y}_{0}}\right) \in {M}^{ + }\left(... | Proof Assume that \( \left( {{X}_{0},{Y}_{0}}\right) \in {M}^{ + }\left( \omega \right) \) . Using the variation of constants formula, we have\n\n\[ X\left( t\right) = X\left( \tau \right) {\mathrm{e}}^{A\left( {t - \tau }\right) + {\int }_{\tau }^{t}z\left( \mu \right) \mathrm{d}\mu } + {\int }_{\tau }^{t}{\mathrm{e}}... | Yes |
Lemma 3.3 Assume that H1-H3 hold and \( \gamma \in \left( {{\gamma }_{2},{\gamma }_{1}}\right) \) . Let \( \Phi \left( t\right) = \Phi \left( {t,\omega ,{Y}_{0}}\right) \in {C}_{\gamma }^{ + } \), be the unique solution of the system (3.3). Then for every \( {Y}_{0},{\widetilde{Y}}_{0} \in {H}_{2} \), we have the follo... | Proof By the same arguments as in Lemma 3.2, we have\n\n\[ \begin{Vmatrix}{\mathfrak{F}}_{1}\left( \Phi \right) \end{Vmatrix}}_{{C}_{\gamma }^{1, + }} \leq \mathcal{L}\frac{1}{{\gamma }_{1} - \gamma }\parallel \Phi {\parallel }_{{C}_{\gamma }^{ + }}\]\n\nand\n\n\[ \begin{Vmatrix}{\mathfrak{F}}_{2}\left( \Phi \right) \e... | Yes |
Theorem 3.1 (Invariant manifolds) Assume that H1-H3 hold and let \( \gamma \in \left( {{\gamma }_{2},{\gamma }_{1}}\right) \) , then the manifolds of the system(3.2) exists. Moreover, the invariant manifold is \( {M}^{ + }\left( \omega \right) \) represented as a graph of a Lipschitz function, i.e., \[{M}^{ + }\left( \... | Proof From the definition of the \[ {M}^{ + }\left( \omega \right) = \left\{ {\left( {{X}_{0},{Y}_{0}}\right) \in {H}_{1} \times {H}_{2} \mid \left( {X\left( t\right), Y\left( t\right) }\right) \in {C}_{\gamma }^{ + }}\right\} , \] and Lemma 3.1, we can directly deduce that \( {M}^{ + }\left( \omega \right) = \left\{ {... | Yes |
Lemma 4.1 Assume that H1-H3 hold and \( \gamma \in \left( {{\gamma }_{2},{\gamma }_{1}}\right) \) . Let \( \left( {X\left( {t,\omega }\right), Y\left( {t,\omega }\right) }\right) \) be the solution of system (4.3) with initial date \( \left( {{X}_{0},{Y}_{0}}\right) \) . Take \( \gamma \in \left( {{\gamma }_{2},{\gamma... | \[ \Phi \left( t\right) = \left( {X\left( t\right), Y\left( t\right) }\right) = \left( {X\left( {t,\omega ,{Y}_{0}}\right), Y\left( {t,\omega ,{Y}_{0}}\right) }\right) \in {C}_{\gamma }^{ + }, \] which satisfies \[ \Phi \left( t\right) = \left( \begin{matrix} X\left( {t,\omega }\right) \\ Y\left( {t,\omega }\right) \en... | Yes |
Theorem 4.1(Invariant manifolds) Assume that H1-H3 hold and \( \gamma \in \left( {{\gamma }_{2},{\gamma }_{1}}\right) \) . then the manifolds of the system (4.4) exists. Moreover, the invariant manifold \( {M}^{ + }\left( \omega \right) \) is represented as a graph of a Lipschitz function. Where \( {M}^{ + }\left( \ome... | and\n\[ h\left( {\omega ,{Y}_{0}}\right) = {\int }_{\infty }^{0}{\mathrm{e}}^{-{As}}f\left( {X + {z}_{1}, Y + {z}_{2},,{\theta }_{s}\omega }\right) \mathrm{d}s \]\nis the graph mapping with Lipschitz constant satisfying \( \operatorname{Lip}h\left( {\omega , \cdot }\right) \leq \mathcal{L}\frac{1}{\left( {{\gamma }_{1}... | Yes |
Theorem 2.3 Let Assumption 3 and Assumption 4 hold. Then for any initial value \( \xi \left( s\right) \in C\left( {\left\lbrack {-\tau ,0}\right\rbrack ;{\mathbb{R}}_{ + }^{3}}\right) \), there is a stationary distribution for the system (2.15). | The proof of Theorem 2.3 is similar to that of Theorem 2.2, so it is omitted. | No |
Theorem 3.1 Assume that (H1) and (H2) hold and let \( {\widehat{\alpha }}_{t} \) be given by (1.3). Then\n\n\[ \n{\widehat{\alpha }}_{t}\overset{P}{ \rightarrow }\alpha \land \frac{1}{2}\;\text{ as }\;t \rightarrow T.\n\]\n\nIn particular, when \( \alpha < 1 + \frac{d}{2},{\widehat{\alpha }}_{t}\overset{a.s.}{ \rightar... | In order to prove this theorem, we need to make use of the following three lemmas.\n\nLemma 3.1 \( {}^{\left\ | No |
Lemma 3.2 Suppose that \( - 1 < d < 0 \) and \( \alpha \in \left( {0,1 + d/2}\right) \) . Let \( {Z}_{t} \) be defined in (3.1). Then \( {Z}_{T} \mathrel{\text{:=}} \mathop{\lim }\limits_{{t \rightarrow T}}{Z}_{t} \) exists in \( {L}^{2} \) .Furthermore, there exists a continuous modification of \( {\left\{ {Z}_{t}\rig... | Proof For all \( s \leq t < T \), by Lemma 3.1 and using (2.4), we have\n\n\[ \mathrm{E}\left\lbrack {\left( {Z}_{t} - {Z}_{s}\right) }^{2}\right\rbrack = {\int }_{s}^{t}\mathrm{\;d}u{\left( T - u\right) }^{-\alpha }{\int }_{s}^{t}{\left( T - v\right) }^{-\alpha }\phi \left( {u, v}\right) \mathrm{d}v \]\n\n\[ \leq c{\i... | Yes |
Lemma 3.3 Suppose that \( - 1 < d < 0 \) and \( \alpha \in \left( {0,1 + d/2}\right) \), and Let \( {Z}_{t} \) be defined in (3.1). Then, as \( t \rightarrow T \): 1) if \( 0 < \alpha < \frac{1}{2} \), then \[ {\left( T - t\right) }^{1 - {2\alpha }}{\int }_{0}^{t}{Z}_{u}^{2}{\left( T - u\right) }^{{2\alpha } - 2}\mathr... | Proof 1) By Lemma 3.2, using the \( \left( {\frac{1 + d/2}{2} - \frac{\alpha }{2}}\right) \) -Hölder continuity of \( {Z}_{t} \), we obtain \[ \left| {{\left( T - t\right) }^{1 - {2\alpha }}{\int }_{0}^{t}{Z}_{u}^{2}{\left( T - u\right) }^{{2\alpha } - 2}\mathrm{\;d}u - \frac{{Z}_{T}^{2}}{1 - {2\alpha }}}\right| \] \[ ... | Yes |
Lemma 3.5 Let \( {Y}_{t} \) be defined in (3.2). For any \( t \in \lbrack 0, T) \), we have\n\n\[ \n{Y}_{t} = {\int }_{0}^{t}{\left( T - u\right) }^{\alpha - 1}\mathrm{\;d}{G}_{u} \times {\int }_{0}^{t}{\left( T - s\right) }^{-\alpha }\mathrm{d}{G}_{s} - {\int }_{0}^{t}\delta {G}_{s}{\left( T - s\right) }^{-\alpha }{\i... | Proof Let \( t \in \lbrack 0, T) \) be fixed. By (2.6), \( {Y}_{t} = {\int }_{0}^{t}\mathrm{\;d}{G}_{u}{\left( T - u\right) }^{\alpha - 1}{\int }_{0}^{u}{\left( T - s\right) }^{-\alpha }\mathrm{d}{G}_{s} \) becomes\n\n\[ \n{Y}_{t} = {\int }_{0}^{t}\mathrm{\;d}{\left( T - u\right) }^{\alpha - 1}\mathrm{\;d}{G}_{u} \time... | Yes |
1) If \( \alpha \in \left( {0, - \frac{d}{2}}\right) \) then, as \( t \rightarrow T \) ,\n\n\[ \left( {F,{\left( T - t\right) }^{-\frac{d}{2} - \alpha }{\int }_{0}^{t}{\left( T - u\right) }^{\alpha - 1}\mathrm{\;d}{G}_{u}}\right) \xrightarrow[]{law}\left( {F,\sqrt{{\sigma }_{1G}}N}\right) . \] | Proof For any \( m \geq 1,{s}_{1},{s}_{2},\cdots ,{s}_{m} \in \lbrack 0,\infty ) \), we shall prove that, as \( t \rightarrow T \) ,\n\n\[ \left( {{B}_{{s}_{1}},\cdots ,{B}_{{s}_{m}},{\left( T - t\right) }^{-\frac{d}{2} - \alpha }{\int }_{0}^{t}{\left( T - u\right) }^{\alpha - 1}\mathrm{\;d}{G}_{u}}\right) \xrightarrow... | Yes |
Theorem 1.1 Let \( v > 0 \) and \( u \geq 0 \) be two differentiable functions in a domain \( \Omega \subset {\mathbb{R}}^{N}, N \geq 3 \) . Assume that differentiable functions \( g\left( u\right) \) and \( f\left( v\right) \) satisfy that for \( p > \) \( 1, q > 1,\frac{1}{p} + \frac{1}{q} = 1, \n\n\[ \n\frac{g\left(... | Proof of Theorem 1.1 It is easily derived \n\n\[ \nR\left( {u, v}\right) = {\left| \nabla u\right| }^{p} - \nabla \left( \frac{g\left( u\right) }{f\left( v\right) }\right) {\left| \nabla v\right| }^{p - 2}\nabla v \n\] \n\n\[ \n= {\left| \nabla u\right| }^{p} - \frac{{g}^{\prime }\left( u\right) {\left| \nabla v\right|... | Yes |
Example 3.1 Let \( {k}_{1}\left( x\right) \) and \( {k}_{2}\left( x\right) \) be two continuous functions, \( {k}_{1}\left( x\right) < {k}_{2}\left( x\right) \) on \( \Omega \subset {\mathbb{R}}^{N} \) . If there exists a function \( u \in {C}^{2}\left( \Omega \right) \) satisfying \[ \left\{ \begin{matrix} - {\Delta }... | Proof Assume that \( v \) does not change sign. By (3.1),(3.2) and Theorem 1.1, we have \[ 0 \leq {\int }_{\Omega }L\left( {u, v}\right) \mathrm{d}x = {\int }_{\Omega }R\left( {u, v}\right) \mathrm{d}x \] \[ = {\int }_{\Omega }{\left| \nabla u\right| }^{p}\mathrm{\;d}x - {\int }_{\Omega }\nabla \left( \frac{g\left( u\r... | Yes |
Example 3.2 Let \( \left( {u, v}\right) \in {C}^{2}\left( \Omega \right) \times {C}^{2}\left( \Omega \right) \) be a pair of positive solutions to the elliptic system\n\n\[ \left\{ \begin{array}{ll} - {\Delta }_{p}u = f\left( v\right) , & x \in \Omega , \\ - {\Delta }_{p}v = \frac{{\left\lbrack f\left( v\right) \right\... | Proof For any \( {\phi }_{1} \) and \( {\phi }_{2} \) in \( {W}_{0}^{1, p}\left( \Omega \right) \), we get by (3.3) that\n\n\[ {\int }_{\Omega }{\left| \nabla u\right| }^{p - 2}\nabla u\nabla {\phi }_{1}\mathrm{\;d}x = {\int }_{\Omega }f\left( v\right) {\phi }_{1}\mathrm{\;d}x \]\n\n(3.4)\n\n\n\n\[ {\int }_{\Omega }{\l... | Yes |
Example 3.3 Suppose that a function \( v \in {C}^{2}\left( \Omega \right) \) and \( v > 0 \), satisfying\n\n\[ \n- {\Delta }_{p}v \geq {\lambda k}\left( x\right) f\left( v\right) ,\;\text{ in }\;\Omega , \n\]\n\n(3.6)\n\nwhere \( f\left( v\right) > 0,\lambda > 0 \) and \( k\left( x\right) \) is a nonnegative weight. Th... | Proof By Theorem 1.1 and (3.6), we have\n\n\[ \n0 \leq {\int }_{\Omega }L\left( {u, v}\right) \mathrm{d}x = {\int }_{\Omega }R\left( {u, v}\right) \mathrm{d}x \n\]\n\n\[ \n= {\int }_{\Omega }{\left| \nabla u\right| }^{p}\mathrm{\;d}x - {\int }_{\Omega }\nabla \left( \frac{g\left( u\right) }{f\left( v\right) }\right) {\... | Yes |
Theorem 2.2 For \( T \) sufficiently small, the map \( \left( {\left( {{\beta }_{11},{\beta }_{21}}\right) ,\left( {{\beta }_{12},{\beta }_{22}}\right) }\right) \in U \rightarrow \left( {u, v}\right) = \left( {u, v}\right) \left\lbrack \left( {\left( {{\beta }_{11},{\beta }_{21}}\right) ,\left( {{\beta }_{12},{\beta }_... | Proof By Lemma 2.1, we obtain that for almost any \( t \in \left( {0, T}\right) \) \[ {\int }_{Q}\left| {{u}_{1} - {u}_{2}}\right| \left( {a, t}\right) \mathrm{d}a\mathrm{\;d}t = {\int }_{Q \cap \left( {a < t}\right) }\left| {{u}_{1} - {u}_{2}}\right| \left( {a, t}\right) \mathrm{d}a\mathrm{\;d}t + {\int }_{Q \cap \lef... | Yes |
Theorem 3.2 If \( {\beta }^{ * } = \left( {{\beta }_{1}^{ * },{\beta }_{2}^{ * }}\right) \) is an optimal control for the problem \( \left( \mathrm{{OH}}\right) \) and \( \left( {{u}^{{\beta }^{ * }},{v}^{{\beta }^{ * }}}\right) \) the solution of the system (1.1) corresponding to \( {\beta }^{ * } \), then\n\n\[ \n{\b... | Proof Since \( \left( {{\beta }_{1}^{ * },{\beta }_{2}^{ * }}\right) \) is an optimal control for the problem \( \left( \mathrm{{OH}}\right) \), for any given \( m = \) \( \left( {{m}_{1},{m}_{2}}\right) \in {L}^{\infty }\left( Q\right) \times {L}^{\infty }\left( Q\right) \), such that \( {\beta }^{ * } + {\varepsilon ... | Yes |
Consider a delayed fractional-order neural network as follows:\n\n\[ \left\{ \begin{array}{ll} {}^{c}{D}^{q}{x}_{1}\left( t\right) & = - {3.5}{x}_{1}\left( {t - \tau }\right) + {f}_{1}\left( {{x}_{1}\left( t\right) }\right) - {0.8}{f}_{2}\left( {{x}_{2}\left( t\right) }\right) \\ {}^{c}{D}^{q}{x}_{2}\left( t\right) & =... | Applying the method used in [11], for system (3.1), we establish the following computation\n\nscheme:\n\[ \begin{cases} {x}_{1, n + 1} = & {x}_{10} + \frac{{h}^{q}}{\Gamma \left( {q + 2}\right) }\left\lbrack {-{3.5}{x}_{1, n + 1 - k}^{P} + {f}_{1}\left( {x}_{1, n + 1}^{P}\right) - {0.8}{f}_{2}\left( {x}_{2, n + 1}^{P}\... | Yes |
Example 2.1 Let \( X = Y = {\mathbb{R}}^{2}, C = \left\{ {\left( {{t}_{1},{t}_{2}}\right) \in {\mathbb{R}}^{2} : {t}_{1} \geq 0,{t}_{2} \geq 0}\right\} ,\theta = \left( {1,1}\right) \in \operatorname{int}C \) , \( {\eta }_{0}\left( {{x}_{1},{x}_{2}}\right) = {x}_{1} + {x}_{2}, Q = \left\{ {\left( {{t}_{1},{t}_{2}}\righ... | \[ {\lambda }^{1 + \alpha }\theta + {\lambda F}\left( {x}_{1}\right) + \left( {1 - \lambda }\right) F\left( {x}_{2}\right) \subset F\left( {{x}_{2} + \lambda {\eta }_{0}\left( {{x}_{1},{x}_{2}}\right) }\right) + C. \] Thus, \( F \) is \( \alpha \) -order \( C \) -subpreinvex with respect to \( {\eta }_{0} \) on \( Q \)... | Yes |
Example 3.1 Suppose \( X = \mathbb{R}, Y = {\mathbb{R}}^{2}, C = \left\{ {\left( {{t}_{1},{t}_{2}}\right) \in {\mathbb{R}}^{2} : {t}_{1} \geq 0,{t}_{2} \geq 0}\right\} \) , \( \Theta = \left\{ {\left( {{t}_{1},{t}_{2}}\right) \in {\mathbb{R}}^{2} : {t}_{1} + {t}_{2} = 1,{t}_{1} \geq 0,{t}_{2} \geq 0}\right\} ,\delta = ... | Let \( \varepsilon = \frac{1}{3} \), one has \[ {C}_{\varepsilon }\left( \Theta \right) = \operatorname{cl}\left( {{S}_{\varepsilon }\left( \Theta \right) }\right) = \left\{ {\left( {{t}_{1},{t}_{2}}\right) \in {\mathbb{R}}^{2} : {t}_{2} \geq - \frac{\sqrt{2}}{4}{t}_{1},{t}_{2} \geq - 2\sqrt{2}{t}_{1}}\right\} \] \[ \o... | Yes |
Lemma 3.1 Suppose \( S \subset X \) is an invex set, \( \alpha > 0,\theta \in \operatorname{int}C, F : S \rightarrow {2}^{Y} \) is \( \alpha \) -orde \( C \) -subpreinvex with respect to \( \eta \) on \( S,\left( {{x}_{0},{y}_{0}}\right) \in \operatorname{graph}F,{D}_{g}F\left( {{x}_{0},{y}_{0}}\right) \) exists and\n\... | Proof Let \( y \in F\left( x\right) \), taking \( {x}_{n} \mathrel{\text{:=}} {x}_{0} + \frac{1}{n}\eta \left( {x,{x}_{0}}\right) ,{y}_{n} \mathrel{\text{:=}} {y}_{0} + \frac{1}{n}\left( {y - {y}_{0}}\right) + \frac{1}{{n}^{1 + \alpha }}\theta \), then \( \left( {{x}_{n},{y}_{n}}\right) \rightarrow \left( {{x}_{0},{y}_... | Yes |
Theorem 3.2 Suppose the conditions of Lemma 3.1 are satisfied. \( {x}_{0} \in S,{y}_{0} \in F\left( {x}_{0}\right) \) , and there exists an \( {\varepsilon }_{0} \in \left( {0,\delta }\right) \), such that\n\n\[ \n{D}_{g}F\left( {{x}_{0},{y}_{0}}\right) \left( {\eta \left( {x,{x}_{0}}\right) }\right) \cap \left( {-\ope... | Proof Since \( C + \operatorname{int}{C}_{{\varepsilon }_{0}}\left( \Theta \right) \subset \operatorname{int}{C}_{{\varepsilon }_{0}}\left( \Theta \right) \), it follows from (3.10) that\n\n\[ \n\left( {{D}_{g}F\left( {{x}_{0},{y}_{0}}\right) \left( {\eta \left( {x,{x}_{0}}\right) }\right) + C}\right) \cap \left( {-\op... | Yes |
Example 3.2 Suppose \( X = Y = {\mathbb{R}}^{2}, C = \left\{ {\left( {{t}_{1},{t}_{2}}\right) \in {\mathbb{R}}^{2} : {t}_{1} \geq 0,{t}_{2} \geq 0}\right\} ,\Theta = \) \( \left\{ {\left( {{t}_{1},{t}_{2}}\right) \in {\mathbb{R}}^{2} : {t}_{1} + {t}_{2} = 1,{t}_{1} \geq 0,{t}_{2} \geq 0}\right\} ,\delta = \frac{1}{2},\... | a direct calculation gives\n\n\[ T\left( {\operatorname{epi}F,\left( {{x}_{0},{y}_{0}}\right) }\right) = \left\{ {\left( {x, y}\right) \in {\mathbb{R}}^{2} \times {\mathbb{R}}^{2} : {x}_{1} \in \mathbb{R},{x}_{2} \geq 0,{y}_{2} \geq 0,2{y}_{1} + 3{y}_{2} \geq 0}\right\} ,\]\n\n\[ \widehat{G}\left( {\eta \left( {x,{x}_{... | Yes |
Theorem 3.2 Let \( 1 < p < \infty ,\phi \in {B}_{k} \) and \( f\left( t\right) = \frac{t}{\phi \left( t\right) } \) . Then\n\n\[ \n{\left( p{H}_{1}^{S}, p{H}_{\infty }^{S}\right) }_{f, F} = p{H}_{\phi F}^{S}.\n\] | Proof We define the operator \( T\left( a\right) = {S}^{\left( p\right) }\left( a\right) + R\left( a\right) \) on \( {L}_{1}.T\left( a\right) \) is sublinear, and\n\n\[ \n\parallel {Ta}{\parallel }_{1} = {\begin{Vmatrix}{S}^{\left( p\right) }\left( a\right) + R\left( a\right) \end{Vmatrix}}_{1} \leq 2\parallel a{\paral... | Yes |
Theorem 3.4 If \( \phi \in {B}_{k} \) and \( f\left( t\right) = \frac{t}{\phi \left( t\right) } \) then\n\n\[{\left( P{H}_{1}, P{H}_{\infty }\right) }_{f, F} = P{H}_{\phi F}.\] | Proof The proof is similar to Theorem 3.4, so we omit it. | No |
Lemma 2.1 Let \( u\left( t\right) \in {C}^{1}\left\lbrack {{t}_{0}, + \infty }\right) \) be a positive solution of the following inequality,\n\n\[ \left\{ \begin{array}{l} \frac{\mathrm{d}u\left( t\right) }{\mathrm{d}t} \leq u\left( t\right) \left\lbrack {{a}_{1}\left( t\right) - {b}_{1}\left( t\right) u\left( t\right)... | Proof From the comparison theorem, we can easily obtain this conclusion, so we omit its proof here. | No |
Lemma 2.2 Let \( u\left( t\right) \in {C}^{1}\left\lbrack {{t}_{0}, + \infty }\right) \) be a solution of the following inequality, and assume that it is bounded above,\n\n\[ \left\{ \begin{array}{l} \frac{\mathrm{d}u\left( t\right) }{\mathrm{d}t} \geq u\left( t\right) \left\lbrack {{a}_{2}\left( t\right) - {b}_{2}\lef... | Now we divide the proof into three steps.\n\nStep 1:\n\nClaim 2.1 If \( u\left( t\right) \in {C}^{1}\left\lbrack {{t}_{0}, + \infty }\right) \) is a solution of (2.2), then\n\n\[ u\left( t\right) > 0\text{for}t \in \left\lbrack {{t}_{0}, + \infty }\right) \text{.} \]\n\n(2.5)\n\nProof We use a contradiction. If (2.5) d... | No |
Lemma 2.1 Suppose that (1.8) hold and \( {h}_{1}\left( x\right) \in {H}^{2}\left( \mathbb{R}\right) ,{h}_{2}\left( x\right) \in {H}^{1}\left( \mathbb{R}\right) \) . Then\n\n(i) for any \( \left( {{u}_{0},{n}_{0}}\right) \in V \), the solution of (1.5)-(1.7) belongs to \( {L}^{\infty }\left( {{\mathbb{R}}^{ + };V}\right... | Proof Taking the analogous procedure as in [6-7], we can obtain the following:\n\n\[ \frac{\mathrm{d}}{\mathrm{d}t}\parallel u{\parallel }^{2} + {2\alpha }\parallel u{\parallel }^{2} + {2\beta }{\begin{Vmatrix}{u}_{x}\end{Vmatrix}}^{2} + 2\operatorname{Im}\left( {{h}_{1}, u}\right) = 0. \]\n\n(2.1)\n\nLet\n\n\[ {E}_{1}... | Yes |
There is a constant \( C > 0 \) and an increasing function \( \omega \left( \sigma \right) \) with \( \omega \left( 0\right) = 0 \) such that the solution of (3.2)-(3.4) satisfies\n\n\[ \n{\begin{Vmatrix}{u}_{\sigma }\end{Vmatrix}}_{{H}^{2}},{\begin{Vmatrix}{n}_{\sigma }\end{Vmatrix}}_{{H}^{1}} \leq C,\;\text{ for all ... | Proof Multiplying (3.2) by \( 2{\bar{u}}_{\sigma } \), integrating over \( \mathbb{R} \), and then taking imaginary parts we get\n\n\[ \n\frac{\mathrm{d}}{\mathrm{d}t}{\begin{Vmatrix}{u}_{\sigma }\end{Vmatrix}}^{2} + {2\alpha }{\begin{Vmatrix}{u}_{\sigma }\end{Vmatrix}}^{2} + {2\beta }{\begin{Vmatrix}{u}_{\sigma x}\end... | Yes |
Theorem 4.1 Suppose that (1.8) hold and \( {h}_{1}\left( x\right) \in {H}^{2}\left( \mathbb{R}\right) ,{h}_{2}\left( x\right) \in {H}^{1}\left( \mathbb{R}\right), S\left( t\right) \) be the semigroup generated by (1.5)-(1.7). Then there exists a set \( A \subset V \) satisfying\n\n(i) \( S\left( t\right) A = A,\;t \in ... | To prove the theorem, we need the following compact imbedding lemma.\n\nLemma 4.1 \( {}^{\left\ | No |
Lemma 2.1 Let \( \left( {\mathrm{H}}_{1}\right) - \left( {\mathrm{H}}_{2}\right) \) hold. Then there exists a positive constant \( C > 0 \), such that for any \( \sigma \left( \cdot \right) \in {M}_{\mathrm{{ad}}} \) and \( t,{t}_{1},{t}_{2} \in \left\lbrack {0, T}\right\rbrack \), the following inequalities hold:\n\n\... | Proof According to the state equation (2.4), we have\n\n\[ x\left( t\right) = {x}_{0} + {\int }_{0}^{t}f\left( {s, x\left( s\right) ,\sigma \left( s\right) }\right) \mathrm{d}s + {\int }_{0}^{t}g\left( {s, x\left( s\right) }\right) \mathrm{d}W\left( s\right) .\n\nUsing Itô’s formula to \( {\left| x\left( t\right) \righ... | Yes |
Theorem 3.1(Pontryagin’s maximum principle) Assume \( \left( {\mathrm{H}}_{1}\right) - \left( {\mathrm{H}}_{2}\right) \) hold. If \( \left( {\bar{x}\left( \cdot \right) ,\bar{u}\left( \cdot \right) }\right) \) is a solution to our control problem \( \left( \mathrm{P}\right) \) and \( \left( {p\left( \cdot \right), q\le... | Proof From \( J\left( {{\sigma }^{\alpha }\left( \cdot \right) }\right) \geq J\left( {\delta }_{\bar{u}\left( \cdot \right) }\right) \), and by Lebesgue’s dominated convergence theorem, we obtain\n\n\[ 0 \leq \mathop{\lim }\limits_{{\alpha \rightarrow 0}}\frac{J\left( {\left( {1 - \alpha }\right) {\delta }_{\bar{u}\lef... | Yes |
Theorem 1.1 Let \( \\varphi : \\mathbb{N} \\rightarrow \\left( {0,1}\\right) \) be a positive function. For any \( \\mathbf{y} = {\\left\{ {y}_{n}\\right\} }_{n \\geq 1} \\subset \\left\\lbrack {0,1}\\right\\rbrack \) and any dimension function \( f \) such that \( f\\left( x\\right) /x \) is decreasing, | \[ {\\mathcal{H}}^{f}\\left( {{\\mathbb{E}}_{\\mathbf{y}}\\left( \\varphi \\right) }\\right) = \\left\\{ \\begin{array}{ll} 0, & \\mathop{\\sum }\\limits_{{n = 1}}^{\\infty }{q}_{1}\\cdots {q}_{n} \\cdot f\\left( \\frac{\\varphi \\left( n\\right) }{{q}_{1}\\cdots {q}_{n}}\\right) < \\infty ; \\\\ {\\mathcal{H}}^{f}\\le... | Yes |
Theorem 3.2 If condition \( z > h > {y}_{ * } \) holds, then there exists a \( {p}_{ * } \in \left( {0,1}\right) \) such that\n\n1) When \( p = {p}_{ * } \), system (1.3) has an order-1 periodic solution ;\n\n2) When \( p < {p}_{ * } \), all the trajectories will tend to the equilibrium \( {E}_{ * }\left( {{x}_{ * },{y... | Proof First, we prove the existence of \( {p}_{ * } \in \left( {0,1}\right) \.\n\nIt is known that straight line \( y = h \) intersects with the isoclinic line \( y = \frac{\delta }{\beta }x \) . Denote the intersection point by \( D\left( {{x}_{D}, h}\right) \) . Denote the trajectory passing through the point \( D \)... | Yes |
Theorem 3.3 If \( z > h > {y}_{ * }, p > {p}_{ * } \) and \( \left( {{x}_{0},{y}_{0}}\right) \in {G}_{1}^{0} \), then the system (1.3) has a unique order-1 periodic solution, and this periodic solution is orbital asymptotically stable, where \( {G}_{1}^{0} \) is interior point set of region \( {G}_{1} \) . | Proof It is known that straight line \( y = \left( {1 - p}\right) h \) intersects with the straight line \( x = {x}_{D} \), and denote the intersection point by \( {D}_{1}\left( {{x}_{{D}_{1}},\left( {1 - p}\right) h}\right) \). According to the trajectory trend, the trajectory from the point \( \left( {{x}_{0},{y}_{0}... | Yes |
Lemma 2.3 Given \( h \in C\lbrack 1, + \infty ) \) with \( 0 < {\int }_{1}^{+\infty }h\left( s\right) \frac{\mathrm{d}s}{s} < + \infty \), then the unique solution of the problem\n\n\[ \left\{ \begin{array}{l} {D}_{{1}^{ + }}^{\alpha }x\left( t\right) + h\left( t\right) = 0, \\ x\left( 1\right) = 0,{D}_{{1}^{ + }}^{\al... | Proof By Lemma 2.2, we can see that\n\n\[ x\left( t\right) = - {I}_{1}^{\alpha }h\left( t\right) + {c}_{1}{\left( \ln t\right) }^{\alpha - 1} + {c}_{2}{\left( \ln t\right) }^{\alpha - 2} = - \frac{1}{\Gamma \left( \alpha \right) }{\int }_{1}^{t}{\left( \ln \frac{t}{s}\right) }^{\alpha - 1}h\left( s\right) \frac{\mathrm... | Yes |
Lemma 2.4 The function \( G\left( {t, s}\right) \) defined as (2.2)-(2.4) admits the following properties:\n\n1) \( G\left( {t, s}\right) \) is continuous, for any \( \left( {t, s}\right) \in \lbrack 1, + \infty ) \times \lbrack 1, + \infty ) \) ;\n\n2) \( G\left( {t, s}\right) \geq 0 \), for any \( t, s \in \lbrack 1,... | Proof Obviously, 1) and 2) hold.\n\nNext, we prove 3). For any \( t, s \in \lbrack 1, + \infty ) \), we have\n\n\[ \frac{G\left( {t, s}\right) }{1 + {\left( \ln t\right) }^{\alpha - 1}} = \frac{g\left( {t, s}\right) }{1 + {\left( \ln t\right) }^{\alpha - 1}} + \mathop{\sum }\limits_{{i = 1}}^{{+\infty }}\frac{{\lambda ... | Yes |
Lemma 2.5 \( \left( {E,\parallel \cdot \parallel }\right) \) is a Banach space. | Proof The proof is similar to the proof of Lemma2.7 in [10]. | No |
Lemma 2.6 If there exists a positive function \( l\left( t\right) \) with \( {l}^{ * } = {\int }_{1}^{+\infty }\left\lbrack {1 + {\left( \ln t\right) }^{\alpha - 1}}\right\rbrack l\left( t\right) \frac{\mathrm{d}t}{t} < \) \( + \infty \), such that\n\n\[ \left| {f\left( {t, x}\right) - f\left( {t, y}\right) }\right| \l... | Proof For any \( x \in E \), taking \( y = 0 \), then we have\n\n\[ \left| {f\left( {t, x\left( t\right) }\right) }\right| \leq l\left( t\right) \left| {x\left( t\right) }\right| + \left| {f\left( {t,0}\right) }\right| = l\left( t\right) \left\lbrack {1 + {\left( \ln t\right) }^{\alpha - 1}}\right\rbrack \frac{\left| x... | Yes |
Theorem 2.2 Let \( f \in {L}_{\Phi }^{ * }\lbrack 0,\infty ) \) . Then for some constants \( C,{t}_{0} \), we obtain\n\n\[ \n{C}^{-1}{\omega }_{\varphi }^{r}{\left( f, t\right) }_{\Phi } \leq {\bar{K}}_{r,\varphi }{\left( f,{t}^{r}\right) }_{\Phi } \leq C{\omega }_{\varphi }^{r}{\left( f, t\right) }_{\Phi } \n\] \n\nfo... | Proof Since \( {\bar{K}}_{r,\varphi }{\left( f,{t}^{r}\right) }_{\Phi } \geq {K}_{r,\varphi }{\left( f,{t}^{r}\right) }_{\Phi } \), we only need to prove the upper estimate. We can split the third term of \( {\bar{K}}_{r,\varphi }{\left( f,{t}^{r}\right) }_{\Phi } \) as\n\n\[ \n{\begin{Vmatrix}{g}^{\left( r\right) }\en... | No |
Lemma 3.1 \( {}^{\left\lbrack {21}\right\rbrack } \) Let \( {\delta }_{n}^{2}\left( x\right) = {\varphi }^{2}\left( x\right) + \frac{1}{n},{\varphi }^{2}\left( x\right) = x\left( {1 + x}\right) \) . Then we have | \[ {B}_{n}\left( {1, x}\right) = 1,{B}_{n}\left( {t - x, x}\right) = \frac{{2x} + 1}{n},{B}_{n}\left( {{\left( t - x\right) }^{2}, x}\right) \leq \frac{C}{n}{\delta }_{n}^{2}\left( x\right) ,{B}_{n}\left( {{\left( t - x\right) }^{4}, x}\right) \leq \frac{C{\delta }_{n}^{4}\left( x\right) }{{n}^{2}}. \] | No |
Lemma 3.2 For \( f \in {L}_{\Phi }^{ * }\lbrack 0,\infty ) \), we obtain \( {\begin{Vmatrix}{B}_{n}\left( f\right) \end{Vmatrix}}_{\Phi } \leq 2\parallel f{\parallel }_{\Phi } \) . | Proof By the Jensen inequality and (1.1), one has\n\n\[ \n{\begin{Vmatrix}{B}_{n}\left( f\right) \end{Vmatrix}}_{\Phi } \leq 2{\begin{Vmatrix}{B}_{n}\left( f\right) \end{Vmatrix}}_{\left( \Phi \right) } = 2\mathop{\inf }\limits_{\lambda }\left\{ {\lambda > 0 : {\int }_{0}^{\infty }\Phi \left( {\frac{1}{n + 1}\mathop{\s... | Yes |
Lemma 3.4 \( {}^{\left\lbrack 6\right\rbrack } \) For \( {\varphi }^{2}\left( x\right) = x\left( {1 + x}\right) \) and \( u \) between \( t \) and \( x \), we have | \[ \frac{\left| t - u\right| }{{\varphi }^{2}\left( u\right) } \leq \frac{\left| t - x\right| }{x}\left( {\frac{1}{1 + x} + \frac{1}{1 + t}}\right) . \] | Yes |
Lemma 1.9 \( {}^{\left\lbrack 6 - 7\right\rbrack } \) For (1.4) with \( {X}_{0} = 0 \), we have \( {X}_{0} \leq {X}_{1} \leq \cdots ,\mathop{\lim }\limits_{{k \rightarrow \infty }}{X}_{k} = S \) . Moreover, | \[ \mathop{\limsup }\limits_{{k \rightarrow \infty }}{\begin{Vmatrix}S - {X}_{k}\end{Vmatrix}}^{1/k} \leq \rho \left( {{\left\lbrack I \otimes {A}_{1} + {D}_{1}^{\mathrm{T}} \otimes I\right\rbrack }^{-1}\left\lbrack {I \otimes \left( {{A}_{2} + {SC}}\right) + {\left( {D}_{2} + CS\right) }^{\mathrm{T}} \otimes I}\right\... | Yes |
Theorem 2.1 For the MARE (1.1), if \( K \) in (1.2) is a nonsingular M-matrix or an irreducible singular M-matrix, \( S \) is the minimal nonnegative solution to (1.1), then the matrix sequence \( \left\{ {X}_{k}\right\} \) generated by FP4 is well defined, monotonically increasing and converges to \( S \) . | Proof We first prove by induction that for any \( k \geq 0 \), \[ 0 \leq {X}_{k} \leq {X}_{k + 1},\;{X}_{k} \leq S. \] (2.3) When \( k = 0 \), we have \( A{X}_{1} + {X}_{1}D = B \) . Thus, \( \left\lbrack {\left( {I \otimes A}\right) + \left( {{D}^{\mathrm{T}} \otimes I}\right) }\right\rbrack \operatorname{vec}\left( {... | Yes |
Theorem 2.2 The convergent rate of FP4 is given by\n\n\\[ \n\\mathop{\\limsup }\\limits_{{k \\rightarrow \\infty }}{\\begin{Vmatrix}S - {X}_{k}\\end{Vmatrix}}^{1/k} \\leq \\rho \\left( {{\\left\\[ \\left( I \\otimes \\left( A - SC\\right) \\right) + {D}^{\\mathrm{T}} \\otimes I\\right\\rbrack }^{-1}\\left\\[ {{\\left( ... | Proof From \\( \\left( {A - {X}_{k}C}\\right) \\left( {{X}_{k + 1} - S}\\right) + \\left( {{X}_{k + 1} - S}\\right) D = \\left( {{X}_{k} - S}\\right) {CS} \\), we have by induction that\n\n\\[ \n\\operatorname{vec}\\left( {{X}_{k} - S}\\right) = {\\left\\[ \\left( I \\otimes \\left( A - {X}_{k - 1}C\\right) \\right) + ... | Yes |
Corollary 2.1 Let the convergent rate of FP4 be denoted by \( r\left( {\mathrm{{FP}}4}\right) \), and the convergent rate of FP3 be denoted by \( r\left( {\mathrm{{FP}}3}\right) \). Then for the noncritical case we have \( r\left( {\mathrm{{FP}}4}\right) \leq r\left( {\mathrm{{FP}}3}\right) \). | Proof Let \( S \) be the unique minimal nonnegative solution of (1.1), and \( T = I \otimes (A - \) \( {SC}) + {\left( D - CS\right) }^{\mathrm{T}} \otimes I \). For the noncritical case, \( T \) is a nonsingular M-matrix by Lemma 1.8, thus we have \( {T}^{-1} \geq 0 \). Let\n\n\[ T = {\mathfrak{M}}_{1} - {\mathfrak{N}... | Yes |
Example 3.1 This MARE is taken from Chapter 2.4 in [5], where\n\n\[ D = \\left( \\begin{matrix} 3 & - 1 \\ - 1 & 3 \\end{matrix}\\right) ,\\;C = \\left( \\begin{array}{ll} {0.5} & {0.5} \\ {0.5} & {0.5} \\end{array}\\right) ,\n\n\[ B = \\left( \\begin{matrix} {1.5} & {1.5} \\ {1.5} & {1.5} \\end{matrix}\\right) ,\\;A =... | We summarize the computational results in Table 3.1.\n\nTab. 3.1 Computational results of Example 3.1\n\n<table><thead><tr><th>Methods</th><th>IT</th><th>CPU</th><th>RES</th></tr></thead><tr><td>FP3</td><td>17</td><td>0.005920</td><td>\\( {7.4942}\\mathrm{e} \\) - \\( {07} \\)</td></tr><tr><td>FP4</td><td>11</td><td>0.... | Yes |
Example 3.2 Consider the MARE with\n\n\[ A = \left( \begin{matrix} {4.27} & - 2 \\ - 1 & 6 \end{matrix}\right) ,\;B = \left( \begin{array}{ll} 1 & 1 \\ 2 & 1 \end{array}\right) \]\n\n\[ C = \left( \begin{array}{ll} 3 & 4 \\ 2 & 1 \end{array}\right) ,\;D = \left( \begin{matrix} 5 & - 1 \\ - 1 & 4 \end{matrix}\right) . \... | This example is from [6], where the corresponding \( K \) is a nonsingular M-matrix. We summarize the computational results in Table 3.2.\n\nTab. 3.2 Computational results of Example 3.2\n\n<table><thead><tr><th>Methods</th><th>IT</th><th>CPU</th><th>RES</th></tr></thead><tr><td>FP3</td><td>282</td><td>0.050858</td><td... | Yes |
Example 3.4 This MARE is taken from [7], where\n\n\[ A = D = \left( \begin{matrix} 2 & - 1 & & \\ & 2 & \ddots & \\ & & \ddots & - 1 \\ - 1 & & & 2 \end{matrix}\right) ,\]\n\nand \( B = C = I \) . The corresponding \( K \) is an irreducible singular M-matrix. For \( n = {100} \), we have the following computational res... | Tab. 3.4 Computational results of Example 3.4\n\n<table><thead><tr><th>Methods</th><th>IT</th><th>CPU</th><th>RES</th></tr></thead><tr><td>FP3</td><td>993</td><td>21.041723</td><td>\( {9.9865}\mathrm{e} \) - \( {07} \)</td></tr><tr><td>FP4</td><td>500</td><td>10.508013</td><td>9.9800e-07</td></tr><tr><td>FP5</td><td>50... | Yes |
Example 3.5 Consider the MARE for which \( A, B, C, D \) are generated as follows.\n\nFirstly set \( R = \operatorname{rand}\left( {{100},{100}}\right) \) ; then set \( W = \operatorname{diag}\left( \operatorname{Re}\right) - R \), with \( e = {\left( 1,1,\cdots ,1\right) }^{\mathrm{T}} \) ; finally, define\n\n\[ D = W... | In this case, the corresponding \( M \) is a nonsingular M-matrix. We summarize the computational results in Table 3.5.\n\nTab. 3.5 Computational results of Example 3.5\n\n<table><thead><tr><th>Methods</th><th>IT</th><th>CPU</th><th>RES</th></tr></thead><tr><td>FP3</td><td>33</td><td>0.118302</td><td>\( {9.1695}\mathrm... | Yes |
Theorem 2.1 Given a probability space \( \left( {\Omega ,{\mathcal{F}}_{t},\mathcal{F},\mathcal{P}}\right) ,\xi \in {\mathcal{F}}_{0} \) . It is independent of \( \left\{ {W}_{t}\right\} \) which is an \( N \) -dimensional Brownian motion. The diffusion processes \( {X}_{t} \) and \( {Y}_{t} \) are defined as the solut... | Proof Since \( t \geq r \) ,\n\n\[ {X}_{t}\left( \omega \right) = {X}_{r}\left( \omega \right) + {\int }_{r}^{t}b\left( {X}_{u}\right) \mathrm{d}u + {\int }_{r}^{t}\sigma \left( {X}_{u}\right) \mathrm{d}{B}_{u}. \]\n\nBy uniqueness,\n\n\[ {X}_{t}\left( \omega \right) = {X}_{t}^{r,{X}_{r}}\left( \omega \right) \]\n\nIn ... | Yes |
Let the following two geometric Brownian motion be\n\n\[ \n\\mathrm{d}{X}_{t} = {a}_{1}{X}_{t}\\mathrm{\\;d}t + \\sigma {X}_{t}\\mathrm{\\;d}{B}_{t},{X}_{0} = \\xi ,\n\]\n\nand\n\n\[ \n\\mathrm{d}{Y}_{t} = {a}_{2}{Y}_{t}\\mathrm{\\;d}t + \\sigma {Y}_{t}\\mathrm{\\;d}{B}_{t},{X}_{0} = \\eta ,\n\]\n\nwhere \( {a}_{1} \\l... | Proof For a geometric Brownian motion, we directly calculate its solution by Itô's\n\nformula\n\[ \n{X}_{t} = {X}_{0}\\exp \\left\\lbrack {\\left( {{a}_{1} - \\frac{1}{2}{\\sigma }^{2}}\\right) t + \\sigma {B}_{t}}\\right\\rbrack \n\]\n\nSimilarly,\n\n\[ \n{Y}_{t} = {Y}_{0}\\exp \\left\\lbrack {\\left( {{a}_{2} - \\fra... | Yes |
Proposition 2.1 The same diffusion processes are considered as in Theorem 2.1. If \( {b}_{1} \leq {b}_{2} \) and \( \xi { \leq }_{\mathrm{{icx}}}\eta \), then \( {X}_{t}{ \leq }_{\mathrm{{icx}}}{Y}_{t} \). | Proof Given an increasing convex function \( \phi \), we use the following representation in Lemma 2.1 for increasing convex function \( \phi \) for which \( \phi \left( {-\infty }\right) = 0 \), i.e.\n\n\[ \phi \left( u\right) = {\int }_{-\infty }^{u}g\left( s\right) \mathrm{d}s \]\n\nfor some increasing function \( g... | Yes |
Lemma 3.2 Transformed probability density function \( p\left( {y, t \mid x, s}\right) \) satisfies equation\n\n\[ \frac{\partial p\left( {\widetilde{y}, t \mid \widetilde{x}, s}\right) }{\partial t} = {L}_{\bar{y}}p\left( {\widetilde{y}, t \mid \widetilde{x}, s}\right) . \] | The one-dimensional Fokker-Planck equation is formed as follows:\n\n\[ \frac{\partial p\left( {y, t \mid x, s}\right) }{\partial t} = \frac{1}{2}\frac{{\partial }^{2}\left\lbrack {{\sigma }^{2}\left( {y, t}\right) p\left( {y, t \mid x, s}\right) }\right\rbrack }{\partial {y}^{2}} - \frac{\partial \left\lbrack {a\left( ... | Yes |
Theorem 3.1 \( p\left( {t, x}\right) \) satisfies partial differential equation | \[ \left\{ \begin{array}{l} \frac{\partial p\left( {t, x}\right) }{\partial t} + {Ap}\left( {t, x}\right) + C\left( {t, x}\right) p\left( {t, x}\right) = \varphi \left( x\right) , \\ p\left( {0, x}\right) = f\left( x\right) , \end{array}\right. \] | Yes |
Theorem 2.2 Assume \( g\left( x\right) \) satisfies (1.2), for the eigenvalue problem (1.1), we have 1) \( \left\{ {\lambda }_{k}\right\} \) is a nondecreasing sequence and \( {\lambda }_{k} \leq \gamma \) for all \( k \in \mathbb{N} \) ; 2) If \( {\lambda }_{k} \leq \gamma \), then \( \left\{ {\lambda }_{k}\right\} \)... | The Proof of Theorem 2.2 1) Since \( {\sum }_{1} \supset {\sum }_{2} \supset \cdots \supset {\sum }_{k} \supset \cdots \), we have that \( {\left\{ {\lambda }_{k}\right\} }_{k} \) is a nondecreasing sequence. Let us take \( A \in {\sum }_{k} \), and define \[ {A}_{t} = \left\{ {{v}_{t} : {v}_{t}\left( x\right) = {t}^{\... | No |
Theorem 1.1 Assume \( \mathcal{A} \) satisfies (1.2)-(1.3), \( \psi \in {W}_{\text{loc }}^{1, s}\left( \Omega \right), s \in \left( {p, n}\right) \) and \( f, F \) satisfy (1.4). Then a solution \( u \) to the \( {\mathcal{K}}_{\psi ,\theta } \) -obstacle problem belongs to \( {L}_{\text{loc }}^{{s}^{ * }}\left( \Omega... | Recall that for \( \psi \in {W}_{\text{loc }}^{1, s}\left( \Omega \right), s > p \), a solution to the \( {\mathcal{K}}_{\psi ,\theta } \) -obstacle problem belongs to \( {W}_{\text{loc }}^{1, q}\left( \Omega \right) \) for some \( q > p \), see [2]. This integrability exponent \( q \) is obviously \( \leq s \) . Sobol... | Yes |
Theorem 1.2 If \( \mathrm{E}{\xi }_{1}^{2}{\left( {\log }^{ + }\left| {\xi }_{1}\right| \right) }^{\delta + 1 - 2/\gamma } < \infty \) for some \( \gamma > 0,\delta > - 1,\gamma \left( {\delta + 1}\right) > 2 \), and \( \mathop{\sum }\limits_{{m = 1}}^{\infty }{\rho }^{2/s}\left( {2}^{m}\right) < \infty \) for some \( ... | \[ \mathop{\lim }\limits_{{\varepsilon \searrow 0}}{\varepsilon }^{\gamma \left( {\delta + 1}\right) - 2}\mathop{\sum }\limits_{{n = 3}}^{\infty }\frac{{\left( \log n\right) }^{\delta - 2/\gamma }}{{n}^{2}}\mathrm{E}{S}_{n}^{2}I\left\{ {\left| {S}_{n}\right| \geq {\varepsilon \sigma }\sqrt{n}{\left( \log n\right) }^{1/... | Yes |
Lemma 1.5 \( {}^{\left\lbrack 8\right\rbrack } \) Let \( E \) be a real uniformly convex and \( q \) -uniformly smooth Banach space with constant \( {K}_{q} \) for some \( q \in (1,2\rbrack \) . Let \( C \) be a nonempty closed convex subset of \( E \) . Let \( A : C \rightarrow E \) be an m-accretive mapping and \( B ... | \( {\begin{Vmatrix}{J}_{r}^{A}\left( I - rB\right) x - {J}_{r}^{A}\left( I - rB\right) y\end{Vmatrix}}^{q} \)\n\n\[ \leq \parallel x - y{\parallel }^{q} - r\left( {{q\alpha } - {K}_{q}{r}^{q - 1}}\right) \parallel {Bx} - {By}{\parallel }^{q} - {\varphi }_{q}\left( \begin{Vmatrix}{\left( {I - {J}_{r}^{A}}\right) \left( ... | Yes |
Theorem 2.1 Let \( E \) be a real uniformly convex and \( q \) -uniformly smooth Banach space with constant \( {K}_{q} \) for some \( q \in (1,2\rbrack \) . Let \( C \) be a nonempty, closed and convex sunny nonexpansive retract of \( E \), and \( {Q}_{C} \) be the sunny non-expansive retraction of \( E \) onto \( C \)... | \[ \left\{ \begin{array}{l} {x}_{0} \in C, \\ {u}_{n} = {Q}_{C}\left\lbrack {\left( {1 - {\alpha }_{n}}\right) \left( {{x}_{n} + {e}_{n}^{\prime }}\right) }\right\rbrack , \\ {v}_{n} = {\beta }_{n}{u}_{n} + {\gamma }_{n}\mathop{\sum }\limits_{{i = 1}}^{\infty }{a}_{i}{J}_{{r}_{n, i}}^{{A}_{i}}\left( {{u}_{n} - {r}_{n, ... | No |
Theorem 3.4 If \( \epsilon \equiv 0 \) and \( h\left( x\right) \equiv \widetilde{k} \), were \( \widetilde{k} \) is a constant, then \( u\left( x\right) \equiv \widetilde{k} \) is the unique solution of the curvature system (3.1). Moreover, \( \left\{ {u\left( x\right) \in \mathop{\bigcap }\limits_{{i = 1}}^{\infty }{L... | Proof From Theorem 3.3, we know that (3.1) has a unique solution for this special case. It is easy to check that \( u\left( x\right) \equiv \widetilde{k} \) satisfies (3.1), which implies that \( u\left( x\right) \equiv \widetilde{k} \) is the unique solution of (3.1) for this special case.\n\nNext, we show that \( \ma... | Yes |
Theorem 3.5 Let \( {q}^{\prime } = \mathop{\sup }\limits_{{i \in \mathbb{N}}}\left\{ {q}_{i}^{\prime }\right\}, q = \mathop{\inf }\limits_{{i \in \mathbb{N}}}\left\{ {q}_{i}\right\}, X = {L}^{{q}^{\prime }}\left( \Omega \right) \) and \( E = {L}^{q}\left( \Omega \right) \) . Let \( f : E \rightarrow E \) be a fixed con... | If, in the curvature systems (3.1), \( \epsilon \equiv 0 \), and \( h\left( x\right) \equiv \widetilde{k} \), then two sequences \( \left\{ {{x}_{n}\left( t\right) }\right\} \) and \( \left\{ {{z}_{n}\left( t\right) }\right\} \) converge strongly to the unique solution \( u\left( t\right) \) of (3.1), which is also the... | Yes |
Theorem 2.1 \( {}^{\left\lbrack 4\right\rbrack } \) A function \( x : \left\lbrack {\alpha ,\beta }\right\rbrack \rightarrow {\mathbb{R}}^{n},\left\lbrack {\alpha ,\beta }\right\rbrack \subset \left\lbrack {a, b}\right\rbrack \) is a solution of (1.1) on \( \left\lbrack {\alpha ,\beta }\right\rbrack \) if and only if \... | \[ \frac{\mathrm{d}x}{\mathrm{\;d}\tau } = {DF}\left( {x, t}\right) \] on \( \left\lbrack {\alpha ,\beta }\right\rbrack \), where \( F\left( {x, t}\right) = {\int }_{{t}_{0}}^{t}f\left( {x\left( s\right), s}\right) \mathrm{d}s \) . | Yes |
Lemma 2.2 \( {}^{\left\lbrack 8\right\rbrack } \) Assume that \( F : G \rightarrow {\mathbb{R}}^{n} \) belongs to the class \( F\left( {G, h,\omega }\right) \) . If \( x, y : \left\lbrack {a, b}\right\rbrack \rightarrow \) \( \bar{B} \) are regulated functions, then | \[ \begin{Vmatrix}{{\int }_{a}^{b}D\left\lbrack {F\left( {x\left( \tau \right), t}\right) - F\left( {y\left( \tau \right), t}\right) }\right\rbrack }\end{Vmatrix} \leq {\int }_{a}^{b}\omega \left( {\parallel x\left( t\right) - y\left( t\right) \parallel }\right) \mathrm{d}h\left( t\right) . \] | Yes |
Theorem 3.1 Assume that \( T > 0,{\varepsilon }_{0} > 0, L > 0 \) . Consider function \( f : G \rightarrow {\mathbb{R}}^{n} \) which is bounded, Lipschitz-continuous in the first argument and continuous in the second argument. Denote\n\n\[ \n{f}^{0}\left( x\right) = \frac{1}{T}{\int }_{0}^{T}f\left( {x, s}\right) \math... | Proof Let\n\n\[ \nF\left( {x, t}\right) = {\int }_{0}^{t}f\left( {x, s}\right) \mathrm{d}s.\n\]\n\nGiven an \( \varepsilon \in \left( {0,{\varepsilon }_{0}}\right\rbrack \), the function \( {x}^{\varepsilon } \) satisfies\n\n\[ \n\frac{\mathrm{d}{x}^{\varepsilon }}{\mathrm{d}\tau } = {D\varepsilon F}\left( {{x}^{\varep... | Yes |
Lemma 2.2 Let \( \\left( {X,\\rho }\\right) \) be an ultrametric space. Then\n\n\[ \n\\operatorname{dist}\\left( {B\\left( {x, r}\\right), X \\smallsetminus B\\left( {x, r}\\right) }\\right) \\geq r \n\]\n\n(2.2)\n\nfor each \( x \\in X \) and \( r > 0 \) . | Proof For any \( y \\in B\\left( {x, r}\\right) \) and \( z \\in X \\smallsetminus B\\left( {x, r}\\right) ,\\rho \\left( {x, y}\\right) \\leq r,\\rho \\left( {x, z}\\right) > r \) . Combining (2.1), we get \( \\rho \\left( {y, z}\\right) > r \) . So \( \\operatorname{dist}\\left( {B\\left( {x, r}\\right), X \\smallset... | Yes |
Lemma 2.3 Let \( \\left( {X,\\rho }\\right) \) be an ultrametric space. If \( E \\subset X \) is bounded uniformly disconnected and \( s \) -super homogeneous, then for each \( p \\in E \) and \( 0 < r < R \\leq \\left| E\\right| \), there exists disjoint balls \( B\\left( {{x}_{i}, r}\\right) ,{x}_{i} \\in E \\cap B\\... | Proof Give \( p \\in E \) and \( 0 < r < R \\leq \\left| E\\right| \) . Let \( \\left\\{ {{x}_{1},{x}_{2},\\cdots ,{x}_{m}}\\right\\} \) be the the maximal \( {3r} \) -discrete subset of \( B\\left( {x, R}\\right) \\cap E \) . Then by the \( s \) -super homogeneous of \( E \), we have \( m \\leq \) \( C{\\left( R/3r\\r... | Yes |
Lemma 2.4 Let \( \\left( {Y,\\rho }\\right) \) be a metric space. If \( F \\subset Y \) is bounded \( t \) -lower homogeneous, then for each \( q \\in F \) and \( 0 < r < R \\leq \\left| F\\right| \), there exists disjoint balls \( B\\left( {{x}_{i}, r}\\right) ,{x}_{i} \\in F \\cap \) \( B\\left( {q, R}\\right), i = 1... | Proof Give \( q \\in F \) and \( 0 < r < R \\leq \\left| F\\right| \) . Let \( \\left\\{ {{x}_{1},{x}_{2},\\cdots ,{x}_{n}}\\right\\} \) be the the maximal \( {3r} \) -discrete subset of \( B\\left( {x, R}\\right) \\cap F \) . Then \( B\\left( {{x}_{i}, r}\\right), i = 1,2,\\cdots, n \), are disjoint. Meanwhile, we hav... | Yes |
Theorem 4.1 If \( {R}_{0} > 1 \), then model (2.1) has an unique endemic equilibrium point \( {E}^{ * }\left( {{S}^{ * },{C}^{ * },{I}^{ * },{R}^{ * }}\right) \), where\n\n\[ \n{S}^{ * } = \frac{{N}^{ * }}{{R}_{0}} = \frac{\Lambda - \delta {I}^{ * }}{\mu {R}_{0}},\;{C}^{ * } = \frac{\mu + \delta + \gamma }{\theta }{I}^... | Proof Let \( {E}^{ * }\left( {{S}^{ * },{C}^{ * },{I}^{ * },{R}^{ * }}\right) \) be an endemic equilibrium point of model (2.1). Then \( {S}^{ * },{C}^{ * },{I}^{ * } \), and \( {R}^{ * } \) satisfy the following equations\n\n\[ \n\Lambda - \frac{\beta {S}^{ * }\left( {\varepsilon {C}^{ * } + {I}^{ * }}\right) }{{N}^{ ... | Yes |
Theorem 3.1 Let \( \xi \) be an \( {\mathcal{A}}_{{t}_{0}} \) -measurable, \( {C}_{-\infty ,\infty } \) -valued random variable such that \( \mathrm{E}{D}_{-\infty ,\infty }^{2}\left( {\xi ,\langle 0\rangle }\right) < \infty \), and let \( f : J \times \Omega \times {C}_{-\infty, S} \rightarrow \mathcal{F}\left( {\math... | Let us define a sequence \( {x}^{n} : ( - \infty, T\rbrack \times \Omega \rightarrow \mathcal{F}\left( {\mathbb{R}}^{d}\right), n = 0,1,\cdots \), of successive approximations as follows: \( {x}_{{t}_{0}}^{0} = \xi \), and \( {x}^{0}\left( t\right) = \xi \left( 0\right) \) for every \( t \in J \), and for \( n = 1,2\cd... | Yes |
Lemma 3.2 Let the assumptions of Theorem 3.1 hold. Then, there exists a positive constant \( {C}_{2} \) such that\n\n\[ \mathrm{E}\mathop{\sup }\limits_{{s \in \left\lbrack {{t}_{0}, t}\right\rbrack }}{d}_{\infty }^{2}\left( {{x}^{n + m}\left( s\right) ,{x}^{n}\left( s\right) }\right) \leq {C}_{2}{\int }_{{t}_{0}}^{t}K... | Proof Let \( t \in J \) . Due to Propositions 2.4 and 2.8 we infer that\n\n\[ \mathrm{E}\mathop{\sup }\limits_{{s \in \left\lbrack {{t}_{0}, t}\right\rbrack }}{d}_{\infty }^{2}\left( {{x}^{n + m}\left( s\right) ,{x}^{n}\left( s\right) }\right) \leq 2\mathrm{E}\mathop{\sup }\limits_{{s \in \left\lbrack {{t}_{0}, t}\righ... | Yes |
Lemma 3.3 Let the assumptions of Theorem 3.1 hold. Then there exists a positive constant \( {C}_{3} \) such that\n\n\[ \mathrm{E}\mathop{\sup }\limits_{{s \in \left\lbrack {{t}_{0}, t}\right\rbrack }}{d}_{\infty }^{2}\left( {{x}^{n + m}\left( s\right) ,{x}^{n}\left( s\right) }\right) \leq {C}_{3}\left( {t - {t}_{0}}\ri... | Proof By Lemmas 3.1 and 3.2, we have that\n\n\[ \mathrm{E}\mathop{\sup }\limits_{{s \in \left\lbrack {{t}_{0}, t}\right\rbrack }}{d}_{\infty }^{2}\left( {{x}^{n + m}\left( s\right) ,{x}^{n}\left( s\right) }\right) \leq {C}_{2}{\int }_{{t}_{0}}^{t}K\left( {\mathrm{E}\mathop{\sup }\limits_{{v \in \left\lbrack {{t}_{0}, s... | Yes |
Lemma 3.4 There exists a positive constant \( {t}_{0} \leq {T}_{1} < T \) such that for all \( n, m \geq 1 \) ,\n\n\[ 0 \leq {\varphi }_{n, m}\left( t\right) \leq {\varphi }_{n}\left( t\right) \leq {\varphi }_{n - 1}\left( t\right) \leq \cdots \leq {\varphi }_{1}\left( t\right) \]\n\nfor all \( {t}_{0} \leq t \leq {T}_... | Proof Let \( t \in \left\lbrack {{t}_{0},{T}_{1}}\right\rbrack \) . First of all, by Lemma 3.3,\n\n\[ {\varphi }_{1, m}\left( t\right) = \mathrm{E}\mathop{\sup }\limits_{{v \in \left\lbrack {{t}_{0}, t}\right\rbrack }}{d}_{\infty }^{2}\left( {{x}^{1 + m}\left( v\right) ,{x}^{1}\left( v\right) }\right) \leq {C}_{3}\left... | Yes |
The compressible Euler equations (2.1) with the following initial conditions:\n\n\\[ \n\\rho \\left( {x,0}\\right) = 1 + {0.2}\\sin \\left( {\\pi x}\\right), u\\left( {x,0}\\right) = 1, p\\left( {x,0}\\right) = 1, \n\\]\n\nare discussed. In the calculation we use 2-periodic boundary conditions. | The exact solutions are\n\n\\[ \n\\rho \\left( {x, t}\\right) = 1 + {0.2}\\sin \\left( {\\pi \\left( {x - t}\\right) }\\right), u\\left( {x, t}\\right) = 1, p\\left( {x, t}\\right) = 1.\n\\] | Yes |
The Sod shock wave problem is used. The initial conditions are\n\n\[ \left( {{\rho }_{L},{u}_{L},{p}_{L}}\right) = \left( {{1.0},0,{1.0}}\right) ,\left( {{\rho }_{R},{u}_{R},{p}_{R}}\right) = \left( {{0.125},0,{0.10}}\right) .\n\nThe computational domain is \( \left\lbrack {0,1}\right\rbrack \), and the discontinuity i... | The exact solution of this example consists of a left rarefaction wave, a right-travelling shock wave, and a right moving contact discontinuity. Figure 1 shows the numerical results at \( t = {0.2} \) with 200 uniform cells. There is no oscillation and the shock transition region is grasped sharply. The schemes with Go... | No |
Example 3 We consider the two interacting blast waves problem. \( {}^{\left\lbrack 3\right\rbrack } \) The initial conditions are \[ \rho = 1, u = 1, p = \left\{ \begin{array}{ll} {10}^{3}, & 0 < x < {0.1}, \\ {10}^{-2}, & {0.1} < x < {0.9}, \\ {10}^{2}, & {0.9} < x < {1.0}. \end{array}\right. \] The reflective boundar... | Figure 2 shows the numerical results at \( t = {0.038} \) for 400 uniform cells with different numerical fluxes. We also provide the referenced solutions for this problem, which is obtained with RKCV method \( {}^{\left\lbrack 2\right\rbrack } \) on a mesh with 2000 cells. Numerical results are given in Fig.2. It can b... | No |
Example 5 The problem with high pressure and high density ratios initially is discussed in this example. The initial conditions are given as follows\n\n\[ \left( {{\rho }_{L},{u}_{L},{p}_{L}}\right) = \left( {{1000},0,{1000}}\right) ,0 \leq x < {0.3}, \]\n\n\[ \left( {{\rho }_{R},{u}_{R},{p}_{R}}\right) = \left( {1,0,1... | In our calculation, the CFL condition number is taken as 0.2 and the constant in TVB limiter is 50 . The numerical results with 1000 uniform meshes for energy with different fluxeses are shown in Fig. 4. It is clearly showed that LWCV-LLF gives the worst numerical results. LWCV with Roe fluxes gives the best results. N... | No |
Lemma 3.1 The function \( G\left( {t, s}\right) \) in Lemma 2.3 satisfies the following properties:\n\n(i) \( G\left( {t, s}\right) \) is continuous on \( \lbrack 0, + \infty ) \times \lbrack 0, + \infty ) \) ;\n\n(ii) \( G\left( {t, s}\right) \geq 0 \), for any \( t, s \in \lbrack 0, + \infty ) \) ;\n\n(iii) \( \frac{... | Proof By the definition of \( G \) and \( \left( {\mathrm{H}}_{3}\right) \), it is easy to see that (i),(ii) hold. So we prove that the rest are true. Because\n\n\[ \frac{G\left( {t, s}\right) }{1 + {t}^{q - 1}} \leq \frac{1}{\Gamma \left( q\right) } + \frac{\Gamma \left( {q - 1}\right) t}{\left( {1 + {t}^{q - 1}}\righ... | Yes |
Lemma 3.2 Let \( F\left( t\right) = {\int }_{0}^{t}\frac{{\left( t - s\right) }^{q - 1}}{\Gamma \left( q\right) \left( {1 + {t}^{q - 1}}\right) }g\left( s\right) \mathrm{d}s \), then \( \mathop{\lim }\limits_{{t \rightarrow + \infty }}F\left( t\right) \) exists. | Proof Since\n\n\[ \n{F}^{\prime }\left( t\right) = {\int }_{0}^{t}\frac{\left( {q - 1}\right) \left( {1 + {t}^{q - 1}}\right) {\left( t - s\right) }^{q - 2} - \left( {q - 1}\right) {\left( t - s\right) }^{q - 1}{t}^{q - 2}}{\Gamma \left( q\right) {\left( 1 + {t}^{q - 1}\right) }^{2}}g\left( s\right) \mathrm{d}s \n\]\n\... | Yes |
Theorem 3.1 Suppose conditions \( \left( {\mathrm{H}}_{1}\right) ,\left( {\mathrm{H}}_{2}\right) \) and \( \left( {\mathrm{H}}_{3}\right) \) hold. Let \( 0 < a < b < d = c \) and suppose that \( f \) satisfies the following conditions:\n\n\( \left( {\mathrm{A}}_{1}\right) f\left( {t,\left( {1 + {t}^{q - 1}}\right) u}\r... | Proof We will show that the conditions of the Leggett-Williams fixed point theorem are satisfied for the operator \( T \) defined by (3.1). For \( u \in \overline{{P}_{c}} \), we have \( \parallel u{\parallel }_{{\mathbb{E}}_{\infty }} \leq c \), that is, \( 0 \leq \frac{u\left( t\right) }{1 + {t}^{q - 1}} \leq c \) fo... | Yes |
Example 4.1 Consider the following boundary value problem\n\n\[ \left\{ \begin{array}{l} {}^{c}{D}_{{0}^{ + }}^{q}u\left( t\right) = g\left( t\right) f\left( {t, u\left( t\right) }\right) ,\;t \in \lbrack 0, + \infty ), \\ u\left( 0\right) = {u}^{\left( 2\right) }\left( 0\right) = 0,\;{}^{c}{D}_{{0}^{ + }}^{\frac{3}{2}... | Then \( \frac{1}{4}{\int }_{0}^{+\infty }t\frac{1}{2}{\mathrm{e}}^{-t}\mathrm{\;d}t = \frac{1}{4}\Gamma \left( \frac{3}{2}\right) < \Gamma \left( {q - 1}\right) = \Gamma \left( \frac{3}{2}\right) \). \n\nFor applying Theorem 3.1, let \( a = \frac{4}{5}, b = 1, c = {10}^{5},\theta \left( u\right) = \mathop{\min }\limits... | Yes |
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