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Lemma 3.2 Let \( {\phi }_{1} \) be the eigenfunction corresponding to \( {\lambda }_{1} \) with \( \begin{Vmatrix}{\phi }_{1}\end{Vmatrix} = 1 \) . If \( {f}_{0} < {\lambda }_{1} < l \), then there exists \( m = m\left( {{f}_{0}, q, f, N,\Omega }\right) \) such that for all \( a\left( x\right) \in {L}^{\infty }\left( \... | Proof By \( \left( {\mathrm{h}}_{1}\right) \) and \( \left( {\mathrm{h}}_{2}\right) \), if \( l \in \left( {{\lambda }_{1}, + \infty }\right) \), for any \( \varepsilon > 0 \), there exist \( A = A\left( \varepsilon \right) \geq 0 \) and \( B = B\left( \varepsilon \right) \) such that for all \( \left( {x, s}\right) \i... | Yes |
Lemma 3.5 Suppose \( l = {\lambda }_{k} \) and \( f \) satisfies \( \left( {\mathrm{h}}_{4}\right) \) . Then the functional \( \mathcal{J} \) satisfies the (C) condition. | Proof Suppose \( {u}_{n} \in {X}_{0} \) satisfies\n\n\[ \mathcal{J}\left( {u}_{n}\right) \rightarrow c \in \mathbb{R},\;\left( {1 + \left| \right| {u}_{n}\left| \right| }\right) \left| \right| {\mathcal{J}}^{\prime }\left( {u}_{n}\right) \left| \right| \rightarrow 0\text{ as }n \rightarrow \infty . \]\n\n(3.15)\n\nIn v... | Yes |
Lemma 3.6 Under conditions (a), \( \left( {\mathrm{h}}_{1}\right) ,\left( {\mathrm{h}}_{2}\right) \), then there exists \( m = m\left( {{f}_{0}, q, f, N,\Omega }\right) \) such that for all \( a\left( x\right) \in {L}^{\infty }\left( \Omega \right) \) and \( a\left( x\right) > 0 \) with \( {\left| a\right| }_{\infty } ... | Proof Under our conditions, we still can prove it by using the Ekeland variational principle. Since the proof is completely similar to Theorem 1.1 in [12] and Lemma 2.1 in [14], we omit it. | No |
Lemma 2.2 The probability mass function of \( {M}_{r} = \mathop{\sum }\limits_{{i = 1}}^{r}{Z}_{i} \) is\n\n\[ P\left( {{M}_{r} = i}\right) = \mathop{\sum }\limits_{{Z \in {Q}_{i}^{ * }}}\cdots \sum {m}^{\left( {m}_{r}\right) }{n}^{\left( {n}_{r}\right) }{\mathrm{e}}^{\frac{\left( {m - {m}_{r}}\right) {\lambda }_{1}}{{... | Proof The detail of the Lemma 2.1 and 2.2 can be refer to Balakrishnan and Rasouli \( {}^{\left\lbrack 5\right\rbrack } \) . | No |
Corollary 2.1 The probability mass function of \( {N}_{r} = \mathop{\sum }\limits_{{i = 1}}^{r}\left( {1 - {Z}_{i}}\right) \) is | \[ P\left( {{N}_{r} = i}\right) = \mathop{\sum }\limits_{{Z \in {Q}_{i}^{* * }}}\cdots \sum {m}^{\left( {m}_{r}\right) }{n}^{\left( {n}_{r}\right) }{\mathrm{e}}^{\frac{\left( {m - {m}_{r}}\right) {\lambda }_{1}}{{\sigma }_{1}}}{\mathrm{e}}^{\frac{\left( {n - {n}_{r}}\right) {\lambda }_{2}}{{\sigma }_{2}}}\mathop{\prod ... | Yes |
Lemma 2.3 Let \( {U}_{1},{U}_{2},\cdots ,{U}_{k} \) are iid random variables follow \( \varepsilon \left( 1\right) \) and \( \Theta = \left( {{\theta }_{1},{\theta }_{2},\cdots ,{\theta }_{k}}\right) \) be a vector of distinct positive real numbers. Then, the pdf of \( U = \mathop{\sum }\limits_{{i = 1}}^{k}\frac{{U}_{... | \[ {f}_{u}^{\left( k\right) }\left( {u;\Theta }\right) = \mathop{\sum }\limits_{{j = 1}}^{k}\left( {\mathop{\prod }\limits_{{i = 1, i \neq j}}^{k}\frac{{\theta }_{i}}{{\theta }_{i} - {\theta }_{j}}}\right) {\theta }_{j}{\mathrm{e}}^{-u{\theta }_{j}}, u > 0. \] | Yes |
Lemma 2.4 Let \( {U}_{1},{U}_{2},\cdots ,{U}_{k} \) are iid random variables follow \( \varepsilon \left( 1\right) \) and \( W \sim \Gamma \left( {w,;a,1}\right) \) independently of \( {U}_{i}, i = 1,2\cdots, n \) where \( a \) is a positive integer. Let \( \Theta = \left( {{\theta }_{1},{\theta }_{2},\cdots ,{\theta }... | \[ {f}_{s}^{\left( k\right) }\left( {s;a,\Theta }\right) = \mathop{\sum }\limits_{{j = 1}}^{k}\left( {\mathop{\prod }\limits_{{i = 1, i \neq j}}^{k}\frac{{\theta }_{i}}{{\theta }_{i} - {\theta }_{j}}}\right) \frac{{\theta }_{j}{\mathrm{e}}^{-u{\theta }_{j}}}{{\left( 1 - {\theta }_{j}\right) }^{a}}P\left( {a, s\left( {1... | Yes |
Theorem 3.1 When \( - \infty < x < + \infty \), the PDF of \( {\widehat{\sigma }}_{1} \) is given by\n\n\[ \n{f}_{1}\left( {x;\sum ,\Lambda }\right) = \mathop{\sum }\limits_{{{m}_{r} = 1}}^{{r - 1}}\left\{ {\mathop{\sum }\limits_{{Z \in {Q}_{{m}_{r}}^{ * }}}\cdots \sum {m}^{\left( {m}_{r}\right) }{n}^{\left( {n}_{r}\ri... | Proof\n\n\[ \n{f}_{1}\left( {x;\sum ,\Lambda }\right) = \mathop{\sum }\limits_{{{m}_{r} = 1}}^{{r - 1}}f\left( {x;{\left. \sum ,{\left. \Lambda \right| }_{{M}_{r} = {m}_{r}})P\left( {M}_{r} = {m}_{r}\right. \right| }_{r}}\right) = \mathop{\sum }\limits_{{{m}_{r} = 1}}^{{r - 1}}{m}_{r}{g}_{1}\left( {{m}_{r}x}\right) P\l... | Yes |
Theorem 3.2 When \( - \infty < x < + \infty \), the PDF of \( {\widehat{\sigma }}_{2} \) is given by\n\n\[ \n{f}_{2}\left( {x;\sum ,\Lambda }\right) = \mathop{\sum }\limits_{{{n}_{r} = 1}}^{{r - 1}}\left\{ {\mathop{\sum }\limits_{{Z \in {Q}_{{n}_{r}}^{{ * }_{ * }}}}\cdots \sum {m}^{\left( {n}_{r}\right) }{n}^{\left( {n... | Proof\n\n\[ \n{f}_{2}\left( {x;\sum ,\Lambda }\right) = \mathop{\sum }\limits_{{{m}_{r} = 1}}^{{r - 1}}f\left( {x;\sum ,{\left. \Lambda \right| }_{{N}_{r} = {n}_{r}}}\right) P\left( {{N}_{r} = {n}_{r}}\right) = \mathop{\sum }\limits_{{{n}_{r} = 1}}^{{r - 1}}{n}_{r}{g}_{2}\left( {{n}_{r}x}\right) P\left( {{N}_{r} = {n}_... | Yes |
Corollary 2.1 Let \( G \) be an \( r \) -regular graph with \( n \) vertices. Then\n\n\[ \n{\mu }_{C\left( G\right) }\left( x\right) = {\left( -1\right) }^{n\left( {r + 1}\right) }{\left( \left( x - r\right) \left( x - r - 2\right) \right) }^{\frac{n\left( {r - 2}\right) }{2}}{\mu }_{G}\left( {x\left( {r + 2 - x}\right... | Proof Note that \( s\left( G\right) \) is a \( \left( {2, r}\right) \) -semiregular graph with \( n + \frac{nr}{2} \) vertices and \( {nr} \) edges. It follows from Lemma 2.2 that\n\n\[ \n{\mu }_{C\left( G\right) }\left( x\right) = {\left( -1\right) }^{\frac{n\left( {r + 2}\right) }{2}}{\left( x - \left( r + 2\right) \... | Yes |
Theorem 3.1 Let \( G \) be an \( \left( {r, s}\right) \) -semiregular graph with \( n \) vertices. Then\n\n\[ \n\operatorname{Kf}\left( {\mathcal{L}\left( G\right) }\right) = \frac{{n}^{2}{r}^{2}{s}^{2}}{{\left( r + s\right) }^{3}} - \frac{n\left( {n - 1}\right) {rs}}{{\left( r + s\right) }^{2}} + \frac{nrs}{r + s}\mat... | Proof Let \( m \) be the number of edges of \( G \) . Then \( m = \frac{nrs}{r + s} \) . Note that \( {\mu }_{1} = r + s,{\mu }_{n} = 0 \) . It follows from (1.2) and (2.1) that\n\n\[ \n\operatorname{Kf}\left( {\mathcal{L}\left( G\right) }\right) = \frac{nrs}{r + s}\mathop{\sum }\limits_{{i = 1}}^{{\frac{nrs}{r + s} - ... | Yes |
Corollary 3.1 Let \( G \) be an \( r \) -regular graph with \( n \) vertices and \( m \) edges, and let \( C\left( G\right) \) be the para-line graph of \( G \) . Then\n\n\[ \operatorname{Kf}\left( {C\left( G\right) }\right) = \frac{{nr}\left( {{nr} - {2n} + 2}\right) }{2\left( {r + 2}\right) } + \frac{{n}^{2}\left( {r... | Note that \( s\left( G\right) \) is a \( \left( {2, r}\right) \) -semiregular graph with \( n + \frac{nr}{2} \) vertices and \( {nr} \) edges. It follows from Theorem 3.1 that\n\n\[ \operatorname{Kf}\left( {C\left( G\right) }\right) = \frac{{n}^{2}{r}^{2}}{r + 2} - \frac{{n}^{2}{r}^{2} + 2{n}^{2}r - {2nr}}{2\left( {r +... | Yes |
Theorem 3.2 Let \( G \) be an \( \left( {r, s}\right) \) -semiregular graph with \( n \) vertices. Then\n\n\[ \n\operatorname{Kf}\left( {\mathcal{L}\left( G\right) }\right) \geq \frac{{n}^{2}{r}^{2}{s}^{2}}{{\left( r + s\right) }^{3}} - \frac{n\left( {n - 1}\right) {rs}}{{\left( r + s\right) }^{2}} + \frac{n{\left( n -... | Proof Let \( m \) be the number of edges of \( G \) . Then \( m = \frac{nrs}{r + s} \) . Note that \( \mathop{\sum }\limits_{{i = 1}}^{{n - 1}}{\mu }_{i} = {2m} \) . By the arithmetic-harmonic mean inequality\n\n\[ \n\mathop{\sum }\limits_{{i = 2}}^{{n - 1}}\frac{1}{r + s - {\mu }_{i}} \geq \frac{{\left( n - 2\right) }... | Yes |
Corollary 3.2 Let \( G \) be an \( r \) -regular graph with \( n \) vertices. Then\n\n\[ \operatorname{Kf}(C\left( G\right) \geq \frac{{nr}\left( {{nr} - {2n} + 2}\right) }{2\left( {r + 2}\right) } + \frac{{nr}{\left( nr + 2n - 4\right) }^{2}}{2\left( {r + 2}\right) \left( {{nr} + {2n} - 2}\right) - {8nr}}, \]\n\nthe e... | Proof Note that \( s\left( G\right) \) is a \( \left( {2, r}\right) \) -semiregular graph with \( n + \frac{nr}{2} \) vertices and \( {nr} \) edges. The inequality (3.6) follows from Theorem 3.2 immediately. Meanwhile, it follows from the Theorem that the equality in (3.6) holds if and only if \( s\left( G\right) \cong... | Yes |
Proposition 2.1 Let \( T \) be a \( \Omega \) -set. Then for each \( k \geq 2 \), there are \( {\alpha }_{1},{\alpha }_{2},\cdots ,{\alpha }_{k} \in T \) , such that \( {\alpha }_{1} + {\alpha }_{2} + \cdots + {\alpha }_{k} = 1 \) . | Proof For fixed \( k \geq 2 \), choose \( \beta \in T \) and define\n\n\[ \n{\alpha }_{1} = {\beta }^{k - 1},{\alpha }_{2} = {\beta }^{k - 2}\left( {1 - \beta }\right) ,\cdots ,{\alpha }_{k - 1} = \beta \left( {1 - \beta }\right) ,{\alpha }_{k} = \left( {1 - \beta }\right) .\n\]\n\nThen, it is easy to see by the defini... | Yes |
Proposition 2.2 Each \( \Omega \) -set \( T \) is dense in \( \left\lbrack {0,1}\right\rbrack \) . | Proof Let \( T \) be a \( \Omega \) -set. For any \( {x}_{0} \in \left( {0,1}\right) \), we show that \( {x}_{0} \) is a cluster of \( T \) .\n\nIf \( {x}_{0} = 0 \) or 1, taking \( \alpha \in T \), we have \( {\alpha }^{n},1 - {\alpha }^{n} \in T \) by the definition of \( \Omega \) -sets and \( {\alpha }^{n} \rightar... | Yes |
Example 2 Let \( \mathbb{Q} \) be the set of rationals in \( \mathbb{R} \). Then \( \mathbb{Q} \) is \( \Omega \) -convex with \( T = \left( {0,1}\right) \cap \mathbb{Q} \) or any \( \Omega \) -set contained in \( \mathbb{Q} \), e.g. \( \left\{ {\left. {\frac{l}{{3}^{n}} \mid 1 < l < {3}^{n},}\right| \;n = 1,2,\cdots }... | However, \( \mathbb{Q} \) is clearly not convex. Therefore, the \( \Omega \) -convexity is properly weaker than the classical convexity. | Yes |
For any fixed integer \( n \geq 2 \), the set \( S = \left\{ {\frac{l}{{n}^{m}} \mid m = 1,2,\cdots ,0 \leq l \leq {n}^{m}}\right\} \) is \( \Omega \) -convex with \( T = \left\{ {\frac{l}{{n}^{m}} \mid m = 1,2,\cdots ,1 \leq l < {n}^{m}}\right\} \) . | In fact, for any \( {x}_{1},{x}_{2},\ldots ,{x}_{k} \in S \) and \( {\alpha }_{1},{\alpha }_{2},\ldots ,{\alpha }_{k} \in T \) with \( {\alpha }_{1} + {\alpha }_{2} + \ldots + {\alpha }_{k} = 1 \), writing \( {x}_{i} = \frac{{l}_{i}}{{n}^{{m}_{i}}},0 \leq {l}_{i} \leq {n}^{{m}_{i}},{\alpha }_{i} = \frac{{s}_{i}}{{n}^{{... | Yes |
Proposition 3.1 If \( S \subset V \) is middle-point convex (i.e. \( \frac{1}{2} \)-convex), then\n\n\[ \left\{ {\left. \frac{l}{{2}^{m}}\right| \;m = 1,2,\cdots ,1 \leq l < {2}^{m}}\right\} \subset {\Omega }_{S}. \]\n | Proof We will show \( \frac{l}{{2}^{m}} \in {\Omega }_{S} \) by induction on \( m \) . There is nothing to prove for the case \( m = 1 \) (and so \( l = 1 \) ).\n\nNow suppose \( \frac{l}{{2}^{k}} \in {\Omega }_{S} \) for all \( 1 \leq k \leq m - 1 \) and \( 1 \leq l < {2}^{k} \). Thus, for \( \frac{l}{{2}^{m}} \) with... | Yes |
Proposition 4.1 Let \( K \subset V \) and \( S \subset V \) be \( \Omega \) -convex with \( \Omega \) -sets \( {T}_{K} \) and \( {T}_{S} \) respectively. If \( {T}_{K} \cap {T}_{S} \neq \varnothing \), then \( K \cap S \) is \( \Omega \) -convex (with the \( \Omega \) -set \( {T}_{K} \cap {T}_{S} \) ). | Proof By Proposition 2.3, \( T \mathrel{\text{:=}} {T}_{K} \cap {T}_{S} \) is a \( \Omega \) -set. If \( K \cap S = \varnothing \), then there is nothing to prove. If \( K \cap S \neq \varnothing \), then for arbitrary \( {x}_{1},{x}_{2},\cdots ,{x}_{k} \in K \cap S \) and \( {\alpha }_{1},{\alpha }_{2},\cdots ,{\alpha... | Yes |
If \( K \subset {V}_{1} \) and \( S \subset {V}_{2} \) are \( \Omega \) -convex with \( {T}_{K} \) and \( {T}_{S} \) respectively and \( T \mathrel{\text{:=}} {T}_{K} \cap {T}_{S} \neq \varnothing \), then \( K \times S \) is \( \Omega \) -convex with the convexity-indicating set \( T \) . Conversely, if \( K \times S ... | If \( T \mathrel{\text{:=}} {T}_{K} \cap {T}_{S} \neq \varnothing \), then \( T \) is a \( \Omega \) -set by Proposition 2.3. Furthermore, for arbitrary \( \left( {{x}_{1},{y}_{1}}\right) ,\left( {{x}_{2},{y}_{2}}\right) ,\cdots ,\left( {{x}_{k},{y}_{k}}\right) \in K \times S \) and \( {\alpha }_{1},{\alpha }_{2},\cdot... | Yes |
Theorem 4.2 If \( A : {V}_{1} \rightarrow {V}_{2} \) is an affine mapping and \( S \subset {V}_{1} \) is a \( T \) -convex set, then the image \( A\left( S\right) \) of \( S \) under the mapping \( \mathrm{A} \) is \( T \) -convex. Conversely, if \( K \subset {V}_{2} \) is \( T \) -convex, then the inverse image \( {A}... | Proof Suppose \( A : {V}_{1} \rightarrow {V}_{2} \) is affine and \( S \) is \( T \) -convex, then, for arbitrary \( {y}_{1},{y}_{2},\cdots ,{y}_{k} \in \) \( A\left( S\right) \) and \( {\alpha }_{1},{\alpha }_{2},\cdots ,{\alpha }_{k} \in T \) with \( \mathop{\sum }\limits_{{i = 1}}^{k}{\alpha }_{i} = 1 \), we have \(... | Yes |
Theorem 4.3 Let \( K, S \subset V \) be \( \Omega \) -convex with the convexity-indicating sets \( {T}_{K},{T}_{S} \) respectively. If \( T \mathrel{\text{:=}} {T}_{K} \cap {T}_{S} \neq \varnothing \), then, for any \( \lambda ,\mu \in \mathbb{R},{\lambda K} + {\mu S} \) is \( \Omega \) -convex with the convexity-indic... | Proof Define a mapping \( A : V \times V \rightarrow V \) by \( A\left( \left( {x, y}\right) \right) \mathrel{\text{:=}} {\lambda x} + {\mu y},\left( {x, y}\right) \in V \times V \) . Then \( A \) is affine. Thus by Theorem 4.2, \( {\lambda K} + {\mu S} = A\left( {K \times S}\right) \) is \( T \) -convex since \( K \ti... | Yes |
Theorem 3.1 Let \( \\left\\{ {{X}_{ni};1 \\leq i \\leq n, n \\geq 1}\\right\\} \) be an array of rowwise negatively associated random variables and \( \\left\\{ {{a}_{n};n \\geq 1}\\right\\} \) be a sequence of positive real numbers. Let \( \\left\\{ {{\\psi }_{n}\\left( t\\right) ;n \\geq 1}\\right\\} \) be a sequence... | \[ \\mathop{\\sum }\\limits_{{n = 1}}^{\\infty }\\mathop{\\sum }\\limits_{{i = 1}}^{n}\\mathrm{E}{\\psi }_{i}\\left( \\frac{\\left| {X}_{ni}\\right| }{{a}_{n}}\\right) < \\infty . \] (3.1) Then condition (3.1) implies (1.10) and (2.9). | Yes |
Lemma 2.1 \( {}^{\left\lbrack 6\right\rbrack } \) Let \( {r}_{k}^{h} = r\left( {kh}\right) \) for \( h > 0 \) and \( k \geq 0 \) . Then \( \left\{ {{r}_{k}^{h}, k = 0,1,2,\cdots }\right\} \) is a discrete Markov chain with the one-step transition probability matrix | \[ P\left( h\right) = {\left( {P}_{ij}\left( h\right) \right) }_{N \times N} = {\mathrm{e}}^{h\Gamma }.\] | Yes |
Lemma 3.1 Under the conditions (ii)-(v), for any \( t \in \left\lbrack {0, T}\right\rbrack \) the following inequalities hold:\n\n\[ \mathrm{E}{\left| X\left( t\right) - {\bar{X}}_{1}\left( t\right) \right| }^{2} \leq {Ch},\;\mathrm{E}{\left| X\left( t - \tau \right) - {\bar{X}}_{2}\left( t\right) \right| }^{2} \leq {C... | Proof The proof is similar to that of Lemma 3.2 and Lemma 3.3 in [10]. Here we only prove the inequality:\n\n\[ \mathrm{E}{\left| X\left( t\right) - {\bar{X}}_{1}\left( t\right) \right| }^{4} \leq C{h}^{2},\forall t \in \left\lbrack {0, T}\right\rbrack . \]\n\n(3.2)\n\nFor any \( t \in \left\lbrack {0, T}\right\rbrack ... | No |
Lemma 3.3 Under the conditions (iii),(v), for any \( T > 0 \)\n\n\[ \mathrm{E}{\int }_{0}^{T}{\left| g\left( {\bar{X}}_{1}\left( u\right) ,{\bar{X}}_{2}\left( u\right) ,\bar{r}\left( u\right) \right) - g\left( {\bar{X}}_{1}\left( u\right) ,{\bar{X}}_{2}\left( u\right), r\left( u\right) \right) \right| }^{2}\mathrm{\;d}... | Using Remak 2.4 this can be shown by similar techniques as in the proof of Lemma 3.2. | No |
Lemma 2.1 Let \( \nu \left( \cdot \right) \) be a submeasure. Then \( \nu \left( \cdot \right) \) is continuous if and only if \( \nu \left( \cdot \right) \) is order-continuous. | Proof (only if) Continuity of \( \nu \left( \cdot \right) \) implies upper continuity of \( \nu \left( \cdot \right) \), it is easy to check that \( \nu \left( {A}_{n}\right) \downarrow \nu \left( \phi \right) = 0 \), whenever \( {A}_{n} \downarrow \phi \) .\n\n(if) Let \( {A}_{n} \uparrow A \) . By monotonicity and su... | Yes |
Lemma 2.3 Let \( {X}_{1},{X}_{2},{X}_{3},\cdots ,{X}_{k},\cdots \) be a sequence of random variables. Then \( {X}_{1},{X}_{2},{X}_{3},\cdots ,{X}_{k},\cdots \) are regular relative to \( \omega \left( \cdot \right) \) . | Proof We prove the first equality. Since\n\n\[ \left( {{X}_{k} \geq \beta }\right) \subseteq \left( {{X}_{k} \geq \alpha }\right) \]\n\nwith monotonicity of \( \omega \left( \cdot \right) \), we have\n\n\[ \omega \left( {{X}_{k} \geq \beta }\right) \leq \omega \left( {{X}_{k} \geq \alpha }\right) . \]\n\nOn the other h... | Yes |
Lemma 3.2 Let \( \Omega \) be a compact space, and \( \nu \) a convex and continuous capacity on the \( \sigma \) -algebra \( \mathcal{F} \) of subsets of \( \Omega \) . Let \( {X}_{1},{X}_{2},{X}_{3},\cdots ,{X}_{k},\cdots \) be a sequence of continuous, regular, mutually independent and nonnegative random variables w... | The proofs of Lemmas 3.1 and 3.2 are very similar to those of Lemma 22 and 24 in [2], so we omit them. | No |
Example 2.1 \( w\left( z\right) = \tan \frac{\pi }{4}z \) is a finite order transcendental meromorphic solution of the following complex differential-difference equation\n\n\[ \n{w}^{\prime \prime }\left( {z + 1}\right) + {w}^{\prime \prime }\left( {z - 1}\right) = \frac{2{\pi }^{2}\left\lbrack {{w}^{5}\left( z\right) ... | In this case\n\n\[ \n\max \{ p, q\} = 6,\lambda = 2, t = 2.\n\]\n\nThus\n\n\[ \n\max \{ p, q\} = 6 = \lambda \left( {t + 1}\right) .\n\] | Yes |
Theorem 3.1 If \( \dim \ker {F}_{\left( u, v\right) }\left( {\chi \left( {m, n}\right) ;{u}_{c},\frac{\beta }{\alpha }{u}_{c}}\right) = 1 \), then there exists a positive constant \( \delta \) such that nonconstant solutions of (2.1) near \( \left( {\chi ;u, v}\right) = \left( {\chi \left( {m, n}\right) ;{u}_{c},\frac{... | Proof It follows from \( \dim \ker {F}_{\left( u, v\right) }\left( {\chi \left( {m, n}\right) ;{u}_{c},\frac{\beta }{\alpha }{u}_{c}}\right) = 1 \) that\n\n\[ \ker {F}_{\left( u, v\right) }\left( {\chi \left( {m, n}\right) ;{u}_{c},\frac{\beta }{\alpha }{u}_{c}}\right) = \operatorname{span}\left\{ \left( \begin{matrix}... | Yes |
Theorem 4.1 The function \( \widetilde{\chi }\left( s\right) \) in (3.1) satisfies\n\n\[ \n\lambda {k}_{mn}{u}_{c}\widetilde{\chi }\left( 0\right) \parallel \Phi {\parallel }_{2}^{2} = 2\left( {3{b}_{3}{u}_{c} - {b}_{2}}\right) A - \chi \left( {m, n}\right) \left( {{k}_{mn}B + {\lambda C} - D}\right) .\n\] | To prove Theorem 4.1, we first prove the following Lemma 4.1.\n\nLemma 4.1 \( \left( {A, | No |
Lemma 4.1 \( \\left( {A, B, C, D}\\right) \) satisfy the following algebraic equations:\n\n\[ \n\\left( \\begin{matrix} - {2d\\lambda } - \\left( {3{b}_{3}{u}_{c}^{2} - 2{b}_{2}{u}_{c} - {b}_{1}}\\right) & {2d} & {2\\lambda \\chi }\\left( {m, n}\\right) {u}_{c} & - {2\\chi }\\left( {m, n}\\right) {u}_{c} \\\\ 2{\\lambd... | Proof Using integration by parts, we have\n\n\[ \n\\left\\langle {\\Delta \\widetilde{u}\\left( 0\\right) ,{\\Phi }^{2}}\\right\\rangle = 2\\left\\langle {\\widetilde{u}\\left( 0\\right) ,{\\left| \\nabla \\Phi \\right| }^{2} - \\lambda {\\Phi }^{2}}\\right\\rangle ,\\;\\left\\langle {\\Delta \\widetilde{u}\\left( 0\\r... | Yes |
Theorem 3.1 For any solution \( \left( {x\left( t\right), y\left( t\right), z\left( t\right) }\right) \) of system (1.1), there exists a constant \( G > 0 \), such that \( x\left( t\right) \leq G, y\left( t\right) \leq G \) and \( z\left( t\right) \leq G \) hold for all \( t \) large enough. | Proof Let \( X\left( t\right) = \left( {x\left( t\right), y\left( t\right), z\left( t\right) }\right) \) be a solution of (1.1) with initial value \( \left( {{x}_{0},{y}_{0},{z}_{0}}\right) \) . Define a function \( \psi \left( t\right) = \frac{{\beta }_{2}}{{\beta }_{1}}x\left( t\right) + y\left( t\right) + \frac{{\be... | Yes |
Theorem 3.2 1) Suppose \( \\left( {x\\left( t\\right), y\\left( t\\right), z\\left( t\\right) }\\right) \) is any solution of system (1.1), then the prey and top predator-free periodic solution \( \\left( {0,{y}^{ * }\\left( t\\right) ,0}\\right) \) is locally asymptotically stable provided that | Proof The local stability of periodic solution \( \\left( {0,{y}^{ * }\\left( t\\right) ,0}\\right) \) can be determined by considering the behavior of small amplitude perturbations of the solution. Define \( x\\left( t\\right) = {\\omega }_{1}\\left( t\\right) \) , \( y\\left( t\\right) = {\\omega }_{2}\\left( t\\righ... | Yes |
Theorem 3.4 Subsystem (3.10) is permanent if the following conditions holds,\n\n\[ \n\\frac{2{\\beta }_{4}\\left( {1 - \\theta }\\right) }{{B}_{2}{b}_{2}}\\left( {{\\Lambda }_{1} - {\\Lambda }_{2} + {\\Lambda }_{3} - {\\Lambda }_{4}}\\right) + \\ln \\left( {1 - {\\delta }_{3}}\\right) > {d}_{2}T \n\] \n\nwhere \n\n\[ \... | The proof is similar as Theorem 3.3. We omit it here. | No |
Theorem 3.5 System (1.1) is permanent if the following conditions hold,\n\n\[ \n\\left( {r - \\alpha {g}_{1}}\\right) T + \\ln \\left( {1 - {\\delta }_{1}}\\right) > \\frac{{\\beta }_{1}}{{b}_{1}}\\left( {1 - \\theta }\\right) \\frac{{q}^{m}{A}_{1}}{{B}_{1}}\n\]\n\nand\n\n\[ \n\\frac{2{\\beta }_{4}\\left( {1 - \\theta ... | Proof Consider subsystems (3.9) and (3.10) of system (1.1). It follows from Lemma 2.1 that \( {x}_{1}\\left( t\\right) \\leq x\\left( t\\right) ,{y}_{11}\\left( t\\right) \\geq y\\left( t\\right) ,{y}_{22}\\left( t\\right) \\leq y\\left( t\\right) \) and \( {z}_{2}\\left( t\\right) \\leq z\\left( t\\right) \), where \(... | Yes |
Theorem 1.1 For any \( \alpha > 1 \) and \( \beta > 0,{\dim }_{\mathrm{H}}E\left( {\alpha ,\beta }\right) = 1 \), where \( {\dim }_{\mathrm{H}} \) denotes the Hausdorff dimension. | ## 2. Proof of Main Theorem\n\nBefore the proof of the theorem, we summarize some auxiliary results.\n\nFor any \( n \geq 1 \), denote by \( {\mathrm{L}}_{n} \) the collection of all admissible blocks of order \( n \) and \( {\mathrm{L}}_{n} \) is given as\n\n\[{\mathrm{L}}_{n} = \left\{ {\left( {{d}_{1},{d}_{2},\cdots... | No |
Lemma 2.4 For any integer \( m \geq 2 \) and for any \( \alpha > 1 \) and \( \beta > 0 \), we have \( {E}_{m}\left( {\alpha ,\beta }\right) \subset \) \( E\left( {\alpha ,\beta }\right) \) . | Proof Fix \( y \in {E}_{m}\left( {\alpha ,\beta }\right) \) . For any integer \( n \) large enough, there is an integer \( k \geq 1 \) such that\n\n\[{\left( k + {N}_{0}\right) }^{3} = {n}_{k} \leq n < {n}_{k + 1} = {\left( k + 1 + {N}_{0}\right) }^{3}.\n\]\n\nOn the one hand, we have\n\n\[ \frac{1}{{n}^{\alpha }}\math... | Yes |
Proposition 2.1 Suppose \( M \) and \( {M}_{1} \) are Orlicz functions, \( {L}_{M} \) is \( P\left( {n,\varepsilon }\right) \) -convex, and there exists \( {\varepsilon }^{\prime } \in \left( {0,\frac{\varepsilon }{1 - \varepsilon }}\right) \) such that \( M\left( t\right) \leq {M}_{1}\left( t\right) \leq \left( {1 + {... | Proof Since \( M\left( t\right) \leq {M}_{1}\left( t\right) \leq \left( {1 + {\varepsilon }^{\prime }}\right) M\left( t\right) \) holds for all \( t \in \mathbb{R} \) we have, by Lemma 1.28 in \( \left\lbrack 1\right\rbrack \) ,\n\n\[ \parallel x{\parallel }_{M} \leq \parallel x{\parallel }_{{M}_{1}} \leq \left( {1 + {... | Yes |
Lemma 2.1 \( {l}_{M} \) is \( \mathrm{F} \) -convex if and only if \( M \in {\delta }_{2} \) and \( N \in {\delta }_{2} \) . | Proof (Necessity) Since \( {l}_{M} \) is F-convex and \( {h}_{M} \) is a closed subspace of \( {l}_{M} \), one get \( {h}_{M} \) is F-convex. The definition of F-convex yields \( {l}_{\left( N\right) } \), as the dual space of \( {h}_{M} \), is P-convex. So \( N \in {\delta }_{2} \) and \( M \in {\delta }_{2} \) . (Suf... | Yes |
Lemma 2.3 Suppose \( M \in {\delta }_{2}, N \in {\delta }_{2} \), then for any \( l \geq m > 0 \) and \( w > 0 \), there exists \( r = r\left( {w, m, l}\right) \in \left( {0,1}\right) \) such that for any equi-P-convex Banach spaces \( \left\{ {{X}_{s} : s = 1,2,\cdots }\right\} \) with equi-constant \( {n}_{0} \), and... | Proof Some methods come from [14]. To show the constant \( r \) is independent of \( {X}_{s} \) we shall give part of the proof.\n\nWithout loss of generality, we may assume that \( {\begin{Vmatrix}{k}_{s}^{\left( 1\right) }{x}_{s}^{\left( 1\right) }\end{Vmatrix}}_{s} = \max \left\{ {{\begin{Vmatrix}{k}_{s}^{\left( i\r... | Yes |
Theorem 3.4 \( {L}_{M}\left( {\mu, X}\right) \) (or \( {L}_{\left( M\right) }\left( {\mu, X}\right) \) ) is F-convex if and only if \( M \in {\Delta }_{2}, N \in {\Delta }_{2} \) , and \( X \) is \( \mathrm{F} \) -convex. | Proof We only prove the sufficiency for \( {L}_{M}\left( {\mu, X}\right) \) . Actually, the F-convexity of \( X \) implies that \( {X}^{ * } \) is P-convex. so \( {X}^{ * } \) is reflexive, and so \( {X}^{ * } \) has the Radon-Nikodym property. Thus by Theorem 2 in [3], we have \( {\left( {E}_{M}\left( \mu, X\right) \r... | Yes |
Lemma 2.4 The Green function \( G \in C\left( {\left\lbrack {0,1}\right\rbrack \times \left\lbrack {0,1}\right\rbrack ,{\mathbb{R}}^{ + }}\right) \), and\n\n\[ \n{t}^{\alpha - 1}G\left( {1, s}\right) \leq G\left( {t, s}\right) \leq G\left( {1, s}\right) ,\forall t, s \in \left\lbrack {0,1}\right\rbrack \text{.} \n\] | We can adopt the method of Lemma 3 in [12] to prove (2.4), so we omit the proof. | No |
Example 3.3 We choose \( p\left( v\right) = {v}^{\frac{4}{5}}, q\left( u\right) = {u}^{2},\zeta \left( v\right) = {v}^{2},\eta \left( u\right) = \ln \left( {u + 1}\right) \) for all \( u, v \in {\mathbb{R}}^{ + } \) . | \[ \mathop{\lim }\limits_{{u \rightarrow + \infty }}\frac{p\left( {{\delta q}\left( u\right) }\right) }{u} = \mathop{\lim }\limits_{{u \rightarrow + \infty }}\frac{{\delta }^{\frac{4}{5}}{u}^{\frac{8}{5}}}{u} = + \infty , \] and \[ \mathop{\lim }\limits_{{u \rightarrow {0}^{ + }}}\frac{\zeta \left( {{\delta \eta }\left... | Yes |
Theorem 3.3 Assume that \( f : J \times \mathbb{R} \rightarrow \mathbb{R},{I}_{k} : \mathbb{R} \rightarrow \mathbb{R},{\bar{I}}_{k} : \mathbb{R} \rightarrow \mathbb{R} \) are continuous functions and the following assumptions hold\n\n\( \left( {\mathrm{H}}_{4}\right) \left| {f\left( {t, y}\right) }\right| \leq \mu \lef... | Proof Let \( \eta = \max \left\{ {\parallel \mu \parallel ,{\beta }_{1},{\beta }_{2}}\right\} ,\parallel \mu \parallel = \mathop{\sup }\limits_{{t \in \left\lbrack {0,1}\right\rbrack }}\left| {\mu \left( t\right) }\right| \), and consider \( {B}_{r} = \{ y \in \) \( \mathbb{P}C\left( J\right) : \parallel y\parallel \le... | Yes |
Example 4.1 Consider the following boundary value problem\n\n\[ \left\{ \begin{array}{l} {}^{c}{D}_{{0}^{ + }}^{\frac{3}{2}}\left( {D + 3}\right) y\left( t\right) = {t}^{2} + \cos t + 2 + \frac{3}{100}{\tan }^{-1}y\left( t\right) ,\;t \in \left\lbrack {0,1}\right\rbrack \smallsetminus \left\{ \frac{1}{3}\right\} , \\ {... | Here \( f\left( {t, y}\right) = {t}^{2} + \cos t + 2 + \frac{3}{100}{\tan }^{-1}y,\delta = \frac{3}{2}, k = 1, m = 1,\lambda = 3 \), and\n\n\[ \left| {f\left( {t, y}\right) - f\left( {t, v}\right) }\right| \leq \frac{3}{100}\left| {{\tan }^{-1}y - {\tan }^{-1}v}\right| \leq \frac{3}{100}\left| {y - v}\right| ,\]\n\n\[ ... | Yes |
Theorem 2.1 Let \( {X}_{1},{X}_{2},\cdots ,{X}_{n} \), be the iid random vectors in \( {\mathbb{R}}^{p} \) with \( \mathrm{E}\left( X\right) = \mu \) , \( \mathrm{E}\left( {\begin{Vmatrix}{X}_{1}\end{Vmatrix}}^{2}\right) < \infty \) and \( \operatorname{Var}\left( {X}_{1}\right) = \sum \) of rank \( q > 0 \) . Then at ... | Proof of Theorem 2.1 It suffices for us to consider the case when \( \mathrm{E}{X}_{1}^{2} < \infty \) . We may proceed by the method of Lagrange multipliers. Write\n\n\[ \nG = \mathop{\sum }\limits_{{i = 1}}^{{N\left( t\right) }}\log \left( {N\left( t\right) {p}_{i}}\right) - N\left( t\right) \mathop{\sum }\limits_{{i... | Yes |
Example 1.1 Let\n\[ w\left( z\right) = \frac{\cos {\pi z}}{\sin {\pi z}} \]\n\nthen\n\n\[ w\left( z\right) w\left( {z + \frac{1}{2}}\right) = - 1 \] | The example implies that the poles of \( w\left( z\right) \) change into the zeros of \( w\left( z\right) \) in the process of parallel transformation in \( z \) . So \( w\left( {z + c}\right) w\left( z\right) \) has neither zeros nor poles.\n\nThe above result is impossible in complex differential theory, which also s... | No |
Example 1.2 Consider the following systems of difference equations\n\n\[ \n\\left\\{ \\begin{array}{l} {\\left( {w}_{2}\\left( z + 1\\right) \\right) }^{4} = \\tan \\left( {\\pi z}\\right) {w}_{1}^{3} \\\\ {\\left( {w}_{1}\\left( z + 2\\right) \\right) }^{4} = \\tan \\left( {\\pi z}\\right) {w}_{2}^{3} \\end{array}\\ri... | Obviously, \( \\left( {{w}_{1},{w}_{2}}\\right) = \\left( {\\tan \\left( {\\pi z}\\right) ,\\tan \\left( {\\pi z}\\right) }\\right) \) contains two non-admissible components, and \( p = 3, q = 3,{m}_{1} = 4,{m}_{2} = 4, t = 0, s = 0 \), which satisfies \( p < {m}_{2}, q < {m}_{1}, t < q - 2, s < p - 2 \) and \( {\\bar{... | Yes |
Corollary 1.2 Let \( \\left( {{w}_{1},{w}_{2}}\\right) \) be a finite-order meromorphic solution of Eq.(1.4). If \( p < {m}_{2}, q < {m}_{1},0 \\leq t < p - 1,0 \\leq s < q - 1 \), and \( \\left| {c}_{i}\\right| ,\\left| {d}_{l}\\right| \\notin {S}^{ * } \) | \[ \\bar{N}\\left( {r,\\frac{1}{{w}_{i}}}\\right) \\leq \\left( {\\delta + o\\left( 1\\right) }\\right) T\\left( r\\right) \\;\\left( {r \\in I, i = 1,2}\\right) ,\\] where \( 0 \\leq \\delta < 1, T\\left( r\\right) = \\min \\left\{ {T\\left( {r,{w}_{1}}\\right), T\\left( {r,{w}_{2}}\\right) }\\right\} \), and \( I \) ... | Yes |
Lemma 2.1 \( {}^{\left\lbrack 6\right\rbrack } \) Let \( {g}_{0}\left( z\right) \) and \( {g}_{1}\left( z\right) \) be meromorphic functions in \( \left| z\right| < \infty \) and linearly independent over \( \mathbb{C} \), and put \( {g}_{0}\left( z\right) + {g}_{1}\left( z\right) = \Phi \), then we have\n\n\[ T\left( ... | where \( {S}_{1}\left( r\right) = O\left( 1\right) \), when \( {g}_{1},{g}_{0} \) are rational; \( {S}_{1}\left( r\right) = O\left( {\log r}\right) \), when \( {g}_{1},{g}_{0} \) are of finite order; \( {S}_{1}\left( r\right) = O\left( {\log T\left( {r,{g}_{0}}\right) + \log T\left( {r,{g}_{1}}\right) }\right) + O\left... | Yes |
Lemma 2.4 Let \( \left( {{w}_{1},{w}_{2}}\right) \) be a finite-order meromorphic solution of Eq.(1.4). Putting\n\n\[ \n{A}_{1} = - {a}_{p}\left( z\right) {w}_{1}^{p},{B}_{1} = {\left( {\Omega }_{1}\left( z,{w}_{2}\right) \right) }^{{m}_{1}},{\Phi }_{1} = \mathop{\sum }\limits_{{i = 0}}^{t}{a}_{i}{w}_{1}^{i}, \n\]\n\n\... | Proof We firstly prove that \( {A}_{1},{B}_{1} \) are linearly independent. Suppose \( {A}_{1} \) and \( {B}_{1} \) be linearly dependent, there exist constants \( a \) and \( b \) such that\n\n\[ \na{A}_{1} + b{B}_{1} = 0,\left| a\right| + \left| b\right| \neq 0. \n\]\n\nBy the first equation of Eq.(1.4), we have\n\n\... | Yes |
Lemma 2.5 Let \( \\left( {{w}_{1},{w}_{2}}\\right) \) be a finite-order meromorphic admissible solution of Eq.(1.4). If \( p < {m}_{2}, q < {m}_{1} \) and \( {\\bar{N}}_{i}^{ * }\\left( r\\right) = S\\left( r\\right) \\left( {i = 1,2}\\right) \), then\n\n\[ N\\left( {r,{w}_{1}}\\right) = S\\left( r\\right) ,\\;N\\left(... | Proof Let \( \\left( {{w}_{1},{w}_{2}}\\right) \) be a meromorphic admissible solution of finite order of Eq.(1.4). By Lemma 2.2, we get\n\n\[ {m}_{1}N\\left( {r,{\\Omega }_{1}\\left( {z,{w}_{2}}\\right) }\\right) \\leq {pN}\\left( {r,{w}_{1}}\\right) + \\mathop{\\sum }\\limits_{{i = 0}}^{t}N\\left( {r,{a}_{i}}\\right)... | Yes |
Lemma 2.6 Let \( \left( {{w}_{1},{w}_{2}}\right) \) be of finite-order meromorphic admissible solutions of Eq.(1.4). If \( p < {m}_{2}, q < {m}_{1} \) and \( \left| {c}_{i}\right| ,\left| {d}_{l}\right| \notin {S}^{ * } \), then \[ N\left( {r,{w}_{1}}\right) = S\left( r\right) ,\;N\left( {r,{w}_{2}}\right) = S\left( r\... | In fact, when \( \left| {c}_{i}\right| ,\left| {d}_{l}\right| \notin {S}^{ * } \), we have \( N\left( {r,{w}_{2}}\right) \leq N\left( {r,{\Omega }_{1}\left( {z,{w}_{2}}\right) }\right) \) . Therefore Lemma 2.6 holds. | No |
Lemma 2.7 Let \( p, q < \min \left\{ {{m}_{1},{m}_{2}}\right\} \) and\n\n\[ \n{\bar{N}}_{1}^{ * }\left( r\right) + {\bar{N}}_{2}^{ * }\left( r\right) + \bar{N}\left( {r,\frac{1}{{w}_{1} + c\left( z\right) }}\right) + \bar{N}\left( {r,\frac{1}{{w}_{2} + d\left( z\right) }}\right) = S\left( r\right) .\n\]\n\nIf Eq.(1.4) ... | Proof Assume that Eq.(1.4) does not have the forms of the systems of equations (2.4), (2.5) or (2.6), and admits an admissible solution \( \left( {{w}_{1},{w}_{2}}\right) \) . We rewrite equation (1.4) as\n\n\[ \n\left\{ \begin{array}{ll} {\left( {\Omega }_{1}\left( z,{w}_{2}\right) \right) }^{{m}_{1}} = {a}_{p}{\left(... | Yes |
Lemma 3.3 Let \( f \in {C}^{\alpha }\left\lbrack {a, b}\right\rbrack, g \in {C}^{\beta }\left\lbrack {a, b}\right\rbrack \) with \( \alpha + \beta > 1,{f}_{m},{g}_{m} \in {C}^{1}\left\lbrack {a, b}\right\rbrack, m \geq 1 \) , and \( {\begin{Vmatrix}{f}_{m} - f\end{Vmatrix}}_{{C}^{\alpha }\left\lbrack {a, b}\right\rbrac... | \[ {\int }_{a}^{b}f\left( t\right) \mathrm{d}g\left( t\right) = \mathop{\lim }\limits_{{m \rightarrow \infty }}{\int }_{a}^{b}{f}_{m}\left( t\right) {g}_{m}^{\prime }\left( t\right) \mathrm{d}t \] | Yes |
Theorem 3.1 Let certain \( {\Omega }^{\prime } \subset \Omega \) such that \( P\left( {\Omega }^{\prime }\right) = 1 \) where \( \left( {\Omega ,\mathfrak{F}, P}\right) \) is the complete probability space, and for any \( \omega \in {\Omega }^{\prime } \) the function \( F\left( {t, u,\omega }\right) \) satisfy the con... | Proof For simplicity, we fix \( \omega \in {\Omega }^{\prime } \) and omit \( \omega \) throughout the proof. According to Definition 3.3 and condition 1), the integral \( {\int }_{c}^{d}F\left( {t, u,\omega }\right) \mathrm{d}Z\left( u\right) \) exists, then the iterated integral \( {J}_{1} \) exists according to cond... | Yes |
Lemma 2.10 \( {}^{\left\lbrack 6\right\rbrack } \) Let \( p\left( \cdot \right) \in {\mathcal{P}}^{0}\left( {\mathbb{R}}^{n}\right) \cap {LH}\left( {\mathbb{R}}^{n}\right) \). Then for all \( f \in {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\right) \), \[ \parallel f{\parallel }_{{H}^{p\left( \cdot \right) }} \sim \... | Moreover, suppose that \( 0 < {p}_{ - } \leq {p}_{ + } < q \leq \infty \). If \( q \gg 1 \) and \( p\left( \cdot \right) \in {LH}\left( {\mathbb{R}}^{n}\right) \), then for all \( f \in {\mathcal{S}}^{\prime }\left( {\mathbb{R}}^{n}\right) \)\[ \parallel f{\parallel }_{{H}^{p\left( \cdot \right) }} \sim \parallel f{\pa... | No |
Lemma 3.1 \( {}^{\left\lbrack 8,{14}\right\rbrack } \) Let \( \alpha \left( \cdot \right) \in \mathcal{P}\left( {\mathbb{R}}^{n}\right) \) . If \( \alpha \left( \cdot \right) \) is log-Hölder continuous at the origin, then | \[ {C}^{-1}{\left| x\right| }^{\alpha \left( 0\right) } \leq {\left| x\right| }^{\alpha \left( x\right) } \leq C{\left| x\right| }^{\alpha \left( 0\right) },\;\left| x\right| < 1. \] If \( \alpha \left( \cdot \right) \) is log-Hölder continuous at the infinity, then \[ {C}^{-1}{\left| x\right| }^{\alpha \left( \infty \... | Yes |
Lemma 3.2 Let \( \alpha \left( \cdot \right) \in \mathcal{P}\left( {\mathbb{R}}^{n}\right) ,1 < \alpha \left( \cdot \right) < n \) . Set \( \Omega \left( {x, z}\right) \in {L}^{\infty }\left( {\mathbb{R}}^{n}\right) \times {L}^{r}\left( {S}^{n - 1}\right), r > 1 \) , satisfies the \( {L}^{r} \) -Dini condition. Suppose... | Proof The proof follows the idea of [3]. We just prove the case for \( \frac{1}{2} < R \leq 1 \) . And the others are similar but easier. Since \( \left| y\right| < {\gamma R},\gamma \in \left( {0,\frac{1}{2}}\right) \) and \( R < \left| x\right| < {2R} \), we can easily get that \( \left| {x - y}\right| \sim \left| x\... | No |
Lemma 3.3 \( {}^{\left\lbrack {15} - {16}\right\rbrack } \) Let \( \alpha \left( \cdot \right) \in \mathcal{P}\left( {\mathbb{R}}^{n}\right) ,\alpha \left( \cdot \right) \in {LH}\left( {\mathbb{R}}^{n}\right) \) . Suppose \( \Omega \in {L}^{\infty }\left( {\mathbb{R}}^{n}\right) \times {L}^{r}\left( {S}^{n - 1}\right) ... | \[ {\begin{Vmatrix}{T}_{\Omega ,\alpha \left( \cdot \right) }f\end{Vmatrix}}_{{L}^{q}} \leq C\parallel f{\parallel }_{{L}^{p}} \] | Yes |
Lemma 3.1 There exists a positive constant \( C \) such that\n\n\[ \left| {\varphi \left( t\right) }\right| \leq C\mathcal{E}\left( t\right) ,\forall t \geq 0. \] | Proof With the help of the Cauchy inequality and the Poincaré inequality, we get\n\n\[ \left| {\varphi \left( t\right) }\right| \leq \frac{1}{2}{\int }_{\Omega }{\left| u\right| }^{2}\mathrm{\;d}x + \frac{1}{2}{\int }_{\Omega }{\left| {u}_{t}\right| }^{2}\mathrm{\;d}x \leq \frac{{C}_{p}}{2}{\int }_{\Omega }{\left| \nab... | Yes |
Lemma 3.3 For any \( t \geq 0 \), and for any \( {\varepsilon }_{3},{\varepsilon }_{4},{\delta }_{1},{\delta }_{2},{\delta }_{3} > 0 \), we have\n\n\[{\psi }^{\prime }\left( t\right) \leq \left( {\beta {c}^{2}{\varepsilon }_{3} + \left( {1 - \ell }\right) {\delta }_{1} + {\delta }_{3}}\right) {\int }_{\Omega }{\left| u... | Proof Differentiating \( \psi \left( t\right) \) with respect to \( t \), we have\n\n\[ \frac{\mathrm{d}}{\mathrm{d}t}\psi \left( t\right) = {\int }_{\Omega }\left( {{u}_{t} - \beta {u}_{tt}}\right) {\int }_{0}^{t}g\left( {t - s}\right) \left( {u\left( t\right) - u\left( s\right) }\right) \mathrm{d}s\mathrm{\;d}x + {\i... | Yes |
For the fully observable M/M/1 constant retrial queue with the \( N \) -policy, the expected sojourn times of a tagged customer when he is at the \( j \) th position in the retrial orbit and the server’s state is \( i\left( {i = 0,1,2}\right) \), are respectively given by\n\n\[ T\left( {0, j}\right) = \frac{N - j}{\lam... | Proof First, by analysis we can derive the following equations:\n\n\[ T\left( {1,0}\right) = \frac{1}{\mu } \]\n\n(3.4)\n\n\[ T\left( {1, j}\right) = \frac{1}{\lambda + \mu } + \frac{\lambda }{\lambda + \mu }T\left( {1, j}\right) + \frac{\mu }{\lambda + \mu }T\left( {2, j}\right) ,\;j \geq 1, \]\n\n(3.5)\n\n\[ T\left( ... | Yes |
For the fully observable M/M/1 constant retrial queue with the \( N \) -policy, the state space is \( {\Omega }_{0b}^{1} = \{ \left( {0, n}\right) : 0 \leq n \leq N - 1\} \cup \{ \left( {1, n}\right) : 0 \leq n \leq n\left( 1\right) + 1\} \cup \{ \left( {2, n}\right) : 1 \leq \) \( n \leq n\left( 1\right) + 1\} \), whe... | Proof The balance equations for the stationary distribution are given as follows:\n\n\[ {\lambda p}\left( {0,0}\right) = {\mu p}\left( {1,0}\right) \]\n\n\[ {\lambda p}\left( {0, n}\right) = {\lambda p}\left( {0, n - 1}\right) ,\;1 \leq n \leq N - 1, \]\n\n\[ \left( {\lambda + \mu }\right) p\left( {1,0}\right) = {\thet... | Yes |
Theorem 4.1 For the fully observable M/M/1 constant retrial queue with the \( N \) -policy, the state space is \( {\Omega }_{0b}^{2} = \{ \left( {0, n}\right) : 0 \leq n \leq N - 1\} \cup \{ \left( {1, n}\right) : 0 \leq n \leq N\} \cup \{ \left( {2, n}\right) : 1 \leq n \leq N\} \) , when \( n\left( 1\right) = N - 1 \... | Proof The balance equations for the stationary distribution are given as follows:\n\n\[ {\lambda p}\left( {0,0}\right) = {\mu p}\left( {1,0}\right) \]\n\n(4.4)\n\n\[ {\lambda p}\left( {0, n}\right) = {\lambda p}\left( {0, n - 1}\right) ,\;1 \leq n \leq N - 1, \]\n\n(4.5)\n\n\[ \left( {\lambda + \mu }\right) p\left( {1,... | Yes |
For the fully observable M/M/1 constant retrial queue with the \( N \) -policy, the state space is \( {\Omega }_{0b}^{3} = \{ \left( {0, n}\right) : 0 \leq n \leq N - 1\} \cup \{ \left( {1, n}\right) : 0 \leq n \leq N\} \cup \{ \left( {2, n}\right) : 1 \leq n \leq N\} \) , when \( n\left( 1\right) \leq N - 2 \), the st... | Proof The balance equations for the stationary distribution are given as follows:\n\n\[ {\lambda p}\left( {0,0}\right) = {\mu p}\left( {1,0}\right) \]\n\n\[ {\lambda p}\left( {0, n}\right) = {\lambda p}\left( {0, n - 1}\right) ,\;1 \leq n \leq N - 1, \]\n\n\[ \left( {\lambda + \mu }\right) p\left( {1,0}\right) = {\thet... | Yes |
Lemma 2.2 \( {\sigma }_{\pi } \) is an automorphism of \( \mathcal{O}\left( {{S}_{2}, q}\right) \) . | Proof Each vertex \( \left\lbrack \alpha \right\rbrack \) is written in the standard form. Obviously, \( {\sigma }_{\pi } \) is a bijective mapping on \( V\left( {\mathcal{O}\left( {{S}_{2}, q}\right) }\right) \) . Suppose that \( \left\lbrack \alpha \right\rbrack = \left\lbrack {{a}_{1},{a}_{2}}\right\rbrack \nsim \le... | Yes |
Lemma 2.3 \( {\sigma }_{\pi, y} \) is an automorphism of \( \mathcal{O}\left( {{S}_{n}, q}\right) \) . | Proof Obviously, \( {\sigma }_{\pi, y} \) is a bijective mapping on \( V\left( {\mathcal{O}\left( {{S}_{n}, q}\right) }\right) \) . The following derivation shows that \( {\sigma }_{\pi, y} \) preserves adjacency relation of vertices in both directions.\n\n\[ \left\lbrack {{a}_{1},{a}_{2},\ldots ,{a}_{n}}\right\rbrack ... | Yes |
Lemma 2.4 \( {\sigma }_{\omega } \) is an automorphism of \( \mathcal{O}\left( {{S}_{n}, q}\right) \) . | Proof Obviously, \( {\sigma }_{\omega } \) is a bijective mapping on \( V\left( {\mathcal{O}\left( {{S}_{n}, q}\right) }\right) \) . The procedure\n\n\[ \left\lbrack {{a}_{1},{a}_{2},\ldots ,{a}_{n}}\right\rbrack \nsim \left\lbrack {{b}_{1},{b}_{2},\ldots ,{b}_{n}}\right\rbrack \]\n\n\[ \Leftrightarrow \left( {\mathop{... | Yes |
Lemma 2.5 (i) \( \left\{ {{\beta }_{1},{\beta }_{2},\ldots ,{\beta }_{n}}\right\} \) forms a non-isotropic orthogonal basis of \( {F}_{q}^{n} \) ; | Proof (i) was proved just before this lemma. | No |
Lemma 2.6 Let \( \sigma, a, b,{\pi }_{i} \) be defined as in Lemma 2.3. If \( \sigma \left( \left\lbrack {\alpha }_{i}\right\rbrack \right) = \left\lbrack {\alpha }_{i}\right\rbrack \) for all values of \( i \), then the following assertions hold.\n\n(i) \( {\pi }_{j}{\left( 1\right) }^{2} = 1 \) for \( j = 2,3,\ldots,... | Proof By (iv) of Lemma 2.3, assertion (i) of this lemma is obvious.\n\nFor \( x \in {F}_{q} \) and \( 2 \leq j \leq n - 1 \), applying \( \sigma \) to \( \left\lbrack {{\alpha }_{1} + x{\alpha }_{2} + x{\alpha }_{j}}\right\rbrack \nsim \left\lbrack {{\alpha }_{2} - {\alpha }_{j}}\right\rbrack \), we have \( \left\lbrac... | Yes |
Lemma 3.1 Let \( {S}_{n} = \operatorname{diag}\left( {{I}_{n - 1},\theta }\right) \) with \( \theta = 1 \) or \( z \) . If \( n \) is odd, then for each \( K \in {\mathrm{{GO}}}_{n}\left( {F}_{q}\right) \) with \( K{S}_{n}{K}^{\mathrm{T}} = k{S}_{n}, k \) must be a square element. If \( n \) is even, then there exists ... | Proof Let \( n \) be odd, and suppose \( K \in {\mathrm{{GO}}}_{n}\left( {F}_{q}\right) \) such that \( K{S}_{n}{K}^{\mathrm{T}} = k{S}_{n} \) . Then \( \theta {\left| K\right| }^{2} = {k}^{n}\theta \), showing that \( {k}^{n} = {\left| K\right| }^{2} \), and \( k \) is a square element. Now, assume that \( n \) is eve... | Yes |
Lemma 2.2 \( {}^{\left\lbrack 7,{12}\right\rbrack } \) Let \( X \) be a Banach space, then\n\n1) \( X \) is non- \( {l}_{n}^{\left( 1\right) } \) if and only if for all \( {x}^{\left( 1\right) },{x}^{\left( 2\right) },\cdots ,{x}^{\left( n\right) } \in X \smallsetminus \{ 0\} \), the inequality\n\n\[ \n\frac{\begin{Vma... | From the proof in [12] we know that \( \varepsilon \left( {x}^{\left( 1\right) }\right) \) in Lemma 2.2 2) can be chosen as \( \varepsilon \) in definition. | No |
Assume that \( \mathcal{P} \) is a family of probability measures defined on \( \left( {\Omega ,\mathcal{F}}\right) \). For any random variable \( \xi \), we indicate the upper expectation by \( \widehat{\mathbb{E}}\left( \xi \right) = \mathop{\sup }\limits_{{Q \in \mathcal{P}}}{\mathbf{E}}_{Q}\left( \xi \right) \). Th... | \[ \widehat{\mathbb{E}}\left( {{\varphi }_{1}\left( \mathbf{X}\right) {\varphi }_{2}\left( \mathbf{Y}\right) }\right) = \mathop{\sup }\limits_{{Q \in \mathcal{P}}}{\mathbf{E}}_{Q}\left( {{\varphi }_{1}\left( \mathbf{X}\right) {\varphi }_{2}\left( \mathbf{Y}\right) }\right) = \mathop{\sup }\limits_{{Q \in \mathcal{P}}}{... | Yes |
Lemma 3.2 (Theorem 3.1 in [17]) \( \\left\\{ {{X}_{n};n \\geq 1}\\right\\} \) is a sequence of upper extended negatively dependent random variables in \( \\left( {\\Omega ,\\mathcal{H},\\widehat{\\mathbb{E}}}\\right) ,\\widehat{\\mathbb{E}}{X}_{k} \\leq 0, k = 1,\\ldots, n \), and there exists a constant \( K > 0 \) . ... | \[ \mathbb{V}\\left( {{S}_{n} \\geq x}\\right) \\leq \\mathbb{V}\\left( {\\mathop{\\max }\\limits_{{k \\leq n}}{X}_{k} \\geq y}\\right) + K\\exp \\left\\{ {-\\frac{{x}^{2}}{2\\left( {{xy} + {B}_{n}}\\right) }\\left( {1 + \\frac{2}{3}\\ln \\left( {1 + \\frac{xy}{{B}_{n}}}\\right) }\\right) }\\right\\} ,\] where \( {B}_{... | Yes |
Theorem 3.1 Suppose that \( 0 < p \leq 2 \), and \( \mathbb{V} \) is countably sub-additive. Let \( \left\{ {{X}_{n};n \geq }\right. \) 1\} be a sequence of upper END random variables under sub-linear expectations. There exists a r.v. \( X \) and a constant \( c > 0 \) satisfying\n\n\[ \widehat{\mathbb{E}}\left( {h\lef... | Proof of Theorem 3.1 Without loss of generality, for \( p \geq 1 \), we can suppose that \( \widehat{\mathbb{E}}{X}_{k} = 0 \) . Obviously, \( {C}_{\mathbb{V}}\left( {\left| X\right| }^{p}\right) < \infty \) is equivalent to \( {C}_{\mathbb{V}}\left( {{\left| X\right| }^{p}/{c}^{p}}\right) < \infty \) for any \( c > 0 ... | Yes |
Lemma 3.1 Assume \( 0 < {\beta }_{k} < 1 \) . Then the bifurcation points are finite, that is, there is a non-negative integer \( {\mathbb{N}}_{1} \leq \frac{1}{\sqrt{2\left( {{d}_{11} + {d}_{22}}\right) }} \) and \( {\mathbb{N}}_{1} \in {\mathbb{N}}_{0} \), such that \( {\beta }_{k} \) are (resp. not) bifurcating poin... | Proof When \( 0 \leq \beta \leq 1 \), Max \( {a}_{11}\left( \beta \right) = {a}_{11}\left( 0\right) \), that is, \( {a}_{11}\left( 0\right) < \left( {{d}_{11} + {d}_{22}}\right) {k}^{2} \) when \( k \) is big enough. So there is \( {T}_{k}\left( \beta \right) < 0 \) . Then (2.6) does not exist any purely imaginary root... | Yes |
Lemma 3.2 Assume \( 0 < {\beta }_{k} < 1 \) . Then for any \( 0 \leq k \leq {\mathbb{N}}_{1} \), there is \( 0 < {\beta }_{{\mathbb{N}}_{1}} < \cdots < \) \( {\beta }_{k} < \cdots {\beta }_{0} < 1 \) | Proof From (3.3), we can easily see that the value of \( \beta \) increases with the decrease of \( k \) . This competes the proof. | No |
Lemma 3.3 Assume \( \left( {\mathrm{H}}_{1}\right) \) and \( \left( {\mathrm{H}}_{2}\right) \) hold. Then for any \( 0 < \beta < 1 \), we have \( \operatorname{Re}\left( {{\lambda }^{\prime }\left( \beta \right) }\right) < \) 0. | Proof Suppose that the root of (2.7) has the form \( \lambda \left( \beta \right) = \alpha \left( \beta \right) + {ib}\left( \beta \right) \), where \( \alpha \left( \beta \right) \) , \( b\left( \beta \right) \in \mathbb{R} \) . Since the real part of eigenvalue \( \lambda \) is \( - \frac{{T}_{k}}{2} = - \frac{\left(... | Yes |
Lemma 4.1 When \( \frac{\sqrt{3}}{3} < \beta < 1 \), for all \( k \in {\mathbb{N}}_{0} \), and \( {\mathbb{N}}_{2} \) and \( {\omega }_{k} \) are defined by (4.6) and (4.8), respectively. (2.5) has a pair of purely imaginary roots \( \pm \mathrm{i}{\omega }_{k} \) for each \( k \in \left\{ {0,1\cdots {\mathbb{N}}_{2}}\... | From (4.2), for \( k \in \left\{ {0,1,\cdots {\mathbb{N}}_{2}}\right\} \), we can obtain\n\n\[ \n{\tau }_{kj} = {\tau }_{k0} + \frac{2\pi j}{{\omega }_{k}}\n\]\n\n(4.9)\n\nand\n\n\[ \n{\tau }_{k0} = \frac{1}{{\omega }_{k}}\operatorname{ArcCos}\frac{{\omega }_{k}^{2} - {D}_{k} + {a}_{21}{d}_{12}{k}^{2} - {a}_{12}{a}_{21... | Yes |
Lemma 4.2 For \( k \in \left\{ {0,1,2\cdots {\mathbb{N}}_{0}}\right\} \) and \( j \in {\mathbb{N}}_{0},{\left. \frac{\mathrm{d}\operatorname{Re}\left( \lambda \right) }{\mathrm{d}\tau }\right| }_{\tau = {\tau }_{kj}} > 0 \) . | Proof Differentiating two sides of (2.6) on \( \tau \), we get,\n\n\[ \n{\left( \frac{\mathrm{d}\lambda }{\mathrm{d}\tau }\right) }^{-1} = \frac{{2\lambda } + {T}_{k}}{\lambda \left( {{a}_{21}{d}_{12}{k}^{2} - {a}_{12}{a}_{21}}\right) {\mathrm{e}}^{-{\lambda \tau }}} - \frac{\tau }{\lambda }. \n\]\n\n(4.12)\n\nBy (2.6)... | Yes |
Theorem 4.1 Assume \( \left( {\mathrm{H}}_{1}\right) \) and \( \left( {\mathrm{H}}_{2}\right) \) hold and for each \( k \in {\mathbb{N}}_{0},{T}_{k} > 0,{D}_{k} > 0 \) are valid, \( {\tau }_{00} \) is defined by (4.9).\n\n1) When \( \tau \in \left\lbrack {0,{\tau }_{00}}\right\rbrack \), the system (1.4) has an asympto... | In Case 2, suppose \( \tau \neq 0 \) . Keep \( {d}_{11} = 1,{d}_{12} = 1,{d}_{21} = 1,{d}_{22} = 3,\gamma = 1,\beta = \frac{2}{3} \), then we get \( {\tau }_{00} \approx {0.922} \) . Take \( \tau = {0.01} < {\tau }_{00} \), and the curves in Fig.5.3 tend to be stable. Take \( \tau = 1 > {\tau }_{00} \), and the curves ... | No |
Lemma 3.2 Let \( {Q}_{h} \) be the \( {L}^{2} \) projection onto \( {V}_{h} \) and let \( {C}_{p} \) be the constant in Lemma 3.1. Then the following estimates hold:\n\n\[ \forall {v}_{h} \in {W}_{h},\;{\begin{Vmatrix}{v}_{h} - {Q}_{h}{v}_{h}\end{Vmatrix}}_{{L}^{2}\left( {\mathcal{T}}_{h}\right) } \lesssim {C}_{p}{\lef... | Proof The equation (3.1) follows from \( {\begin{Vmatrix}{v}_{h} - {Q}_{h}{v}_{h}\end{Vmatrix}}_{{L}^{2}\left( {\mathcal{T}}_{h}\right) } \leq {\begin{Vmatrix}{v}_{h} - {\mathcal{I}}_{\mathrm{{Os}}}{v}_{h}\end{Vmatrix}}_{{L}^{2}\left( {\mathcal{T}}_{h}\right) } \) and Lemma 3.1. The equation (3.2) follows from the inve... | Yes |
Lemma 3.5 Let \( {u}_{h}^{q} \in {V}_{h} \) solve (2.3). Then\n\n\[ \n{\begin{Vmatrix}\nabla {u}_{h}^{q}\end{Vmatrix}}_{{L}^{2}\left( \Omega \right) }^{2} - {k}^{2}{\begin{Vmatrix}{u}_{h}^{q}\end{Vmatrix}}_{{L}^{2}\left( \Omega \right) }^{2} \leq 2\parallel f{\parallel }_{{L}^{2}\left( \Omega \right) }{\begin{Vmatrix}{... | \[ \n\mathop{\sum }\limits_{{j = 1}}^{q}\mathop{\sum }\limits_{{e \in {\mathcal{E}}_{h}^{I}}}{\gamma }_{j, e}{\left( \frac{{h}_{e}}{{p}^{2}}\right) }^{{2j} - 1}{\begin{Vmatrix}\left\lbrack \frac{{\partial }^{j}{u}_{h}^{q}}{\partial {n}_{e}^{j}}\right\rbrack \end{Vmatrix}}_{{L}^{2}\left( e\right) }^{2} + k{\begin{Vmatri... | Yes |
Theorem 3.1 Let \( {u}_{h}^{q} \in {V}_{h} \) solve (2.3) and suppose \( {h}_{K},{h}_{e} \eqsim h,{\gamma }_{j, e} \simeq {\gamma }_{j}, j = 1,2,\cdots, q \) . Then\n\n\[ k{\begin{Vmatrix}{u}_{h}^{q}\end{Vmatrix}}_{{L}^{2}\left( \Omega \right) } + {\begin{Vmatrix}{u}_{h}^{q}\end{Vmatrix}}_{1, h, q} \lesssim {C}_{\mathr... | Proof We divide the proof into three steps.\n\nStep 1 Derivation of a representation identity for \( {\begin{Vmatrix}{u}_{h}^{q}\end{Vmatrix}}_{{L}^{2}\left( \Omega \right) } \) . Define \( {v}_{h} \) by \( {\left. {v}_{h}\right| }_{K} = \) \( \alpha \cdot {\left. \nabla {u}_{h}^{q}\right| }_{K} \) for every \( K \in {... | Yes |
Lemma 3.2 As \( \left\{ {t}_{n}\right\} \) is a bounded sequence in \( \lbrack 0,\infty ) \), we have\n\n\[ I\left( {{t}_{n}{u}_{n}}\right) \leq I\left( {u}_{n}\right) + o\left( 1\right) \;\text{ as }n \rightarrow \infty . \]\n\nMoreover, if \( {t}_{n} \rightarrow 0 \) as \( n \rightarrow \infty \), we have\n\n\[ \math... | Proof By (5), we have\n\n\[ I\left( {{t}_{n}{u}_{n}}\right) - I\left( {u}_{n}\right) = \frac{\left( {t}_{n}^{2} - 1\right) }{2}{\begin{Vmatrix}{u}_{n}\end{Vmatrix}}^{2} - \left( {{t}_{n}^{4} - 1}\right) \Psi \left( {u}_{n}\right) . \]\n\nSince \( \left\{ {u}_{n}\right\} \) is a Cerami sequence, we have\n\n\[ \Psi \left... | Yes |
Lemma 3.4 There exists \( \alpha > 0 \) such that \( \inf \left\{ {I\left( u\right) : u \in {H}^{1}\left( {\mathbb{R}}^{2}\right) : \parallel u\parallel = \beta }\right\} > 0 \) and \( \inf \left\{ {{I}^{\prime }\left( u\right) u : u \in {H}^{1}\left( {\mathbb{R}}^{2}\right) : \parallel u\parallel = \beta }\right\} > 0... | Proof By the Young inequality and the Sobolev embedding theorem, we have\n\n\[ I\left( u\right) = \frac{\parallel u{\parallel }^{2}}{2} - \Psi \left( u\right) \]\n\n\[ \geq \frac{1}{2}\parallel u{\parallel }^{2} - {C}_{0}{\left| {K}_{0, m}\right| }_{{p}^{\prime }}{\left| u\right| }_{p}^{4} \]\n\n\[ \geq \frac{1}{2}\par... | Yes |
Lemma 3.5 Letting \( u \in {H}^{1}\left( {\mathbb{R}}^{2}\right) \smallsetminus \{ 0\} \), we obtain that the function \( {\varphi }_{u} : \mathbb{R} \rightarrow \mathbb{R} \) , \( {\varphi }_{u}\left( t\right) = I\left( {tu}\right) \) is even, and there exists a unique \( {t}_{u} \in \left( {0,\infty }\right) \) such ... | Proof This conclusion follows from the fact\n\n\[ \frac{{\varphi }_{u}^{\prime }\left( t\right) }{t} = \parallel u{\parallel }^{2} - 4{t}^{2}\Psi \left( u\right) \text{ as }t > 0. \] | No |
Lemma 3.1 If (1.3) and\n\n\[ \frac{\left( {{\mu \eta } - {\delta }_{3}g}\right) \left( {1 - \theta }\right) }{h{\delta }_{3}}{\bar{x}}_{m} > \frac{k}{h} \]\n\n(3.1)\n\nhold, then there exists a unique positive equilibrium \( \bar{E}\left( {{\bar{x}}_{i},{\bar{x}}_{m},\bar{y}}\right) \), where\n\n\[ {\bar{x}}_{i} = \fra... | Proof Lemma 3.1 can be proved by simple direct calculation, thus it is omitted here. | No |
Lemma 2.4 (i) The mass matrix \( {\left( \left( {l}_{k}^{-2,\beta },{l}_{m}^{-2,\beta }\right) \right) }_{k, m \geq 0} \) is symmetric pentadiagonal, i.e., | \[ {\left( {l}_{k}^{-2,\beta },{l}_{m}^{-2,\beta }\right) }_{k, m \geq 2} = \left\{ \begin{array}{ll} 6{\beta }^{-1}, & k = m, \\ - 4{\beta }^{-1}, & k = m \pm 1, \\ {\beta }^{-1}, & k = m \pm 2, \\ 0, & \text{ otherwise,} \end{array}\right. \] (2.16) and \[ \left( {{l}_{0}^{-2,\beta },{l}_{0}^{-2,\beta }}\right) = {\b... | Yes |
Lemma 2.5 According to (2.16) and (2.19), for any \( \gamma > 0 \), we have\n\n\[ \n{\left( \left( {\partial }_{x}{l}_{k}^{-2,\beta },{\partial }_{x}{l}_{m}^{-2,\beta }\right) + \gamma \left( {l}_{k}^{-2,\beta },{l}_{m}^{-2,\beta }\right) \right) }_{k, m \geq 2} = \left\{ \begin{array}{ll} {6\gamma }{\beta }^{-1} + \fr... | (2.22) | No |
Theorem 3.1 Assume that \( {S}_{k}^{\beta }\left( x\right) \in {X}_{N}^{0,\beta },2 \leq k \leq N \), whose leading coefficient is the same as the Laguerre function \( {l}_{k}^{-2,\beta }\left( x\right) \), satisfies the orthogonality:\n\n\[ {A}_{\gamma }\left( {{S}_{k}^{\beta },{S}_{m}^{\beta }}\right) = {\xi }_{k}{\d... | Proof Set \( {S}_{k}^{\beta }\left( x\right) = {l}_{k}^{-2,\beta }\left( x\right) - {b}_{k - 1}{S}_{k - 1}^{\beta }\left( x\right) - {c}_{k - 2}{S}_{k - 2}^{\beta }\left( x\right) + \mathop{\sum }\limits_{{m = 2}}^{{k - 3}}{d}_{m}{S}_{m}^{\beta }\left( x\right) \) . We first check that \( {d}_{m} = 0 \) for \( 2 \leq m... | Yes |
For all \( \varphi \in C\left( {\mathbb{R},\mathbb{R}}\right) ,\varphi \) non-negative, then \( {T\varphi } \) is bounded and differentiable, with\n\n\[ 0 \leq {T\varphi }\left( t\right) \leq K,\;\left| {{\left( T\varphi \right) }^{\prime }\left( t\right) }\right| \leq {hK}, t \in \mathbb{R}. \]\n\nMoreover, if \( \var... | Proof Recall that \( {h\varphi }\left( s\right) + g\left( {\varphi \left( \cdot \right) }\right) \) is non-decreasing function. Consider any non-negative \( \varphi \in {\left\lbrack 0, K\right\rbrack }_{C} \) . Then, for \( t \in \mathbb{R} \) ,\n\n\[ 0 \leq {T\varphi }\left( t\right) \leq \left\lbrack {{hK} + g\left(... | Yes |
(i) \( T{\phi }_{ * }\left( t\right) \geq {\phi }_{ * }\left( t\right) \), for all \( t \in \mathbb{R} \) ; | Proof Define \( {\phi }_{1} \mathrel{\text{:=}} T{\phi }_{ * } \) . We have\n\n\[ \n{\phi }_{1}^{\prime } + h{\phi }_{1} - \left\lbrack {h{\phi }_{ * } + g\left( {{\phi }_{ * }\left( {\cdot - \tau }\right) }\right) }\right\rbrack = 0. \n\]\n\n(2.11)\n\nLet \( w\left( t\right) = {\phi }_{1}\left( t\right) - {\phi }_{ * ... | Yes |
Lemma 2.4 The set \( S \) is \( \parallel \cdot {\parallel }_{\rho } \) -closed, convex and non-empty. | Proof From Lemma 2.1, we have \( {\phi }_{ * }\left( t\right) \leq {\mathrm{e}}^{{\lambda }_{1}}t \) and \( {\phi }_{ * }\left( t\right) \leq {u}_{2}^{ * }\left( t\right), t \in \mathbb{R} \), thus \( {\phi }_{ * }\left( t\right) \in S \) . It is clear that \( S \) is convex and \( \parallel \varphi {\parallel }_{\rho ... | Yes |
Lemma 2.6 For \( S \) defined in (2.12), the set \( T\left( S\right) \) is relatively compact in \( (C\left( {\mathbb{R},\mathbb{R}}\right) ,\parallel \) . \( {\left. \right| }_{\rho } \) ). | Proof For any compact interval \( I \in S \) and \( {\varphi }_{n} \in I \), let \( {\psi }_{n} = T{\varphi }_{n}, n \in \mathbb{N} \) . From Lemma 2.2, \( \left( {\psi }_{n}\right) \) is uniformly bounded on \( \mathbb{R} \) and equicontinuous. By Ascoli-Arzelà theorem, there is a subsequence of \( \left( {\psi }_{n}\... | Yes |
Theorem 2.1 Assume that conditions \( \left( {\mathrm{H}}_{1}\right) - \left( {\mathrm{H}}_{4}\right) \) are satisfied. Then, there is a positive solution \( u\left( t\right) \) of (2.2), defined on \( \mathbb{R} \) and satisfying \( u\left( {-\infty }\right) = 0 \) and \( u\left( t\right) = O\left( {\mathrm{e}}^{{\lam... | Proof Consider \( S \) as in (2.12). From Lemmas 2.1-2.3, \( T\left( S\right) \subset S \) . From Lemma 2.4 and Lemma 2.5, \( T : S \rightarrow S \) is \( \parallel \cdot {\parallel }_{\rho } \) completely continuous. Lemma 2.6 allows us to use the Schauder’s fixed-point theorem to conclude that there is \( u \in S \) ... | Yes |
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