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Lemma 3.1 If the penalty parameter \( {\mu }_{k} \) is chosen as Step 4 of Algorithm A, the inequality (2.4) is always hold. | Proof If \( {c}_{k} = 0 \), the first block equation in (2.5) and positive definition of \( {W}_{k} \) imply\n\n\[ \Delta {m}_{k}\left( {{d}_{k},{\mu }_{k}}\right) = - {g}_{k}^{T}{d}_{k} = {d}_{k}^{T}{W}_{k}{d}_{k} + {d}_{k}^{T}{A}_{k}^{T}\left( {{\lambda }_{k} + {\delta }_{k}}\right) \]\n\n\[ = {d}_{k}^{T}{W}_{k}{d}_{... | Yes |
Lemma 3.2 The directional derivative of the merit function \( \Phi \left( {x,\mu }\right) \) along a step \( d \) satisfies \( {\Phi }^{\prime }\left( {d,\mu }\right) = {g}^{T}d - \mu \parallel c\parallel \) . | Proof We can obtain from Taylor expansion that\n\n\[ \Phi \left( {x + {\alpha d},\mu }\right) - \Phi \left( {x,\mu }\right) = f\left( {x + {\alpha d}}\right) - f\left( x\right) + \mu (\parallel c\left( {x + {\alpha d}}\right) \parallel - \parallel c\left( x\right) \parallel \]\n\n\[ \leq \alpha {g}^{T}d + {K}_{1}\mu {\... | Yes |
Lemma 3.3 There exists a constant \( {K}_{2} > 0 \) independent of the iterates such that\n\n\[ \begin{Vmatrix}{v}_{k}\end{Vmatrix} \leq {K}_{2}\begin{Vmatrix}{c}_{k}\end{Vmatrix} \] | Proof From \( {A}_{k}{v}_{k} = - {c}_{k} \) and the fact \( {v}_{k} \) lies in the range space of \( {A}_{k}^{T} \), it follows that\n\n\[ {v}_{k} = {A}_{k}^{T}{\left( {A}_{k}{A}_{k}^{T}\right) }^{-1}{A}_{k}{v}_{k} = {A}_{k}^{T}{\left( {A}_{k}{A}_{k}^{T}\right) }^{-1}\left( {-{c}_{k}}\right) ,\]\n\nand so \( \begin{Vma... | Yes |
Lemma 3.4 The sequence of penalty parameters \( \left\{ {\mu }_{k}\right\} \) is bounded above and there exists an integer \( K \geq 0 \) such that \( {\mu }_{k} = {\mu }_{K} \) for all \( k \geq K \) . | Proof The penalty parameter is increased during iteration \( k \) of Algorithm A only when (2.6) is invoked. From Lemma 3.1, we know that \( {\mu }_{k} \) is chosen to satisfy the inequality (2.4), namely\n\n\[ \Delta {m}_{k}\left( {{d}_{k},{\mu }_{k}}\right) - \frac{1}{2}{d}_{k}^{T}{W}_{k}{d}_{k} \geq \sigma {\mu }_{k... | Yes |
Lemma 3.5 Suppose that there exists a constant \( \varepsilon > 0 \) such that\n\n\[ \begin{Vmatrix}{{g}_{k} + {A}_{k}^{T}{\lambda }_{k}}\end{Vmatrix} + \begin{Vmatrix}{c}_{k}\end{Vmatrix} \geq \varepsilon \]\n\nThen, for some constant \( {K}_{4} > 0 \) independent of \( \varepsilon ,{\alpha }_{k} \geq {\alpha }_{\min ... | Proof Suppose that the line search fails for some \( \bar{\alpha } > 0 \), then\n\n\[ \Phi \left( {{x}_{k} + \bar{\alpha }{d}_{k},{\mu }_{k}}\right) - \Phi \left( {{x}_{k},{\mu }_{k}}\right) > \bar{\alpha }\eta {\Phi }^{\prime }\left( {{d}_{k},{\mu }_{k}}\right) .\n\nA Taylor expansion of \( \Phi \left( {x,\mu }\right)... | Yes |
Theorem 3.1 Under Assumptions 3.1-3.4, if Algorithm A does not terminate finitely, then \( \mathop{\lim }\limits_{{k \rightarrow \infty }}\begin{Vmatrix}{c}_{k}\end{Vmatrix} = 0 \) . | Proof Assume that \( {c}_{k} \) does not tend to zero, i.e., \( \mathop{\limsup }\limits_{{k \rightarrow \infty }}\begin{Vmatrix}{c}_{k}\end{Vmatrix} > 0 \) . Let\n\n\[ \varepsilon \mathrel{\text{:=}} \frac{1}{2}\mathop{\limsup }\limits_{{k \rightarrow \infty }}\begin{Vmatrix}{c}_{k}\end{Vmatrix} \]\n\n\[ {I}_{\varepsi... | Yes |
Theorem 2.2 Suppose that \( \left( {A\left( z\right) ,\widetilde{A}\left( z\right) }\right) \) be a pair of \( 1 \times 3 \) Laurent polynomial vectors such that\n\n\[ A\left( z\right) {\widetilde{A}}^{T}\left( {z}^{-1}\right) = 1 \]\n\nand\n\n\[ {A}_{1}\left( z\right) = {A}_{1}\left( {z}^{-1}\right) ,{A}_{2}\left( z\r... | Proof Let us recall the theory of symmetric polynomial operations.\n\nAssume that Laurent polynomials \( {f}_{1}\left( z\right) ,{f}_{2}\left( z\right) \) are symmetric about the points \( {l}_{1},{l}_{2} \) , respectively, and \( {f}_{3}\left( z\right) ,{f}_{4}\left( z\right) \) are antisymmetric about the points \( {... | Yes |
Example 3.1 Let the lengths of scaling filters be \( \left( {{15},9}\right) \) . Assume that the scaling symbols \( \left( {{H}_{0}\left( z\right) ,{\widetilde{H}}_{0}\left( z\right) }\right) \) have the following form | \[ {H}_{0}\left( z\right) = {\left( \frac{1 + z + {z}^{2}}{3}\right) }^{5}Q\left( z\right) \] \[ {\widetilde{H}}_{0}\left( z\right) = {\left( \frac{1 + z + {z}^{2}}{3}\right) }^{3}\widetilde{Q}\left( z\right) \] where \( \left( {Q\left( z\right) ,\widetilde{Q}\left( z\right) }\right) \) are symmetric Laurent polynomial... | Yes |
Example 3.2 Let the lengths of scaling filters be \( \left( {{21},{15}}\right) \) . Assume that the scaling symbols \( \left( {{H}_{0}\left( z\right) ,{\widetilde{H}}_{0}\left( z\right) }\right) \) have the following form | \[{H}_{0}\left( z\right) = {\left( \frac{1 + z + {z}^{2}}{3}\right) }^{7}Q\left( z\right) \]\n\n\[{\widetilde{H}}_{0}\left( z\right) = {\left( \frac{1 + z + {z}^{2}}{3}\right) }^{5}\widetilde{Q}\left( z\right) \]\n\nwhere \( \left( {Q\left( z\right) ,\widetilde{Q}\left( z\right) }\right) \) are symmetric Laurent polyno... | Yes |
Lemma 2.2 (see [15]) There exists a unique solution for equation (2.1) in \( {\mathcal{S}}^{2} \times {\mathcal{L}}^{2}\left( W\right) \times \) \( {\mathcal{L}}^{2}\left( \widetilde{\mu }\right) \) provided the generator \( f \) satisfies the following conditions:\n\n(i) \( E\left\lbrack {{\int }_{0}^{T}{\left| f\left... | The last condition (iii) implies that \( f \) is Lipschitz continuous in \( u \), with Lipschitz constant denoted by \( A \), and\n\n\[ \left| {f\left( {t, y, z, u}\right) - f\left( {t, y, z,{u}^{\prime }}\right) }\right| \leq C{\int }_{{R}^{ * }}\left| {u\left( x\right) - {u}^{\prime }\left( x\right) }\right| \left( {... | Yes |
Lemma 2.5 Let, for any \( \xi \in {L}^{2}\left( {\Omega ,{\mathcal{F}}_{T}, P}\right), f \) satisfies the conditions in Definition 2.4. Then there exists a unique random variable \( \eta \in {L}^{2}\left( {\Omega ,{\mathcal{F}}_{T}, P}\right) \), such that | This \( \eta \) ic called the conditional \( f \) -expectation of \( \xi \) and is denoted by \( {\varepsilon }_{f}\left\lbrack {\xi \mid {\mathcal{F}}_{t}}\right\rbrack \) . Moreover, \( {\varepsilon }_{f}\left\lbrack {\xi \mid {\mathcal{F}}_{t}}\right\rbrack = {Y}_{t}^{\xi, f, T} \) is the solution of equation (2.1) ... | No |
Theorem 3.4 Let \( {f}_{1}\left( {t, y, p, q}\right) \) and \( {f}_{2}\left( {t, y, p, q}\right) \) satisfy the conditions in Lemma 2.2 and in Definition 2.4. \( \forall \eta \in {L}^{2}\left( {\mathcal{F}}_{T}\right) \), denote \( {Y}_{t}^{1}\left( \eta \right) = {Y}_{t}^{\eta ,{f}_{1}, T},{Y}_{t}^{2}\left( \eta \righ... | Proof For any fixed \( \left( {t, y, p, q}\right) \), define\n\n\[ \n{\eta }_{n} \mathrel{\text{:=}} y + p\left( {{B}_{t + \frac{1}{n}} - {B}_{t}}\right) + q{\int }_{t}^{t + \frac{1}{n}}{\int }_{{R}^{ * }}1 \land \left| x\right| \widetilde{\mu }\left( {{ds},{dx}}\right) .\n\]\n\nBy Lemma 3.3, we know \( n\left( {{Y}_{t... | Yes |
Lemma 2.4 \( L\left( P\right) \subset {P}_{0} \) . | Proof By (R2) of Lemma 2.2, we have\n\n\[ \left( {Lu}\right) \left( t\right) = {\int }_{0}^{1}G\left( {t, s}\right) u\left( s\right) \mathrm{d}s \leq \left( {\alpha - 1}\right) {\int }_{0}^{1}q\left( s\right) u\left( s\right) \mathrm{d}s. \]\n\nOn the other hand, from (2.3), we find\n\n\[ {\int }_{0}^{1}\left( {Lu}\rig... | Yes |
Theorem 3.3 Suppose that (H1), (H3) and (H5) hold, then (1.1) has at least two positive solutions. | Proof By (H5), we have\n\n\[ \parallel {Au}\parallel = \mathop{\max }\limits_{{t \in \left\lbrack {0,1}\right\rbrack }}{\int }_{0}^{1}G\left( {t, s}\right) f\left( {s, u\left( s\right) }\right) \mathrm{d}s \]\n\n\[ \leq \left( {\alpha - 1}\right) {\zeta \rho }{\int }_{0}^{1}q\left( s\right) \mathrm{d}s = \frac{\left( {... | Yes |
Example 4.1 Let \( f\left( {t, u}\right) = {u}^{\alpha }, t \in \left\lbrack {0,1}\right\rbrack, u \in {\mathbb{R}}^{ + } \), where \( \alpha \in \left( {0,1}\right) \cup \left( {1,\infty }\right) \). If \( \alpha \in \) \( \left( {1,\infty }\right) \), then \( \left( {\mathrm{H}1}\right) \) and \( \left( {\mathrm{H}2}... | By Theorems 3.1 or 3.2, (1.1) has at least one positive solution. | No |
Example 4.2 Let\n\n\[ f\left( {t, u}\right) = \left\{ \begin{matrix} \frac{{\lambda }_{2}}{2}{u}^{\alpha }, & 0 \leq u \leq 1, \\ 2{\lambda }_{1}{u}^{\alpha } - 2{\lambda }_{1} + \frac{{\lambda }_{2}}{2}, & u \geq 1, \end{matrix}\right. \]\n\nwhere \( \alpha \geq 1 \) . | Now (H1) and (H2) are satisfied. By Theorem 3.1,(1.1) has at least one positive solution. | No |
Example 4.3 Let\n\n\[ f\left( {t, u}\right) = \left\{ \begin{matrix} 2{\lambda }_{1}{u}^{\beta }, & 0 \leq u \leq 1 \\ \frac{{\lambda }_{2}}{2}{u}^{\beta } + 2{\lambda }_{1} - \frac{{\lambda }_{2}}{2}, & u \geq 1 \end{matrix}\right. \]\n\nwhere \( 0 < \beta \leq 1 \) . | Now (H3) and (H4) are satisfied. By Theorem 3.2,(1.1) has at least one positive solution. | No |
Example 4.4 Let \( f\left( {t, u}\right) = \lambda \left( {{u}^{a} + {u}^{b}}\right) \), where \( 0 < a < 1 < b,0 < \lambda < {\left( \alpha - 1\right) }^{-1}\Gamma \left( {\alpha + 2}\right) \) . (H1),(H3) and (H5) are satisfied with \( \rho = 1 \) . | By Theorem 3.3,(1.1) has at least two positive solutions. | No |
Example 1.3 A non-*- \( A\left( k\right) \) and quasi-*- \( A\left( k\right) \) operator. | Take \( A \) and \( B \) as\n\n\[ A = \left( \begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right) \;B = \left( \begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right) .\n\]\n\nThen\n\n\[ {B}^{2} - {A}^{2} = \left( \begin{array}{ll} 1 & 2 \\ 2 & 2 \end{array}\right) \ngeq 0.\n\]\n\nHence \( {T}_{A, B} \) is not a \( * - A\le... | Yes |
Example 1.4 A non-quasi-*- \( A \) and quasi-*- \( A\left( 2\right) \) operator. | Let \( T \) be a unilateral weighted shift operator with weighted sequence \( \left( {\alpha }_{i}\right) \), given \( {\alpha }_{1} = \) \( 3,{\alpha }_{2} = 1,{\alpha }_{3} = 8,{\alpha }_{3} = {\alpha }_{4} = {\alpha }_{5} = \cdots \) . Simple calculations show that \( T \) is quasi-*- \( A\left( 2\right) \) and a no... | Yes |
Example 1.5 A non-quasi-*- \( A\left( 2\right) \) and quasi-*- \( A \) operator. | Let \( T \) be a unilateral weighted shift operator with weighted sequence \( \left( {\alpha }_{i}\right) \), given \( {\alpha }_{1} = \) \( 1,{\alpha }_{2} = \frac{1}{2},{\alpha }_{3} = 2,{\alpha }_{4} = \frac{1}{8},{\alpha }_{5} = {64},{\alpha }_{5} = {\alpha }_{6} = \cdots \) . Simple calculations show that \( T \) ... | "No" |
Lemma 2.2 For each \( k > 0 \), every quasi-*- \( A\left( k\right) \) operator is a quasi-absolute-*-k-paranormal operator. | Proof Suppose that \( T \) belongs to quasi-*- \( A\left( k\right) \) for \( k > 0 \), i.e., \[ {T}^{ * }{\left| {T}^{ * }\right| }^{2}T \leq {T}^{ * }{\left( {T}^{ * }{\left| T\right| }^{2k}T\right) }^{\frac{1}{k + 1}}T. \] Then, for every \( x \in H \) , \[ {\begin{Vmatrix}{T}^{ * }Tx\end{Vmatrix}}^{2\left( {k + 1}\r... | Yes |
Lemma 2.5 Let \( T \) be a quasi-*- \( A\left( k\right) \) operator for \( 0 < k \leq 1 \) and \( \lambda \neq 0 \) . Then \( {Tx} = {\lambda x} \) implies \( {T}^{ * }x = \bar{\lambda }x \) . | Proof Let \( \lambda \neq 0 \) and suppose \( x \in N\left( {T - \lambda }\right) \), we get \( {Tx} = {\lambda x} \) . Since \( T \) is quasi-*- \( A\left( k\right) \) , for every unit vector \( x \in H,{\begin{Vmatrix}{T}^{ * }Tx\end{Vmatrix}}^{k + 1} \leq \parallel {\left| T\right| }^{k}{T}^{2}x\parallel \parallel {... | Yes |
Theorem 2.8 If \( T \) is quasi-*- \( A\left( k\right) \) for \( 0 < k \leq 1 \), then \( T - \lambda \) has finite ascent for each \( \lambda \in \mathbb{C} \) . | Proof If \( \lambda \neq 0 \), then \( N\left( {T - \lambda }\right) \subseteq N\left( {{T}^{ * } - \bar{\lambda }}\right) \) by Lemma 2.5, thus \( N\left( {T - \lambda }\right) = N{\left( T - \lambda \right) }^{2} \) . If \( \lambda = 0 \), let \( x \in N\left( {T}^{2}\right) \), since \( T \) is quasi-*- \( A\left( k... | Yes |
Theorem 2.9 If \( T \) is quasi-*- \( A\left( k\right) \) for \( 0 < k \leq 1 \), then \( T \) has SVEP. | Proof Clearly by Theorem 2.8 and [14, Proposition 1.8]. | No |
Corollary 2.10 If \( T \) is quasi-*- \( A\left( k\right) \) for \( 0 < k \leq 1 \), then\n\n(i) \( {\sigma }_{ea}\left( {f\left( T\right) }\right) = f\left( {{\sigma }_{ea}\left( T\right) }\right) \) for every \( f \in H\left( {\sigma \left( T\right) }\right) \), where \( H\left( {\sigma \left( T\right) }\right) \) is... | Proof Note that \( T \) has SVEP, Corollary 2.10 follows by [1]. | No |
Example 3.1 Let \( \Omega = X = \left\lbrack {0,1}\right\rbrack, T : \Omega \times X \rightarrow {2}^{X} \) such that \( T\left( {w, x}\right) = \left\lbrack {x,1}\right\rbrack \) for each \( \left( {w, x}\right) \in \Omega \times X \) . Let \( y : \Omega \rightarrow X \) be a measurable function such that \( y\left( w... | For each \( n = 1,2,\cdots \), define a set-valued mapping \( {T}_{n} : \Omega \times X \rightarrow {2}^{X} \) such that for each \( \left( {w, x}\right) \in \Omega \times X \)\n\n\[ \n{T}_{n}\left( {w, x}\right) = \left\lbrack {x + \frac{1 - x}{n},1}\right\rbrack ,\forall \left( {w, x}\right) \in \Omega \times X. \n\]... | Yes |
Theorem 3.2 For each \( T \in {CB}\left( X\right) \), then the set \( {EF}\left( T\right) \), consisting of all essential fixed points of \( T \), is closed. | Proof Let \( {\xi }_{t} \in {EF}\left( T\right) \) with \( {\xi }_{t} \rightarrow \xi \) . Then for each \( t = 1,2,\cdots ,{\xi }_{t}\left( w\right) \in T\left( {w,{\xi }_{t}\left( w\right) }\right) \) , \( \forall w \in \Omega \) . Since \( F\left( T\right) \) is closed, we have that \( \xi \) is a random fixed point... | Yes |
Theorem 3.3 For each \( T \in {CB}\left( X\right) \), each random fixed point of \( T \) is essential if and only if the set-valued mapping \( F \) is continuous at \( T \) . | Proof Assume that \( F \) is continuous at \( T \), then, for any \( \varepsilon > 0 \) there exists \( \delta > 0 \) such that for each \( S \in M \) with \( \rho \left( {T, S}\right) < \delta \) satisfying that \( H\left( {F\left( T\right), F\left( S\right) }\right) < \varepsilon \) . Therefore, for each \( \xi \in F... | Yes |
Theorem 3.4 The metric space \( \left( {{CB}\left( X\right) ,\rho }\right) \) is complete. | Proof Let \( {\left\{ {T}_{n}\right\} }_{n = 1}^{\infty } \subset {CB}\left( X\right) \) be a Cauchy sequence. Then for each \( \varepsilon > 0 \), there is a number \( N \) such that \( \rho \left( {{T}_{n},{T}_{m}}\right) < \varepsilon \) for any \( n, m > N \) . That is, \( H\left( {{T}_{n}\left( {w, x}\right) ,{T}_... | Yes |
Theorem 3.5 Each random fixed point for most random set-valued mappings in \( {CB}\left( X\right) \) is essentially stable. | Proof Noting that the complete metric space \( {CB}\left( X\right) \) is a Baire space, from Theorem 3.1 and Fort’s Lemma 2.2 (or Theorem 4.2 in [20]), there is a dense residual subset \( Q \) of \( {CB}\left( X\right) \), such that \( F \) is continuous on \( Q \) . By Theorem 3.3, we have that each random fixed point... | Yes |
Lemma 3.1 Let \( \mathrm{F} \) be a family of holomorphic mappings of a domain \( D \) in \( {\mathbb{C}}^{m} \) into \( {P}^{n}\left( \mathbb{C}\right) \) . The family \( \mathrm{F} \) is not normal on \( D \) if and only if there exists a compact set \( K \subset D \) and sequences \( \left\{ {f}_{i}\right\} \subset ... | For the proof of Lemma 3.1, see Theorem 3.1 in reference [1], Theorem 6.5 in ref. [7] (Cf. reference [14]). | No |
Theorem 1.1 Suppose that \( V\left( x\right) \) satisfies \( \left( {\mathbf{V}}_{\mathbf{1}}\right) - \left( {\mathbf{V}}_{\mathbf{2}}\right) \) and \( f \) satisfies \( \left( {\mathbf{f}}_{\mathbf{1}}\right) - \left( {\mathbf{f}}_{\mathbf{4}}\right) \), then there exists \( {\varepsilon }_{0} > 0 \) such that proble... | To verify Theorem 1.1, we mainly employ the framework used in [4]. We first exploit the truncation method to modify the nonlinearity \( f\left( u\right) \) in order to obtain the existence of a nodal solution. Furthermore, to show the phenomenon of concentration, we establish an upper estimate of the energy for the sol... | No |
Lemma 3.1 Given \( \varepsilon > 0 \), the function \( {u}_{\varepsilon } \) satisfies\n\n\[ \mathop{\limsup }\limits_{{\varepsilon \rightarrow 0}}{\varepsilon }^{-3}{J}_{\varepsilon }\left( {u}_{\varepsilon }\right) \leq {c}_{{V}_{0}}^{ + } + {c}_{{V}_{0}}^{ - } \] | Proof Let \( {x}_{0} \in \operatorname{int}\left( \Lambda \right) \) be such that \( V\left( {x}_{0}\right) = {V}_{0} \) . Choosing \( r > 0 \) such that \( {B}_{r}\left( {x}_{0}\right) \subset \) \( \operatorname{int}\left( \Lambda \right) \) and \( \eta \) is a smooth function, \( 0 \leq \eta \leq 1,\left| {\nabla \e... | Yes |
Lemma 4.1 The positive local maximum and negative local minimum points of \( {u}_{\varepsilon } \) are both in \( \Lambda \) . | Proof Let \( {x}_{\varepsilon } \) be a positive local maximum of \( {u}_{\varepsilon } \) . Suppose by contradiction that \( {x}_{\varepsilon } \in {\Lambda }^{c} \) . Since \( \Delta {u}_{\varepsilon }\left( {x}_{\varepsilon }\right) \leq 0 \), using the definition of \( g \), we have\n\n\[ \alpha {u}_{\varepsilon }\... | Yes |
Lemma 4.2 Let \( {P}_{\varepsilon }^{1} \) be a local maximum of \( {u}_{\varepsilon }^{ + } \) and \( {P}_{\varepsilon }^{2} \) a local minimum of \( {u}_{\varepsilon }^{ - } \), then\n\n(1) \( {u}_{\varepsilon }\left( {P}_{\varepsilon }^{1}\right) \geq {a}_{1},{u}_{\varepsilon }\left( {P}_{\varepsilon }^{2}\right) \l... | Proof By Lemma 4.1, then \( {P}_{\varepsilon }^{1},{P}_{\varepsilon }^{2} \in \Lambda \) . Moreover, from the definition of \( g \), then for \( i = 1,2 \), we have\n\n\[ \left( {{\varepsilon }^{2}a + {\varepsilon b}{\int }_{{\mathbb{R}}^{3}}{\left| \nabla {u}_{\varepsilon }\right| }^{2}{dx}}\right) \Delta {u}_{\vareps... | No |
Lemma 4.3 If \( {\varepsilon }_{n} \downarrow 0 \) and \( {x}_{n}^{i} \in \bar{\Lambda }, i = 1,2 \) are such that\n\n\[ \n{u}_{{\varepsilon }_{n}}\left( {x}_{n}^{1}\right) \geq b > 0,\;{u}_{{\varepsilon }_{n}}\left( {x}_{n}^{2}\right) \leq - b < 0, \]\n\nthen\n\n\[ \n\mathop{\lim }\limits_{{n \rightarrow \infty }}V\le... | Proof The proof is similar to Proposition 4.1 in [4]. | No |
Lemma 4.4 If \( {m}_{\varepsilon }^{ + } = \mathop{\max }\limits_{{x \in \partial \Lambda }}{u}_{\varepsilon }^{ + }\left( x\right) ,{m}_{\varepsilon }^{ - } = \mathop{\min }\limits_{{x \in \partial \Lambda }}{u}_{\varepsilon }^{ - }\left( x\right) \), then\n\n\[ \mathop{\lim }\limits_{{\varepsilon \rightarrow 0}}{m}_{... | Proof The proof is similar to Corollary 4.1 in [4]. | No |
Theorem 3.1 All the solutions of system (2.1) are positive. | Proof From the first equation of system (2.1) we can get \( \frac{dS}{S} = \left\lbrack {r\left( {1 - \frac{S + I}{K}}\right) - {\beta I}}\right\rbrack {dt} \) , which implies \( \frac{dS}{S} = \phi \left( {S, I}\right) {dt} \), where \( \phi \left( {S, I}\right) = r\left( {1 - \frac{S + I}{K}}\right) - {\beta I} \) .\... | Yes |
Theorem 3.2 All the solutions of system (2.1) are bounded above. | Proof From the third equation of system (2.1) we may conclude that \( I \geq \frac{d}{qp},\forall t \) . Again, from the first equation of the system (2.1), we write \( S + I \leq K,\forall t \) or \( S \leq K - I \leq K - \frac{d}{qp},\forall t \) . Using above equation in the second equation we can get \( Y \leq {\be... | Yes |
Theorem 4.1 The boundary equilibrium \( {A}_{0}\left( {K,0,0}\right) \), is locally asymptotically when \( {\beta K} < c \) . | Proof The characteristic equation of system (2.1) at \( {A}_{0}\left( {K,0,0}\right) \) can be written as \( \left( {\lambda + r}\right) \left( {\lambda + d}\right) \left( {\lambda + c - {\beta K}}\right) = 0 \) . Therefore the eigenvalues to system (2.1) at \( {A}_{0}\left( {K,0,0}\right) \) are given by\n\n\[ \lambda... | Yes |
Theorem 4.2 The sufficient conditions for system (2.1) to be locally stable, at its predator free equilibrium \( {A}_{1}\left( {{S}_{1},{I}_{1},0}\right) \), are \( {qp}{I}_{1} < d \) . | Proof The characteristic equation of system (2.1) at \( {A}_{1}\left( {{S}_{1},{I}_{1},0}\right) \) can be written as \( \left( {\lambda - {qp}{I}_{1} + d}\right) \left( {{\lambda }^{2} + \frac{r{S}_{1}}{K}\lambda + \frac{r\beta }{K}{I}_{1} + {\beta }^{2}{S}_{1}{I}_{1}}\right) = 0 \) . Clearly, if \( {qp}{I}_{1} < d \)... | Yes |
Theorem 4.3 The interior equilibrium \( {A}^{ * } \) is always locally asymptotically stable. | Proof Clearly, the Jacobian matrix of system (2.1) at \( {A}^{ * } \) is\n\n\[ J\left( {A}^{ * }\right) = \left( \begin{matrix} - \frac{rS}{K} & - \frac{rS}{K} - {\beta S} & 0 \\ {\beta I} & 0 & - {pI} \\ 0 & {qpY} & 0 \end{matrix}\right) . \]\n\n(4.2)\n\nThe characteristic equation of the system (2.1) around its inter... | Yes |
Theorem 2 Let \( X \) be a Banach space, \( 2 \leq q < \infty, q \leq r < \infty \) . Then the following statements are equivalent:\n\n(1) \( X \) is \( q \) convexifiable;\n\n(2) There exists a constant \( c > 0 \) only depending on \( p \) and \( r \) such that for every \( X \) -valued quasi-martingale \( f = \left(... | Proof (1) \( \Rightarrow \) (2) Suppose \( X \) is \( q \) -convexifiable, by Remark there exists a constant \( c > 0 \) such that for every X-valued quasi-martingale \( f = \left( {{f}_{n},{\sum }_{n}, n \geq 0}\right) \)\n\n\[{\begin{Vmatrix}{S}^{\left( q\right) }\left( f\right) \end{Vmatrix}}_{r} \leq c\left( {{\beg... | Yes |
Lemma 2.1 For \( n = {2m} \), the conjugacy classes of the dihedral group \( {D}_{2n} \) are the following.\n\n\[ \n{C}_{0} = \{ 1\} ;{C}_{i} = \left\{ {{a}^{i},{a}^{{2m} - i}}\right\}, i = 1,2,\cdots, m - 1;{C}_{m} = \left\{ {a}^{m}\right\} \n\] \n\nand \n\n\[ \n{C}_{m + 1} = \left\{ {b, b{a}^{2},\cdots, b{a}^{{2m} - ... | Proof For any \( {a}^{j} \in {D}_{2n} \) and \( b{a}^{j} \in {D}_{2n}, j = 0,1,\cdots ,{2m} - 1 \), pick \( {a}^{i} \in {D}_{2n}, i = \) \( 1,2,\cdots, m - 1 \) . We have \( {a}^{j}{a}^{i}{\left( {a}^{j}\right) }^{-1} = {a}^{j + i - j} = {a}^{i} \) and \( \left( {b{a}^{j}}\right) {a}^{i}{\left( b{a}^{j}\right) }^{-1} =... | Yes |
Lemma 2.2 For \( n = {2m} \), let \( {D}_{2n} \) be a dihedral group. Suppose that \( \left( {x, y}\right) \in {R}_{i} \Leftrightarrow \) \( y{x}^{-1} \in {C}_{i}, i = 0,1,\cdots, m + 2 \) . Then\n\n(1) \( {R}_{0} = \left\{ {\left( {x, x}\right) \mid x \in {D}_{2n}}\right\} \) .\n\n(2) \( {R}_{i} = \left\{ {\left( {{a}... | Proof Let \( {b}^{k}{a}^{l} \in {D}_{2n} \) . If \( {b}^{k}{a}^{l}{x}^{-1} \in {C}_{i}, i = 1,\cdots, m - 1 \), then \( {b}^{k}{a}^{l}{x}^{-1} = {a}^{i} \) or \( {b}^{k}{a}^{l}{x}^{-1} = {a}^{{2m} - i} \) . Therefore\n\n\[ x = {a}^{{2m} - i}{b}^{k}{a}^{l} = {b}^{k}{a}^{{\left( -1\right) }^{k}\left( {{2m} - i}\right) + ... | Yes |
Proposition 3.1 Suppose that \( n = {2m} \) . Then the set of the adjacency matrices of the association scheme \( \chi \) is given by the following matrices. | (1) \( {A}_{0} = I \) ;\n\n(2) \( {A}_{i} = \left\lbrack \begin{matrix} {S}^{n - i} + {S}^{i} & 0 \\ 0 & {S}^{n - i} + {S}^{i} \end{matrix}\right\rbrack, i = 1,\cdots, m - 1 \) ;\n\n(3) \( {A}_{m} = \left\lbrack \begin{matrix} {S}^{m} & 0 \\ 0 & {S}^{m} \end{matrix}\right\rbrack \) ;\n\n(4) \( {A}_{m + 1} = \left\lbrac... | Yes |
Proposition 3.2 Suppose that \( n = {2m} \) . Then the intersection numbers of the association scheme \( \chi \) are as following. | Proof By Proposition 3.1, we obtain the results as following. If \( 0 \leq i \leq m + 2 \), we have\n\n\[ {A}_{0}{A}_{i} = {A}_{i}{A}_{0} = {A}_{i} \]\n\nIf \( 1 \leq i, j \leq m - 1 \), we have\n\n\[ \begin{array}{l} = \left\{ \begin{matrix} {A}_{i}{A}_{j} = {A}_{j}{A}_{i} \\ \left\lbrack \begin{matrix} {S}^{n - \left... | Yes |
Lemma 2.1 Let \( C \) be a nonempty subset of a Banach space \( E \) and \( T : C \rightarrow C \) be an asymptotically hemi-pseudocontractive mapping with the sequence \( \left\{ {k}_{n}\right\} \subset \lbrack 1,\infty ) \) , \( \mathop{\lim }\limits_{{n \rightarrow \infty }}{k}_{n} = 1 \) . Then\n\n\[ \parallel x - ... | Proof Since \( T \) is an asymptotically hemi-pseudocontractive mapping with the sequence \( \left\{ {k}_{n}\right\} \), for all \( x \in C \) and \( p \in F\left( T\right) \), there exists \( j\left( {x - p}\right) \in J\left( {x - p}\right) \), such that\n\n\[ \left\langle {{T}^{n}x - p, j\left( {x - p}\right) }\righ... | Yes |
Lemma 2.2 Let \( E \) be a real Banach space and \( C \) be a nonempty convex subset of \( E \) with \( C + C \subset C \) and \( T : C \rightarrow C \) be a uniformly \( L \) -Lipschitzian asymptotically hemi-pseudocontractive mapping with the sequence \( \left\{ {k}_{n}\right\} \subset \lbrack 1,\infty ),\mathop{\lim... | Proof (1) Let \( p \in F\left( T\right) \), by (1.1), we have\n\n\[ {x}_{n} = {x}_{n + 1} + {\alpha }_{n}{x}_{n} - {\alpha }_{n}{T}^{n}{y}_{n} - {u}_{n} \]\n\n\[ = \left( {1 + {\alpha }_{n}}\right) {x}_{n + 1} + {\alpha }_{n}\left( {{k}_{n}I - {T}^{n}}\right) {x}_{n + 1} - \left( {1 + {k}_{n}}\right) {\alpha }_{n}{x}_{... | Yes |
Theorem 2.1 Let \( E \) be a real Banach space and \( C \) be a nonempty closed convex subset of \( E \) with \( C + C \subset C \) and \( T : C \rightarrow C \) be a uniformly \( L \) -Lipschitzian asymptotically hemi-pseudocontractive mapping with the sequence \( {k}_{n} \subset \lbrack 1,\infty ),\mathop{\lim }\limi... | Proof The necessary of Theorem 2.1 is obvious. We just need to prove the sufficiency. Assume that \( \mathop{\liminf }\limits_{{n \rightarrow \infty }}d\left( {{x}_{n}, F\left( T\right) }\right) = 0 \), by Lemma 1.2, then \( \mathop{\lim }\limits_{{n \rightarrow \infty }}d\left( {{x}_{n}, F\left( T\right) }\right) = 0 ... | Yes |
Corollary 2.1 Under the assumptions of Theorem 2.1. Then \( \left\{ {x}_{n}\right\} \) converges strongly to a fixed point \( p \) of \( T \) if and only if there exists a subsequence \( \left\{ {x}_{{n}_{k}}\right\} \) of \( \left\{ {x}_{n}\right\} \) which converges strongly to \( p \) . | Proof It follows from \( \mathop{\liminf }\limits_{{n \rightarrow \infty }}d\left( {{x}_{n}, F}\right) \leq \mathop{\liminf }\limits_{{k \rightarrow \infty }}d\left( {{x}_{{n}_{k}}, F}\right) \leq \mathop{\lim }\limits_{{k \rightarrow \infty }}\begin{Vmatrix}{{x}_{{n}_{k}} - p}\end{Vmatrix} = 0 \) and Theorem 2.1 that ... | Yes |
Theorem 2.1 The first-order optimality system (KKT condition) of problem (2.6) | Because optimality condition and admissible condition are not ordinary differential equations, we couldn't solve the KKT system above using ordinary differential solver directly. Here, we discretize the primal equation and the adjoint equation firstly. Then we solve the KKT system from the perspective of nonlinear equa... | Yes |
Lemma 2.2 [8] Given \( y \in C\left\lbrack {0,1}\right\rbrack \) . The problem\n\n\[ \left\{ \begin{array}{l} {\left\lbrack {\varphi }_{p}\left( {D}_{0 + }^{\alpha }u\left( t\right) \right) \right\rbrack }^{\prime } + y\left( t\right) = 0,0 < t < 1, \\ {u}^{\left( i\right) }\left( 0\right) = 0\left( {i = 0,1,\cdots, N ... | is equivalent to \( u\left( t\right) = {\int }_{0}^{1}G\left( {t, s}\right) {\varphi }_{q}\left( {{\int }_{0}^{s}y\left( r\right) {dr}}\right) {ds} \), where\n\n\[ G\left( {t, s}\right) = \left\{ \begin{array}{l} \frac{{t}^{\alpha - 1}{\left( 1 - s\right) }^{\alpha - \beta - 1} - {\left( t - s\right) }^{\alpha - 1}}{\G... | Yes |
Theorem 3.1 If \( {f}_{\infty }{C}_{2} > {F}_{0}{C}_{1} \) holds, then for each \( \lambda \in \left( {\frac{1}{{f}_{\infty }{C}_{2}},\frac{1}{{F}_{0}{C}_{1}}}\right) \) ,(1.1) has at least one positive solution. Here we impose \( \frac{1}{{f}_{\infty }{C}_{2}} = 0 \) if \( {f}_{\infty } = + \infty \) and \( \frac{1}{{... | Proof For \( \lambda \in \left( {\frac{1}{{f}_{\infty }{C}_{2}},\frac{1}{{F}_{0}{C}_{1}}}\right) \), let \( \varepsilon > 0 \) be such that \( \frac{1}{\left( {{f}_{\infty } - \varepsilon }\right) {C}_{2}} \leq \lambda \leq \frac{1}{\left( {{F}_{0} + \varepsilon }\right) {C}_{1}} \) . There exists \( {r}_{1} > 0 \) suc... | Yes |
Theorem 3.2 If \( {f}_{0}{C}_{2} > {F}_{\infty }{C}_{1} \) holds, then for each \( \lambda \in \left( {\frac{1}{{f}_{0}{C}_{2}},\frac{1}{{F}_{\infty }{C}_{1}}}\right) \) ,(1.1) has at least one positive solution. Here we impose \( \frac{1}{{f}_{0}{C}_{2}} = 0 \) if \( {f}_{0} = + \infty \) and \( \frac{1}{{F}_{\infty }... | Proof For \( \lambda \in \left( {\frac{1}{{f}_{0}{C}_{2}},\frac{1}{{F}_{\infty }{C}_{1}}}\right) \), let \( \varepsilon > 0 \) be such that \( \frac{1}{\left( {{f}_{0} - \varepsilon }\right) {C}_{2}} \leq \lambda \leq \frac{1}{\left( {{F}_{\infty } + \varepsilon }\right) {C}_{1}} \) . There exists \( {r}_{1} > 0 \) suc... | Yes |
Theorem 3.3 Suppose there exist \( {r}_{2} > {r}_{1} > 0 \) or \( {\gamma }_{0}{r}_{1} > {r}_{2} > 0 \) such that\n\n\[ \n\\mathop{\\max }\\limits_{{0 \\leq u \\leq {r}_{2}}}f\\left( u\\right) \\leq {\\varphi }_{p}\\left( \\frac{{r}_{2}}{\\lambda {C}_{1}}\\right) ,\\mathop{\\min }\\limits_{{{\\gamma }_{0}{r}_{1} \\leq ... | The proof of Theorem 3.3 is similar to that of Theorem 3.1, we omit it here. | No |
Theorem 3.4 Assume \( \left( {\mathrm{H}}_{1}\right) \) holds. If \( {f}_{0} = + \infty \) and \( {f}_{\infty } = + \infty \), then (1.1) has at least two positive solutions for each \( \lambda \in \left( {0,{\lambda }_{1}}\right) \) . | Proof Define \( a\left( r\right) = \frac{r}{{C}_{1}\mathop{\max }\limits_{{0 < u < r}}{\varphi }_{q}\left( {f\left( u\right) }\right) }.a\left( r\right) : \left( {0, + \infty }\right) \rightarrow \left( {0, + \infty }\right) \) is continuous and \( \mathop{\lim }\limits_{{r \rightarrow 0}}a\left( r\right) = \mathop{\li... | Yes |
Theorem 3.5 Assume \( \left( {\mathrm{H}}_{1}\right) \) holds. If \( {F}_{0} = 0 \) and \( {F}_{\infty } = 0 \), then (1.1) has at least two positive solutions for each \( \lambda \in \left( {{\lambda }_{2}, + \infty }\right) \) . | Proof Define \( b\left( r\right) = \frac{r}{{C}_{2}\mathop{\min }\limits_{{{\gamma }_{0}r \leq u \leq r}}{\varphi }_{q}\left( {f\left( u\right) }\right) }.b\left( r\right) : \left( {0, + \infty }\right) \rightarrow \left( {0, + \infty }\right) \) is continuous and \( \mathop{\lim }\limits_{{r \rightarrow 0}}b\left( r\r... | Yes |
Lemma 2.3 Under condition (2.2), the imbedding \( {W}_{0}^{1, p}\left( \Omega \right) \hookrightarrow C\left( \bar{\Omega }\right) \) is compact. | Proof Using Lemma D. 2 in [14], it follows that \( {W}_{0}^{1, p}\left( \Omega \right) \) is continuously embedded in \( {W}^{1,{p}^{ - }}\left( \Omega \right) \) . On the other hand, since we assume \( {p}^{ - } > N \), we deduce that \( {W}_{0}^{1,{p}^{ - }}\left( \Omega \right) \) is compactly embedded in \( C\left(... | Yes |
Lemma 3.2 Given \( \delta > 0 \), there is \( j \in N \) such that for all \( u \in {X}_{j},{\left| u\right| }_{H} \leq \delta \parallel u\parallel \) . | Proof We prove the lemma by contradiction. Suppose that there exist \( \delta > 0 \) and \( {u}_{j} \in {X}_{j} \) for every \( j \in N \) such that \( {\left| {u}_{j}\right| }_{H} \geq \delta \begin{Vmatrix}{u}_{j}\end{Vmatrix} \) . Taking \( {v}_{j} = \frac{{u}_{j}}{{\left| {u}_{j}\right| }_{H}} \), we have \( {\left... | Yes |
Lemma 3.3 Suppose \( f \) satisfy \( \left( {f}_{ * }\right) \), then there exist \( j \in N \) and \( \rho ,\alpha > 0 \), such that \( {\left. I\right| }_{\partial {B}_{\rho } \cap {X}_{j}} \geq \alpha \) | Proof Now suppose that \( \parallel u\parallel > 1 \), from \( \left( {f}_{ * }\right) \), we know that\n\n\[ I\left( u\right) = {\int }_{\Omega }P\left( \left| {\nabla u}\right| \right) {dx} - {\int }_{\Omega }F\left( {x, u}\right) {dx} \]\n\n\[ \geq \parallel u{\parallel }^{{p}^{ - }} - {C}_{1}{\left| u\right| }_{H}^... | Yes |
Lemma 3.4 Suppose \( f \) satisfy \( \left( {f}_{2}\right) \), then for each finite dimensional subspace \( \widetilde{E} \subset E \) , there is an \( R = R\left( \widetilde{E}\right) \) such that \( I \leq 0 \) on \( \widetilde{E} \smallsetminus {B}_{R\left( \widetilde{E}\right) } \) . | Proof From condition \( \left( {f}_{2}\right) \), given \( L > 0 \), there is a \( C > 0 \) such that for every \( t \in R \) , a.e. \( x \in \Omega \) ,\n\n\[ F\left( {x, t}\right) \geq L{\left| t\right| }^{{p}^{ + }} - C. \]\n\nNow let \( \widetilde{E} \) be a finite dimensional subspace, suppose that \( u \in \widet... | Yes |
Lemma 3.5 Suppose \( f \) satisfies \( \left( {f}_{1}\right) \), then \( I \) satisfies (PS) condition. | Proof We suppose that \( \begin{Vmatrix}{u}_{n}\end{Vmatrix} > 1 \) ,\n\n\[ \nM + o\left( 1\right) \begin{Vmatrix}{u}_{n}\end{Vmatrix} \geq I\left( {u}_{n}\right) - \frac{1}{\eta }{I}^{\prime }\left( {u}_{n}\right) {u}_{n} \n\]\n\n\[ \n= {\int }_{\Omega }P\left( \left| {\nabla {u}_{n}}\right| \right) {dx} - \frac{1}{\e... | Yes |
Lemma 1.3 (see [8]) Let \( \mathfrak{p} \in \mathcal{Q} \) with \( \phi \left( 0\right) = a \) and let the function \( q\left( z\right) = a + {a}_{n}{z}^{n} + \cdots \) be analytic in \( \mathbb{U} \) with \( q\left( z\right) \neq a \) and \( n \in \mathbb{N} \) . If \( q \) is not subordinate to \( \mathfrak{p} \), th... | \[ {z}_{0} = {r}_{0}{e}^{i\theta } \in \mathbb{U}\text{ and }{\xi }_{0} \in \partial \mathbb{U} \smallsetminus E\left( f\right) , \] for which \[ q\left( {\mathbb{U}}_{{r}_{0}}\right) \subset \mathfrak{p}\left( \mathbb{U}\right), q\left( {z}_{0}\right) = \mathfrak{p}\left( {z}_{0}\right) \text{ and }{z}_{0}{q}^{\prime ... | Yes |
Lemma 1.5 (see [15]) The function \( L\left( {z, t}\right) = {a}_{1}\left( t\right) z + {a}_{2}\left( t\right) {z}^{2} + \cdots \) with \( {a}_{1}\left( t\right) \neq 0 \) and \( \mathop{\lim }\limits_{{t \rightarrow \infty }}\left| {{a}_{1}\left( t\right) }\right| = \infty \) is a subordination chain if and only if | \[ \operatorname{Re}\left\{ \frac{z\partial L\left( {z, t}\right) /\partial z}{\partial L\left( {z, t}\right) /\partial t}\right\} > 0\;\left( {z \in \mathbb{U};0 \leq t < \infty }\right) . \] | Yes |
Theorem 2.1 Let \( f, g \in \mathcal{A}\left( p\right) \) and suppose that\n\n\[ \n\operatorname{Re}\left\{ {1 + \frac{z{\phi }^{\prime \prime }\left( z\right) }{{\phi }^{\prime }\left( z\right) }}\right\} > - \delta \left( {\phi \left( z\right) = \left( {1 - \lambda }\right) {\left( \frac{{Q}_{\beta, p}^{\alpha }g\lef... | Proof Let us define the functions \( F \) and \( G \), respectively, by\n\n\[ \nF\left( z\right) = {\left( \frac{{Q}_{\beta, p}^{\alpha }f\left( z\right) }{{z}^{p}}\right) }^{\mu }\text{ and }G\left( z\right) = {\left( \frac{{Q}_{\beta, p}^{\alpha }g\left( z\right) }{{z}^{p}}\right) }^{\mu }.\n\]\n\nWe first prove that... | Yes |
Theorem 2.2 Let \( f, g \in \mathcal{A}\left( p\right) \) and suppose that\n\n\[ \operatorname{Re}\left\{ {1 + \frac{z{\phi }^{\prime \prime }\left( z\right) }{{\phi }^{\prime }\left( z\right) }}\right\} > - \delta \left( {\phi \left( z\right) = \left( {1 - \lambda }\right) {\left( \frac{{Q}_{\beta, p}^{\alpha }g\left(... | Proof Let us define the functions \( F \) and \( G \) just as (2.4). We first observe that, if the function \( q \) is defined by (2.5), then we obtain from (2.6) that\n\n\[ \phi \left( z\right) = G\left( z\right) + \frac{{\lambda z}{G}^{\prime }\left( z\right) }{\mu \left( {\alpha + \beta + p - 1}\right) } = \varphi \... | Yes |
Corollary 3.1 Let \( f,{g}_{j} \in \mathcal{A}\left( p\right) \left( {j = 1,2}\right) \). Suppose that the condition (2.14) is satisfied and\n\n\[ \operatorname{Re}\left\{ {1 + \frac{z{\psi }^{\prime \prime }\left( z\right) }{{\psi }^{\prime }\left( z\right) }}\right\} > - \delta \left( {\psi \left( z\right) = \left( {... | Proof To prove our result, we have to show that the condition (3.1) implies the univalence of \( \psi \) and \( F\left( z\right) = {\left( \frac{{Q}_{\beta, p}^{\alpha }f\left( z\right) }{{z}^{p}}\right) }^{\mu } \). Since \( \delta \) given by (2.2) in Theorem 2.1 satisfies the inequality \( 0 < \delta \leq \frac{1}{2... | Yes |
Corollary 3.3 Let \( f,{g}_{j} \in \mathcal{A}\left( p\right) \left( {j = 1,2}\right) \) and suppose that\n\n\[ \operatorname{Re}\left\{ {1 + \frac{z{\phi }_{j}^{\prime \prime }\left( z\right) }{{\phi }_{j}^{\prime }\left( z\right) }}\right\} > - \delta \left( {{\phi }_{j}\left( z\right) = \left( {1 - \lambda }\right) ... | implies that\n\n\[ {\left( \frac{{J}_{c}\left( {g}_{1}\right) \left( z\right) }{{z}^{p}}\right) }^{\mu } \prec {\left( \frac{{J}_{c}\left( f\right) \left( z\right) }{{z}^{p}}\right) }^{\mu } \prec {\left( \frac{{J}_{c}\left( {g}_{2}\right) \left( z\right) }{{z}^{p}}\right) }^{\mu }.\]\n\nMoreover, the functions \( {\le... | Yes |
Lemma 2.1 The solution of (1.1) and (1.2) satisfies the following estimates:\n\n\[ \n{\\begin{Vmatrix}y\\left( t;{y}_{0}, u\\right) \\end{Vmatrix}}_{C\\left( {\\left\\lbrack {0, T}\\right\\rbrack ;{L}^{2}\\left( \\Omega \\right) }\\right) } \\leq C\\left( {{\\begin{Vmatrix}{y}_{0}\\end{Vmatrix}}_{{L}^{2}\\left( \\Omega... | We can deduce this lemma by the classical semigroup theory, see [2, 8, 14]. | No |
For any \( \phi \in {L}^{2}\left( \Omega \right) \) and any \( t \in \left\lbrack {0, T}\right\rbrack \) ,\n\n\[ \n{S}_{\alpha }\left( t\right) \phi \rightarrow S\left( t\right) \phi \text{ strongly in }{L}^{2}\left( \Omega \right) \;\text{ as }\alpha \rightarrow 1.\n\] | Proof For any \( \phi \in {L}^{2}\left( \Omega \right) \) ,\n\n\[ \n{S}_{\alpha }\left( t\right) \phi = \mathop{\sum }\limits_{{i = 1}}^{\infty }{e}^{-{\lambda }_{i}^{\alpha }t} < \phi ,{e}_{i} > {e}_{i}\n\]\n\nand\n\n\[ \nS\left( t\right) \phi = \mathop{\sum }\limits_{{i = 1}}^{\infty }{e}^{-{\lambda }_{i}t} < \phi ,{... | Yes |
Lemma 2.3 For any \( t \in \left\lbrack {0, T}\right\rbrack \) and \( u \in {\mathcal{U}}_{ad} \) ,\n\n\[ \n{y}_{\alpha }\left( {t;{y}_{0}, u}\right) \rightarrow y\left( {t;{y}_{0}, u}\right) \text{ strongly in }{L}^{2}\left( \Omega \right) \text{ as }\alpha \rightarrow 1.\n\] | Proof Indeed,\n\n\[ \ny\left( {t;{y}_{0}, u}\right) = S\left( t\right) {y}_{0} + {\int }_{0}^{t}S\left( {t - s}\right) {\chi }_{\omega }u\left( s\right) {ds}\n\]\n\nand\n\n\[ \n{y}_{\alpha }\left( {t;{y}_{0}, u}\right) = {S}_{\alpha }\left( t\right) {y}_{0} + {\int }_{0}^{t}{S}_{\alpha }\left( {t - s}\right) {\chi }_{\... | Yes |
Lemma 2.4 The adjoint operator of \( {\Lambda }_{1}^{\alpha } \) is \( {\Lambda }_{2}^{\alpha } \). For any \( f\left( \cdot \right) \in {L}^{2}\left( {0, T;{L}^{2}\left( \Omega \right) }\right) \), we have\n\n\[ \left( {{\Lambda }_{2}^{\alpha }f}\right) \left( s\right) \rightarrow \left( {{\Lambda }_{2}f}\right) \left... | Proof For any \( f\left( \cdot \right), g\left( \cdot \right) \in {L}^{2}\left( {0, T;{L}^{2}\left( \Omega \right) }\right) \), \n\n\[ < {\Lambda }_{1}^{\alpha }f\left( t\right), g\left( t\right) { > }_{{L}^{2}\left( {0, T;{L}^{2}\left( \Omega \right) }\right) } \]\n\n\[ = {\int }_{0}^{T} < g\left( t\right) ,{\int }_{0... | Yes |
Theorem 1.1 Let \( \\left( {{M}^{n},\\langle \\rangle ,\\rangle }\\right) \) be an \( n\\left( { \\geq 2}\\right) \) -dimensional complete Riemannian manifold isometrically immersed in the Euclidean space \( {\\mathbb{R}}^{N} \) by \( \\Psi ,\\Omega \) is a bounded domain in \( {M}^{n} \) . \( A \) is a positive defini... | ## 2 Proof of Theorem 1.1\n\nFirst, we give the following lemma.\n\nLemma 2.1 Let \( {\\lamb | No |
Lemma 2.1 Let \( {\lambda }_{i} \) be the \( i \) -th eigenvalue of problem (1.2) and \( {u}_{i} \) the orthonormal eigenfunction corresponding to \( {\lambda }_{i} \) (i.e. \( {\int }_{\Omega }{u}_{i}{u}_{j} = {\delta }_{ij} \) ). For any function \( {m}_{\alpha } \in {C}^{2}\left( \bar{\Omega }\right) \) satisfying \... | Proof We consider the function \( {T}_{\alpha } = {m}_{\alpha }{u}_{1} - {u}_{1}{\int }_{\Omega }{m}_{\alpha }{u}_{1}^{2} \) . Then, it is easy to check that \( {\left. {T}_{\alpha }\right| }_{\partial \Omega } = 0 \) and \[ {\int }_{\Omega }{T}_{\alpha }{u}_{1} = 0 \] Hence \[ {\int }_{\Omega }{T}_{\alpha }{u}_{l} = 0... | Yes |
Lemma 2.1 \( \left\lbrack {3,5}\right\rbrack \) Let \( \left( {T, * }\right) \) be a regular \( * \) -semigroup and \( a \in T \) . | (1) \( {F}_{T}^{2} \subseteq E\left( T\right) \) and \( a{F}_{T}^{1}{a}^{ * },{a}^{ * }{F}_{T}^{1}a \subseteq {F}_{T} \). (2) \( {F}_{T} = \left\{ {{x}^{ * }x \mid x \in T}\right\} = \left\{ {e \in E\left( T\right) \mid {e}^{ * } = e}\right\} \). (3) If \( x, y,{xy} \in {F}_{T} \), then \( {xy} = {yx} \). (4) If \( x, ... | Yes |
Lemma 3.4 [2] If \( S \) is a regular semigroup, then the mapping \( \theta : a \mapsto \left( {{\rho }_{a},{\delta }_{a}}\right) \) is a homomorphism from \( S \) to \( {\mathcal{T}}_{l}\left( {S/\mathcal{R}}\right) \times {\mathcal{T}}_{r}\left( {S/\mathcal{L}}\right) \), where \( {\rho }_{a} \) and \( {\delta }_{a} ... | \[ {\rho }_{a}\left( {R}_{x}\right) = {R}_{ax},\;{L}_{x}{\delta }_{a} = {L}_{xa}\;\left( {x \in S}\right) . \] | Yes |
Lemma 3.1 Let \( \\left( {X, d}\\right) \) be complete cone metric space, where the cone is normal. \( {E}_{1},{E}_{2},\\cdots ,{E}_{i},\\cdots \) are the subsets of \( \\left( {X, d}\\right) \) such that \( {E}_{1} \\supset {E}_{2} \\supset \\cdots \\supset {E}_{i} \\supset \\cdots \) and \( \\operatorname{diam}\\left... | Proof Suppose \( {E}_{1},{E}_{2},\\cdots ,{E}_{i},\\cdots \) are the subsets of \( \\left( {X, d}\\right) \) such that \( {E}_{1} \\supset {E}_{2} \\supset \\cdots \\supset \) \( {E}_{i} \\supset \\cdots \) and \( \\operatorname{diam}\\left( {E}_{i}\\right) \\rightarrow 0,\\left( {i \\rightarrow \\infty }\\right) \). L... | Yes |
Theorem 3.1 Let \( \left( {X, d}\right) \) be complete cone metric space, and \( Y \) is the \( n \) -dimensional Euclidean space ordered by a normal cone \( C \), whose normal constant is \( l \) . Let \( {e}^{ * } : Y \rightarrow R \) be a linear functional and \( f : X \times X \rightarrow Y \) . Suppose the followi... | Proof Without loss of generality, we consider the case \( \varepsilon = 1 \) . Let\n\n\[ F\left( x\right) = \{ y \in \left( {X, d}\right) : f\left( {x, y}\right) + d\left( {x, y}\right) \in - C\} \]\n\nfor each \( x \in \left( {X, d}\right) \) . From the conditions (i),(iv) and Lemma 2.3, we have \( F\left( x\right) \)... | Yes |
Theorem 3.2 Let \( \left( {X, d}\right) \) be a sequentially compact complete cone metric space whose cone is normal. Suppose that the function \( f : \left( {X, d}\right) \times \left( {X, d}\right) \rightarrow Y \) and \( {e}^{ * } : Y \rightarrow R \) satisfy the following conditions:\n\n(i) \( f\left( {t, t}\right)... | Proof It follows from the conditions (i-v) that the Theorem 3.1 holds. Taking \( \varepsilon = \frac{1}{n} \) , from Theorem 3.1 (b), we can find a sequence \( \left\{ {x}_{n}\right\} \) such that\n\n\[ f\left( {{x}_{n}, y}\right) + \frac{1}{n}d\left( {{x}_{n}, y}\right) \notin - C,\forall y \neq {x}_{n}. \]\n\nSince \... | Yes |
Lemma 2.3 Each \( \widetilde{\mathcal{L}} \) -class of \( S \) contains a unique idempotent. | Proof Suppose \( \left( {e, f}\right) \in \widetilde{\mathcal{L}} \) for \( e, f \in E \) . Then, we have \( e\mathcal{L}f \) . This leads to \( {ef} = e \) and \( {fe} = f \) . Since \( f \) is a left central idempotent, we immediately have \( {ef} = {fef} \) . Thereby\n\n\[\ne = {ef} = {fef} = f.\n\] | No |
Lemma 2.4 The relation \( \widetilde{\mathcal{L}} \) is a congruence on \( S \) . | Proof Let \( \left( {a, b}\right) \in \widetilde{\mathcal{L}} \) for \( a, b \in S \) . In order to show that \( \left( {{ca},{cb}}\right) \in \widetilde{\mathcal{L}} \) for any \( c \in S \) , we suppose that cae \( = {ca} \) for any \( e \in E \) . By Lemma 2.3, then there exists a unique idempotent \( {c}^{ * } \) i... | Yes |
Lemma 2.5 \( {\left( ab\right) }^{ * } = {a}^{ * }{b}^{ * } \) for any \( a, b \in S \) . | Proof It is trivial that \( \left( {b,{b}^{ * }}\right) \in \widetilde{\mathcal{L}} \) for any \( b \in S \) . Since \( \widetilde{\mathcal{L}} \) is a congruence on a \( U \) - abundant semigroup with left central idempotents, we have \( \left( {{ab}, a{b}^{ * }}\right) \in \widetilde{\mathcal{L}} \) for any \( a \in ... | Yes |
Lemma 2.6 Define a relation \( \sigma \) on \( S \) by \( {a\sigma b} \) if and only if \( {a}^{ * }{b}^{ * } = {b}^{ * } \) and \( {b}^{ * }{a}^{ * } = {a}^{ * } \) for any \( a, b \in S \) . Then the relation \( \sigma \) is a semilattice congruence on \( S \) . | Proof It is easy to see that \( \sigma \) is an equivalent relation on \( S \) . Suppose that \( {a\sigma b} \) for \( a, b \in \) \( S \) . Then \( {a}^{ * }{b}^{ * } = {b}^{ * } \) and \( {b}^{ * }{a}^{ * } = {a}^{ * } \) . By applying Lemma 2.5, we get \( {\left( ac\right) }^{ * }{\left( bc\right) }^{ * } = {a}^{ * ... | Yes |
Theorem 3.2 Let \( S \) be a semigroup. Then the following statements are equivalent:\n\n(i) \( S \) is a regular semigroup with left central idempotents;\n\n(ii) \( S \) is a strong semilattice of right groups;\n\n(iii) \( S \) is a right Clifford semigroup and \( E \) is a right normal band. | Proof We first note that \( \widetilde{\mathcal{L}} = \mathcal{L} \) and \( \widetilde{\mathcal{R}} = \mathcal{R} \) on regular semigroups. If \( S \) is a regular semigroup with left central idempotents, then by the results of Lemma 2.7, we know immediately that \( \mathcal{L} = \mathcal{H} \subseteq \mathcal{R} = \ma... | No |
Theorem 1.3 Let \( {\Omega }_{1},{\Omega }_{2} \) be two equidimensional irreducible bounded symmetric domains. Define\n\n\[ \n{X}_{1}\left( {z, w}\right) \mathrel{\text{:=}} \frac{\parallel w{\parallel }^{2}}{{N}_{{\Omega }_{1}}{\left( z,\bar{z}\right) }^{{\mu }_{1}}}\left( {\left( {z, w}\right) \in {\Omega }_{1}^{{B}... | Theorem 1.3 obviously implies the following corollaries. | "No" |
Theorem 1 Let \( x \) and \( y \) be two arbitrary fault-free vertices of enhanced hypercube \( {Q}_{n, k}\left( {n \geq 3,1 \leq k \leq n - 1}\right) \) with \( \left| {F}_{v}\right| = {f}_{v} \leq n - 1 \) . There exists a fault-free path \( P\left( {x, y}\right) \) of length of at least \( {2}^{n} - 2{f}_{v} - 1 \) ... | Proof We show this lemma by induction on \( n \geq 3 \) . For \( n = 3,\left| {F}_{v}\right| = 2 \), now we consider \( {Q}_{3,1} \) and \( {Q}_{3,2} \) . Since \( h\left( {x, y}\right) \) is odd, it is easy to know that \( h\left( {x, y}\right) = 1 \) or \( h\left( {x, y}\right) = 3 \) in \( {Q}_{3,1} \) and \( {Q}_{3... | Yes |
In this example, we use the non-uniform space mesh to discrete the space domain. We set time step corresponding to the minimal space mesh step \( {h}_{f} \) . | The numerical result of Example 4.1 are present in Figures 4 and 5 . The \( {L}^{2} \) error show in Figure 6 . There is a huge spurious reflection after the wave propagated the interface between the fine and coarse mesh. The spurious reflection is harmful to the convergence and stability of the numerical simulation. W... | No |
Example 4.2 In this example, we use Szeftel boundary condition as the interface condition. We set the local time step corresponding to fine mesh and coarse mesh respectively. | The results of Example 4.2 present in Figures 7 and 8, and the error norm is in Figure 9 , we can see from the figures, the interface condition decrease the spurious reflection significantly. | No |
Corollary 2.1 Let \( A, B, C \in {\mathcal{M}}_{n} \) be involutiory matrices and they are mutually commuting, \( a, b, c \in {\mathbb{C}}^{ * } \) and \( \left( {a + b + c}\right) \left( {a - b + c}\right) \left( {a + b - c}\right) \left( {a - b - c}\right) \neq 0 \), then\n\n\[ \n{\left( aA + bB + cC\right) }^{-1} = ... | Proof In Lemma 2.1, put \( D = 0 \) and \( d = 0 \), we will obtain Corollary 2.1. | Yes |
Lemma 2.2 (see [7], Theorem 2.2) Let \( {T}_{1},{T}_{2},{T}_{3} \in {\mathcal{M}}_{n} \smallsetminus \{ 0\} \) be three mutually commuting tripotent matrices and \( {c}_{1},{c}_{2},{c}_{3} \in {\mathbb{C}}^{ * } \) such that \( \left( {{c}_{i} + {c}_{j}}\right) \left( {{c}_{i} - {c}_{j}}\right) \neq 0(i, j = 1,2,3 \) a... | (2.4) | Yes |
Lemma 2.2 If an ambiguous ideal \( \mathcal{A} \) of \( K \) does not have rational factors, it must be of the form\n\n\[ \mathcal{A} = \mathop{\prod }\limits_{{i = 1}}^{s}{\mathfrak{P}}_{i}^{{s}_{i}},\;\text{ where }{\mathfrak{P}}_{i} \mid {P}_{i},{s}_{i} \in \{ 0,1,\cdots, l - 1\} . \] | Proof Let \( \mathcal{A} \) be an unprincipal ambiguous ideal without rational factors. It factors as\n\n\[ \mathcal{A} = \mathop{\prod }\limits_{{i \in I}}{\mathfrak{P}}_{i}^{{a}_{i}}\mathop{\prod }\limits_{{j \in J}}{\mathfrak{Q}}_{j}^{{b}_{j}} \]\n\nwhere \( {\mathfrak{P}}_{i} \) ’s are ramified prime ideals and \( ... | Yes |
Lemma 2.4 Let \( \varphi \) be a local automorphism of \( {T}_{n}\left( R\right) \) satisfying \( \varphi \left( {e}_{ii}\right) = {e}_{ii} \) for \( i = \) \( 1,2,\cdots, n \) . Then \( \varphi \left( {e}_{ij}\right) = {a}_{ij}{e}_{ij} \), where \( 1 \leq i < j \leq n \) and \( {a}_{ij} \in {R}^{ * } \) . | Proof For \( 1 \leq i < j \leq n \), there exists an automorphism \( {\varphi }_{{e}_{ii} + {e}_{jj}} \) which agree with \( \varphi \) at \( {e}_{ii} + {e}_{ij} \) . By Lemma 2.1, we know there exist \( {\beta }_{ij} \in \Upsilon ,{\tau }_{ij} \in F \) and \( {u}_{ij} \in {T}_{n}^{ * }\left( R\right) \) such that \( {... | Yes |
Lemma 2.5 Let \( \varphi \) be a local automorphism of \( {T}_{n}\left( R\right) \) satisfying \( \varphi \left( {e}_{ii}\right) = {e}_{ii} \) for \( i = \) \( 1,2,\cdots, n \) . Then there exists an inner automorphism \( {\theta }_{d} \) such that \( {\theta }_{d}\varphi \left( {e}_{i, i + 1}\right) = {e}_{i, i + 1} \... | Proof By Lemma 2.4, we have \( \varphi \left( {e}_{i, i + 1}\right) = {a}_{i, i + 1}{e}_{i, i + 1} \) with \( {a}_{i, i + 1} \in {R}^{ * } \) . Let\n\n\[ d = \operatorname{diag}\left( {1,{a}_{12}^{-1},{\left( {a}_{12}{a}_{23}\right) }^{-1},\cdots ,{\left( {a}_{12}{a}_{23}\cdots {a}_{n - 1, n}\right) }^{-1}}\right) . \]... | Yes |
Lemma 3.2 Let \( \delta \) be a local derivation of \( {T}_{n}\left( R\right), n \geq 2 \) . If \( \delta \left( {e}_{11}\right) = 0 \), then there exist an inner derivation ad \( m = \mathop{\sum }\limits_{{j = 2}}^{n}\operatorname{ad}{m}_{j} \) and a central derivation \( {\eta }_{\sigma } \) such that \( (\delta - \... | Proof By the definition of \( \delta \) and Lemma 3.1, there exists a derivation \( {\delta }_{{e}_{22}} = \operatorname{ad}{t}_{2} + {\eta }_{{\sigma }_{2}} \) , corresponding to \( {e}_{22} \), such that\n\n\[ \delta \left( {{e}_{11} + {e}_{22}}\right) = \delta \left( {e}_{11}\right) + \delta \left( {e}_{22}\right) =... | Yes |
Lemma 3.3 Let \( \delta \) be a local derivation of \( {T}_{n}\left( R\right) \) satisfying \( \delta \left( {e}_{ii}\right) = 0 \) for \( i = 1,2,\cdots, n \) . Then \( \delta \left( {e}_{ij}\right) = {a}_{ij}{e}_{ij} \) for some \( {a}_{ij} \in R \) and \( 1 \leq i < j \leq n \) . | Proof For \( {e}_{ii} + {e}_{ij}, j \neq i \), since \( \delta \) is a local derivation, from Lemma 3.1 we know there exist an inner derivation ad \( {x}_{ij} \) and a central derivation \( {\eta }_{{\gamma }_{ij}} \), depending on \( {e}_{ii} + {e}_{ij} \), such that\n\n\[ \delta \left( {{e}_{ii} + {e}_{ij}}\right) = ... | Yes |
Lemma 3.4 Let \( \delta \) be a local derivation of \( {T}_{n}\left( R\right) \) . If \( \delta \left( {e}_{ii}\right) = 0 \) for \( i = 1,2,\cdots, n \), then there exists some \( h \in {T}_{n}\left( R\right) \) such that \( \left( {\delta - \operatorname{ad}h}\right) \left( {e}_{i, i + 1}\right) = 0 \) for \( i = 1,2... | Proof By Lemma 3.3, we have \( \delta \left( {e}_{i, i + 1}\right) = {a}_{i, i + 1}{e}_{i, i + 1} \) for some \( {a}_{i, i + 1} \in R \) . Let\n\n\[ h = \operatorname{diag}\left( {0, - {a}_{12}, - \left( {{a}_{12} + {a}_{23}}\right) ,\cdots , - \left( {{a}_{12} + {a}_{23} + \cdots + {a}_{n - 1, n}}\right) }\right) .\n\... | Yes |
Lemma 3.5 Let \( \delta \) be a local derivation of \( {T}_{n}\left( R\right) \) satisfying \( \delta \left( {e}_{ii}\right) = 0 \) for \( i = 1,2,\cdots, n \) , and \( \delta \left( {e}_{i, i + 1}\right) = 0 \) for \( i = 1,2,\cdots, n - 1 \) . Then we have \( \delta \left( {e}_{i, i + k}\right) = 0 \) for any \( {e}_... | Proof We will prove this lemma by induction on \( k, k \geq 2 \) .\n\nWhen \( k = 2 \), for \( {e}_{i, i + 1} + {e}_{i + 1, i + 2} + {e}_{i, i + 2} + {e}_{i + 1, i + 1} \), since \( \delta \) is a local derivation, by Lemma 3.1, there exist an inner derivation ad \( {q}_{i}^{\left( 2\right) } \) and a central derivatio... | Yes |
Theorem 3.1 Let \( R \) be a commutative ring with identity, \( {T}_{n}\left( R\right) \) the Lie algebra consisting of all upper triangular \( n \times n \) matrices over \( R \) . Then every local derivation \( \delta \) of \( {T}_{n}\left( R\right) \) is a derivation. | Proof Let \( \delta \) be a local derivation of \( {T}_{n}\left( R\right) \) . When \( n \geq 2 \), for \( {e}_{11} \in {T}_{n}\left( R\right) \), there exists a derivation \( {\delta }_{{e}_{11}} \), depending on \( {e}_{11} \), such that \( \delta \left( {e}_{11}\right) = {\delta }_{{e}_{11}}\left( {e}_{11}\right) \)... | Yes |
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