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Theorem 3.3 Let \( \\left( {H,\\beta }\\right) \) be a Hom-bialgebra, \( \\left( {A,\\alpha }\\right) \) a left \( \\left( {H,\\beta }\\right) \) -module Hom-algebra with module structure \( \\vartriangleright : H \\otimes A \\rightarrow A \) and a left \( \\left( {H,\\beta }\\right) \) -comodule Hom-coalgebra with com... | Proof By a tedious computation we can prove it. | No |
Proposition 3.4 Let \( \left( {H,\beta ,{S}_{H}}\right) \) be a Hom-Hopf algebra, and \( \left( {A,\alpha }\right) \) ba a Hom-algebra and a Hom-coalgebra. Assume that \( \left( {A{\diamond }_{n}^{m}H,\alpha \otimes \beta }\right) \) is a generalized Radford biproduct Hom-bialgebra defined as above, and \( {S}_{A} : A ... | Proof For all \( a \in A, h \in H \), we have\n\n\[ \left( {{S}_{A{\diamondsuit }_{n}^{m}H} * {\operatorname{Id}}_{A{\diamondsuit }_{n}^{m}H}}\right) \left( {a \otimes h}\right) \]\n\n\[ {S}_{A{\diamondsuit }_{n}^{m}H}\left( {{a}_{1} \otimes {\beta }^{n}\left( {a}_{2\left( {-1}\right) }\right) {\beta }^{-1}\left( {h}_{... | No |
Proposition 4.2 When \( \left( {H,\beta }\right) \) is a Hom-Hopf algebra, \( \left( {HYD}\right) \) is equivalent to\n\n\[ \rho \left( {{\beta }^{m + 3}\left( h\right) \vartriangleright u}\right) = \left( {{\beta }^{-n - 3}\left( {h}_{11}\right) {\beta }^{-1}\left( {u}_{\left( -1\right) }\right) }\right) S\left( {{\be... | Proof \( \left( {HYD}\right) \Rightarrow {\left( HYD\right) }^{\prime } \) . We have\n\n\[ \left( {{\beta }^{-n - 3}\left( {h}_{11}\right) {\beta }^{-1}\left( {u}_{\left( -1\right) }\right) }\right) S\left( {{\beta }^{-n - 1}\left( {h}_{2}\right) }\right) \otimes {\beta }^{m + 2}\left( {h}_{12}\right) \vartriangleright... | Yes |
Proposition 4.4 Let \( \left( {H,\beta }\right) \) be a Hom-bialgebra and \( \left( {U,{ \vartriangleright }_{U},{\rho }^{U},{\alpha }_{U}}\right) ,\left( {V,{ \vartriangleright }_{V},{\rho }^{V},{\alpha }_{V}}\right) \) \( { \in }_{H}^{H}\mathbb{{YD}} \) . Define the linear map \[ {\tau }_{U, V} : U \otimes V \rightar... | Proof It is easy to prove the first equality, so we only check the second one. For all \( u \in U, v \in V \) and \( w \in W \), we have \[ \left( {{\alpha }_{W} \otimes {\tau }_{U, V}}\right) \circ \left( {{\tau }_{U, W} \otimes {\alpha }_{V}}\right) \circ \left( {{\alpha }_{U} \otimes {\tau }_{V, W}}\right) ≔ \left( ... | Yes |
Lemma 4.6 Let \( \\left( {H,\\beta }\\right) \) be a Hom-bialgebra, and\n\n\\[\\left( {U,{ \\vartriangleright }_{U},{ \\rho }^{U},{ \\alpha }_{U}}\\right) ,\\left( {V,{ \\vartriangleright }_{V},{ \\rho }^{V},{ \\alpha }_{V}}\\right) ,\\left( {W,{ \\vartriangleright }_{W},{ \\rho }^{W},{ \\alpha }_{W}}\\right) \\in {}_{... | Proof Same to the proof of [9, Proposition 3.2]. | No |
Corollary 3.1 (The Uniform Boundedness Theorem) Let \( X \) be an \( F \) -space, \( Y \) a quasi-normed linear space and \( {\left\{ {T}_{\lambda }\right\} }_{\lambda \in \Lambda } \) a family of continuous mappings defined on \( X \) into \( Y \) . Assume that\n\n\( 1)\;\| {T}_{\lambda }\left( {x + y}\right) \| \leq ... | Proof By Assumption 1), we have\n\n\[ \frac{1}{n}\begin{Vmatrix}{{T}_{\lambda }\left( x\right) }\end{Vmatrix} \leq \begin{Vmatrix}{{T}_{\lambda }\left( {\frac{1}{n}x}\right) }\end{Vmatrix}\;\text{ for each }n \in \mathbb{N}. \]\n\nTherefore, by Assumption 2), we obtain\n\n\[ \mathop{\lim }\limits_{{n \rightarrow \infty... | Yes |
Corollary 3.2 (The Generalized Closed Graph Theorem) Let \( X \) and \( Y \) be \( F \) -spaces. Let \( T : \mathcal{D}\left( T\right) \subset X \rightarrow Y \) be a mapping which satisfies the following conditions\n\n1) \( \mathcal{D}\left( T\right) \) is a closed subspace of \( X \) ;\n\n2) \( \parallel T\left( {-x}... | Proof Define \( p : \mathcal{D}\left( T\right) \rightarrow \mathbb{R} \) as follows \( p\left( x\right) \mathrel{\text{:=}} \parallel T\left( x\right) \parallel \) . It is not difficult to see that \( p \) satisfies all the conditions in Theorem 1.1. Hence, by Theorem 1.1, \( p \) is continuous on \( \mathcal{D}\left( ... | Yes |
Corollary 3.3 (The Open Mapping Theorem) Let \( X \) and \( Y \) be \( F \) -spaces. Let \( T \) : \( \mathcal{D}\left( T\right) \subset X \rightarrow Y \) be a closed linear operator with \( \mathcal{R}\left( T\right) = Y \) . Then \( T \) is an open mapping, i.e., \( T\left( U\right) \) is open in \( Y \) whenever \(... | Proof Define \( p : Y \rightarrow \mathbb{R} \) as follows\n\n\[ p\left( y\right) \mathrel{\text{:=}} \mathop{\inf }\limits_{{x \in {T}^{-1}\left( y\right) }}\parallel x\parallel \]\n\n(3.3)\n\nThen \( p \) is obviously with nonnegativity, symmetry, and absorbability on \( Y \) . We now verify that \( p \) is with coun... | Yes |
Lemma 4.1 Let \( Y \) be an \( F \) -space, \( \left\{ {Y}_{n}\right\} \) a sequence of closed subspaces of \( Y \) . Let \( \mathfrak{Y} \) be the set of all vectors \( \xi = \left\{ {y}_{n}\right\} \) with \( {y}_{n} \in {Y}_{n} \) (for every \( n \in \mathbb{N} \) ) and \( \mathop{\lim }\limits_{{n \rightarrow \inft... | Proof It is easy to show that \( \mathfrak{Y} \) is a linear space and \[ \parallel \xi \parallel \geq 0\;\mathrm{{and}}\;\parallel \xi \parallel = 0 \Leftrightarrow \xi = \mathbf{0},\;\parallel \xi + \eta \parallel \leq \parallel \xi \parallel + \parallel \eta \parallel ,\;\parallel - \xi \parallel = \parallel \xi \pa... | Yes |
Let \( T,{T}_{n} \) be all self-adjoint operators in a Hilbert space \( H \), and let\n\n\[ \underset{n \rightarrow \infty }{s\text{-}\lim }{R}_{\lambda }\left( {T}_{n}\right) = {R}_{\lambda }\left( T\right) \;\left( {\forall \lambda \in \mathbb{C} \smallsetminus \mathbb{R}}\right) ,\]\n\nwhere \( {R}_{\lambda }\left( ... | See \( \left\lbrack {6\text{, pp.}{152} - {153}}\right\rbrack \) or \( \left\lbrack {4\text{, pp.}{148} - {149}}\right\rbrack \) for the proof of the statement. | No |
Let \( T \sim \operatorname{Erlang}\left( {2,2}\right) ,\lambda = 1, c = {0.8},\delta = {0.05} \) and \( \beta = 1 \) . The solution of eq.(4.1) are | \[ {\rho }_{0} = {0.4511},\; - {R}_{0} = - {0.1386},\;{\rho }_{\gamma }^{ * } = {6.1374},\; - {R}_{\gamma }^{ * } = - {0.8249}, \] and we have \[ {V}_{2}\left( {u, b}\right) = {c}_{1}{e}^{-{0.4511u}} + {c}_{2}{e}^{0.1386u} + {c}_{3}{e}^{-{6.1374u}} + {c}_{4}{e}^{0.8249u}. \] The coefficients \( {\left\{ {c}_{i}\right\}... | Yes |
Theorem 2 For any positive integer \( n \geq 1 \), we have the identities\n\n\[ B\left( {n, x}\right) = \frac{1}{{e}^{x}}\mathop{\sum }\limits_{{m = 0}}^{\infty }\frac{{x}^{m} \cdot {m}^{n}}{m!} = \frac{1}{{e}^{x}}\left( {\frac{{x}^{1} \cdot {1}^{n}}{1!} + \frac{{x}^{2} \cdot {2}^{n}}{2!} + \frac{{x}^{3} \cdot {3}^{n}}... | From Theorem 1 and the recurrence formula of \( B\left( {n, x}\right) \), we may immediately deduce the following congruence. | No |
When \( e = 2, f = 1, q = 3 = 2 \times 1 + 1 \) . In \( \operatorname{GF}\left( q\right) ,{D}_{0} = \{ 1\} ,{D}_{1} = \{ 2\} \) , so \[ \begin{aligned} G = & \mathrm{{GF}}\left( q\right) \times \mathrm{{GF}}\left( q\right) \\ = & \{ \left( {0,0}\right) ,\left( {0,1}\right) ,\left( {0,2}\right) ,\left( {1,0}\right) ,\le... | The corresponding strongly regular graph is Figure 1, which is disconnected. | Yes |
Example 2 When \( f = 2, q = 5 = 2 \times 2 + 1 \) . In \( \operatorname{GF}\left( q\right) ,{D}_{0} = \{ 1,4\} ,{D}_{1} = \{ 2,3\} \), so\n\n\[ D = \{ \left( {1,1}\right) ,\left( {1,4}\right) ,\left( {4,1}\right) ,\left( {4,4}\right) ,\left( {2,2}\right) ,\left( {2,3}\right) ,\left( {3,2}\right) ,\left( {3,3}\right) \... | It is easy to check that \( D \) is a \( \left( {{25},8,3,2}\right) \) partial difference set. The corresponding strongly regular graph is Figure 2. | Yes |
Theorem 2.1 Let \( D = \mathop{\bigcup }\limits_{{i = 0}}^{{e - 1}}\left( {{D}_{i} \times {D}_{i}}\right) \), where \( {D}_{i} \times {D}_{i} \) stands for \( \left\{ {\left( {x, y}\right) \mid x, y \in {D}_{i}}\right\} \) . Then \( D \) is a partial difference set in \( \left( {\mathrm{{GF}}\left( q\right) \oplus \mat... | We will prove the theorem by using character theory and a property of Gauss periods.\n\nLet \( \operatorname{GF}\left( q\right) \) be the finite field of order \( q \), where \( q = {p}^{t}, p \) is a prime. Let \( q = {ef} + 1 \) , where \( e > 1 \), and let \( g \) be a primitive element of \( \mathrm{{GF}}\left( q\r... | Yes |
Lemma 2.2 Let \( G \) be an abelian group of order \( v \) and \( D \) be a subset of \( G \) with \( \{ {d}^{-1} : \;d \in D\} = D \) . Then \( D \) is a \( \left( {v, k,\lambda ,\mu }\right) \) partial difference set in \( G \) if and only if, for any character \( \chi \) of \( G \) ,\n\n\[ \mathop{\sum }\limits_{{d ... | Proof of Theorem 2.1. Let \( {\psi }_{a} \otimes {\psi }_{b} \) be a character of \( \left( {\mathrm{{GF}}\left( q\right) \oplus \mathrm{{GF}}\left( q\right) , + }\right) \) . Then\n\n\[ {\psi }_{a} \otimes {\psi }_{b}\left( D\right) = \mathop{\sum }\limits_{{i = 0}}^{{e - 1}}{\psi }_{a}\left( {D}_{i}\right) {\psi }_{b... | No |
Proposition 3.6 Let \( {\mathcal{C}}^{\prime },\mathcal{C} \) and \( {\mathcal{C}}^{\prime \prime } \) be triangulated categories, let diagram (2.2) be an upper recollement of \( \mathcal{C} \) relative to \( {\mathcal{C}}^{\prime } \) and \( {\mathcal{C}}^{\prime \prime } \), and let \( \left( {{\mathcal{C}}^{ \leq 0}... | Proof (i) For \( X \in {\mathcal{C}}^{ \leq 0}, Y \in {\mathcal{C}}^{ \geq 1} \), since \( \left( {{i}^{ * },{i}_{ * }}\right) \) is an adjoint pair and \( {i}_{ * }{i}^{ * } \) is left \( t \) -exact, we have \( {\operatorname{Hom}}_{{\mathcal{C}}^{\prime }}\left( {{i}^{ * }X,{i}^{ * }Y}\right) \cong {\operatorname{Ho... | Yes |
Theorem 1.1 Let \( E \) be a Lebesgue measurable subset of \( \mathbb{R} \) with \( \left| E\right| < \infty \) and let \( H \) be the Hilbert transform. For all \( 1 < p < \infty \) , | \[ {\int }_{E}{\left| H\left( {\chi }_{E}\right) \left( x\right) \right| }^{p}{dx} = \left( {2 - \frac{1}{{2}^{p - 2}}}\right) \frac{\left| E\right| }{{\pi }^{p}}\zeta \left( p\right) \Gamma \left( {p + 1}\right) ,\] (1.5) \[ {\int }_{\mathbb{R} \smallsetminus E}{\left| H\left( {\chi }_{E}\right) \left( x\right) \right... | Yes |
Theorem 1.2 Let \( E \) be a Lebesgue measurable subset of \( \mathbb{R} \) with \( \left| E\right| < \infty \) and let \( H \) be the Hilbert transform. For any \( \lambda > 0 \) , \[ \left| \left\{ {x \in E : \left| {H\left( {\chi }_{E}\right) \left( x\right) }\right| > \lambda }\right\} \right| = \frac{2\left| E\rig... | ## 2 Proof of Theorem 1.2 We first recall the following result in [8]. Lemma 2.1 Let \( E \) b | No |
Lemma 2.3 Let \( E \) as in the Lemma 2.1. Denote \( g\left( x\right) = f{\left( x\right) }^{-1} = \mathop{\prod }\limits_{{k = 1}}^{n}\frac{x - {b}_{k}}{x - {a}_{k}} \), we have\n\n\[ \left| {\{ x \in E : g\left( x\right) > \xi \} }\right| = \left| {\{ x \in \mathbb{R} \smallsetminus E : g\left( x\right) < - \xi \} }\... | \[ \left| {\{ x \in \mathbb{R} \smallsetminus E : g\left( x\right) > \xi \} }\right| = \left| {\{ x \in \mathbb{R} : g\left( x\right) > \xi \} }\right| = \frac{1}{\xi - 1}\mathop{\sum }\limits_{{i = 1}}^{n}\left( {{b}_{i} - {a}_{i}}\right) \text{ for any }\xi > 1; \]\n\n\[ \left| {\{ x \in E : g\left( x\right) < - \xi ... | Yes |
Theorem 4.1 If \( 0 < \lambda < {\lambda }_{1}\left( f\right) \), then \( \mathop{\inf }\limits_{{u \in {\mathcal{N}}^{ - }\left( \lambda \right) }}{J}_{\lambda }\left( u\right) > 0 \) . | Proof By (3.2) and the structure of \( {\mathcal{N}}^{ - }\left( \lambda \right) \), we easily obtain \( {J}_{\lambda }\left( u\right) > 0 \) whenever \( u \in {\mathcal{N}}^{ - }\left( \lambda \right) \) and so \( {J}_{\lambda }\left( u\right) \) is bounded from below by 0 on \( {\mathcal{N}}^{ - }\left( \lambda \righ... | Yes |
Theorem 4.4 Suppose \( {\int }_{\Omega }g\left( \xi \right) {\phi }_{1}^{r}{d\xi } > 0 \) . Then\n\n(i) \( \mathop{\lim }\limits_{{\lambda \rightarrow {\lambda }_{1}^{ - }\left( f\right) }}\mathop{\inf }\limits_{{u \in {\mathcal{N}}^{ - }\left( \lambda \right) }}{J}_{\lambda }\left( u\right) = 0 \) ;\n\n(ii) if \( {\la... | Proof (i) We may assume, without loss of generality, that \( \begin{Vmatrix}{\phi }_{1}\end{Vmatrix} = 1 \) . Since \( {\int }_{\Omega }g\left( \xi \right) {\phi }_{1}^{r}{d\xi } > \) 0 and \( \lambda < {\lambda }_{1}\left( f\right) \), we have \( {\phi }_{1} \in {\mathcal{L}}^{ + } \cap {\mathcal{B}}^{ + } \) . Hence ... | Yes |
Lemma 5.1 Suppose \( {\int }_{\Omega }g\left( \xi \right) {\phi }_{1}^{r}{d\xi } < 0 \) . Then there exists \( \delta > 0 \) such that \( \overline{{\mathcal{L}}^{ - }} \cap \overline{{\mathcal{B}}^{ + }} = \varnothing \) whenever \( {\lambda }_{1}\left( f\right) \leq \lambda < {\lambda }_{1}\left( f\right) + \delta \)... | Proof Suppose that the result is false. Then there exist sequences \( \left\{ {\lambda }_{m}\right\} \) and \( \left\{ {u}_{m}\right\} \) such that \( \begin{Vmatrix}{u}_{m}\end{Vmatrix} = 1,{\lambda }_{m} \rightarrow {\lambda }_{1}^{ + }\left( f\right) \) and\n\n\[ \n{\int }_{\Omega }\left( {{\left| {\nabla }_{\mathbb... | Yes |
Theorem 5.2 Suppose \( \overline{{\mathcal{L}}^{ - }} \cap \overline{{\mathcal{B}}^{ + }} = \varnothing \) . Then\n\n(i) every minimizing sequence for \( {J}_{\lambda } \) on \( {\mathcal{N}}^{ - }\left( \lambda \right) \) is bounded;\n\n(ii) \( \mathop{\inf }\limits_{{u \in {\mathcal{N}}^{ - }\left( \lambda \right) }}... | Proof (i) Suppose that \( \left\{ {u}_{m}\right\} \in {\mathcal{N}}^{ - }\left( \lambda \right) \) is a minimizing sequence of \( {J}_{\lambda } \) . Then\n\n\[{\int }_{\Omega }\left( {{\left| {\nabla }_{\mathbb{H}}{u}_{m}\right| }^{p} - {\lambda f}\left( \xi \right) {\left| {u}_{m}\right| }^{p}}\right) {d\xi } = {\int... | Yes |
Corollary 5.1 Suppose \( {\int }_{\Omega }g\left( \xi \right) {\phi }_{1}^{r}{d\xi } < 0 \) and \( \delta \) is as in Lemma 5.1. Then equations (1.1)-(1.2) have at least two positive solutions whenever \( {\lambda }_{1}\left( f\right) < \lambda < {\lambda }_{1}\left( f\right) + \delta \) . | Proof Since \( \lambda > {\lambda }_{1}\left( f\right) \), we have that \( {\phi }_{1} \in {\mathcal{L}}^{ - } \) . By Theorems 5.2 and 5.3, there exist minimizers \( {u}_{\lambda }^{ + } \) and \( {u}_{\lambda }^{ - } \) of \( {J}_{\lambda }\left( u\right) \) on \( {\mathcal{N}}^{ + }\left( \lambda \right) \) and \( {... | Yes |
Lemma 2.3 Let \( \varphi : R \rightarrow S \) be a ring extension with (C1) and (C4). If \( M \) is a left \( S \) -module, then we have\n\n1. \( M \) is a projective left \( R \) -module if and only if \( M \) is a projective left \( S \) -module;\n\n2. \( M \) is a flat left \( R \) -module if and only if \( M \) is ... | Proof (1) If \( M \) is a projective left \( R \) -module, then \( S{ \otimes }_{R}M \) is a projective \( S \) -module and hence \( M \) is a projective left \( S \) -module since \( {\left. {}_{S}M\right| }_{S}\left( {S{ \otimes }_{R}M}\right) \) . Conversely if \( M \) is a projective left \( S \) -module, then \( M... | Yes |
Lemma 2.4 Let \( \varphi : R \rightarrow S \) be a ring extension with (C1) and (C4). If \( M \) is a left \( S \) -module, then we have\n\n1. \( {pd}\left( {{}_{S}M}\right) = {pd}\left( {{}_{R}M}\right) = {pd}\left( {{}_{S}\left( {S{ \otimes }_{R}M}\right) }\right) \) ;\n\n2. \( {fd}\left( {{}_{S}M}\right) = {fd}\left... | Proof (1) By Lemma 2.3 (1) we have \( {pd}\left( {{}_{S}M}\right) \geq {pd}\left( {{}_{R}M}\right) \) . By (C1), we get \( {pd}\left( {{}_{S}\left( {S{ \otimes }_{R}}\right. }\right. \) \( M)) \geq {pd}\left( {{}_{S}M}\right) \) . If \( {pd}\left( {{}_{R}M}\right) = n < \infty \), then there exists a projective resolut... | Yes |
Theorem 2.5 Let \( \varphi : R \rightarrow S \) be a ring extension with (C1),(C2) and (C4). Then \( {lD}\left( R\right) = {lD}\left( S\right) \) and \( {wD}\left( R\right) = {wD}\left( S\right) \) . | Proof By Lemma 2.4, we have \( {lD}\left( R\right) \geq {lD}\left( S\right) \) and \( {wD}\left( R\right) \geq {wD}\left( S\right) \) . We now prove \( {lD}\left( R\right) \leq {lD}\left( S\right) \) . For any left \( R \) -module \( M \), we have \( {\left. {}_{R}M\right| }_{R}\left( {S{ \otimes }_{R}M}\right) \) by (... | Yes |
1. If \( M \in R \) -Mod is Gorenstein projective (resp. Gorenstein flat), then \( S{ \otimes }_{R}M \) is Gorenstein projective (resp. Gorenstein flat). | Proof (1) If \( M \in R \) -Mod is Gorenstein projective, then we have an exact sequence \( \xi = \cdots {F}^{-1} \rightarrow {F}^{0} \rightarrow {F}^{1} \rightarrow \cdots \) of projective \( R \) -modules with \( M = \ker \left( {{F}^{0} \rightarrow {F}^{1}}\right) \) and such that it remains exact whenever \( {\oper... | Yes |
Lemma 2.8 Let \( \varphi : R \rightarrow S \) be a ring extension with (C1),(C2) and (C4). Then \( R \) is right coherent if and only if \( S \) is right coherent. | Proof Let \( R \) be right coherent and \( \left\{ {{M}_{i} \mid i \in I}\right\} \) a family of flat left \( S \) -modules. Then every \( {M}_{i} \) is a flat left \( R \) -module by (C4) and \( \prod {M}_{i} \) is a flat left \( R \) -module. And hence \( S{ \otimes }_{R}\prod {M}_{i} \) is a flat left \( S \) -modul... | Yes |
1. \( M \) is a Gorenstein projective left \( R \) -module if and only if \( M \) is a Gorenstein projective left \( S \) -module. | Proof (1) Assume that \( {}_{R}M \) is Gorenstein projective, then \( S{ \otimes }_{R}M \) is a Gorenstein projective \( S \) -module by Lemma 2.7. And hence \( {}_{S}M \) is Gorenstein projective by [15, Theorem 2.5] and (C1). Conversely, it is immediate by Lemma 2.7. | Yes |
Theorem 2.11 Let \( \varphi : R \rightarrow S \) be a ring extension. Then\n\n1. If \( \varphi \) satisfies (C1),(C3) and (C4), then \( l.{Ggl}\dim \left( S\right) \leq l.{Ggl}\dim \left( R\right) \). Moreover, if \( \varphi \) also satisfies (C2), then \( l.{Ggl}\dim \left( S\right) = l.{Ggl}\dim \left( R\right) \). I... | Proof (1) Suppose that \( l.{Ggl}\dim \left( R\right) = m < \infty \) . Let \( {}_{S}M \) be an \( S \) -module. Then \( l.{Gpd}\left( {{}_{R}M}\right) \leq m \) . So there exists a Gorenstein projective resolution of \( {}_{R}M \) :\n\n\[ 0 \rightarrow {G}_{k} \rightarrow \cdots \rightarrow {G}_{0}{ \rightarrow }_{R}M... | Yes |
Theorem 2.13 Let \( \varphi : R \rightarrow S \) be a ring extension with (C1)-(C4). If \( R \) and \( S \) are Artin algebras, then\n\n1. \( R \) is \( {CM} \) -free if and only if \( S \) is \( {CM} \) -free.\n\n2. \( R \) is of finite representation type if and only if \( S \) is of finite representation type.\n\n3.... | Proof (1) Let \( R \) be \( {CM} \) -free and let \( M \in S \) -mod be Gorenstein projective. Then \( {}_{R}M \) is Gorenstein projective by Lemma 2.7. So \( {}_{R}M \) is projective and hence \( {}_{S}M \) is also projective by Lemma 2.3 (1). Thus \( S \) is \( {CM} \) -free. Conversely, Let \( S \) be \( {CM} \) -fr... | Yes |
Theorem 1.1 Assume \( \\frac{1}{p} + \\frac{1}{q} = 1, p = \\frac{\\beta + 2}{\\beta + 1},\\frac{2}{m + 2} \\geq \\frac{n\\beta }{2 + \\beta } \) and \( n - \\left( {\\beta + 1}\\right) p\\left( {\\frac{2}{2 + m} - }\\right. \) \( \\left. \\frac{n\\beta }{2 + \\beta }\\right) = 1 \) . Given initial data \( {\\varphi }_... | \[ \\mathop{\\sup }\\limits_{{t \\in \\left\\lbrack {0, T}\\right\\rbrack }}\\left( {\\parallel u\\left( {t, \\cdot }\\right) {\\parallel }_{{\\dot{H}}_{q}^{\\frac{2}{2 + m} - \\frac{n\\beta }{2 + \\beta }}\\left( {\\mathbb{R}}^{n}\\right) } + {\\begin{Vmatrix}{\\partial }_{t}u\\left( t, \\cdot \\right) \\end{Vmatrix}}... | Yes |
Theorem 1.2 Assume \( \\frac{1}{p} + \\frac{1}{q} = 1, p = \\frac{\\beta + 2}{\\beta + 1},\\frac{m + 4}{2\\left( {m + 2}\\right) } \\geq \\frac{\\left( {n + 1}\\right) \\beta }{2\\left( {2 + \\beta }\\right) }, n - \\left( {\\beta + 1}\\right) p\\left( {\\frac{2}{2 + m} - \\frac{n\\beta }{2 + \\beta }}\\right) = 1 \) | Given initial data \( {\\varphi }_{i}\\left( x\\right) \\in {C}_{c}^{\\infty }\\left( {\\mathbb{R}}^{n}\\right), i = 1,2 \) with \[ {\\begin{Vmatrix}{\\varphi }_{1}\\end{Vmatrix}}_{{\\dot{H}}_{p}^{\\frac{2}{m + 2}}\\left( {\\mathbb{R}}^{n}\\right) } + {\\begin{Vmatrix}{\\varphi }_{2}\\end{Vmatrix}}_{{L}^{p}\\left( {\\m... | Yes |
Lemma 2.3 Let \( P \) be real, \( {C}^{\infty } \) in neighborhood of the support of \( v \in {C}_{0}^{\infty } \) . Assume that the rank of \( {H}_{p}\left( y\right) = {\partial }_{{y}_{k}{y}_{l}}^{2}P\left( y\right) \) is at least \( \rho \) on the support of \( v \) . Then for some integer \( M \) , | \[ {\begin{Vmatrix}{\left( {e}^{itP}v\right) }^{ \land }\end{Vmatrix}}_{\infty } \leq C{\left( 1 + \left| t\right| \right) }^{-\frac{\rho }{2}}\mathop{\sum }\limits_{{\left| \alpha \right| \leq M}}{\begin{Vmatrix}{D}^{\alpha }v\end{Vmatrix}}_{1} \] where \( C \) depends on bounds of the derivatives of \( P \) on \( \op... | Yes |
Lemma 3.1 The functions \[ {V}_{1}\left( {t,\left| \xi \right| }\right) = {e}^{-\frac{z}{2}}\Phi \left( {\frac{m}{2\left( {2 + m}\right) },\frac{m}{2 + m};z}\right) ,\;{V}_{2}\left( {t,\left| \xi \right| }\right) = t{e}^{-\frac{z}{2}}\Phi \left( {\frac{4 + m}{2\left( {2 + m}\right) },\frac{4 + m}{2 + m};z}\right) \] on... | Unlike the method of using maximum principle to solve the degenerate parabolic equation [16], here the hypergeometric functions play an important role. According to Lemma 2.1, it is easy to verify that \( {\left( {V}_{1}\left( 1,\left| \xi \right| \right) \right) }^{ \vee } \) and \( {\left( {V}_{2}\left( 1,\left| \xi ... | No |
Lemma 3.3 Assume \( {\varphi }_{1}\left( x\right) \in {\dot{H}}_{p}^{{s}_{0} + \frac{2}{m + 2}}\left( {\mathbb{R}}^{n}\right) ,{\varphi }_{2}\left( x\right) \in {\dot{H}}_{p}^{{s}_{0}}\left( {\mathbb{R}}^{n}\right) \), then \( {u}_{0}\left( {t, x}\right) = \) \( {\left( {V}_{1}\left( t,\left| \xi \right| \right) {v}_{1... | Proof In terms of the representation of \( {u}_{0}\left( {t, x}\right) \) defined in (3.8), by use of the inequalities (3.3)-(3.6) with the value of \( s \) fixed on the right endpoint number separately, it is easy to verify that (3.9) holds. And (3.10) can be derived by choosing the value of \( s \) with the correspon... | Yes |
Lemma 3.4 If \( f\left( {t, x}\right) \in C\left( {\lbrack 0,\infty ),{\dot{H}}_{p}^{{s}_{0}}\left( {\mathbb{R}}^{n}\right) }\right) \), then\n\n\[ \parallel {t}^{\frac{m}{4}}{Ef}\left( {t, \cdot }\right) {\parallel }_{{\dot{H}}_{q}^{{s}_{0} + \frac{m + 4}{2\left( {2 + m}\right) } - \frac{n + 1}{2}\left( {\frac{1}{p} -... | Proof By a direct computation, we obtain\n\n\[ {\left( {\partial }_{t}Ef\right) }^{ \land }\left( {t,\xi }\right) = {\int }_{0}^{t}\left( {{\partial }_{t}{V}_{2}\left( {t,\left| \xi \right| }\right) {V}_{1}\left( {\tau ,\left| \xi \right| }\right) - {\partial }_{t}{V}_{1}\left( {t,\left| \xi \right| }\right) {V}_{2}\le... | Yes |
Theorem 3.1 Let \( 1 \leq p, q < \infty \) and \( \varphi \) satisfy conditions (1.1)-(1.3). (I) Suppose that \( \omega \in \bar{W}\left( \left\lbrack {0,\ell }\right\rbrack \right) ,\omega \left( {2t}\right) \leq {C\omega }\left( t\right) ,\frac{1}{\omega } \in \underline{W}\left( \left\lbrack {0,\ell }\right\rbrack \... | Proof Put \( s = 1 \) and \( \nu \left( t\right) = \omega \left( t\right) \) in Lemma 2.1. Then \[ \left| {{\mathrm{H}}_{\omega }^{\alpha }f\left( y\right) }\right| \leq {C\omega }\left( \left| y\right| \right) {\left| y\right| }^{\alpha - Q}{\int }_{0}^{\left| y\right| }\frac{{t}^{\frac{Q}{{p}^{\prime }} - 1}{\varphi ... | Yes |
Theorem 3.2 Let \( \mathcal{L} \) be a subLaplacian on the homogeneous Carnot group \( \mathbb{G} \) and \( d \) be an \( \mathcal{L} \) -gauge. Suppose \( 0 < \alpha < Q,1 < p < \frac{Q}{\alpha },\frac{1}{q} = \frac{1}{p} - \frac{\alpha }{Q} \) and \( \varphi ,\psi \in \beth \) . If\n\n\[ \n{C}_{\delta } \mathrel{\tex... | Proof As is well known, \( {M}^{\alpha }f \leq C{I}^{\alpha }\left( \left| f\right| \right) \), and we only consider the case for \( {I}^{\alpha } \) . At first we divide the function \( f \) into the expression \( f = {f}_{1} + {f}_{2} \) so that \( {I}^{\alpha }f = {I}^{\alpha }{f}_{1} + {I}^{\alpha }{f}_{2} \) , whe... | Yes |
Theorem 3.3 Let \( \mathcal{L} \) be a subLaplacian on the homogeneous Carnot group \( \mathbb{G} \) of homogenous dimension \( Q \) and \( d \) be an \( \mathcal{L} \) -gauge. Suppose \( 0 < \alpha < Q \) and \( \varphi ,\psi \in \beth \) . If\n\n\[ \n{C}_{\delta } \mathrel{\text{:=}} {\int }_{\delta }^{\infty }\frac{... | Proof Applying the representation formula of solution of Dirichlet problem for sub-Laplacian to \( u \in {\mathcal{C}}_{0}^{\infty }\left( {\mathbb{G},\mathbb{R}}\right) \), by integrating by parts, we see that\n\n\[ \nu\left( x\right) = {\int }_{\mathbb{G}}\left( {{\nabla }_{\mathcal{L}}\Gamma }\right) \left( {{x}^{-1... | Yes |
Lemma 3.1 For \( p \geq 7 \) and \( 0 \leq s < p - 4 \) . Then the fourth Greek letter element \( {\widetilde{\delta }}_{s + 4} \in {\operatorname{Ext}}_{A}^{s + 4,{t}_{1}\left( s\right) }\left( {{\mathbb{Z}}_{p},{\mathbb{Z}}_{p}}\right) \) is represented by | \[ {a}_{4}^{s}{h}_{4,0}{h}_{3,1}{h}_{2,2}{h}_{1,3} \in {E}_{1}^{s + 4,{t}_{1}\left( s\right) , * } \] in the \( {E}_{1} \) -term of the May spectral sequence, where \( {\widetilde{\delta }}_{s + 4} \) is actually \( {\widetilde{\alpha }}_{s + 4}^{\left( 4\right) } \) described in [6] and \( {t}_{1}\left( s\right) = q\l... | Yes |
Lemma 3.2 Let \( p \geq 7,0 \leq s < p - 5 \) . Then we have the May \( {E}_{1} \) -term\n\n\[ {E}_{1}^{s + 9, t\left( s\right) , * } = {\mathbb{Z}}_{p}\left\{ {{G}_{1},{G}_{2},\cdots ,{G}_{11}}\right\} \]\n\nwhere \( t\left( {s, n}\right) = q\left\lbrack {\left( {s + 1}\right) + \left( {s + 5}\right) p + \left( {s + 3... | Proof The proof of this lemma is divided into the following six cases. Consider\n\n\[ h = {x}_{1}{x}_{2}\cdots {x}_{m} \in {E}_{1}^{s + 9, t\left( s\right) , * } \]\n\nin the MSS, where \( {x}_{i} \) is one of \( {a}_{k},{h}_{r, j} \) or \( {b}_{u, z},0 \leq k \leq 4,0 \leq r + j \leq 4,0 \leq u + z \leq 3, r > 0 \) , ... | Yes |
Lemma 3.3 (1) \( {b}_{0}^{3}{\widetilde{\delta }}_{s + 4} \in {\operatorname{Ext}}_{A}^{s + {10}, t\left( s\right) }\left( {{\mathbb{Z}}_{p},{\mathbb{Z}}_{p}}\right) \) is represented by \( {b}_{1,0}^{3}{a}_{4}^{s}{h}_{4,0}{h}_{3,1}{h}_{2,2}{h}_{1,3} \in \) \( {E}_{1}^{s + {10}, t\left( s\right) , * } \) in the MSS, wh... | Proof (1) Since it is known that \( {b}_{1, i} \) and \( {a}_{4}^{s}{h}_{4,0}{h}_{3,1}{h}_{2,2}{h}_{1,3} \in {E}_{1}^{*,*, * } \) are all permanent cycles in the MSS as [7] and converge nontrivially to \( {b}_{i},{\widetilde{\delta }}_{s + 4} \in {\operatorname{Ext}}_{A}^{*, * }\left( {{\mathbb{Z}}_{p},{\mathbb{Z}}_{p}... | Yes |
Lemma 2.4 Suppose that \( \beta \geq \left( {p - 1}\right) \max \left\{ {{\mu }_{1},{\mu }_{2}}\right\} \) . Then the following system\n\n\[ \left\{ \begin{array}{l} d + g \leq {d}_{0} + {d}_{0}, \\ {l}_{1}\left( {d, g}\right) \geq 0,{l}_{2}\left( {d, g}\right) \geq 0, \\ d > 0, g \geq 0,\left( {d, g}\right) \neq \left... | Proof See [1, Lemmas 2.1, 2.2, 2.3, 2.4]. | No |
Lemma 2.5 Assume that \( - {\lambda }_{1}\left( \mathbb{B}\right) < \lambda < 0 \), and then (1.2) has a positive least energy solution \( \varphi \in {\mathcal{H}}_{2,0}^{1,\frac{N}{2}}\left( \mathbb{B}\right) \) with energy \[ {A}_{1} \mathrel{\text{:=}} \frac{1}{N}{\int }_{\mathbb{B}}\left( {{\left| {\nabla }_{\math... | Proof Let \( {S}_{\lambda }\left( {u;\mathbb{B}}\right) = \frac{{\begin{Vmatrix}{\nabla }_{\mathbb{B}}u\end{Vmatrix}}_{{L}_{2}^{\frac{N}{2}}}^{2} + \lambda \parallel u{\parallel }_{{L}_{2}^{\frac{N}{2}}}^{2}}{\parallel u{\parallel }_{{L}_{{2}^{ * }}^{\frac{N}{2}}}^{2}} \) and \( {S}_{\lambda }\left( \mathbb{B}\right) =... | Yes |
For \( - \infty < \beta < 0 \), if \( B \) is attained by a couple \( \left( {u, v}\right) \in \mathcal{M} \), then this couple is a critical point of \( E\left( {u, v}\right) \) in (1.7). | The proof is analogous to that in [1, Lemma 2.5]. So we omit it here. | No |
Lemma 3.2 For all \( R > 0 \), we have \( {B}^{\prime }\left( R\right) \equiv {B}^{\prime } \) . | Proof Let \( {R}_{1} > {R}_{2} \), since \( {\mathcal{M}}^{\prime }\left( {R}_{2}\right) \subset {\mathcal{M}}^{\prime }\left( {R}_{1}\right) \), we get \( B\left( {R}_{1}\right) \leq {B}^{\prime }\left( {R}_{2}\right) \) . For any \( \left( {u, v}\right) \in {\mathcal{M}}^{\prime }\left( {R}_{1}\right) \), we define\n... | Yes |
Lemma 3.3 For \( 0 < \varepsilon < p - 1 \), there holds\n\n\[ \n{B}_{\varepsilon } < \min \left\{ {\mathop{\inf }\limits_{{\left( {u,0}\right) \in {\mathcal{M}}_{\varepsilon }^{\prime }}}{E}_{\varepsilon }\left( {u,0}\right) ,\mathop{\inf }\limits_{{\left( {0, v}\right) \in {\mathcal{M}}_{\varepsilon }^{\prime }}}{E}_... | The proof is analogous to that in [1, Lemma 2.7]. So we omit it here. | No |
For any \( 0 < \varepsilon < p - 1 \) ,(3.14) has a classical least energy solution \( \left( {{u}_{\varepsilon },{v}_{\varepsilon }}\right) \) , and \( {u}_{\varepsilon },{v}_{\varepsilon } \) are both partly radially symmetric decreasing. | Fix any \( 0 < \varepsilon < p - 1 \), and then it is easy to see that \( {B}_{\varepsilon } > 0 \) . Let \( \left( {u, v}\right) \in {\mathcal{M}}_{\varepsilon }^{\prime } \) with \( u \geq 0, v \geq 0 \), and \( \left( {{u}^{ * },{v}^{ * }}\right) \) be its cone Schwartz symmetrization. Then we have\n\n\[ \n{\int }_{... | Yes |
Theorem 3.1 The solution \( x\left( t\right) \) of system (1.2) satisfies that \( \forall t \in \left\lbrack {0,{t}_{0}}\right), x\left( t\right) > 0 \), if it is defined in the interval \( \left\lbrack {0,{t}_{0}}\right) \left( {0 < {t}_{0} \leq + \infty }\right) \) and the initial value satisfies that\n\n\[ x\left( 0... | Proof By the definition of \( I\left( {x}_{k}\right) \), there exists \( \delta > 0\left( {\delta < {ET}}\right) \), such that when \( \left| {x}_{k}\right| < \delta \) , \( I\left( {x}_{k}\right) = 0 \) holds. Consider \( x\left( t\right) \) with \( x\left( 0\right) > 0 \) and \( \parallel x\left( t\right) \parallel <... | Yes |
Theorem 3.2 Assume that the following assumptions hold for system (1.1).\n\n(1) The system is uniformly persistent.\n\n(2) Solutions are uniformly ultimately bounded.\n\nThen system (1.2) has one positive equilibrium at least, if the suitable \( {ET} \) is chosen. | Proof Using the Theorem 5.1 of Li and Shuai [6], we obtain that system (1.1) has one equilibrium at least. Let \( {x}^{ * } = \left( {{x}_{1}^{ * },{x}_{2}^{ * },\cdots ,{x}_{n}^{ * }}\right) \) denote the positive equilibrium of system (1.1) . Then by choosing \( {ET} \geq {x}_{k}^{ * } \), we have \( \{ 0\} \in \bar{... | Yes |
Proposition 2.1 Let \( G\\left( z\\right) \) be a function analytic in the right half-plane \( {\\mathbb{C}}_{ + } \) which is defined in (2.4) and vanishes exactly on the sequence \( \\Lambda = \\left\\{ {{\\lambda }_{n} = \\left| {\\lambda }_{n}\\right| {e}^{i{\\theta }_{n}} : n = 1,2,\\cdots }\\right\\} \) , where \... | Proof Since the zeros of \( G\\left( z\\right) \) have a gap, we can choose a sequence of strict positive numbers \( \\left\\{ {t}_{n}\\right\\} \) such that the disks \( D\\left( {{\\lambda }_{n},{t}_{n}}\\right) \) are mutually disjoint and (2.3) is satisfied. We can also select strict positive number \( {d}_{n} \) s... | Yes |
Lemma 3.1 (see [11]) Let \( \\mathbf{A} \\in \\mathbf{L}\\left( {\\mathbf{c},\\mathbf{D}}\\right) ,\\mathbf{B} \\in {\\mathbf{A}}_{\\alpha ,\\beta } \) and \( \\Lambda = {\\left\\{ {\\lambda }_{n},{\\mu }_{n}\\right\\} }_{n = 1}^{\\infty } \) be its \( \\{ \\lambda ,\\mu \\} \) reordering. Then the function\n\n\\[ \nG\... | With \( r = \\left| z\\right| \) and \( x = \\Re z \), for some positive constant \( A \), we have\n\n\\[ \n\\left| {G\\left( z\\right) }\\right| \\leq \\exp \\left\\{ {x{\\sigma }_{\\Lambda }\\left( r\\right) + {Ax}}\\right\\}\n\\]\n\n(3.3)\n\nwhere \( {\\sigma }_{\\Lambda }\\left( r\\right) = 2\\mathop{\\sum }\\limit... | Yes |
Proposition 3.1 Let \( G\left( z\right) \) be a function analytic in the right half-plane \( {\mathbb{C}}_{ + } \) which is defined in (3.2) and vanishes exactly on the sequence \( \Lambda = {\left\{ {\lambda }_{n},{\mu }_{n}\right\} }_{n = 1}^{\infty } \), which is the \( \{ \lambda ,\mu \} \) reordering of some posit... | Proof The proof is a modification of the one for Proposition 2.1. By the properties of the zeros of \( G\left( z\right) \), we can choose a sequence of strict positive numbers \( \left\{ {t}_{n}\right\} \) such that the disks \( D\left( {{\lambda }_{n},{t}_{n}}\right) \) are mutually disjoint and (3.4) is satisfied. We... | Yes |
Theorem 1 If \( \\left( {{x}^{ * },{t}^{ * }}\\right) \) is a global optimal solution of EP, then \( {x}^{ * } \) is also a global optimal solution of problem MLFP, and \( {t}^{ * } \) is the optimal value of EP and MLFP. | Proof Readers can refer to [1]. | No |
Theorem 2 For all \( x \in {X}^{k} = \left\lbrack {l, u}\right\rbrack \), let \( {\Delta x} = u - l \), consider the functions \( {\Phi }_{i}^{l}\left( {x, t}\right) \) and \( {\Phi }_{i}\left( {x, t}\right) \). Then we have\n\n\[ \mathop{\lim }\limits_{{{\Delta x} \rightarrow 0}}\left( {{\Phi }_{i}\left( {x, t}\right)... | Proof From the definitions \( {\Phi }_{i}\left( {x, t}\right) \) and \( {\Phi }_{i}^{l}\left( {x, t}\right) \), we have\n\n\[ \left| {{\Phi }_{i}\left( {x, t}\right) - {\Phi }_{i}^{l}\left( {x, t}\right) }\right| = \left| {\frac{{n}_{i}\left( x\right) }{{d}_{i}\left( x\right) } - \left( {\frac{{n}_{i}\left( x\right) }{... | Yes |
Theorem 3 If the algorithm terminates finitely, then upon termination, \( {x}^{k} \) is a global \( \epsilon \) -optimal solution for problem MLFP; else, it will generate an infinite sequence \( \left\{ {x}^{k}\right\} \) of iterations such that along any infinite branch of the branch and bound tree, and any accumulati... | Proof When the algorithm terminates finitely, the conclusion is obvious. When the algorithm terminates infinitely, as stated in [25], a sufficient condition for the algorithm to be convergent to a global optimum is that the bounding operation must be consistent and the selection operation is bound improving.\n\nA bound... | Yes |
\( \min \max \left\{ {\frac{2{x}_{1} + 2{x}_{2} - {x}_{3} + {0.9}}{{x}_{1} - {x}_{2} + {x}_{3}},\frac{3{x}_{1} - {x}_{2} + {x}_{3}}{8{x}_{1} + 4{x}_{2} - {x}_{3}}}\right\} , \) | s.t. \( \;{x}_{1} + {x}_{2} - {x}_{3} \leq 1 \), \n\n\[ \n- {x}_{1} + {x}_{2} - {x}_{3} \leq - 1 \n\] \n\n\[ \n{12}{x}_{1} + 5{x}_{2} + {12}{x}_{3} \leq {34.8} \n\] \n\n\[ \n{12}{x}_{1} + {12}{x}_{2} + 7{x}_{3} \leq {29.1} \n\] \n\n\[ \n- 6{x}_{1} + {x}_{2} + {x}_{3} \leq - {4.1} \n\] \n\n\[ \n{1.0} \leq {x}_{1} \leq {... | Yes |
\( \min \max \left\{ {\frac{3{x}_{1} + {x}_{2} - 2{x}_{3} + {0.8}}{2{x}_{1} - {x}_{2} + {x}_{3}},\frac{4{x}_{1} - 2{x}_{2} + {x}_{3}}{7{x}_{1} + 3{x}_{2} - {x}_{3}},\frac{3{x}_{1} + 2{x}_{2} - {x}_{3} + {1.9}}{{x}_{1} - {x}_{2} + {x}_{3}},\frac{4{x}_{1} - {x}_{2} + {x}_{3}}{8{x}_{1} + 4{x}_{2} - {x}_{3}}}\right\} , \) | s.t. \( {x}_{1} + {x}_{2} - {x}_{3} \leq 1 \)\n\n\[ \n- {x}_{1} + {x}_{2} - {x}_{3} \leq - 1 \n\]\n\n\[ \n{12}{x}_{1} + 5{x}_{2} + {12}{x}_{3} \leq {34.8} \n\]\n\n\[ \n{12}{x}_{1} + {12}{x}_{2} + 7{x}_{3} \leq {29.1} \n\]\n\n\[ \n- 6{x}_{1} + {x}_{2} + {x}_{3} \leq - {4.1} \n\]\n\n\( {1.0} \leq {x}_{1} \leq {1.2},{0.55... | Yes |
\( \min \max \left\{ {\frac{{2.1}{x}_{1} + {2.2}{x}_{2} - {x}_{3} + {0.8}}{{1.1}{x}_{1} - {x}_{2} + {1.2}{x}_{3}},\frac{{3.1}{x}_{1} - {x}_{2} + {1.3}{x}_{3}}{{8.2}{x}_{1} + {4.1}{x}_{2} - {x}_{3}}}\right\} , \) | s.t. \( \;{x}_{1} + {x}_{2} - {x}_{3} \leq 1 \), \n\n\[ \n- {x}_{1} + {x}_{2} - {x}_{3} \leq - 1 \n\] \n\n\[ \n{12}{x}_{1} + 5{x}_{2} + {12}{x}_{3} \leq {40} \n\] \n\n\[ \n{12}{x}_{1} + {12}{x}_{2} + 7{x}_{3} \leq {50} \n\] \n\n\[ \n- 6{x}_{1} + {x}_{2} + {x}_{3} \leq - 2 \n\] \n\n\( {1.0} \leq {x}_{1} \leq {1.2},{0.55... | No |
\( \min \max \left\{ {\frac{5{x}_{1} + 4{x}_{2} - {x}_{3} + {0.9}}{3{x}_{1} - {x}_{2} + 2{x}_{3} + {0.5}},\frac{3{x}_{1} - {x}_{2} + 4{x}_{3} + {0.5}}{9{x}_{1} + 3{x}_{2} - {x}_{3} + {0.5}},\frac{4{x}_{1} - {x}_{2} + 6{x}_{3} + {0.5}}{{12}{x}_{1} + 7{x}_{2} - {x}_{3} + {0.9}},\frac{7{x}_{1} - {x}_{2} + 7{x}_{3} + {0.5}... | s.t. \( \;2{x}_{1} + 2{x}_{2} - {x}_{3} \leq 3 \), \( - 2{x}_{1} + {x}_{2} - 3{x}_{3} \leq - 1 \)\n\n\[ \n{11}{x}_{1} + 7{x}_{2} + {12}{x}_{3} \leq {47} \]\n\n\[ \n{13}{x}_{1} + {13}{x}_{2} + 6{x}_{3} \leq {56} \]\n\n\[ \n- 6{x}_{1} + {x}_{2} + 3{x}_{3} \leq - 1 \]\n\n\[ \n{1.0} \leq {x}_{1} \leq 2,{0.35} \leq {x}_{2} ... | Yes |
Lemma 2.2 For any \( \varphi ,\psi \in {\mathcal{P}}_{T}\left( {P, L}\right) \) , \[ \begin{Vmatrix}{{\varphi }^{\left\lbrack n\right\rbrack } - {\psi }^{\left\lbrack n\right\rbrack }}\end{Vmatrix} \leq \mathop{\sum }\limits_{{j = 0}}^{{n - 1}}{L}^{j}\parallel \varphi - \psi \parallel, n = 1,2,\cdots . \] | The result can be obtained by the definition of \( {\mathcal{P}}_{T}\left( {P, L}\right) \) . | No |
Lemma 2.3 Suppose \( {c}_{1} \neq 0 \) . If \( x \in {\mathcal{P}}_{T} \), then \( x\left( t\right) \) is a solution of equation (1.1) if and only if\n\n\[ x\left( t\right) = {c}_{2}{\int }_{t}^{t + T}{x}^{\left\lbrack 2\right\rbrack }\left( s\right) G\left( {t, s}\right) {ds} + {\int }_{t}^{t + T}F\left( s\right) G\le... | Proof Let \( x\left( t\right) \in {\mathcal{P}}_{T}\left( {P, L}\right) \) be a solution of (1.1), multiply both sides of the resulting equation with \( {e}^{-{c}_{1}t} \) and integrate from \( t \) to \( t + T \) to obtain\n\n\[ x\left( {t + T}\right) {e}^{-{c}_{1}\left( {t + T}\right) } - x\left( t\right) {e}^{-{c}_{... | Yes |
Lemma 2.4 Operator \( A \) is continuous and compact on \( {\mathcal{P}}_{T}\left( {P, L}\right) \) . | Proof Take \( \varphi ,\psi \in {\mathcal{P}}_{T}\left( {P, L}\right), t \in \mathbb{R} \), use (2.1) and (2.4), \[ \left| {\left( {A\varphi }\right) \left( t\right) - \left( {A\psi }\right) \left( t\right) }\right| \leq \left| {c}_{2}\right| {\int }_{t}^{t + T}\left| {{\varphi }^{\left\lbrack 2\right\rbrack }\left( s\... | Yes |
Lemma 2.5 Operator \( B \) is a contraction mapping on \( {\mathcal{P}}_{T}\left( {P, L}\right) \) . | Proof Take \( \varphi ,\psi \in {\mathcal{P}}_{T}\left( {P, L}\right) \) , \[ \parallel {B\varphi } - {B\psi }\parallel = \mathop{\max }\limits_{{t \in \left\lbrack {0, T}\right\rbrack }}\left| {{\int }_{t}^{t + T}F\left( s\right) G\left( {t, s}\right) {ds} - {\int }_{t}^{t + T}F\left( s\right) G\left( {t, s}\right) {d... | Yes |
Theorem 3.1 In addition to the assumption of Theorem 2.6, suppose that\n\n\[ \left| {c}_{2}\right| {MT}\left( {1 + L}\right) < 1 \]\n\nthen (1.1) has a unique solution in \( {\mathcal{P}}_{T}\left( {P, L}\right) \) . | Proof Define an operator \( H \) from \( {\mathcal{P}}_{T}\left( {P, L}\right) \) into \( {\mathcal{P}}_{T} \),\n\n\[ \left( {Hx}\right) \left( t\right) = \left( {Ax}\right) \left( t\right) + \left( {Bx}\right) \left( t\right) = {c}_{2}{\int }_{t}^{t + T}{x}^{\left\lbrack 2\right\rbrack }\left( s\right) G\left( {t, s}\... | Yes |
Theorem 2.1 Under some regular conditions, \[ {n}^{-1/2}{\widetilde{U}}_{n, G}\left( {\theta }_{0}\right) = {n}^{-1/2}{\bar{U}}_{n, G}\left( {\theta }_{0}\right) + {o}_{p}\left( 1\right) \] \( {\widehat{\theta }}_{n} \) is strong consistency, and \( \sqrt{n}\left( {{\widehat{\theta }}_{n} - {\theta }_{0}}\right) \) con... | The regularity conditions and the proof of Theorem 2.1 can be founded in [15]. | No |
Theorem 1.1 Let \( \left( {{M}^{n},{g}_{i\bar{j}}}\right) \) be an \( n \) -dimensional Kähler manifold with \( n \geq 2 \) . If there exist two smooth real-valued functions \( f,\lambda \) satisfying the equation\n\n\[ \n{R}_{i\bar{j}} + {f}_{i\bar{j}} = \lambda {g}_{i\bar{j}} \n\]\n\n(1.4)\n\nthen \( \lambda \) must ... | Therefore, by virtue of Theorem 1.4 of Chen and Zhu in [8], we obtain the following. | No |
Corollary 1.2 Let \( \left( {{M}^{n},{g}_{i\bar{j}}}\right) \) be an \( n \) -dimensional \( \left( {n \geq 2}\right) \) complete Kähler manifold with harmonic Bochner tensor. If there exist two smooth real-valued functions \( f,\lambda \) satisfying (1.4) with \( {f}_{ij} = 0 \) (that is, \( \nabla f \) is a holomorph... | Using the concepts as in [8], under the Kähler metric \( g = \left( {g}_{i\bar{j}}\right) \), the Ricci curvature and the scalar curvature defined by\n\n\[ \n{R}_{i\bar{j}} = {R}_{i\bar{j}k\bar{k}},\;R = {R}_{i\bar{i}} = {R}_{i\bar{i}j\bar{j}}, \n\]\n\nrespectively. By the first Bianchi identity, we have\n\n\[ \n{R}_{i... | Yes |
Lemma 2.1 (see [15]) Assume \( x \in {B}_{v}^{\prime } \), then for \( t \in J,{x}_{t} \in {B}_{v} \) . Moreover, | \[ l\parallel x\left( t\right) \parallel \leq {\begin{Vmatrix}{x}_{t}\end{Vmatrix}}_{{B}_{v}} \leq \parallel \phi {\parallel }_{{B}_{v}} + l\mathop{\sup }\limits_{{s \in \left\lbrack {0, t}\right\rbrack }}\parallel x\left( s\right) \parallel \] where \[ l = {\int }_{-\infty }^{0}v\left( t\right) {dt} < + \infty \] | Yes |
Lemma 2.2 (see [16-18]) If \( X \) is a real Banach space and \( \mathcal{B},\mathcal{D} \subset X \) are bounded, then the following properties are satisfied:\n\n(1) monotone: if for all bounded subsets \( \mathcal{B},\mathcal{D} \) of \( X,\mathcal{B} \subseteq \mathcal{D} \) implies \( \beta \left( \mathcal{B}\right... | \[ \beta \left( {{\int }_{0}^{t}W\left( s\right) {ds}}\right) \leq {\int }_{0}^{t}\beta \left( {W\left( s\right) }\right) {ds}\text{ for all }t \in J \]\n\nwhere\n\n\[ {\int }_{0}^{t}W\left( s\right) {ds} = \left\{ {{\int }_{0}^{t}u\left( s\right) {ds} : \text{ for all }u \in W, t \in J}\right\} ; \] | Yes |
Lemma 2.4 (see [20]) If \( A \) is a sectorial operator of type \( \left( {M,\theta ,\alpha ,\mu }\right) \), then\n\n\[ \n{S}_{\alpha }\left( t\right) = \frac{1}{2\pi i}{\int }_{c}{e}^{\lambda t}{\lambda }^{\alpha - 1}R\left( {{\lambda }^{\alpha }, A}\right) {d\lambda } = {E}_{\alpha ,1}\left( {A{t}^{\alpha }}\right) ... | \[ \n{T}_{\alpha }\left( t\right) = \frac{1}{2\pi i}{\int }_{c}{e}^{\lambda t}R\left( {{\lambda }^{\alpha }, A}\right) {d\lambda } = {t}^{\alpha - 1}{E}_{\alpha ,\alpha }\left( {A{t}^{\alpha }}\right) = {t}^{\alpha - 1}\mathop{\sum }\limits_{{k = 0}}^{\infty }\frac{{\left( A{t}^{\alpha }\right) }^{k}}{\Gamma \left( {\a... | Yes |
Theorem 3.1 If \( f \) satisfies a uniform Hölder condition with exponent \( \beta \in (0,1\rbrack \) and \( A \) is a sectorial operator of type \( \left( {M,\theta ,\alpha ,\mu }\right) \), then the Cauchy problem (1.1) has a solution, given by \[ x\left( t\right) = \left\{ \begin{array}{l} {x}_{0} = \phi \in {B}_{v}... | Proof For all \( t \in \left( {{t}_{k},{t}_{k + 1}}\right\rbrack \) where \( k = 0,1,\cdots, m \) by Lemma 2.7, we obtain \[ {}^{c}{D}_{t}^{\alpha }\left\lbrack {x\left( t\right) - g\left( {t,{x}_{t}}\right) }\right\rbrack \] \[ = {}^{c}{D}_{t}^{\alpha }\left\lbrack {{S}_{\alpha }\left( t\right) \left\lbrack {\phi \lef... | Yes |
Lemma 4.2 There exists positive constant \( {T}_{0} \) such that the solution \( \left( {x\left( t\right), y\left( t\right) }\right) \) of (1.2) satisfies\n\n\[ 0 < x\left( t\right) \leq {M}_{1}\;\text{ and }\;0 < y\left( t\right) \leq {M}_{2}\;\text{ for }t \geq {T}_{0}, \]\n\nwhere\n\n\[ {M}_{1} = \frac{{r}_{1}^{U}}{... | Proof If follows from system (1.2) that\n\n\[ {x}^{\prime }\left( t\right) \leq x\left( t\right) \left\lbrack {{r}_{1}^{U} - {b}^{L}x\left( {t - {\tau }_{1}\left( t\right) }\right) }\right\rbrack \]\n\nFrom Lemma 4.1 yield that there exists a positive constant \( {T}_{1} \) such that \( x\left( t\right) \leq {M}_{1} \)... | Yes |
Lemma 4.3 If \( {\Delta }_{1} > 0 \) then there exists a positive constant \( {T}^{ * } \) such that the solution \( \left( {x\left( t\right), y\left( t\right) }\right) \) of system (1.2) satisfies\n\n\[ x\left( t\right) \geq {m}_{1}\;\text{ and }\;y\left( t\right) \geq {m}_{2}\;\text{ for }\;t \geq {T}^{ * }, \]\n\nwh... | Proof If follows from Lemma 4.2 and system (1.2) that for \( t \geq {T}_{0} \) ,\n\n\[ \left\{ \begin{array}{l} {x}^{\prime }\left( t\right) \geq x\left( t\right) \left\lbrack {{\Delta }_{1} - {b}^{U}x\left( {t - {\tau }_{1}\left( t\right) }\right) }\right\rbrack \\ {y}^{\prime }\left( t\right) \geq y\left( t\right) \l... | Yes |
Theorem 4.1 If \( {\Delta }_{1} > 0 \), then system (1.2) is permanent. | Similar to the proofs of Lemma 4.2 and Lemma 4.3, we have | No |
Consider the following equation\n\n\[ \n\\left\\{ \\begin{array}{l} {x}^{\\prime }\\left( t\\right) = x\\left( t\\right) \\left\\lbrack {{r}_{1}\\left( t\\right) - b\\left( t\\right) x\\left( {t - {\\tau }_{1}}\\right) - \\frac{{a}_{1}\\left( t\\right) x\\left( t\\right) y\\left( t\\right) }{{x}^{2}\\left( t\\right) + ... | It is easy to calculation, and all the conditions in Theorems 3.1, 3.2 and 4.1 hold. So we know system (4.3) has at least one positive periodic solution and permanent (see Figures 1, 2, we take \( x\\left( 0\\right) = 1, y\\left( 0\\right) = 5 \) and \( x\\left( 0\\right) ) = 4, y\\left( 0\\right) = 5). | No |
Theorem 1.4 Let \( x : {M}^{n} \rightarrow {\mathbb{R}}_{1}^{n + 1} \) be a complete space-like \( \lambda \) -hypersurface with weight \( s = \langle x, x\rangle \) . Suppose that (1.4) and (1.5) are satisfied. If\n\n\[ \n{\left( \sqrt{S - \frac{1}{n}{H}^{2}} - \left| \lambda \right| \frac{n - 2}{2\sqrt{n\left( {n - 1... | Proof If condition (1.7) is satisfied for a hypersurface \( {\mathbb{H}}^{n}\left( {-{r}^{-2}}\right) \) with \( r > 0 \), then by the fact that \( x = {rN} \), we have \( \lambda = H - \langle x, x\rangle \langle x, N\rangle = \frac{n}{r} - {r}^{3} \) . If follows that\n\n\[ \n0 \leq {\left( \sqrt{S - \frac{1}{n}{H}^{... | Yes |
Corollary 1.5 Let \( x : {M}^{n} \rightarrow {\mathbb{R}}_{1}^{n + 1} \) be a complete space-like \( \lambda \) -hypersurface with weight \( s = \langle x, x\rangle \) . Suppose \( S - \frac{{H}^{2}}{n} \) is constant. If (1.5) and (1.7) are satisfied, then \( x \) is isometric to either the hyperbolic space \( {\mathb... | Proof Since \( S - \frac{{H}^{2}}{n} \) is constant, condition (1.4) in Theorem 1.3 is trivially satisfied. Then Corollary 1.5 follows direct from Theorem 1.4. | Yes |
Theorem 1.6 Let \( x : {M}^{n} \rightarrow {\mathbb{R}}_{1}^{n + 1} \) be a complete space-like self-shrinker with weight \( s = \langle x, x\rangle \) . Suppose that (1.4) and (1.5) are satisfied, then \( x \) is isometric to one of the following two embedded hypersurfaces:\n\n1. the hyperbolic space \( {\mathbb{H}}^{... | Proof When \( \lambda = 0 \), it is clear that (1.7) is trivially satisfied. Furthermore, for a hyperbolic space \( {\mathbb{H}}^{n}\left( {-{r}^{-2}}\right) \subset {\mathbb{R}}_{1}^{n + 1},\lambda = 0 \) also implies that \( {r}^{2} = \sqrt{n} \) . | No |
Corollary 1.7 Let \( x : {M}^{n} \rightarrow {\mathbb{R}}_{1}^{n + 1} \) be a complete space-like self-shrinker with weight \( s = \langle x, x\rangle \) . If \( S - \frac{{H}^{2}}{n} \) is constant and (1.5) is satisfied, then \( x \) is isometric to the either the hyperbolic space \( {\mathbb{H}}^{n}\left( {-\frac{1}... | Proof The assumption that \( S - \frac{{H}^{2}}{n} \) is constant directly means that (1.4) is trivially satisfied. | No |
Lemma 2.1 [5] Let \( x : {M}^{n} \rightarrow {\mathbb{R}}_{1}^{n + 1} \) be a complete space-like hypersurface for which \( \langle x, x\rangle \) does not change its sign. Then, for any \( {C}^{1} \) -function \( u \) on \( {M}^{n} \) with compact support, it holds that\n\n\[{\int }_{{M}^{n}}u\left( {\mathcal{L}v}\rig... | Proof By (2.5), we find\n\n\[{\int }_{{M}^{n}}u\left( {\mathcal{L}v}\right) {e}^{-\frac{{\epsilon a}\langle x, x\rangle }{2}}d{V}_{{M}^{n}} = {\int }_{{M}^{n}}u\left( {{e}^{\frac{{\epsilon a}\langle x, x\rangle }{2}}\operatorname{div}\left( {{e}^{-\frac{{\epsilon a}\langle x, x\rangle }{2}}\nabla v}\right) }\right) {e}... | Yes |
Corollary 2.2 Let \( x : {M}^{n} \rightarrow {\mathbb{R}}_{1}^{n + 1} \) be a complete space-like hypersurface. If \( u, v \) are \( {C}^{2} \) -functions satisfying\n\n\[ \n{\int }_{{M}^{n}}\left( {\left| {u\nabla v}\right| + \left| {\nabla u}\right| \left| {\nabla v}\right| + \left| {u\mathcal{L}v}\right| }\right) {e... | Proof We will use square brackets \( \left\lbrack \cdot \right\rbrack \) to denote weighted integrals\n\n\[ \n\left\lbrack f\right\rbrack = {\int }_{{M}^{n}}f{e}^{-\frac{{\epsilon a}\langle x, x\rangle }{2}}d{V}_{{M}^{n}}\n\]\n\n(2.9)\n\nGiven any \( \phi \) that is \( {C}^{1} \) -with compact support, we can apply Lem... | Yes |
Lemma 2.3 If \( x : {M}^{n} \rightarrow {\mathbb{R}}_{1}^{n + 1} \) is a complete space-like hypersurface, \( u \) is a \( {C}^{1} \) -function with compact support, and \( v \) is a \( {C}^{2} \) -function, then\n\n\[ \n{\int }_{{M}^{n}}u\left( {\widetilde{\mathcal{L}}v}\right) {e}^{-\frac{\langle x, x{\rangle }^{2}}{... | Proof Using (2.15), we have\n\n\[ \n{\int }_{{M}^{n}}u\left( {\widetilde{\mathcal{L}}v}\right) {e}^{-\frac{{\left( x, x\right) }^{2}}{4}}d{V}_{{M}^{n}} = {\int }_{{M}^{n}}u\left( {{e}^{\frac{{\left( x, x\right) }^{2}}{4}}\operatorname{div}\left( {{e}^{-\frac{{\left( x, x\right) }^{2}}{4}}\nabla v}\right) }\right) {e}^{... | Yes |
Corollary 2.4 Let \( x : {M}^{n} \rightarrow {\mathbb{R}}_{1}^{n + 1} \) be a complete space-like hypersurface. If \( u, v \) are \( {C}^{2} \) -functions satisfying \[ {\int }_{{M}^{n}}\left( {\left| {u\nabla v}\right| + \left| {\nabla u}\right| \left| {\nabla v}\right| + \left| {u\widetilde{\mathcal{L}}v}\right| }\ri... | Proof The proof is the same as that of Corollary 2.2 and is omitted. | No |
Lemma 1.4 [10] Let \( E \) be a real strictly convex Banach space and \( C \) be its nonempty closed and convex subset. Let \( {B}_{m} : C \rightarrow C \) be a nonexpansive mapping for each \( m \geq 1 \) . Let \( \left\{ {a}_{m}\right\} \) be a real number sequence in \( \left( {0,1}\right) \) such that \( \mathop{\s... | \[ \operatorname{Fix}\left( {\mathop{\sum }\limits_{{m = 1}}^{\infty }{a}_{m}{B}_{m}}\right) = \mathop{\bigcap }\limits_{{m = 1}}^{\infty }\operatorname{Fix}\left( {B}_{m}\right) \] | Yes |
Theorem 2.1 Suppose \( E \) is a real uniformly smooth and uniformly convex Banach space, \( C \) is a nonempty, closed and convex sunny nonexpansive retract of \( E \), and \( {Q}_{C} \) is the sunny nonexpansive retraction of \( E \) onto \( C \) . Let \( {f}_{i} : E \rightarrow E \) be a contractive mapping with coe... | \[ {W}_{t}x \mathrel{\text{:=}} {t\eta }\mathop{\sum }\limits_{{i = 1}}^{\infty }{a}_{i}{f}_{i}\left( x\right) + \left( {I - t\mathop{\sum }\limits_{{i = 1}}^{\infty }{b}_{i}{F}_{i}}\right) B{Q}_{C}x \] then \( {W}_{t} \) has a fixed point \( {x}_{t} \), for each \( 0 < t \leq {\left( \mathop{\sum }\limits_{{i = 1}}^{\... | Yes |
Theorem 2.2 Let \( E, C,{Q}_{C},{f}_{i},{F}_{i}, k \) and \( \bar{\gamma } \) be the same as those in Theorem 2.1, and let \( {A}_{i} : C \rightarrow C \) be \( m \) -accretive operator for \( i \in {N}^{ + } \) . Let \( \left\{ {\alpha }_{n}\right\} ,\left\{ {\delta }_{n}\right\} ,\left\{ {\beta }_{n}\right\} ,\left\{... | Proof We shall split the proof into five steps.\n\nStep \( 1\left\{ {x}_{n}\right\} \) is well-defined.\n\nIn fact, it suffices to show that \( \left\{ {z}_{n}\right\} \) is well-defined.\n\nFor \( t, s \in \left( {0,1}\right) \), define \( {U}_{t, s} : C \rightarrow C \) by \( {U}_{t, s}x \mathrel{\text{:=}} {tu} + {s... | Yes |
Considering the functions \( \varphi : \left\lbrack {0,1}\right\rbrack \rightarrow \mathbb{R}, b : \left\lbrack {0,1}\right\rbrack \times \left\lbrack {0,1}\right\rbrack \rightarrow {\mathbb{R}}_{ + } \) and \( \rho : \left\lbrack {0,1}\right\rbrack \times \left\lbrack {0,1}\right\rbrack \rightarrow \mathbb{R} \), defi... | \( b\left( {x, u}\right) \left\lbrack {\varphi \left( u\right) - \varphi \left( x\right) }\right\rbrack = {x}^{4}{u}^{2},\;{\varphi }^{ + }\left( {{H}_{x, u}\left( 0\right) }\right) + \rho \left( {x, u}\right) = 0, \)\n\nit is obvious that for \( x \in \left\lbrack {0,1}\right\rbrack \) ,\n\n\[ b\left( {x, u}\right) \l... | Yes |
Example 2 Consider the functions \( \varphi : \left\lbrack {0,1}\right\rbrack \mapsto \mathbb{R}, b : \left\lbrack {0,1}\right\rbrack \times \left\lbrack {0,1}\right\rbrack \mapsto {\mathbb{R}}_{ + } \) and \( \rho : \) \( \left\lbrack {0,1}\right\rbrack \times \left\lbrack {0,1}\right\rbrack \rightarrow \mathbb{R} \),... | Assuming that \( {\varphi }^{ + }\left( {{H}_{x, u}\left( 0\right) }\right) + \rho \left( {x, u}\right) \geq 0 \), that is\n\n\[{\varphi }^{ + }\left( {{H}_{x, u}\left( 0\right) }\right) + \rho \left( {x, u}\right) = {u}^{2} - 2{x}^{2} \geq 0,\forall u \in \left\lbrack {0,1}\right\rbrack ,\]\n\nwe derive \( f\left( u\r... | Yes |
Example 4 Consider the functions \( f : \left\lbrack {0,1}\right\rbrack \rightarrow {\mathbb{R}}^{2},{\rho }_{i} : \left\lbrack {0,1}\right\rbrack \times \left\lbrack {0,1}\right\rbrack \rightarrow \mathbb{R}, i = 1,2 \), defined by \( f\left( x\right) = {\left( {f}_{1}\left( x\right) ,{f}_{2}\left( x\right) \right) }^... | \[ {f}_{1}^{ + }\left( {{H}_{u, x}\left( 0\right) }\right) = {f}_{2}^{ + }\left( {{H}_{u, x}\left( 0\right) }\right) = 0. \] Thus \[ {\left( {\rho }_{1}\left( u, x\right) + {f}_{1}^{ + }\left( {H}_{u, x}\left( 0\right) \right) ,{\rho }_{2}\left( u, x\right) + {f}_{2}^{ + }\left( {H}_{u, x}\left( 0\right) \right) \right... | Yes |
Theorem 3.1 Let \( u \) be arbitrary vector in \( X,{b}_{i} : X \times X \rightarrow {\mathbb{R}}_{+ + },{\rho }_{i} : X \times X \rightarrow \mathbb{R}, i \in P \) and for each \( i \in P,{f}_{i} : X \rightarrow \mathbb{R} \) is \( \left( {{\rho }_{i},{b}_{i}}\right) \) -right differential arcwise connected with respe... | Proof Suppose that \( x \) is not an efficient solution of (MOP), then there exists \( u \in X \) such that\n\n\[ \n{f}_{i}\left( u\right) - {f}_{i}\left( x\right) \leq 0,\forall i \in P \n\]\n\n(3.1)\n\nwith strict inequality for at least one \( i \) . Because, for each \( i \in P,{f}_{i} \) is \( \left( {{\rho }_{i},... | Yes |
Theorem 3.2 Let \( u \) be arbitrary vector in \( X,{b}_{i} : X \times X \rightarrow {\mathbb{R}}_{+ + },{\rho }_{i} : X \times X \rightarrow \) \( {\mathbb{R}}_{ + }, i \in P \) and for each \( i \in P, - {f}_{i} : X \rightarrow \mathbb{R} \) is strictly \( \left( {{\rho }_{i},{b}_{i}}\right) \) -right differential ar... | Proof We proceed by contradiction. Assume that \( x \) is a weak efficient solution of (MOP), but does not solve (SVVI). Then there exists \( u \in X \) such that\n\n\[{\left( {\rho }_{1}\left( x, u\right) + {f}_{1}^{ + }\left( {H}_{x, u}\left( 0\right) \right) ,\cdots ,{\rho }_{p}\left( x, u\right) + {f}_{p}^{ + }\lef... | Yes |
Theorem 3.3 Let \( u \) be arbitrary vector in \( X,{b}_{i} : X \times X \rightarrow {\mathbb{R}}_{+ + },{\rho }_{i} : X \times X \rightarrow \mathbb{R}, i \in P \) and for each \( i \in P,{f}_{i} : X \rightarrow \mathbb{R} \) is pseudo \( \left( {{\rho }_{i},{b}_{i}}\right) \) -right differential arcwise connected wit... | Proof We proceed by contradiction. Assume that \( x \) is not a weak efficient solution of (MOP), namely, there exists \( u \in X \) such that \( {f}_{i}\left( u\right) - {f}_{i}\left( x\right) < 0,\forall i \in P \) . Since, for each \( i \in P,{f}_{i} \) is pseudo \( \left( {{\rho }_{i},{b}_{i}}\right) \) -right diff... | Yes |
Theorem 3.4 Let \( u, x \in X \) be arbitrary vectors, \( {b}_{i} : X \times X \rightarrow {\mathbb{R}}_{+ + },{\rho }_{i} : X \times X \rightarrow \) \( \mathbb{R}, i \in P \) and for each \( i \in P,{f}_{i} : X \rightarrow \mathbb{R} \) is \( \left( {{\rho }_{i},{b}_{i}}\right) \) -right differential arcwise connecte... | Proof Suppose, contrary to the result, that \( x \) does not solve (MVVI), then there exists \( u \in X \), such that\n\n\[{\left( {\rho }_{1}\left( u, x\right) + {f}_{1}^{ + }\left( {H}_{u, x}\left( 0\right) \right) ,\cdots ,{\rho }_{p}\left( u, x\right) + {f}_{p}^{ + }\left( {H}_{u, x}\left( 0\right) \right) \right) ... | Yes |
Theorem 3.5 Let \( u, x \in X \) be arbitrary vectors, \( {\rho }_{i} : X \times X \rightarrow \mathbb{R}, i \in P \) and for each \( i \in P \), the right derivative of \( {f}_{i} : X \rightarrow \mathbb{R} \) is monotone on \( X \). If \( x \in X \) solves (WSVVI), then \( x \) solves (WMVVI). | Proof Suppose \( x \) solves (WSVVI), then there exists no \( u \in X \), such that\n\n\[{\left( {\rho }_{1}\left( x, u\right) + {f}_{1}^{ + }\left( {H}_{x, u}\left( 0\right) \right) ,\cdots ,{\rho }_{p}\left( x, u\right) + {f}_{p}^{ + }\left( {H}_{x, u}\left( 0\right) \right) \right) }^{T} < 0,\]\n\ni.e., there exists... | Yes |
Theorem 3.6 Let \( u, x \in X \) be arbitrary vectors, \( {b}_{i} : X \times X \rightarrow {\mathbb{R}}_{+ + },{\rho }_{i} : X \times X \rightarrow \) \( \mathbb{R}, i \in P \) and for each \( i \in P,{f}_{i} : X \rightarrow \mathbb{R} \) is strictly \( \left( {{\rho }_{i},{b}_{i}}\right) \) -right differential arcwise... | Proof We proceed by contradiction. Suppose that \( x \) is a weak efficient solution of (MOP) but does not solve (MVVI), then there exists \( u \in X \), satisfying\n\n\[ \n{\left( {\rho }_{1}\left( u, x\right) + {f}_{1}^{ + }\left( {H}_{u, x}\left( 0\right) \right) ,\cdots ,{\rho }_{p}\left( u, x\right) + {f}_{p}^{ + ... | Yes |
Theorem 3.7 Let \( u, x \in X \) be arbitrary vectors, \( {b}_{i} : X \times X \rightarrow {\mathbb{R}}_{+ + },{\rho }_{i} : X \times X \rightarrow \) \( \mathbb{R}, i \in P \) and for each \( i \in P,{f}_{i} : X \rightarrow \mathbb{R} \) is \( \left( {{\rho }_{i},{b}_{i}}\right) \) -right differential arcwise connecte... | Proof The proof is similar to that of Theorem 3.4 and therefore being omitted. | No |
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