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Lemma 2.2 Let \( T \) be a quasi-contraction of Definition 1.3, then \( \left\{ {x}_{n}\right\} \) is a Cauchy sequence. | Proof For \( n > m > 1 \), denote \( C\left( {m, n}\right) = \left\{ {d\left( {{x}_{i},{x}_{j}}\right) : m \leq i, j \leq n}\right\} \) . It follows that, for each \( u \in C\left( {m, n}\right) \), there exists \( v \in C\left( {m - 1, n}\right) \) such that \( u \preccurlyeq {kv} \) . Then by Lemma 1.3, one has\n\n\[... | Yes |
Let \( \mathrm{A} = {C}_{\mathbf{R}}^{1}\left\lbrack {0,1}\right\rbrack \) with \( \parallel x\parallel = \parallel x{\parallel }_{\infty } + \begin{Vmatrix}{x}^{\prime }\end{Vmatrix}\infty \) . Define multiplication in the usual way: \( \left( {xy}\right) \left( t\right) = x\left( t\right) y\left( t\right) \) . This m... | \[ {Tx} = u\left( t\right) x\left( t\right) + v\left( t\right) {\int }_{0}^{1}x\left( t\right) \mathrm{d}t, \] where \( u, v \in X \) . Let \( k\left( t\right) = k\left( {u\left( t\right), v\left( t\right) }\right) = \parallel u\parallel + \parallel v{\parallel }_{\infty } \) . Let \( {k}_{1} = k\left( t\right) \) and ... | Yes |
Lemma 4.1 (Local existence) For \( s \geq 1\left( {d = 1}\right) \) and \( s \geq 2\left( {d = 2,3}\right) \), there exists a \( {T}_{0} \) \( > 0 \) such that (3.1) with \( u\left( {\cdot ,0}\right), v\left( {\cdot ,0}\right) \in {H}^{s} \) has a unique solution \( w\left( {\cdot, t}\right) \) on \( \left( {0,{T}_{0}}... | \[ \parallel \mathbf{w}\left( t\right) {\parallel }_{{H}^{s}} \leq C\parallel \mathbf{w}\left( 0\right) {\parallel }_{{H}^{s}},0 < t < {T}_{0}, \] where \( C \) is a positive constant depending on \( {d}_{1},{d}_{2},\bar{U},\gamma ,\chi \) . | Yes |
Lemma 4.2 Let \( \left\lbrack {u\left( {\mathbf{x}, t}\right), v\left( {\mathbf{x}, t}\right) }\right\rbrack \) be a solution of (3.1). Then\n\n\[ \n\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\mathop{\sum }\limits_{{\left| \alpha \right| = 2}}{\int }_{{T}^{d}}\left\{ {{\left| {D}^{\alpha }u\right| }^{2} + \frac{{\left\lb... | Proof Notice that if \( \mathbf{w}\left( {\mathbf{x}, t}\right) \) is a solution of (3.1) on \( {T}^{d} \), then the even extension of \( \mathbf{w}\left( {\mathbf{x}, t}\right) \) on \( 2{T}^{d} = {\left( -\pi ,\pi \right) }^{d}\left( {d = 1,2,3}\right) \) is also the solution of (3.1) which satisfies homogeneous Neum... | Yes |
Lemma 4.3 Let \( w\left( {\mathbf{x}, t}\right) \) be a solution of (2.1) such that for \( 0 \leq t \leq T \)\n\n\[ \parallel \mathbf{w}\left( {\cdot, t}\right) {\parallel }_{{H}^{2}} + \parallel \mathbf{w}\left( {\cdot, t}\right) {\parallel }_{{H}^{2}}^{2} \leq \frac{1}{{\bar{C}}_{2}}\min \left\{ {\frac{{d}_{1}}{4},\f... | Proof By (4.9), one can obtain\n\n\[ \parallel \nabla \mathbf{w}\left( {\cdot, t}\right) {\parallel }^{2} \leq {C}_{9}^{2}\mathop{\sum }\limits_{{\left| \alpha \right| = 2}}{\begin{Vmatrix}{D}^{\alpha }\widetilde{\mathbf{w}}\left( \cdot, t\right) \end{Vmatrix}}^{2}. \]\n\n(4.17)\n\nThus\n\n\[ \parallel \mathbf{w}\left(... | Yes |
Theorem 5. 1 implies that the dynamics of a general perturbation is characterized by such linear dynamics over a long time period of \( \varepsilon {T}^{\delta } \leq t \leq {T}^{\delta } \), for any \( \varepsilon > 0 \) . It’s special that we choose a function \( {w}_{0}\left( \mathbf{x}\right) \in {H}^{2}\left( {T}^... | \[ {\begin{Vmatrix}{\mathbf{w}}_{0}\left( \mathbf{x}\right) \end{Vmatrix}}_{{H}^{2}} = {\left( 1 + {\left| {\mathbf{q}}_{0}\right| }^{2} + {\left| {\mathbf{q}}_{0}\right| }^{4}\right) }^{1/2} \equiv C\left( {\mathbf{q}}_{0}\right) . \] (5.19) If \( 0 \leq t \leq {T}^{\delta } \), it follows from Theorem 5.1 that \[ \be... | Yes |
Lemma 4 (i) Suppose that conditions (C. 1)-(C. 4) and (C. 5)-(C. 6) hold, then, for any \( \varepsilon > 0 \), it holds that\n\n\[ \n{P}_{\theta }\left( {\sqrt{\frac{{nI}\left( \theta \right) }{2\mathrm{{LL}}n}}\left( {{\bar{\theta }}_{n} - \theta }\right) \geq 1 - \varepsilon ,\text{ i. o. }}\right) = 1, \n\]\n\n\[ \n... | Proof Similar to the proof of Lemma 3, we only consider the proof of (23). From (3) and (15), we also have\n\n\[ \n{P}_{\theta }\left( {\sqrt{\frac{{nI}\left( \theta \right) }{2\mathrm{{LL}}n}}\left( {{\underline{\theta }}_{n} - \theta }\right) \geq 1 - \varepsilon ,\text{ i. o. }}\right) \n\]\n\n\[ \n\geq {P}_{\theta ... | Yes |
Example 1 Let \( \\left\\{ {{X}_{n}, n \\geq 1}\\right\\} \) be a sequence of i. i. d. random variables with the common distribution function\n\n\[ f\\left( {x;\\theta }\\right) = \\left\\{ \\begin{array}{ll} \\frac{1}{\\theta }, & x \\in \\left\\lbrack {0,\\theta }\\right\\rbrack , \\\\ 0, & \\text{ otherwise,} \\end{... | And for any \( \\varepsilon > 0,\\lambda \\left( n\\right) > \\varepsilon /\\theta \) ,\n\n\[ {P}_{\\theta }\\left( {\\lambda \\left( n\\right) \\left| {{\\widehat{\\theta }}_{n} - \\theta }\\right| \\geq \\varepsilon }\\right) = {P}_{\\theta }\\left( {{\\widehat{\\theta }}_{n} \\leq - \\frac{\\varepsilon }{\\lambda \\... | Yes |
Consider the reliability growth model which was discussed by Dubman and Sherman \( {}^{\left\lbrack 2\right\rbrack } \) . Assume random variables \( {\left\{ {Y}_{k}\right\} }_{k \geq 1} \) are the waiting times between failures \( \left( {k - 1}\right) \) and \( k \), then the \( {Y}_{k} \) ’s are independent not iden... | Let \( Y \) stand for the \( n \) -dimensional random point \( Y = \left( {{Y}_{0},{Y}_{1},\cdots ,{Y}_{n - 1}}\right) \) . It can be seen that the likelihood function of \( Y \) is denoted by\n\n\[ L\left( {Y, p,\beta }\right) = \mathop{\sum }\limits_{{k = 0}}^{{n - 1}}\left\lbrack {\log \left( {p{\beta }^{k}}\right) ... | Yes |
Lemma 2. \( {\mathbf{1}}^{\left\lbrack {22}\right\rbrack } \) Let \( X \) be a nonempty and compact subset of a Hausdorff topological vector space \( E \), and \( V \) be any open neighborhood of zero in \( E.Q : X \rightarrow {2}^{X} \) is lower semicontinuous on \( X \) . Define \( {Q}_{V} : X \rightarrow {2}^{X} \) ... | Proof For any fixed \( y \in X \) and any \( x \in {Q}_{V}^{-1}\left( y\right) \), we have \( Q\left( x\right) \cap \left( {y - V}\right) \neq \varnothing \) . Since \( Q : X \rightarrow {2}^{X} \) is lower semicontinuous on \( X \), then there exists an open neighborhood \( U \) of \( x \) in \( X \) such that \( Q\le... | Yes |
Theorem 3.1 Let \( X \) be a nonempty and compact subset of a Hausdorff topological vector space \( E \), and \( X \) has the fixed point property. Suppose that \( F : X \rightarrow {2}^{X} \) and \( H : X \rightarrow {2}^{X} \) are two set-valued mapping with the following conditions:\n\n(i) for any \( x \in X, H\left... | Proof By condition (ii), we have \( X = \mathop{\bigcup }\limits_{{y \in X}}\operatorname{int}{H}^{-1}\left( y\right) \) . Since \( X \) is nonempty and compact, there exists a finite number of \( \operatorname{int}{H}^{-1}\left( {y}_{1}\right) ,\cdots ,\operatorname{int}{H}^{-1}\left( {y}_{n}\right) \) such that \( X ... | Yes |
Theorem 3.2 Let \( X \) be a nonempty and compact subset of a locally convex topological vector space \( E \), and \( X \) has the fixed point property. Suppose that \( F : X \rightarrow {2}^{X} \) is a set-valued mapping with the following conditions:\n\n(i) \( F \) is continuous with nonempty compact values;\n\n(ii) ... | Proof For any nonempty open convex neighborhood \( V \) of zero in \( E \), define \( {F}_{V} : X \rightarrow {2}^{X} \) by \( {F}_{V}\left( x\right) = F\left( x\right) + V \) .\n\nSince \( F \) is continuous with nonempty compact values, by Lemma 2. 1, \( {F}_{V}^{-1}\left( y\right) \) is open in \( X \) for any \( y ... | Yes |
Theorem 4.1 For each \( i = 1,\cdots, n \), let \( {X}_{i} \) be a nonempty and compact subset of a Hausdorff topological vector space \( E \), and \( X \) has the fixed point property. Let\n\n\[ \nX = \mathop{\prod }\limits_{{i = 1}}^{n}{X}_{i},{X}_{-i} = \mathop{\prod }\limits_{{1 \leq j \leq n, j \neq i}}{X}_{j}.\n\... | Proof Define two set-valued mappings \( F : X \rightarrow {2}^{X} \) and \( H : X \rightarrow {2}^{X} \) by\n\n\[ \nF\left( x\right) = \mathop{\prod }\limits_{{i = 1}}^{n}{F}_{i}\left( {x}_{-i}\right), H\left( x\right) = \mathop{\prod }\limits_{{i = 1}}^{n}{H}_{i}\left( {x}_{-i}\right) ,\n\]\n\nwhere \( {F}_{i}\left( {... | Yes |
Theorem 4.3 Consider the \( n \) -person non-cooperative game \( \Gamma \left\{ {I,{X}_{i},{f}_{i}}\right\} \) satisfying the following conditions:\n\n(i) for each \( i \in I,{X}_{i} \) is a nonempty and compact subset of a Hausdorff topological vector space \( {E}_{i} \), and \( X \) has the fixed point property;\n\n(... | Proof Define the function \( \varphi : X \times X \rightarrow \mathbf{R} \) by\n\n\[ \varphi \left( {x, y}\right) = \mathop{\sum }\limits_{{i = 1}}^{n}\left\lbrack {{f}_{i}\left( {{y}_{i},{x}_{-i}}\right) - {f}_{i}\left( {{x}_{i},{x}_{-i}}\right) }\right\rbrack ,\forall y = \left( {{y}_{i},{y}_{-i}}\right), x = \left( ... | Yes |
Theorem 4.4 Consider the \( n \) -person non-cooperative game \( \Gamma \left\{ {I,{X}_{i},{f}_{i}}\right\} \) satisfying the following conditions:\n\n(i) for each \( i \in I,{X}_{i} \) is a nonempty and compact subset of a Hausdorff topological vector space \( {E}_{i} \), and \( X \) has the fixed point property;\n\n(... | Proof For any \( k = 1,2,\cdots \), and for each \( i \in I \), define \( {A}_{i}^{k} \subset X \) by\n\n\[ \n{A}_{i}^{k} = \left\{ {x \in X \mid {f}_{i}\left( {{x}_{i},{x}_{-i}}\right) > \mathop{\max }\limits_{{{u}_{i} \in {X}_{i}}}{f}_{i}\left( {{u}_{i},{x}_{-i}}\right) - \frac{1}{k}}\right\} .\n\]\n\nFor any \( k = ... | Yes |
Theorem 2. 1 Under the assumptions of Theorem 1. 1, if \( \left| u\right| \leq M \), then \( {\nabla }_{x}u \in \) \( {L}_{\text{loc }}^{r}\left( {Q}_{T}\right) \) for some \( r : p < r < \min \{ q,\gamma \} \) . Moreover, for any \( \left( {{x}_{0},{t}_{0}}\right) \in {Q}_{T} \) and \( {Q}_{R}\left( {{x}_{0},{t}_{0}}\... | Proof Introduce the Steklov averagings of \( w \in {V}_{p}\left( {Q}_{T}\right) \) , \n\n\[ \n{w}_{h}\left( {x, t}\right) = \left\{ \begin{array}{ll} \frac{1}{h}{\int }_{t}^{t + h}w\left( {x, s}\right) \mathrm{d}s, & t \in (0, T - h\rbrack , \\ 0, & t > T - h. \end{array}\right. \n\] \n\n\[ \n{w}_{h}\left( {x, t}\right... | Yes |
Lemma 3. \( {\mathbf{1}}^{\left\lbrack 4,9\right\rbrack } \) Let \( v\left( {x, t}\right) \in {V}_{p}\left( {Q}_{T}\right) \) be the weak solution of (3.1). Then \( {\nabla }_{x}v \) is locally and essentially bounded, i. e. | \[ \mathop{\sup }\limits_{{\left( {x, t}\right) \in {Q}_{R/2}}}\left| {{\nabla }_{x}v\left( {x, t}\right) }\right| \leq C{\left( \frac{1}{{R}^{n + 2}}{\int }_{{Q}_{R}}{\left| {\nabla }_{x}v\right| }^{p}\mathrm{\;d}x\mathrm{\;d}t\right) }^{1/2}\text{, for}p \geq 2\text{,} \] (3.2) with \( 0 < R < \operatorname{dist}\lef... | Yes |
Assuming \( \left( {\mathrm{H}}_{1}\right) \) and \( \left( {\mathrm{H}}_{4}\right) \) hold, and letting \( v\left( {x, t}\right) \in {V}_{p}\left( {Q}_{T}\right) \left( {2 \leq p < \infty }\right) \) be the weak solution of (3.1) and \( u\left( {x, t}\right) \in {V}_{p}\left( {Q}_{T}\right) \cap {L}^{\infty }\left( {Q... | Proof Let \( {\phi }_{h} = {v}_{h} - {u}_{h} \in {V}_{p}\left( {Q}_{R}\right) \) . By (2.4) and (3.6), it holds that\n\n\[ {\int }_{{Q}_{R}^{c}}{\partial }_{t}\left( {{v}_{h} - {u}_{h}}\right) \cdot {\phi }_{h}\mathrm{\;d}x\mathrm{\;d}t + {\int }_{{Q}_{R}^{c}}\left\langle \left\lbrack {{\left\langle {A}_{R}{\nabla }_{x... | Yes |
Lemma 2.1 Let \( \\left\\{ {U}_{m}\\right\\} \\subset {H}_{0}^{1}\\left( \\Omega \\right) \\times {H}_{0}^{1}\\left( \\Omega \\right) ,{U}_{m} \\rightharpoonup U \) in \( {H}_{0}^{1}\\left( \\Omega \\right) \\times {H}_{0}^{1}\\left( \\Omega \\right), K\\left( x\\right) \) is continuous in \( \\Omega \\), then up to a ... | Proof of Lemma 2.2 (ii) is the well known results of Brezis-Lieb Lemma in [1]. (i) can be proved easily by Vitali's Theorem. | No |
Lemma 2.2 Let \( \\left\\{ {U}_{m}\\right\\} \\subset {H}_{0}^{1}\\left( \\Omega \\right) \\times {H}_{0}^{1}\\left( \\Omega \\right) \) be a (PS) sequence for \( {I}_{K\\left( x\\right) } \) at level \( \\beta \), that is, \( {I}_{K\\left( x\\right) }\\left( {U}_{m}\\right) \\rightarrow \\beta, D{I}_{K\\left( x\\right... | Proof of Lemma 2.2 The main strategy of the proof is similar to that in [4] and here we only give the sketch. By\n\n\[ \n\\frac{1}{2}{\\int }_{\\Omega }\\left( {{\\left| \\nabla {u}_{m}\\right| }^{2} + {\\left| \\nabla {v}_{m}\\right| }^{2}}\\right) - \\frac{1}{{2}^{ \\star }}{\\int }_{\\Omega }K\\left( x\\right) {\\le... | Yes |
Corollary 2.1 Suppose that\n\n\[ c < \frac{1}{N}\frac{{S}_{\alpha ,\beta }^{N/2}}{K{\left( {x}_{0}\right) }^{\left( {N - 2}\right) /2}}, \]\n\nthen the functional (I) satisfies (PS), condition. | Proof of Corollary 2. 1 Suppose \( \left( {U}_{m}\right) \) is a (PS) sequence for \( I \) . For any \( y \in {\mathbf{R}}^{N} \) and \( U \in \) \( \left( {{D}^{1,2}\left( {\mathbf{R}}^{N}\right) \times {D}^{1,2}\left( {\mathbf{R}}^{N}\right) }\right) \smallsetminus \{ \left( {0,0}\right) \} \) satisfying \( \left( {P... | Yes |
Theorem 3.2 Assume that (S1)-(S4) hold and there exists \( U \in {H}_{0}^{1}\left( \Omega \right) \times {H}_{0}^{1}\left( \Omega \right), U \geq \) 0 on \( \Omega, U \neq 0 \) such that\n\n\[ \mathop{\sup }\limits_{{t \geq 0}}I\left( {tU}\right) < \frac{1}{N}\frac{{S}_{\alpha ,\beta }^{N/2}}{K{\left( {x}_{0}\right) }^... | Proof It is easy to check that \( I \) satisfies the geometric structure of Mountain Pass Theorem under the assumptions (S1)-(S4). The proof is similar to that of Theorem 3.1, and we omit it. | No |
Lemma 2.8 Let \( \\left( {X, d}\\right) \) be a complete metric space. If \( A \) is an expansive mapping on \( X \) and it is also surjective, then \( A \) has a unique fixed point. | Proof Since \( A \) is expansive, we can obtain that \( A \) is injective. We can proof this result by contradiction. If \( A \) is not injective, then there exists \( {x}_{0} \\neq {y}_{0} \\in X \), such that \( A\\left( {x}_{0}\\right) = A\\left( {y}_{0}\\right) \) . Since \( {x}_{0} \\neq \) \( {y}_{0} \), then \( ... | Yes |
Theorem 3.3 Let \( M \) and \( N \) be two complete bounded subsets of Banach space \( E \) with \( M \) \( \subset N \) and \( A \) be an expansive and surjective mapping from \( M \) to \( N \) . Let \( \left\{ {x}_{n}\right\} \) be a sequence generated by \[ \left\{ \begin{array}{l} \forall {x}_{0} \in M, \\ T\left(... | Proof Since \( M \) is a bounded subset of \( E \), then we have that for all \( x \in M \), there exists \( Q \) \( > 0 \), such that \( \parallel x\parallel \leq Q \) . And also, for all \( \varepsilon > 0 \), there exists \( N > 0,\forall n > N \), such that \( {\alpha }_{n} \) \( \leq \varepsilon /\left( {2Q}\right... | Yes |
Lemma 2.4 If \( \left( {\mathrm{H}}_{1}\right) \) hold, let \( H, K \) be as defined in (2.1) and (2.3). Then\n\n1) \( H\left( {t, s}\right) \in C\left( {\left\lbrack {0,1}\right\rbrack \times \left\lbrack {0,1}\right\rbrack }\right) \) and \( H\left( {t, s}\right) > 0 \) on \( \left( {0,1}\right) \times \left( {0,1}\r... | Proof By \( \left( {\mathrm{H}}_{1}\right) ,\left( {2.1}\right) \left( {2.3}\right) \) and Lemma 2.3, we have (1)-(3) hold. Since\n\n\[ \frac{{e}_{2}\left( {1 - t}\right) {e}_{2}\left( s\right) }{\Gamma \left( \alpha \right) } \leq {G}_{2}\left( {t, s}\right) \leq K\left( {t, s}\right) \leq \frac{\left( {\alpha - 1}\ri... | Yes |
Lemma 3.1 Let \( \alpha \in \left( {1,2\rbrack ,\sigma \in \lbrack 0,\alpha - 1}\right) ,{t}^{\sigma }f\left( {t, u}\right) \in C\left( {\left\lbrack {0,1}\right\rbrack \times {\mathbf{R}}^{ + },{\mathbf{R}}^{ + }}\right) \) and \( f\left( {t, u}\right) \in \) \( C\left( {(0,1\rbrack \times {\mathbf{R}}^{ + },{\mathbf{... | Proof In fact, \( T : P \rightarrow E \) is well-defined.\n\n\[ T\left( u\right) \left( t\right) = {\int }_{0}^{1}H\left( {t, s}\right) {\phi }_{q}\left( {{\int }_{0}^{1}K\left( {s,\tau }\right) f\left( {\tau, u\left( \tau \right) }\right) \mathrm{d}\tau }\right) \mathrm{d}s \]\n\n\[ = {\int }_{0}^{1}H\left( {t, s}\rig... | Yes |
Corollary 3.1 Assume that \( \left( {\mathrm{H}}_{1}\right) \) holds, and the following conditions are true:\n\n\( \left( {{\mathrm{H}}_{2}{}^{\prime }}\right) \) there exists a sequence \( {\left\{ {u}_{i}\right\} }_{i = 0}^{n} \) such that \( 0 < {u}_{0} < {u}_{1} < \cdots < {u}_{n} \leq \infty \) and\n\n(I) \( {t}^{... | Proof Considering (I) and choosing \( d = {u}_{0} \), we have \( {t}^{\sigma }f\left( {t, u}\right) \geq {\left( \widetilde{M}u\right) }^{p - 1} \) for \( 0 \leq u \leq d \) , \( t \in \left\lbrack {0,1}\right\rbrack \) . With similar argument to achieve (3.11), we get \( {Tu} \neq u \) . By Lemma 2. 5, we get\n\n\[ i\... | Yes |
Corollary 3.2 Assume that \( \left( {\mathrm{H}}_{1}\right) \) holds, and the following conditions are true:\n\n\( \left( {{\mathrm{H}}_{4}{}^{\prime }}\right) \) there exists a sequence \( {\left\{ {u}_{i}\right\} }_{i = 0}^{n} \) such that \( 0 < {u}_{0} < {u}_{1} < \cdots < {u}_{n} \leq \infty \) and\n\n(I) \( {t}^{... | The proof of Corollary 3.2 is similar to that of Corollary 3.1 and so is omitted. | No |
Consider the following boundary value problem\n\n\[ \left\{ \begin{array}{l} {D}_{0 + }^{u}\left( {{\phi }_{p}\left( {{u}^{\prime \prime }\left( t\right) }\right) }\right) = \frac{u\left( t\right) {\mathrm{e}}^{u\left( t\right) } + 1}{{t}^{1/2}},0 < t < 1, \\ u\left( 0\right) = u\left( 1\right) = {\int }_{0}^{1}u\left(... | Obviously, \( f\left( {t, u}\right) = \left( {u{\mathrm{e}}^{u} + 1}\right) {t}^{-1/2} \) is singular at \( t = 0 \) . Let \( \sigma = 1/2 \), then \( {t}^{\sigma }f\left( {t, u}\right) = u{\mathrm{e}}^{u} + 1 \) is continuous on \( \left\lbrack {0,1}\right\rbrack \times {\mathbf{R}}^{ + } \) . It is easy to see that \... | Yes |
Theorem 1 Let \( A \) and \( B \) be Hermitian matrices of order \( n \), and \( W \) be the complex matrix of order \( n \) such that \( B - {W}^{H}{BW} \geq O \) . Then, for \( k = 1,2,\cdots, n \) , \n\n\[ \n\mathop{\sum }\limits_{{i = 1}}^{k}{\lambda }_{i}\left( {A - B}\right) \leq \mathop{\sum }\limits_{{i = 1}}^{... | Proof We first show the inequalities (2.1). Notice that \( B - {W}^{H}{BW} \geq O \) . Hence, by Lemma 2 we have \n\n\[ \n\mathop{\sum }\limits_{{i = 1}}^{k}{\lambda }_{i}\left( {A - B}\right) = \mathop{\sum }\limits_{{i = 1}}^{k}{\lambda }_{i}\left( {A - {W}^{H}{BW} + {W}^{H}{BW} - B}\right) \n\] \n\n\[ \n\leq \mathop... | Yes |
Theorem 2 Let \( A \) and \( B \) be positive semidefinite matrices of the same size, and \( W \) be the complex matrix as shown in Theorem 1. Then, for \( k = 1,2,\cdots, n \) ,\n\n\[ \left| {\mathop{\sum }\limits_{{i = 1}}^{k}{\lambda }_{i}\left( {A - {W}^{H}{BW}}\right) }\right| \leq \mathop{\sum }\limits_{{i = 1}}^... | Proof From (1.3), we get\n\n\[ \mathop{\sum }\limits_{{i = 1}}^{k}{\lambda }_{i}\left( {A - {W}^{H}{BW}}\right) = \mathop{\sum }\limits_{{i = 1}}^{k}{\lambda }_{i}\left( {A + B - B - {W}^{H}{BW}}\right) \]\n\n\[ \leq \mathop{\sum }\limits_{{i = 1}}^{k}{\lambda }_{i}\left( {A + B}\right) + \mathop{\sum }\limits_{{i = 1}... | Yes |
Theorem 3 Let \( A, B \) and \( W \) be the matrices of the same size as shown in Theorem 2 . Then\n\n\[ \left| {\operatorname{tr}\left( {A - {W}^{H}{BW}}\right) }\right| \leq \operatorname{tr}\left| {A - {W}^{H}{BW}}\right| \leq \operatorname{tr}\left( {A + B}\right) . \]\n\n(2.8)\n\nwhere \( \left| X\right| = {\left(... | Proof We first show that if \( A \geq O \), for any unitary matrix \( U,\left| {\operatorname{tr}\left( {AU}\right) }\right| \leq \operatorname{tr}A \), and \( \operatorname{Retr}\left( {A \pm {AU}}\right) \geq 0 \) . To see these, let \( V \) be a unitary matrix such that \( A = {V\sum }{V}^{H} \), where \( \sum = \op... | Yes |
Theorem 3.1 Assume that there exist functions \( a \in \mathcal{{VK}}, b \in \mathcal{{CK}}, V \in {\nu }_{0} \) and a positive number \( p > 0 \) such that (i) for all \( \left( {t, x}\right) \in \left\lbrack {{t}_{0} - \tau ,\infty }\right) \times {\mathbb{R}}^{n} \) ,\n\n\[ a\left( {\left| x\right| }^{p}\right) \leq... | Proof Let \( x\left( t\right) = x\left( {t;{t}_{0},\varphi }\right) \) denote the solution through \( \left( {{t}_{0},\varphi }\right) \) . From Itô differential formula, we have\n\n\[ \mathrm{d}V\left( {t, x\left( t\right) }\right) = \mathcal{L}V\left( {t,{x}_{t}}\right) \mathrm{d}t + \left( \cdots \right) \mathrm{d}B... | Yes |
Theorem 3.2 Assume that there exist functions \( a \in \mathcal{V}\mathcal{K}, b \in \mathcal{C}\mathcal{K}, V \in {\nu }_{0} \), positive number \( p > 0 \) and \( q > 1 \) such that\n\n(i) for all \( \left( {t, x}\right) \in \left\lbrack {{t}_{0} - \tau ,\infty }\right) \times {\mathbb{R}}^{n} \),\n\n\[ a\left( {\lef... | Proof Let \( {\widehat{\lambda }}^{ + }\left( s\right) = \max \{ \widehat{\lambda }\left( s\right) ,0\} \) . We can see \( {\widehat{\lambda }}^{ + }\left( s\right) = {\lambda }^{ + }\left( s\right) \) and \( \widehat{\lambda }\left( s\right) = {\widehat{\lambda }}^{ + }\left( s\right) - {\widehat{\lambda }}^{ - }\left... | Yes |
Example 4.1 Consider a scalar ISFDs of the form\n\n\\[ \n\\left\\{ \\begin{array}{ll} \\mathrm{d}x\\left( t\\right) = {\\lambda }_{1}\\left( t\\right) x\\left( t\\right) \\mathrm{d}t + \\frac{x\\left( {t - \\left| {\\sin t}\\right| }\\right) }{\\sqrt{1 + {t}^{2}}}\\mathrm{\\;d}w\\left( t\\right) , & t \\neq {t}_{k}, t ... | Let \\( V\\left( {t, x}\\right) = {x}^{2} \\) . Then we can compute that\n\n\\[ \n\\mathrm{E}V\\left( {{t}_{k}, x\\left( {t}_{k}\\right) }\\right) = \\mathrm{E}{\\left( \\frac{2}{3}x\\left( {t}_{k}^{ - }\\right) + \\frac{1}{3}x\\left( {t}_{k}^{ - } - \\tau \\right) \\right) }^{2} \n\\]\n\n\\[ \n= {\\left( \\frac{2}{3}\... | Yes |
Lemma 2.2 Let \( E \subset {\mathbb{R}}^{n} \) be a self-similar set satisfying OSC and \( {\dim }_{H}\left( E\right) = s \) . Then the following equalities hold:\n\n(i) \( {H}^{s}\left( E\right) = \inf \left\{ {\sum {\left| E \cap {U}_{i}\right| }^{s} : \bigcup {U}_{i} \supset E,{U}_{i}\text{ closed in }{\mathbb{R}}^{... | Proof By Lemma 2.1 and the definition of Hausdorff measure it is easy to see\n\n\[ {H}^{s}\left( E\right) = \inf \left\{ {\mathop{\sum }\limits_{i}{\left| {U}_{i}\right| }^{s} : \mathop{\bigcup }\limits_{i}{U}_{i} \supset E,{U}_{i}\text{ closed in }{\mathbb{R}}^{n}\text{ for any }i}\right\} . \]\n\n(2.1)\n\nSet\n\n\[ {... | Yes |
Lemma 2.3 Let \( E \subset {\mathbb{R}}^{n} \) be a self-similar set satisfying OSC and \( {\dim }_{H}\left( E\right) = s \) . Then the following hold:\n\n(i) \( \sup \left\{ {\frac{{H}^{s}\left( U\right) }{{\left| U\right| }^{s}} : U \subset E, U\text{is compact in}{\mathbb{R}}^{n}}\right\} = 1 \) . | Proof (i) Set\n\n\[ L = \sup \left\{ {\frac{{H}^{s}\left( U\right) }{{\left| U\right| }^{s}} : U \subset E, U\text{ is compact in }{\mathbb{R}}^{n}}\right\} . \]\n\n\( \left( {2.2}\right) \)\n\nNow we prove \( L = 1 \) . On the one hand, by Lemma 2.1 and the definition of \( L \) we have \( 0 < L \leq 1 \) . On the oth... | Yes |
Corollary 2.1 Let \( E \subset {\mathbb{R}}^{n} \) be a self-similar set satisfying OSC, \( {E}_{0} \) be the closed convex hull of \( E \) and \( {\dim }_{H}\left( E\right) = s \) . Then the following equality holds:\n\n\[ \sup \left\{ {\frac{{H}^{s}\left( {E \cap U}\right) }{{\left| U\right| }^{s}} : U \subset {E}_{0... | Proof Set\n\n\[ H = \sup \left\{ {\frac{{H}^{s}\left( {E \cap U}\right) }{{\left| U\right| }^{s}} : U \subset {E}_{0}, U\text{ is compact in }{\mathbb{R}}^{n}}\right\} . \]\n\nNow we prove \( H = 1 \) . By Lemma 2.3(ii) we easily know \( H \geq 1 \) since \( E \subset {E}_{0} \) . Besides, we can get \( H \leq 1 \) by ... | Yes |
Lemma 2.4 Let \( E \subset {\mathbb{R}}^{n} \) be a self-similar set satisfying SSC and \( {\dim }_{H}\left( E\right) = s \) . Then there exists a real positive number \( \varepsilon = \min \left\{ {d\left( {{S}_{i}\left( E\right) ,{S}_{j}\left( E\right) }\right) : i \neq j, i, j, = 1,2,\cdots, m}\right\} > 0 \) , such... | Proof Let \( E \subset {\mathbb{R}}^{n} \) be the self-similar set for the IFS \( {S}_{1},{S}_{2},\cdots ,{S}_{m} \) satisfying SSC. Set \[ \varepsilon = \min \left\{ {d\left( {{S}_{i}\left( E\right) ,{S}_{j}\left( E\right) }\right) : i \neq j, i, j = 1,2,\cdots, m}\right\} > 0, \] \[ h = \sup \left\{ {\frac{{H}^{s}\le... | Yes |
Lemma 2.5 \( {}^{\left\lbrack 7\right\rbrack }\; \) Let \( \left\{ {A}_{n}\right\} \) be a sequence of nonempty compact subsets of \( {\mathbb{R}}^{n} \) and \( A \subset {\mathbb{R}}^{n} \). If \( \left\{ {A}_{n}\right\} \) converges to \( A \) in Hausdorff metric, then\n\n(i) \( \mathop{\lim }\limits_{{n \rightarrow ... | Proof We only prove (ii). One can see the proofs for (i) and (iii) in [7]. Since \( \left\{ {A}_{n}\right\} \) converges to \( A \) in Hausdorff metric, it is easy to know that for each for any \( a \in A \), there exists a Cauchy’s sequence \( \left\{ {x}_{n}\right\} \) in \( {\mathbb{R}}^{n} \) with \( {x}_{n} \in {A... | No |
Theorem 2.1 Let \( E \subset {\mathbb{R}}^{n} \) be a self-similar set satisfying SSC and \( {\dim }_{H}\left( E\right) = s \) . Then there exists a closed convex set \( V \subset {E}_{0} \) with \( \left| V\right| > 0 \), where \( {E}_{0} \) denotes the closed convex hull of \( E \) such that \( {H}^{s}\left( {E \cap ... | Proof By Lemma 2.4, there exists a real number \( \varepsilon \) with \( 0 < \varepsilon < 1 \) and a sequence \( \left\{ {U}_{i}\right\} \) consisting of compact sets in \( {\mathbb{R}}^{n} \), such that \( \left| {U}_{i}\right| \geq \varepsilon > 0,{U}_{i} \subset E\left( {\forall i \geq 1}\right) \) and \( \frac{{H}... | Yes |
Theorem 2.2 Let \( E \subset {\mathbb{R}}^{n} \) be a self-similar set satisfying SSC and \( {\dim }_{H}\left( E\right) = s \) . Then there exists a closed convex set \( V \subset {E}_{0} \) with \( \left| V\right| > 0 \), where \( {E}_{0} \) denotes the closed convex hull of \( E \) such that \( \sup \left\{ {d\left( ... | Proof Let \( V \) denote the closed convex subset as in Theorem 2.1. It is easy to see from Corollary 2.1 and Theorem 2.1 that\n\n\[ \sup \left\{ {\frac{{H}^{s}\left( {E \cap U}\right) }{{\left| U\right| }^{s} \cdot {H}^{s}\left( E\right) } : U \subset {E}_{0}\text{ is closed }}\right\} = \frac{{H}^{s}\left( {E \cap V}... | Yes |
Corollary 2.3 Let \( E \subset {\mathbb{R}}^{n} \) be a self-similar \( s \) -set satisfying OSC. Suppose that \( E \) satisfying SSC, then \( E \) has a best \( {H}^{s} \) -a.e.-closed-set covering. | Proof By the proof of Theorem 2.1, there exists a closed set \( U \subset E \) with \( \left| U\right| > 0 \) such that \( {H}^{s}\left( U\right) = {\left| U\right| }^{s} \) . So for any \( k \geq 1 \) and \( \left( {{i}_{1},\cdots ,{i}_{k}}\right) \in {J}_{k} \), we have\n\n\[
{H}^{s}\left( {{S}_{{i}_{1}} \circ \cdots... | Yes |
Theorem 1 An optimal replacement policy \( {N}^{ * } \) is determined by\n\n\[ \n{N}^{ * } = \min \{ N : g\left( N\right) \geq 1\} .\n\] | This theorem shows the existence of the optimal replacement policy and if \( g\left( {N}^{ * }\right) > 1 \) , the optimal replacement policy is also unique.In order to use Theorem 1 to find the optimal replacement policy, we consider the following these three cases:\n\nCase 1 If \( g\left( 1\right) > 1 \), it implies ... | No |
Theorem 3.2 Let \( \left( {X, Y,\Phi }\right) \) be a GFC-space, \( Z \) a topological space, \( s \in C\left( {X, Z}\right) \) a continuous map, \( K \subset Z \) a nonempty compact subset and \( \gamma \in \mathbb{R} \) a real number. Suppose \( f : Y \times Z \rightarrow \overline{\mathbb{R}} \) satisfies\n\n1) \( f... | Proof Define \( F : Y \rightarrow {2}^{Z} \) by \( F\left( y\right) \mathrel{\text{:=}} \{ z \in Z : f\left( {y, z}\right) \leq \gamma \} \) for each \( y \in Y \) . Then by 1) and Lemma 2.1, we have \( F \) is a GFs-KKM mapping, and then \( {\mathrm{{cl}}}_{Z}F \) is a GFs-KKM mapping. By 2) and Lemma 2.2 of \( {\math... | Yes |
Theorem 3.3 Let \( \left( {X, Y,\Phi }\right) \) be a GFC-space, \( Z \) a topological space, \( s \in C\left( {X, Z}\right) \) a continuous map, \( K \subset Z \) a nonempty compact subset \( Z \), and \( C \) a nonempty subset \( Y \times Z \) satisfying\n\n1) \( C \) is weakly transfer compactly closed relative to \... | Proof Define \( f : Y \times Z \rightarrow \overline{\mathbb{R}} \) by\n\n\[ f\left( {y, z}\right) \mathrel{\text{:=}} \left\{ \begin{array}{ll} 1, & \text{ if }\left( {y, z}\right) \notin C, \\ 0, & \text{ if }\left( {y, z}\right) \in C, \end{array}\right. \]\n\nand \( F : Y \rightarrow {2}^{Z} \) by \( F\left( y\righ... | Yes |
Theorem 3.4 Let \( \left( {X, Y,\Phi }\right) \) be a GFC-space, \( Y \) topological spaces, \( Z \) a nonempty set, \( s \in C\left( {X, Y}\right) \) a continuous map, \( K \subset Y \) a nonempty compact subset \( Y, f : Y \rightarrow Z \) a surjective and \( F : Y \rightarrow {2}^{Z} \smallsetminus \{ \varnothing \}... | Proof Suppose that the conclusion is false. Then for each \( y \in Y, y \notin \left( {{f}^{-1}F}\right) \left( y\right) \) . Define\n\n\[ C \mathrel{\text{:=}} \left\{ {\left( {y, z}\right) \in Y \times Y : y \notin \left( {{f}^{-1}F}\right) \left( z\right) }\right\} \subset Y \times Y. \]\n\nThen for each \( y \in Y,... | Yes |
Theorem 3.3 For any initial datum \( {f}_{0}\left( v\right) \) in the set of probability density with zero bulk velocity and finite temperature, then the unique solution \( f\left( {v, t}\right) \) to (1.1) converges strongly to the equilibria \( {f}_{\infty }\left( v\right) \), i.e.\n\n\[ \mathop{\lim }\limits_{{t \ri... | Proof Applying the above lemma and Formula (3.4), we have\n\n\[ {\begin{Vmatrix}f\left( v, t\right) - {f}_{\infty }\left( v\right) \end{Vmatrix}}_{{L}^{1}\left( {\mathbb{R}}^{3}\right) }\n\n\leq C\left( {3,1}\right) {\left( {\int }_{{\mathbb{R}}^{3}}{\left| f\left( v, t\right) - {f}_{\infty }\left( v\right) \right| }^{... | Yes |
Theorem 1.1 Suppose that \( g\left( t\right) \) is a nondecreasing positive function with respect to \( \left| t\right| \), and the condition \( \left( {\mathrm{V}}_{1}\right) ,\left( {\mathrm{h}}_{0}\right) - \left( {\mathrm{h}}_{3}\right) \) hold. Then (1.1) has a positive solution. | The main difficulty in treating this class of quasilinear equations (1.1) is the possible lack of compactness due to the unboundedness of the domain besides the presence of the second order nonhomogeneous term which prevents us to work directly with the energy functional \( I\left( u\right) \) . To overcome these diffi... | Yes |
Lemma 2.1 The function \( h\left( t\right), g\left( t\right) \) and \( H\left( t\right) = {\int }_{0}^{t}h\left( \tau \right) \mathrm{d}\tau, G\left( t\right) = {\int }_{0}^{t}g\left( \tau \right) \mathrm{d}\tau \) enjoy the following properties under the assumptions \( \left( {\mathrm{h}}_{0}\right) - \left( {\mathrm{... | Proof The conclusion 1) are immediately by the definition of \( G\left( t\right) \) and the differential mean value theorem.\n\n\[ {\left( \frac{{G}^{-1}\left( t\right) }{t}\right) }_{t}^{\prime } = {\left( \frac{{G}^{-1}\left( t\right) }{t}\right) }_{s}^{\prime }\frac{1}{g\left( s\right) } = \frac{G\left( s\right) - g... | Yes |
Lemma 3.1 There exist \( {\rho }_{0} \) and \( {a}_{0} \) such that \( J\left( v\right) \geq {a}_{0} \) for all \( \parallel v\parallel = {\rho }_{0} \) . | Proof If \( g \) is bounded, by the properties 3) and 6) of Lemma 2.1, we have\n\n\[ J\left( v\right) = \frac{1}{p}{\int }_{{\mathbb{R}}^{N}}{\left| \nabla v\right| }^{p}\mathrm{\;d}x + \frac{1}{p}{\int }_{{\mathbb{R}}^{N}}v\left( x\right) {\left| {G}^{-1}\left( v\right) \right| }^{p}\mathrm{\;d}x - {\int }_{{\mathbb{R... | Yes |
Lemma 3.2 There exist \( v \in {W}^{1, p}\left( {\mathbb{R}}^{N}\right) \) such that \( J\left( v\right) < 0 \) . | Proof Given \( \varphi \in {C}_{0}^{\infty }\left( {{\mathbb{R}}^{N},\left\lbrack {0,1}\right\rbrack }\right) \) with supp \( \varphi = {\bar{B}}_{1} \) . We will prove that \( J\left( {t\varphi }\right) \rightarrow - \infty \) as \( t \rightarrow \infty \), which will prove the result if we take \( v = {t\varphi } \) ... | Yes |
Lemma 2.1 Let \( \mathrm{m} \) be a positive integer, then we have the summation formulas \( {}^{\left\lbrack 9\right\rbrack } \) :\n\n\[ \mathop{\sum }\limits_{{k = 1}}^{\infty }\frac{1}{{k}^{2m}} = \frac{{2}^{{2m} - 1}{\pi }^{2m}}{\left( {2m}\right) !}{B}_{m} \] | (2.4)\n\nwhere the \( {B}_{{m}^{\prime }s} \) are the Bernoulli numbers, viz. \( {B}_{1} = \frac{1}{6},{B}_{2} = \frac{1}{30},{B}_{3} = \frac{1}{42},{B}_{4} = \frac{1}{30},{B}_{5} = \) \( \frac{5}{66},{B}_{6} = \frac{691}{2730},{B}_{7} = \frac{7}{6} \), etc. . | No |
Lemma 2.2 If \( \beta > 0 \), we have \( {}^{\left\lbrack 8\right\rbrack } \): | \[ \mathop{\sum }\limits_{{k = 0}}^{\infty }\frac{1}{{\left( 2k + 1\right) }^{\beta + 1}} = \frac{1}{{2}^{\beta + 1}}\zeta \left( {\beta + 1,\frac{1}{2}}\right) = \left( {1 - \frac{1}{{2}^{\beta + 1}}}\right) \zeta \left( {\beta + 1}\right) ,\] | Yes |
Lemma 2.3 If \( p > 1,\frac{1}{p} + \frac{1}{q} = 1,\alpha \geq 0,\beta > 0 \), we define weight function as:\n\n\[ \n\omega \left( {\alpha ,\beta, x}\right) \mathrel{\text{:=}} {\int }_{0}^{\infty }{\mathrm{e}}^{-{\alpha xy}}\operatorname{csch}\left( {\beta xy}\right) \frac{{y}^{\beta }}{{x}^{\frac{p\beta }{q}}}\mathr... | Proof Setting \( {\beta xy} = u \), then we have:\n\n\[ \n\omega \left( {\alpha ,\beta, x}\right) = {\int }_{0}^{\infty }{\mathrm{e}}^{-{\alpha xy}}\operatorname{csch}\left( {\beta xy}\right) \frac{{y}^{\beta }}{{x}^{\frac{p\beta }{q}}}\mathrm{\;d}y \n\]\n\n\[ \n= \frac{2}{{\beta }^{\beta + 1}}{x}^{-{p\beta } - 1}{\int... | Yes |
Lemma 2.4 If\n\n\[ p > 1,\frac{1}{p} + \frac{1}{q} = 1,\alpha \geq 0,\beta > 0,0 < \varepsilon < \min \{ p, q\} ,\]\n\nand \( \varepsilon \) small enough, let us define the real functions as follows:\n\n\[ \widetilde{f}\left( x\right) \mathrel{\text{:=}} \left\{ {\begin{array}{ll} 0, & x \in \left( {0,1}\right) , \\ {x... | Proof We easily obtain:\n\n\[ \widetilde{J}\varepsilon = {\left\lbrack {\int }_{0}^{\infty }{x}^{-{p\beta } - 1}{\widetilde{f}}^{p}\left( x\right) \mathrm{d}x\right\rbrack }^{\frac{1}{p}}{\left\lbrack {\int }_{0}^{\infty }{y}^{-{q\beta } - 1}{\widetilde{g}}^{q}\left( y\right) \mathrm{d}y\right\rbrack }^{\frac{1}{q}}\va... | Yes |
Let \( \alpha = 0 \), then the following equivalent inequalities hold:\n\n\[ \n{\int }_{0}^{\infty }{\int }_{0}^{\infty }\operatorname{csch}\left( {\beta xy}\right) f\left( x\right) g\left( y\right) \mathrm{d}x\mathrm{\;d}y < C\left( \beta \right) \parallel f{\parallel }_{p,\varphi }\parallel g{\parallel }_{q,\psi }, \... | Further, let \( \beta = 1, p = q = 2 \), by (2.9), \( C\left( 1\right) = \frac{{\pi }^{2}}{4} \) , If \( \varphi \left( x\right) = {x}^{-3}, f, g \in {L}_{\varphi }^{2}\left( {0,\infty }\right) ,\parallel f{\parallel }_{2,\varphi },\parallel g{\parallel }_{2,\varphi } > 0 \), then we have (1.3) and its equivalent is gi... | Yes |
Example 3.2 Let \( \alpha = \beta = 1, p = q = 2 \), by(2.11), we get \( C\left( {1,1}\right) = \frac{{\pi }^{2}}{12} \), If \( \varphi \left( x\right) = \) \( {x}^{-3}, f, g \in {L}_{\varphi }^{2}\left( {0,\infty }\right) ,\parallel f{\parallel }_{2,\varphi },\parallel g{\parallel }_{2,\varphi } > 0 \) . Hence we have... | \[ {\int }_{0}^{\infty }{y}^{3}{\left\lbrack {\int }_{0}^{\infty }{\mathrm{e}}^{-{xy}}\operatorname{csch}\left( xy\right) f\left( x\right) \mathrm{d}x\right\rbrack }^{2}\mathrm{\;d}y < \frac{{\pi }^{4}}{144}\parallel f{\parallel }_{2,\varphi }^{2}, \] where the constant factor \( \frac{{\pi }^{4}}{144} \) is the best p... | Yes |
If \( \varphi \left( x\right) = {x}^{-5}, f, g \in {L}_{\varphi }^{2}\left( {0,\infty }\right) ,\parallel f{\parallel }_{2,\varphi },\parallel g{\parallel }_{2,\varphi } > 0 \), we have the following equivalent inequalities: | \[ {\int }_{0}^{\infty }{\int }_{0}^{\infty }{\mathrm{e}}^{-{xy}}\operatorname{csch}\left( {2xy}\right) f\left( x\right) g\left( y\right) \mathrm{d}x\mathrm{\;d}y < \frac{\zeta \left( {3,\frac{3}{4}}\right) }{16}\parallel f{\parallel }_{2,\varphi }\parallel g{\parallel }_{2,\varphi }, \] (3.10) \[ {\int }_{0}^{\infty }... | Yes |
Lemma 3.1 Suppose \( B \) is a bounded base of \( C,\varnothing \neq Q \subset X, F \) is a set-valued function from \( Q \) into \( Y,\bar{x} \in Q,\bar{y} \in F\left( \bar{x}\right), T \in L\left( {X, Y}\right) \) . If there exists a \( \varphi \in {C}^{\bigtriangleup }\left( B\right) \) such that\n\n\[ \varphi \left... | Proof It suffices to show that \( \bar{y} - T\left( \bar{x}\right) \in \varepsilon - {SE}\left( {\mathop{\bigcup }\limits_{{x \in Q}}\left( {F\left( x\right) - T\left( x\right) }\right), C}\right) \) . By contradiction, suppose that there is a neighborhood \( \widetilde{V} \) of 0 in \( Y \), such that for any neighbor... | Yes |
Example 3.1 Let \( \mathbb{R} \) be the set of real numbers, \( X = \mathbb{R}, Y = {\mathbb{R}}^{2}, C = {\mathbb{R}}_{ + }^{2} = \left\{ {\left( {{t}_{1},{t}_{2}}\right) : }\right. \) \( \left. {{t}_{1} \geq 0,{t}_{2} \geq 0}\right\}, B = \left\{ {\left( {{t}_{1},{t}_{2}}\right) : {t}_{1} + {t}_{2} = 1,{t}_{1} \geq 0... | Let \( a = \left( {3,3}\right) ,\bar{V} = \left( {-{0.5},{0.5}}\right) \), then \( F\left( {\bar{x} + \bar{V}}\right) \subset a - C \), the conditions of Theorem 3.1 are satisfied. Let \( T : X \rightarrow Y, T\left( x\right) = \left( {\frac{1}{2}x,\frac{1}{2}x}\right) \), a direct calculation gives \( T \in {\partial ... | Yes |
Theorem 2.1 Under the assumptions \( \left( {\mathrm{A}}_{b}\right) \) and \( \left( {\mathrm{A}}_{\alpha }\right) \), let \( {x}_{0} \in {L}^{p}\left( \Omega \right) \) for some \( p \geq 1 \) , then the auxiliary equation (2.1) has an unique adapted solution \( y\left( t\right) \) . Moreover, for every \( T > 0 \), t... | \[ {\left| y\left( t\right) \right| }^{p} \leq \left( {{C}_{1, p} + {C}_{2, p}\left( T\right) {\int }_{0}^{t}{\mathrm{e}}^{-p{\int }_{0}^{s}\alpha \left( u\right) \mathrm{d}W\left( u\right) }\mathrm{d}s}\right) {\mathrm{e}}^{{C}_{3, p}\left( T\right) t}\text{ a.s. } \] | Yes |
Theorem 2.2 Under the assumption \( \left( {\mathrm{A}}_{b}\right) \) and \( \left( {\mathrm{A}}_{\alpha }\right) \), suppose that \( {x}_{0} \in {L}^{2}\left( \Omega \right) \), then \( x\left( t\right) \) is a solution of equation (1.1) if and only if there exists a solution \( y\left( t\right) \) of equation (2.1) s... | The proof of the theorem essentially is due to the Itô formula. So we omit it. | No |
Theorem 3.1 Let \( p \) be a positive integer. The sample paths of solution \( y\left( {t;{x}_{0}}\right) \) of the equation (2.1) is exponentially \( p \) -stable for \( t \geq 0 \) if there exists a function \( V\left( y\right) \) of class \( {C}^{1}\left( {\mathbb{R},{\mathbb{R}}^{ + }}\right) \) such that\n\n\[ \n{... | Proof Let \( y\left( t\right) = y\left( {t;{x}_{0}}\right) \) for simplicity. Note that \( y\left( t\right) \) is differentiable a.s., so \( V\left( {y\left( t\right) }\right) \) does. Let \( w\left( u\right) = {\int }_{0}^{u}\alpha \left( v\right) \mathrm{d}W\left( v\right) \) . Then \n\n\[ \nV\left( {y\left( t\right)... | Yes |
Corollary 3.1 Let \( p \) be a positive integer. The solution \( y\left( {t;{x}_{0}}\right) \) of the equation (2.1) is exponentially \( p \) -stable for \( t \geq 0 \) if there exists a function \( V\left( x\right) \) of class \( {C}^{1}\left( {\mathbb{R},{\mathbb{R}}^{ + }}\right) \) which satisfies condition (3.1) a... | Proof From (3.4), we have\n\n\[ \mathrm{E}\left\lbrack {\left| y\left( t\right) \right| }^{p}\right\rbrack \leq \frac{{k}_{2}}{{k}_{1}}\mathrm{E}\left\lbrack {\left| {x}_{0}\right| }^{p}\right\rbrack {\mathrm{e}}^{-\frac{{k}_{3}}{{k}_{2}}t}. \]\n\n(3.5)\n\nThe proof is completed. | Yes |
Theorem 3.2 Let \( {x}_{0} \in \mathbb{R} \) . The solution \( x\left( {t;{x}_{0}}\right) \) of the equation (1.1) is exponentially p-stable for \( t \geq 0 \) if there exists a function \( V\left( x\right) \) of class \( {C}^{1}\left( {\mathbb{R},{\mathbb{R}}^{ + }}\right) \) which satisfies conditions (3.1) and (3.2)... | Proof Denote \( x\left( {t;{x}_{0}}\right) \) by \( x\left( t\right) \) for simplicity. By Theorem 2.2, there exists \( y\left( t\right) \) which solve the auxiliary equation (2.1) such that \( x\left( t\right) = y\left( t\right) {\mathrm{e}}^{{\int }_{0}^{t}\alpha \left( u\right) \mathrm{d}W\left( u\right) } \) . Due ... | Yes |
Consider the following stochastic differential equation, \( {x}_{0},\lambda \in \mathbb{R} \) , \[ x\left( t\right) = {x}_{0} + {\int }_{0}^{t}\left( {{\gamma x}\left( s\right) + \ln \left( {\left| {x\left( s\right) }\right| + 1}\right) }\right) \mathrm{d}s + {\int }_{0}^{t}x\left( s\right) \mathrm{d}W\left( s\right) .... | Let \[ {L}^{w} = \left( {{\gamma y} + \ln \left( {\left| y\right| {\mathrm{e}}^{w} + 1}\right) {\mathrm{e}}^{-w} - \frac{1}{2}y}\right) \frac{\mathrm{d}}{\mathrm{d}y}. \] Since \( \ln \left( {\left| y\right| + 1}\right) \leq \left| y\right| \), when taking \( V\left( y\right) = {\left| y\right| }^{2} \), we have \[ {L}... | Yes |
Theorem 4.1 Suppose that the sample paths of solution of the equation (2.1) is exponentially \( p \) -stable. Then the solution of the equation (1.1) is almost surely exponentially stable. | Proof Let \( y\left( t\right) = y\left( {t;{x}_{0}}\right) \) of the equation (2.1), then \( x\left( t\right) = y\left( t\right) {\mathrm{e}}^{{\beta W}\left( t\right) } \) is the solution of equation (1.1). Since there exists a random variable \( T\left( \omega \right) < \infty \) and positive constants \( {C}_{1} \) ... | Yes |
Theorem 4.2 Suppose that the assumption \( \left( {{\mathrm{A}}_{b} - \left( \mathrm{{ii}}\right) }\right) \) is satisfied and, for some constant \( K > 0 \) ,\n\n\[ \left| {b\left( {u, x}\right) }\right| < K\left| x\right| ,\text{ for all }\left( {u, x}\right) \in {\mathbb{R}}^{ + } \times \mathbb{R}. \]\n\n(4.3)\n\nT... | Proof Let \( V\left( y\right) = {\left| y\right| }^{2} \) . Then, for some \( \beta > 0 \) large enough such that \( {2K} - {\beta }^{2} < 0 \), we have\n\n\[ {L}^{w}V\left( y\right) = 2\left( {b\left( {t, y{\mathrm{e}}^{w}}\right) y{\mathrm{e}}^{-w} - \frac{1}{2}{\beta }^{2}{y}^{2}}\right) \]\n\n\[ \leq \left( {{2K} -... | Yes |
Corollary 1 For given scalars \( {\tau }_{1},{\mu }_{1} \), if there exist symmetric positive-definite matrices \( X,{S}_{1},{T}_{1},{Z}_{1} \), appropriately dimensioned matrices \( {Y}_{0},{Y}_{1},{U}_{1j},{V}_{1j}\left( {j = 1,2,3}\right) \) and positive scalars \( {\varepsilon }_{1},{\varepsilon }_{2} \) and \( \rh... | Then, the uncertain nonlinear single time-varying delay system (3.14) is robustly stabilizable, in this case, an appropriate non-fragile state feedback controller can be chosen by\n\n\[ u\left( t\right) = {Y}_{0}{X}^{-1}x\left( t\right) + {Y}_{1}{X}^{-1}x\left( {t - {\tau }_{1}\left( t\right) }\right) . \] | Yes |
Lemma 2.5 For sufficiently small \( \left| \xi \right| \) ,\n\n\[{\lambda }_{ \pm } \mp \sqrt{-1}\left| \xi \right| = - \frac{1}{2}k{\left| \xi \right| }^{2} + \mathop{\sum }\limits_{{j = 2}}^{\infty }{a}_{j}{\left| \xi \right| }^{{2j} - 1},\]\n\n(2.2)\n\n\[ \frac{\sqrt{-1}\left| \xi \right| }{{\lambda }_{ + } - {\lamb... | Proof Recall that \( {\lambda }_{ + } = \frac{-k{\left| \xi \right| }^{2} + \sqrt{{\left( k{\left| \xi \right| }^{2}\right) }^{2} - 4{\left| \xi \right| }^{2}}}{2} \) . When \( \left| \xi \right| \) is sufficiently small, using Taylor expansion, we get\n\n\[{\lambda }_{ + } - \sqrt{-1}\left| \xi \right| = \frac{-k{\lef... | Yes |
Lemma 2.6 Suppose that for sufficiently small \( \left| \xi \right|, f\left( \xi \right) = \mathop{\sum }\limits_{{j = 2}}^{\infty }{f}_{j}{\left| \xi \right| }^{j} \), where \( {f}_{j} \in \mathbb{C} \) .\n\nThen\n\[ \left| {{D}_{\xi }^{\beta }{\mathrm{e}}^{f\left( \xi \right) t}}\right| \leq {C}_{\beta }\left( {\math... | The proof of this lemma can be found in [12], here we omit it for brief. | No |
Lemma 2.7 For sufficiently small \( \epsilon \) , \[ \left| {{\partial }_{t}^{l}{D}_{\xi }^{\beta }\left( {{\xi }^{\alpha }{\widehat{K}}_{1,1}\left( {\xi, t}\right) }\right) }\right| \leq C{\left| \xi \right| }^{\left| \alpha \right| - \left| \beta \right| + {2l}}{\left( 1 + \left( {\left| \xi \right| }^{2}t\right) \ri... | Proof We just give the proof of (2.5) and (2.7), the other inequalities can be proved by the same method. \[ \left| {{\partial }_{t}^{l}{D}_{\xi }^{\beta }\left( {{\xi }^{\alpha }{\widehat{K}}_{2,1}}\right) }\right| \] \[ \leq C\mathop{\sum }\limits_{{\left| {\beta }_{1}\right| + \left| {\beta }_{2}\right| + \left| {\b... | No |
Proposition 2.1 For any positive integer \( N \), if \( \epsilon \) is small enough, there exists constant \( C \), such that\n\n\[ \left| {{D}_{x}^{\alpha }{E}_{1}\left( {x, t}\right) }\right| \leq C{t}^{-\frac{2 + \left| \alpha \right| }{2}}{A}_{N}\left( {x, t}\right) \] | Proof (2.10) and Lemma 2.1 imply\n\n\[ \left| {{\omega }_{t} * \left( {{D}^{\alpha }{K}_{1,1}}\right) }\right| = \mathop{\sum }\limits_{{0 \leq \left| \gamma \right| \leq 1}}C{t}^{\left| \gamma \right| }{\int }_{\left| y\right| \leq 1}\frac{{D}^{\gamma }\left( {{D}^{\alpha }{K}_{1,1}}\right) }{\sqrt{1 - {\left| y\right... | Yes |
Proposition 2.2 When \( \epsilon, R \) are fixed, for any positive integer \( N \), there exists positive constants \( h \) and \( C \), such that\n\n\[ \left| {{D}_{x}^{\alpha }{E}_{2}\left( {x, t}\right) }\right| \leq C{t}^{-\frac{2 + \left| \alpha \right| }{2}}{\mathrm{e}}^{-{ht}}{B}_{N}\left( {x, t}\right) .\n\] | Proof Recall that \( {\lambda }_{ \pm } = \frac{-k{\left| \xi \right| }^{2} \pm \sqrt{{\left( k{\left| \xi \right| }^{2}\right) }^{2} - 4{\left| \xi \right| }^{2}}}{2} \) . If \( \epsilon < \left| \xi \right| < {2R} \), we have \( \operatorname{Re}{\lambda }_{ \pm } \leq - {2\eta }{\left| \xi \right| }^{2} \) for some ... | Yes |
Theorem 2.1 For \( x \in {\mathbb{R}}^{2}, t > 0 \), and \( \left| \alpha \right| \leq {2l} \), we have\n\n\[ \left| {{D}_{x}^{\alpha }\left( {E - {\chi }_{3}\left( D\right) {F}_{\left| \alpha \right| }}\right) \left( {x, t}\right) }\right| \leq C{\left( 1 + t\right) }^{-\frac{2 + \left| \alpha \right| }{2}}\left( {{B}... | Proof We can write\n\n\[ {D}^{\alpha }\left( {E\left( {x, t}\right) - {\chi }_{3}\left( D\right) {F}_{\left| \alpha \right| }\left( {x, t}\right) }\right) \]\n\n\[ = {D}^{\alpha }{E}_{1}\left( {x, t}\right) + {D}^{\alpha }{E}_{2}\left( {x, t}\right) + {D}^{\alpha }{E}_{3}^{ - }\left( {x, t}\right) + {D}^{\alpha }\left(... | No |
Lemma 3.1 By Theorem 3.1, we know that there is a function which satisfies\n\n\[ \n\parallel K\left( {\cdot, t}\right) {\parallel }_{{L}^{2}} + \parallel K\left( {\cdot, t}\right) {\parallel }_{{L}^{\infty }} \leq C\widetilde{Q}, \n\]\n\n(3.1)\n\nsuch that\n\n\[ \n\left| {{D}_{x}^{\alpha }u\left( {x, t}\right) }\right|... | We list the following lemma which has been proved in [7]. It is important for the propositions in the following. | No |
Proposition 3.4 For any \( \alpha, N \geq n \) and \( \left( {t, x}\right) \in {\mathbb{R}}^{ + } \times {\mathbb{R}}^{2} \), suppose\n\n\[ \left| {{D}_{x}^{\alpha }W\left( {x, t}\right) }\right| \leq C{\left( 1 + t\right) }^{-\frac{2 + \left| \alpha \right| + 1}{2}}\left( {{B}_{N}\left( {x, t}\right) + {A}_{N}\left( {... | Since the proof of these propositions can be found in [9], we omit it here. | No |
Theorem 3.2 Suppose \( {u}_{0},\widetilde{Q}, l \) are given as in Theorem 1.1, then for \( t \) is large enough and \( \left| \alpha \right| \leq {2l} \), we have\n\n\[ \left| {{D}_{x}^{\alpha }E * {u}_{0}}\right| \leq C\widetilde{Q}{\left( 1 + t\right) }^{-\frac{2 + \left| \alpha \right| }{2}}\left( {{B}_{1}\left( {x... | Proof We rewrite \( {D}_{x}^{\alpha }G * {u}_{0} \) as\n\n\[ {D}_{x}^{\alpha }E * {u}_{0} = {D}_{x}^{\alpha }\left( {E - {\chi }_{3}\left( D\right) {F}_{\left| \alpha \right| }\left( {t - s}\right) }\right) * {u}_{0} + {\chi }_{3}\left( D\right) {F}_{\left| \alpha \right| }\left( {t - s}\right) * {D}_{y}^{\alpha }{u}_{... | Yes |
Theorem 3.3 Suppose that the conditions in Theorem 1.1 satisfy, if \( \left| \alpha \right| \leq {2l} - 2 \) and \( t \) is large enough, then\n\n\[ \left| {W}_{2,1}^{\alpha }\right| \leq C\left( {{M}^{2}\left( t\right) + \widetilde{Q}M\left( t\right) }\right) {\left( 1 + t\right) }^{-1 - \nu \left( \left| \alpha \righ... | Proof We have from (3.17),\n\n\[ {W}_{2,1}^{\alpha } = {\int }_{0}^{t}\left( {{f}_{1} + {f}_{2} + {C}_{0}\delta }\right) {\mathrm{e}}^{-b\left( {t - s}\right) } * {D}_{y}^{\alpha }H\left( u\right) \mathrm{d}s.\n\]\nIf we choose \( m = \max \left\{ {{N}^{\prime },1 + l}\right\} \), then \( \left| {{D}_{x}^{\alpha }{f}_{... | Yes |
Lemma 2.1 Let \( \left( {T\left( t\right) ,{T}^{ * }\left( t\right), V\left( t\right), E\left( t\right) }\right) \) be the solution of system (2.3) with initial condition (2.4). Then \( T\left( t\right) ,{T}^{ * }\left( t\right), V\left( t\right), E\left( t\right) > 0,\forall t \geq 0 \) and are ultimately bounded. | Proof From system (2.3), we have\n\n\[ \left\{ \begin{array}{l} T\left( t\right) = T\left( 0\right) {\mathrm{e}}^{-{\int }_{0}^{t}\left( {d + {kV}\left( \xi \right) }\right) \mathrm{d}\xi } + {\int }_{0}^{t}{\lambda }_{T}{\mathrm{e}}^{-{\int }_{\eta }^{t}\left( {d + {kV}\left( \xi \right) }\right) \mathrm{d}\xi }\mathr... | Yes |
(i) If \( {R}_{0} < 1 \), then disease-free equilibrium \( {P}_{0} \) of system (2.3) is locally asymptotically stable for any time delay \( {\tau }_{1},{\tau }_{2} \geq 0 \) . | Proof For the disease-free equilibrium \( {P}_{0} = \left( {{T}_{0},0,0,{E}_{0}}\right) \), Eq.(3.3) reduces to\n\n\[ \left( {\lambda + d}\right) \left( {\lambda + {\delta }_{E}}\right) \left\lbrack {{\lambda }^{2} + \left( {c + \delta + m{E}_{0}}\right) \lambda + c\left( {\delta + m{E}_{0}}\right) - {N\delta }{k}_{1}{... | Yes |
Theorem 3.2 If \( {R}_{0}^{ * } < 1 \), that is, \( {R}_{0} < \delta {\delta }_{E}{\left( \delta {\delta }_{E} + m{\lambda }_{E}\right) }^{-1} \), the disease-free equilibrium \( {P}_{0} \) of system (2.3) is globally attractive for any time delay \( {\tau }_{1} \geq 0,{\tau }_{2} \geq 0 \) . | Proof We show that the disease-free equilibrium \( {P}_{0} \) attracts the nonnegative solutions of (2.3). By the first equation of system (2.3), we have\n\n\[ \dot{T}\left( t\right) \leq {\lambda }_{T} - {dT} \]\n\nthen for any \( \varepsilon > 0 \), there exists \( {t}_{0} > 0 \) such that\n\n\[ T\left( t\right) \leq... | Yes |
Assume that Eq.(3.13) has at least one simple positive root and \( {v}_{1}^{ * } \) is the last such root. Then, \( \mathrm{i}v\left( {\tau }_{1}^{ * }\right) = \mathrm{i}{v}_{1}^{ * } \) is a simple root of Eq.(3.9) and \( u\left( {\tau }_{1}\right) + \mathrm{i}v\left( {\tau }_{1}\right) \) is differentiable with resp... | By differentiating Eq.(3.11) with respect to \( {\tau }_{1} \) . Setting \( u = 0 \) and \( v = {v}_{1}^{ * } \), we have\n\n\[{\left. {A}_{1}\frac{\mathrm{d}u}{\mathrm{\;d}{\tau }_{1}}\right| }_{{\tau }_{1} = {\tau }_{1}^{ * }} - {\left. {B}_{1}\frac{\mathrm{d}v}{\mathrm{\;d}{\tau }_{1}}\right| }_{{\tau }_{1} = {\tau ... | Yes |
Lemma 3.2 \( {}^{\left\lbrack {11}\right\rbrack } \) Suppose that \( {m}_{4} > 0 \). (i) If \( Q \geq 0 \), then Eq.(3.21) has positive roots iff \( {\Omega }_{1} > 0 \) and \( g\left( {\Omega }_{1}\right) < 0 \). (ii) If \( Q < 0 \), then Eq.(3.21) has positive roots iff there exists at least one \( \bar{\Omega } \in ... | \[ {m}_{4} = {p}_{4}^{2} - {q}_{4}^{2} \] \[ = {\left( {a}_{4} + {l}_{4}\right) }^{2} - {q}_{4}^{2} \] \[ = {\left\lbrack c{\delta }_{E}\left( d + k{V}_{1}\right) \left( \delta + m{E}_{1}\right) + c\left( d + k{V}_{1}\right) m{T}_{1}^{ * }{C}_{E}\right\rbrack }^{2} - {\left( d{\delta }_{E}N\delta {k}_{1}{T}_{1}\right) ... | No |
Theorem 3.1 Suppose assumptions (C1)-(C6) hold and \( m \geq q \), define \( {\varepsilon }_{i} = {X}_{i} - \) \( \mu \left( {{Z}_{i},{T}_{i}}\right) \), if \( \mathbf{\Phi } = \mathrm{E}\left\lbrack {{\varepsilon }_{i}{\varepsilon }_{i}^{\mathrm{T}}}\right\rbrack \) is positive definite. Then \( \sqrt{n}\left( {{\wide... | \[ \mathbf{\Omega } = \mathrm{E}{\left\lbrack \left( e - {U}^{\mathrm{T}}\mathbf{\beta }\right) \left( X - {\mathrm{E}}_{\mathcal{F}}\left( \eta \left( Z, T\right) \right) \right) \right\rbrack }^{{ \otimes }^{2}} + \mathrm{E}{\left\lbrack U{U}^{\mathrm{T}}{e}^{2}\rbrack + \mathrm{E}\left( U{U}^{\mathrm{T}} - {\mathbf{... | Yes |
Lemma 5.5 Suppose that Assumptions (C1) - (C6) hold. Then, as \( n \rightarrow \infty \) , \[ \frac{1}{n}{\mathbf{W}}^{\mathrm{T}}\left( {{I}_{n} - {P}_{\mathbf{V}}}\right) \mathbf{W}\overset{P}{ \rightarrow }\mathbf{\Phi } + {\mathbf{\sum }}_{U} \] | Proof of Theorem 3.1 Let \( {\Delta }_{n} = \left\lbrack {{\mathbf{W}}^{\mathrm{T}}\left( {{I}_{n} - {P}_{\mathbf{V}}}\right) \mathbf{W} - n{\mathbf{\sum }}_{U}}\right\rbrack /n \), then \[ {\widehat{\mathbf{\beta }}}_{n} - \mathbf{\beta } \triangleq {\left( n{\Delta }_{n}\right) }^{-1}\left\lbrack {\left( {{\mathbf{\v... | Yes |
Theorem 3.1 Suppose that the assumptions (A1)-(A6) hold, the profile likelihood estimator of \( \beta \) is asymptotically normal, i.e., \[ \sqrt{n}\left( {\widehat{\beta } - \beta }\right) \rightarrow N\left( {0,\Delta }\right) \] | where \( \Delta = {\sigma }^{2}\mathrm{E}{\left\{ \mathrm{E}\left\lbrack \bar{Z}{\bar{Z}}^{\mathrm{T}}\right\rbrack - \mathrm{E}\left\lbrack {R}_{11}^{\mathrm{T}}\left( \mathbf{u}\right) {S}_{11}^{-1}\left( \mathbf{u}\right) {R}_{11}\left( \mathbf{u}\right) \right\rbrack \right\} }^{-1} \) | Yes |
Lemma 5.1 Let \( \left( {{X}_{1},{Y}_{1}}\right) ,\ldots ,\left( {{X}_{n},{Y}_{n}}\right) \) be independent and identically distributed random vectors, where the \( {Y}_{i} \) are scalar random variables. Furthermore, assume that \( \mathrm{E}{\left| y\right| }^{s} < \infty \) and \( \mathop{\sup }\limits_{x}\int {\lef... | This follows immediately from the result obtained by Mack and Sliverman \( {}^{\left\lbrack {15}\right\rbrack } \). | No |
Lemma 5.2 If assumptions (A1)-(A6) hold and each element converges, then | \[ S\left( \mathbf{u}\right) = \frac{1}{n}\left( {{D}^{\mathrm{T}}{WD}}\right) = f\left( \mathbf{u}\right) S\left( \mathbf{u}\right) \left\{ {1 + {o}_{p}\left( 1\right) }\right\} , \] \[ {S}^{-1}\left( \mathbf{u}\right) = {\left( {D}^{\mathrm{T}}WD\right) }^{-1} = \frac{1}{n}{f}^{-1}\left( \mathbf{u}\right) {S}^{-1}\le... | Yes |
Lemma 5.4 If assumptions (A1)-(A6) hold, then\n\n\[ \n{n}^{-1}\widetilde{Z}{\widetilde{Z}}^{\mathrm{T}} \rightarrow \mathrm{E}\left( {\bar{Z}{\bar{Z}}^{\mathrm{T}}}\right) - \mathrm{E}\left\lbrack {{R}_{11}^{\mathrm{T}}\left( \mathbf{u}\right) {S}_{11}^{-1}\left( \mathbf{u}\right) {R}_{11}\left( \mathbf{u}\right) }\rig... | This follows immediately from the result obtained by FAN Jianqin and HUANG Tao \( {}^{\left\lbrack 2\right\rbrack } \). | No |
Corollary 1 Suppose that \( F\left( {t, x}\right) \) satisfies assumption (A) and that there exists \( g \in {L}^{1}\left( {\left\lbrack {0, T}\right\rbrack ,{\mathbb{R}}^{ + }}\right) \), such that \( \left| {\nabla F\left( {t, x}\right) }\right| \leq g\left( t\right) \) for all \( x \in {\mathbb{R}}^{N} \) and a.e. \... | Remark 2 Corollary 1 was first proved by Mawhin and Willem in [1] when \( p = 2 \) . It follows from Theorem 1 easily. | No |
Theorem 2 Assume that \( F\left( {t, x}\right) = {F}_{1}\left( {t, x}\right) + {F}_{2}\left( {t, x}\right) ,{F}_{1} \) and \( {F}_{2} \) satisfy assumption (A) and \( {F}_{1}\left( {t, x}\right) \rightarrow + \infty \) as \( \left| x\right| \rightarrow \infty \) uniformly for a.e. \( t \in \left\lbrack {0, T}\right\rbr... | The sobolev space \( {W}_{T}^{1, p} \) is defined by\n\n\[ {W}_{T}^{1, p} = \left\{ {u : \left\lbrack {0, T}\right\rbrack \rightarrow {\mathbb{R}}^{N}, u}\right. \text{is absolutely continuous,}\]\n\n\[ \left. {u\left( 0\right) = u\left( T\right) \text{ and }\dot{u} \in {L}^{p}\left( {\left\lbrack {0, T}\right\rbrack ,... | Yes |
Lemma 1.2 Let any two of \( f, g \) and \( h \) be weakly compatible self maps of a set \( X \) . If \( f \) , \( g \) and \( h \) have a unique point of coincidence \( w = {fx} = {gx} = {hx} \), then \( w \) is the unique common fixed point of \( f, g \) and \( h \) . | Proof Suppose \( \{ f, g\} \) and \( \{ f, h\} \) are weakly compatible, by \( w = {fx} = {gx} = {hx} \) , we deduce \( {fw} = f\left( {gx}\right) = g\left( {gx}\right) = {gw} \) and \( {fw} = f\left( {hx}\right) = h\left( {hx}\right) = {hw} \), that is to say \( {fw} = {gw} = {hw} \) is a point of coincidence of \( f,... | Yes |
Theorem 2.2 Let \( \left( {X, d}\right) \) be a cone metric space. Suppose that \( {\left\{ {f}_{i}\right\} }_{i \in \mathbb{N}} \) is a family of self-mappings, \( T : X \rightarrow { \cap }_{i \in \mathbb{N}}{f}_{i}\left( X\right) \) and that \( T\left( X\right) \) or \( { \cap }_{i \in \mathbb{N}}{f}_{i}\left( X\rig... | Proof By Theorem 2.1, for any \( i, j, m \in \mathbb{N}, i \neq j, i \neq m,{f}_{i},{f}_{j} \) and \( T \) have a unique common fixed point \( {x}_{ij},{f}_{i},{f}_{m} \) and \( T \) have a unique common fixed point \( {x}_{im} \) . If \( {x}_{ij} \neq {x}_{im} \) , from the definition of \( \varphi \) -contraction, we... | Yes |
Corollary 2.2 Let \( \\left( {X, d}\\right) \) be a cone metric space. Suppose that \( {\\left\{ {f}_{i}\\right\} }_{i \\in \\mathbb{N}} \) is a family of self-mappings, \( T : X \\rightarrow { \\cap }_{i \\in \\mathbb{N}}{f}_{i}\\left( X\\right) \) and that \( T\\left( X\\right) \) or \( { \\cap }_{i \\in \\mathbb{N}}... | \[ d\\left( {{Tx},{Ty}}\\right) \\preccurlyeq \\varphi \\left( \\frac{d\\left( {{f}_{i}x,{Ty}}\\right) + d\\left( {{f}_{j}y,{Tx}}\\right) }{2}\\right) ,\] for every \( x, y \\in X \) . Then \( \\left\{ {{f}_{i}, T}\\right\} \) and \( \\left\{ {{f}_{j}, T}\\right\} \) have a unique point of coincidence in \( X \) . More... | Yes |
Theorem 2.4 Let \( \left( {X, d}\right) \) be a cone metric space. Suppose that \( {\left\{ {f}_{i}\right\} }_{i \in \mathbb{N}} \) is a family of self-mappings, \( T : X \rightarrow { \cap }_{i \in \mathbb{N}}{f}_{i}\left( X\right) \) and that \( T\left( X\right) \) or \( { \cap }_{i \in \mathbb{N}}{f}_{i}\left( X\rig... | Proof By Theorem 2.3 for any \( i, j, m \in \mathbb{N}, i \neq j, i \neq m,{f}_{i},{f}_{j} \) and \( T \) have a unique common fixed point \( {x}_{ij},{f}_{i},{f}_{m} \) and \( T \) have a unique common fixed point \( {x}_{im} \) . If \( {x}_{ij} \neq {x}_{im} \) , from the definition of weakly \( {\varphi }_{m} \) -co... | Yes |
Example 2.1 Let \( X = \lbrack 0, + \infty ), E = {C}_{\mathbb{R}}^{2}\left( \left\lbrack {0,1}\right\rbrack \right) \) with \( \parallel x\parallel = \parallel x{\parallel }_{\infty } + {\begin{Vmatrix}{x}^{\prime }\end{Vmatrix}}_{\infty } \) and \( P = \{ x \in \) \( E : x\left( t\right) \geq 0, t \in \left\lbrack {0... | Taking the function \( \varphi \left( x\right) = \frac{1}{3}x \), for \( x \in X \), all the conditions of Corollary 2.6 are fulfilled. Indeed, since \( T\left( X\right) = \left\lbrack {0,\frac{1}{32}}\right\rbrack ,{ \cap }_{i \in \mathbb{N}}{f}_{i}\left( X\right) = \left\lbrack {0,\frac{1}{4}}\right\rbrack \), we hav... | Yes |
Theorem 2.1 For \( L \) near \( \pi \), there are nontrivial steady state solution branches of (2.2) bifurcated from the trivial solution: | \[ \left\{ \begin{array}{l} u\left( \varepsilon \right) = \varepsilon \sin {\pi x} + {\varepsilon }^{3}\frac{\sigma - \mu }{32\sigma }\sin {3\pi x} + o\left( {\varepsilon }^{3}\right) , \\ v\left( \varepsilon \right) = {\varepsilon }^{2}\left( {\frac{\mu }{2\sigma }\cos {2\pi x} - \frac{\mu }{2\sigma }}\right) + o\left... | Yes |
Lemma 2.1 Let \( 0 < \alpha < n \), then there exists \( C > 0 \) such that\n\n\[{\begin{Vmatrix}{\Theta }_{\bar{h}}\left\lbrack V\right\rbrack w\end{Vmatrix}}_{{L}^{1}} \leq C\left( {\parallel w{\parallel }_{{L}^{1}} + \parallel w{\parallel }_{W\left( {L}^{1}\right) }}\right) \parallel w{\parallel }_{{L}^{1}}\]\n\nand... | Proof Using (2.3), properties of the Fourier transform, the Young inequality for convolution with respect to the \( v \) variable and the Hölder inequality with respect to the \( x \) variable:\n\n\[{\begin{Vmatrix}{\Theta }_{\bar{h}}\left\lbrack V\right\rbrack w\end{Vmatrix}}_{{L}^{1}} \leq C{\begin{Vmatrix}{\mathcal{... | Yes |
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