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Theorem 3.2 Assume that there exists two positive constants \( {r}_{2} > {r}_{1} > 0 \) such that\n\n\[ \left( {\mathrm{H}}_{1}\right) \;f\left( {t, x, y}\right) \leq {L}_{1}{r}_{2},\;\left( {t, x, y}\right) \in \left\lbrack {0,1}\right\rbrack \times \left\lbrack {0,{r}_{2}}\right\rbrack \times \left\lbrack {0,\frac{{r... | Proof Let \( {P}_{1} = \left\{ {x \in E : x\left( t\right) \geq {t}^{\alpha - 1}\parallel x\parallel ,\forall t \in \left\lbrack {0,1}\right\rbrack }\right\} \), it is clear that \( {P}_{1} \) is a cone in \( E \) . From Lemma 4 in [12], we can know that if \( \lambda = 0,{t}^{\alpha - 1}K\left( s\right) \leq G\left( {... | Yes |
Theorem 3.3 Assume that there exist constants \( 0 < a < b < d \leq c \) such that the following assumptions hold:\n\n\( \left( {\mathrm{G}}_{1}\right) \;f\left( {t, x, y}\right) < {L}_{3}a\; \) for \( \;\left( {t, x, y}\right) \in \left\lbrack {0,1}\right\rbrack \times \left\lbrack {0, a}\right\rbrack \times \left\lbr... | Proof Define a nonnegative continuous concave function \( \theta \) on \( P \) as\n\n\[ \theta \left( x\right) = \mathop{\min }\limits_{{\frac{1}{4} \leq x \leq \frac{3}{4}}}x\left( t\right) \]\n\nIf \( x \in \overline{{P}_{c}} = \{ x \in P : \parallel x\parallel \leq c\} \), then \( \parallel x\parallel \leq c \), and... | Yes |
Example 4.1 Let us consider the following differential equation boundary value problem:\n\n\[ \left\{ \begin{array}{l} {D}_{0 + }^{\frac{5}{2}}x\left( t\right) + f\left( {t, x\left( t\right) ,{I}_{0 + }^{\frac{1}{2}}x\left( t\right) }\right) = 0,\;t \in \left( {0,1}\right) , \\ x\left( 0\right) = {x}^{\prime }\left( 0\... | We take\n\n\[ \alpha = \frac{5}{2},\;\beta = \frac{1}{2},\;p = 1,\;q = \frac{1}{2}, \]\n\n\[ \lambda = 0,\;{a}_{1} = \frac{1}{2},\;{a}_{2} = \frac{1}{4},\;{a}_{3} = \frac{1}{3}, \]\n\n\[ {\xi }_{1} = \frac{1}{4},\;{\xi }_{2} = \frac{1}{3},\;{\xi }_{3} = \frac{1}{2}. \]\n\nAfter a simple computation, we have\n\n\[ A = \... | Yes |
Theorem 3.1 Let \( \left( {{z}^{ * },{y}^{ * },{\lambda }^{ * },{p}^{ * }}\right) \) be an efficient solution of (MWCD) and the conditions of Lemma 2.1 hold. Suppose that the following conditions hold,\n\n(i) \( h\left( {{z}^{ * },0}\right) = 0, k\left( {{z}^{ * },0}\right) = 0 \) ;\n\n(ii) \( \left\{ {\nabla {\varphi ... | Proof In view of the generalized Fritz John theorem in [14], since \( \left( {{z}^{ * },{y}^{ * },{\lambda }^{ * },{p}^{ * }}\right) \) is an efficient solution to (MWCD), there exist \( \alpha \in {\mathbb{R}}^{l},\beta \in {\mathbb{R}}^{n},\gamma \in {C}^{ * } \) and \( \delta \in {\mathbb{R}}^{l} \) such that\n\n\[ ... | Yes |
Lemma 1.4 Let \( n \geq 2,\alpha > \frac{n}{n + 2} - \frac{m}{2} - \frac{p}{2} + 1,{\theta }_{1} = \frac{2\left( {k + p - 1}\right) }{p + {2\alpha } + m - 2},{\theta }_{2} = \frac{2\left( {{m}_{1} - 1}\right) \left( {k + p - 1}\right) }{k + p - 3} \) and \( {\kappa }_{i} = \frac{\frac{{m}_{1}}{2} - \frac{{m}_{1}}{{\the... | Proof Since \( 2{m}_{1} > {\theta }_{i}{\kappa }_{i} \) is equivalent to \( {m}_{1} > \frac{{\theta }_{i}}{2} - \frac{2}{n} \), if \( m > \frac{2n}{n + 2} - {2\alpha } - p + 2 \), there exists \( m \) satisfying \( 2{m}_{1} > {\theta }_{i}{\kappa }_{i} \) and \( 2{m}_{1} > k + 1 \), which can be achieved by the fact th... | Yes |
Lemma 3.1 Let \( \varrho \left( \mu \right) = \begin{Vmatrix}{{u}_{\mu }^{\delta }\left( {x, T}\right) - {g}^{\delta }\left( x\right) }\end{Vmatrix} \) and \( 0 < h\left( \delta \right) < \begin{Vmatrix}{g}^{\delta }\end{Vmatrix} \), then we have the following conclusions: (i) For \( \mu \in \left( {0, + \infty }\right... | Proof We easily can prove this Lemma by taking\n\n\[ \varrho \left( u\right) = \begin{Vmatrix}\frac{\mu {\left( 1 + {\left| \xi \right| }^{2}\right) }^{p}{\mathrm{e}}^{{2T}{\left| \xi \right| }^{2\alpha }}\widehat{{g}^{\delta }}\left( \xi \right) }{1 + \mu {\left( 1 + {\left| \xi \right| }^{2}\right) }^{p}{\mathrm{e}}^... | Yes |
Lemma 3.2 Assume that the a-priori bound condition (2.5) is valid, then the regularized solution (2.14) combining with a-posteriori selection rule (3.1) determine that the regularization parameter \( \mu = \mu \left( {\delta ,{g}^{\delta }}\right) \) satisfies \( \frac{1}{\mu } \leq \frac{{E}^{2}}{4{\left( h\left( \del... | Proof From (3.1), there holds\n\n\[ h\left( \delta \right) = \begin{Vmatrix}\frac{\mu {\left( 1 + {\left| \xi \right| }^{2}\right) }^{p}{\mathrm{e}}^{{2T}{\left| \xi \right| }^{2\alpha }}\widehat{{g}^{\delta }}\left( \xi \right) }{1 + \mu {\left( 1 + {\left| \xi \right| }^{2}\right) }^{p}{\mathrm{e}}^{{2T}{\left| \xi \... | Yes |
Theorem 3.1 Suppose that \( u \) given by (2.4) is the exact solution of problem (1.1), \( {u}_{\mu }^{\delta } \) defined by (2.14) is the regularization solution, let the exact data \( g \) and measured data \( {g}^{\delta } \) satisfy (2.9), and the a priori bound (2.5) is satisfied.\n\n(i) If the regularization par... | Proof Using the Parseval theorem, it is clear that\n\n\[ \begin{Vmatrix}{{u}_{\mu }^{\delta }\left( {\cdot, t}\right) - u\left( {\cdot, t}\right) }\end{Vmatrix} \leq \begin{Vmatrix}{\widehat{{u}_{\mu }^{\delta }}\left( {\cdot, t}\right) - \widehat{{u}_{\mu }}\left( {\cdot, t}\right) }\end{Vmatrix} + \begin{Vmatrix}{\wi... | Yes |
Theorem 2.1 Assume that the solution of Problem (2.1) is \( u\left( {x, t}\right) = {\mathrm{e}}^{\lambda t}\cos \left( {nx}\right) \) , where \( \lambda \in \mathbb{C}, n \in {\mathbb{N}}^{ + }, x \in \left\lbrack {0,\pi }\right\rbrack \) and \( t > 0 \) . Then the analytic solution of Problem (2.1) is asymptotically ... | Proof Let \( X = B\left\lbrack {0,\pi }\right\rbrack \) be the Banach space equipped with the maximum norm. Define \( D\left( A\right) = \{ y \in X : \ddot{y} \in X,\dot{y}\left( 0\right) = \dot{y}\left( \pi \right) = 0\} \) and \( {Ay} = \ddot{y} \) for \( y \in D\left( A\right) \) .\n\nLet \( - {r}_{1}{n}^{2}\left( {... | Yes |
Lemma 2.1 Let \( {f}_{\bar{x},\lambda }^{\left( z\right) } \) and \( {f}_{\lambda }^{\left( z\right) } \) be defined as in (2.8) and (2.9) respectively. Then\n\n\[ \n{\begin{Vmatrix}{f}_{\bar{x},\lambda }^{\left( z\right) } - {f}_{\lambda }^{\left( z\right) }\end{Vmatrix}}_{s, q} \leq \frac{1}{\lambda }\parallel {\int ... | Proof (2.11) can be obtained from (1.7) of [16] by taking \( y = {f}^{ * }\left( x\right) \) and \( V\left( t\right) = \) \( \sqrt{1 + {t}^{2}} - 1 \) or it can be obtained from (5.14) in Chapter 5 of [3]. | Yes |
Corollary 2.1 Let \( {f}_{\bar{x},\lambda }^{\left( z\right) } \) be the solution of (2.8) and \( {f}^{ * } \in {L}^{2}\left( {S}^{q}\right) \) . Then there holds almost everywhere the following convergence | \[ \mathop{\lim }\limits_{{n \rightarrow + \infty }}{f}_{\bar{x},\lambda }^{\left( z\right) }\left( x\right) = {f}^{ * }\left( x\right) ,\;x \in {S}^{q}. \] (2.19) Proof (2.19) can be obtained by (2.10) and the fact that \( V\left( t\right) = \sqrt{1 + {t}^{2}} - 1 \rightarrow {0}^{ + } \Leftrightarrow \) \( \left| t\r... | Yes |
Theorem 3.1 Let \( {f}_{\bar{x},\lambda } \) be the solution of (3.7) and \( {f}^{ * } \in {L}^{2}\left( {S}^{q}\right) \) . Then there exists a constant \( C = C\left( {s, q, k}\right) > 0 \) such that\n\n\[ \n{\int }_{{S}^{q}}\frac{{\left( {f}_{\bar{x},\lambda }\left( x\right) - {f}^{ * }\left( x\right) \right) }^{2}... | Proof (3.8) can be obtained by replacing \( \mathcal{B}\left( {z;\gamma }\right) \) in (2.11) and (2.18) with \( {S}^{q} \) . | No |
Theorem 3.1 Let \( {\mathbf{B}}^{n} \in {L}^{\infty }\left( {0, T;\left( {{\mathbf{H}}^{1}\left( {\operatorname{curl};{\Omega }_{k}}\right) }\right) }\right), k = 1,2,3 \), be the solution of the equation (2.4) at time \( t = {t}_{n}, n = 0,1,2,\cdots, N \) . Let \( {\mathbf{B}}_{h}^{n} \) be the finite element solutio... | Proof Denoting \( {\mathbf{B}}^{n} - {\mathbf{B}}_{h}^{n} = {\mathbf{B}}^{n} - {\widetilde{I}}_{h}{\mathbf{B}}^{n} + {\widetilde{I}}_{h}{\mathbf{B}}^{n} - {\mathbf{B}}_{h}^{n} = {\rho }^{n} + {\theta }^{n},{R}_{1}^{n} = {D}_{\tau }{\mathbf{B}}^{n} - {\mathbf{B}}_{t}^{n} \) , taking \( {\mathbf{v}}_{h} = {\theta }^{n} \... | Yes |
Theorem 3.2 Under the assumption of Theorem 3.1, if the thickness parameter is taken by \( \delta = O\left( {h}^{2}\right) \), then there holds\n\n\[ \mathop{\max }\limits_{{1 \leq n \leq N}}{\begin{Vmatrix}{\mathbf{B}}^{n} - {\mathbf{B}}_{h}^{n}\end{Vmatrix}}_{{\mathbf{L}}^{2}\left( \Omega \right) } + \mathop{\sum }\l... | Proof Taking the thickness parameter is taken by \( \delta = O\left( {h}^{2}\right) \), from Lemma 1 and triangle inequality, we can complete the proof. | No |
Lemma 3.1 The complex number \( \lambda \) is an eigenvalue of BVP (2.1)-(2.3) if and only if the equality\n\n\[ \Delta \left( \lambda \right) = \left| {{A}_{1} + {B}_{1}{\Phi }_{1}\left( {{b}_{1},\lambda }\right) \;{A}_{2} + {B}_{2}{\Phi }_{2}\left( {{b}_{2},\lambda }\right) }\right| = 0 \]\n\nholds. | Proof Let\n\n\[ y\left( {x,\lambda }\right) = \left\{ \begin{array}{ll} {c}_{1}{\varphi }_{1} + {c}_{2}{\phi }_{1} + {c}_{3}{\chi }_{1}, & x \in \left\lbrack {{a}_{1},{b}_{1}}\right\rbrack , \\ {c}_{4}{\varphi }_{2} + {c}_{5}{\phi }_{2} + {c}_{6}{\chi }_{2}, & x \in \left\lbrack {{a}_{2},{b}_{2}}\right\rbrack , \end{ar... | Yes |
Theorem 3.1 Let \( {\omega }_{0} = \left( {{a}_{{1}_{0}},{b}_{{1}_{0}},{a}_{{2}_{0}},{b}_{{2}_{0}},{A}_{{1}_{0}},{B}_{{1}_{0}},{A}_{{2}_{0}},{B}_{{2}_{0}},\frac{1}{{p}_{0}},{s}_{0},{q}_{0},{w}_{0}}\right) \in \Omega \), and suppose that \( \mu = \lambda \left( \omega \right) \) is the eigenvalue of BVP (2.1)-(2.3). The... | Proof From Lemma 3.1, \( \mu \) is an eigenvalue of BVP (2.1)-(2.3) if and only if \( \Delta \left( {{\omega }_{0},\mu }\right) = 0 \) . According to the theory of one interval, we know that for any \( \omega \in \Omega ,{\Phi }_{1}\left( {{b}_{1},\lambda \left( \omega \right) }\right) \) and \( {\Phi }_{2}\left( {{b}_... | Yes |
Lemma 3.2 Let \( {\omega }_{0} = \left( {{a}_{{1}_{0}},{b}_{{1}_{0}},{a}_{{2}_{0}},{b}_{{2}_{0}},{A}_{{1}_{0}},{B}_{{1}_{0}},{A}_{{2}_{0}},{B}_{{2}_{0}},\frac{1}{{p}_{0}},{s}_{0},{q}_{0},{w}_{0}}\right) \) . Let \( \lambda = \lambda \left( \omega \right) \) be an eigenvalue of BVP (2.1)-(2.3). If \( \lambda \left( {\om... | Proof The proof can be given similarly as in [9], only to note the case should be extended from one interval case to two-interval case. | No |
Theorem 3.2 Let the hypotheses and notation of Theorem 3.1 hold, and the multiplicity of \( \lambda \left( \omega \right) \) is the highest multiplicity of \( \lambda \left( {\omega }_{1}\right) \) and \( \lambda \left( {\omega }_{2}\right) \) . Let \( M \subset \Omega \) be a neighborhood of \( {\omega }_{0} \) , and ... | Proof 1) Suppose \( \lambda \left( {\omega }_{0}\right) \) is simple, then by Lemma 3.2, there exists a neighborhood \( M \) of \( {\omega }_{0} \) such that \( \lambda \left( \omega \right) \) is simple for all \( \omega \in M \) . For all \( \omega \in M \), choose an eigenfunction \( u = u\left( {\cdot ,\omega }\rig... | No |
Lemma 4.1 Let \( \lambda = \mu \) and \( \lambda = \tau \) be the eigenvalues of BVP (2.1)-(2.3), \( u \) and \( v \) are the eigenfunctions corresponding to \( \mu \) and \( \tau \), respectively, then\n\n\[ \n{\left\lbrack {u}_{1},{v}_{1}\right\rbrack }_{{a}_{1}}^{{b}_{1}} + {\left\lbrack {u}_{2},{v}_{2}\right\rbrack... | Proof According to the method of integration by parts\n\n\[ \n\left( {\mu - \tau }\right) \left( {{\int }_{{a}_{1}}^{{b}_{1}}{u}_{1}\overline{{v}_{1}}{w}_{1} + {\int }_{{a}_{2}}^{{b}_{2}}{u}_{2}\overline{{v}_{2}}{w}_{2}}\right) \n\]\n\n\[ \n= {\int }_{{a}_{1}}^{{b}_{1}}\left\lbrack {\mathrm{i}{\left( {p}_{1}{\left( {p}... | Yes |
Theorem 4.1 Let \( \lambda \left( \omega \right) \) be an eigenvalue of BVP (2.1)-(2.3) with \( \omega \in \Omega \), and \( u = u\left( {\cdot ,\omega }\right) \) be a normalized eigenfunction for \( \lambda \left( \omega \right) \) . Then \( \lambda \) is differentiable with respect to all parameters in \( \omega \),... | 1) Fix all parameters of \( \omega \) except \( {\alpha }_{1} \) and let \( \lambda = \lambda \left( {\alpha }_{1}\right) \) be the eigenvalue of BVP (2.1)- (2.3), and \( u = u\left( {\cdot ,{\alpha }_{1}}\right) \) the normalized eigenfunction. Then\n\n\[ \n{\lambda }^{\prime }\left( {\alpha }_{1}\right) = - {\left| u... | Yes |
Example 4.3 In this example, we consider an irreducible singular M-matrix\n\n\\[ \nA = \\left\\lbrack \\begin{matrix} 3 & - 1 & - 1 & - 1 \\\\ - 1 & 3 & - 1 & - 1 \\\\ - 1 & - 1 & 3 & - 1 \\\\ - 1 & - 1 & - 1 & 3 \\end{matrix}\\right\\rbrack \n\\] | In this case, the square root of \\( A \\) is also unique and both methods have sublinear convergence rate. Numerical results are reported in Tab. 4.3. | No |
Theorem 1.1 Assume that (1.2)-(1.5), \( \left( {\mathrm{H}}_{1}\right) \) and \( \left( {\mathrm{H}}_{2}\right) \) are valid. Moreover, suppose that\n\n\[ s \leq \frac{m}{2} + \frac{1}{2}\;\text{ and }\;{C}_{S} \in (0,1\rbrack .\n\]\n\n(1.9)\n\nIf \( \sigma \) satisfies\n\n\[ \sigma \in \left\lbrack {0,\min \left\{ {\f... | In our paper, when \( m > 3 - \frac{4}{n}, s \leq \frac{m}{2} + \frac{1}{2} \), we have\n\n\[ \frac{S\left( u\right) }{D\left( u\right) } = {\left( u + 1\right) }^{s - m + 1} \leq {\left( u + 1\right) }^{\frac{3}{2} - \frac{m}{2}} < {\left( u + 1\right) }^{\frac{2}{n}}.\n\]\n\n(1.11)\n\nObviously, Theorem 1.1 covers th... | Yes |
Lemma 2.2 Let \( \\left( {u, v, w}\\right) \) be the solution of system (1.1). Then we have\n\n\[ \n{\\int }_{\\Omega }u\\left( {x, t}\\right) \\mathrm{d}x \\leq {m}^{ * } \\mathrel{\\text{:=}} \\max \\left\\{ {{\\int }_{\\Omega }{u}_{0},\\frac{a + b}{b}\\left| \\Omega \\right| }\\right\\}, t \\in \\left( {0,{T}_{\\max... | Proof Integrating the first equation of the system (1.1) over \( \\Omega \), we deduce\n\n\[ \n\\frac{\\mathrm{d}}{\\mathrm{d}t}{\\int }_{\\Omega }u\\mathrm{\\;d}x = a\\left| \\Omega \\right| - b{\\int }_{\\Omega }{u}^{\\eta }\\mathrm{d}x.\n\]\n\n(2.4)\n\nDue to \( \\eta > 1 \) and Young’s inequality, we derive\n\n\[ \... | Yes |
Lemma 3.2 Under Assumptions (1.2)-(1.5),(1.9),( \( {\mathrm{H}}_{1} \) ) and \( \left( {\mathrm{H}}_{2}\right) \), we claim \( \phi \left( y\right) \in \) \( {C}^{1}\left\lbrack {{D}_{1},\infty }\right) \), which is nonnegative (where \( {D}_{1} \) is defined in Lemma 2.5), and there exists a constant \( G > 0 \) satis... | Proof Multiplying the both sides of the second equation in (1.1) by \( {\left( u + 1\right) }^{k + {2s} - m - 1}\phi \left( v\right) \) and integrate over \( \Omega \), we obtain\n\n\[ \n{\int }_{\Omega }{\left( u + 1\right) }^{k + {2s} - m - 1}\phi \left( v\right) \left( {{\Delta v} - {\alpha v} + {\beta u}}\right) \m... | Yes |
1) If \( \chi \left( v\right) \) satisfies\n\n\[ \chi \left( v\right) \leq \frac{{\chi }_{0}}{v},\;v \geq {D}_{1} > 0 \]\n\n(3.13)\n\nthen for \( k \in \left( {m - {2s} + 3,\frac{1}{{\chi }_{0}}}\right) \), there exists a constant \( {C}_{4} \) such that\n\n\[ \frac{\mathrm{d}}{\mathrm{d}t}{\int }_{\Omega }{\left( u + ... | Proof Since \( s \leq \frac{m}{2} + \frac{1}{2} \), we deduce that \( m - {2s} + 3 > 0 \). \n\n1) Let \( k \in \left( {m - {2s} + 3,\frac{1}{{\chi }_{0}}}\right) \), so that \( {\chi }_{0} < \frac{1}{k} \), then we pick \( \mu \) and \( {\chi }_{0} \) such that\n\n\[ \mu < {C}_{S}^{2}{C}_{D}k\left( {k - 1}\right) ,\;{\... | Yes |
Lemma 3.4 Under Assumptions (1.2)-(1.5),(1.9),( \( {\mathrm{H}}_{1} \) ) and \( \left( {\mathrm{H}}_{2}\right) \) . From Lemma 3.3, we have\n\n\[ \frac{\mathrm{d}}{\mathrm{d}t}{\int }_{\Omega }{\left( u + 1\right) }^{k}\mathrm{\;d}x + {D}_{3}{\int }_{\Omega }{\left| \nabla {\left( u + 1\right) }^{\frac{k + m - 1}{2}}\r... | Proof If \( \eta \leq m \), when \( \sigma \in \lbrack 0,\eta ) \), by Lemma and Gagliardo-Nirenberg inequality, there exist \( {C}_{5},{C}_{6} > 0 \) such that\n\n\[ {\int }_{\Omega }{\left( u + 1\right) }^{k + m - 1}\mathrm{\;d}x = {\begin{Vmatrix}{\left( u + 1\right) }^{\frac{k + m - 1}{2}}\end{Vmatrix}}_{{L}^{2}\le... | Yes |
Lemma 3.10 Under Assumptions (1.2)-(1.5),(1.9), \( \left( {\mathrm{H}}_{1}\right) \) and \( \left( {\mathrm{H}}_{2}\right) \). From Lemma 3.3, we have\n\n\[ \n\frac{\mathrm{d}}{\mathrm{d}t}{\int }_{\Omega }{\left( u + 1\right) }^{k}\mathrm{\;d}x + {D}_{3}{\int }_{\Omega }{\left| \nabla {\left( u + 1\right) }^{\frac{k +... | Proof By the Gagliardo-Nirenberg inequality, there exist \( {C}_{11},{C}_{12} > 0 \) such that\n\n\[ \n\int {\left( u + 1\right) }^{k + \sigma - 1}\mathrm{\;d}x = {\begin{Vmatrix}{\left( u + 1\right) }^{\frac{k + m - 1}{2}}\end{Vmatrix}}_{{L}^{\frac{2\left( {k + \sigma - 1}\right) }{k + m - 1}}\left( \Omega \right) }^{... | Yes |
Lemma 2.2 Let \( u \) be a radially symmetric \( {C}_{0}^{1} \) function on \( {W}_{F} = {W}_{1}\left( 0\right) \) which is the unit Wulff ball with center at 0 . Then one has\n\n(i) \( \left| {u\left( x\right) }\right| \leq \frac{{\left| \log {F}^{0}\left( x\right) \right| }^{\frac{1 - \beta }{2}}}{\sqrt{2{k}_{2}\left... | Proof (i) Let \( u\left( x\right) = v\left( {{F}^{0}\left( x\right) }\right) \) . Then\n\n\[ \parallel u{\parallel }_{{\omega }_{0}} = {\left( {\int }_{{W}_{F}}{\left| F\left( \nabla u\left( x\right) \right) \right| }^{2}{\omega }_{0}\left( x\right) \mathrm{d}x\right) }^{1/2} \]\n\n\[ = {\left( {\int }_{{W}_{F}}{\left|... | Yes |
Lemma 3.1(Leckband’s inequality) \( {}^{\left\lbrack {24}\right\rbrack } \) Let \( f \in {L}^{N}\left( {\lbrack 0, + \infty }\right) ) \) such that \( \parallel f{\parallel }_{N} = 1,\varphi \) : \( {\mathbb{R}}^{ + } \rightarrow {\mathbb{R}}^{ + } \) with \( \varphi \geq 0 \) is locally integrable, and set\n\n\[ \nG\l... | Now we complete the proof of the critical case \( \bar{\alpha } = 1 \), i.e. \( \alpha = {\alpha }_{\beta } \), by applying Leckband’s inequality to \( f\left( t\right) = {\psi }^{\prime }\left( t\right) {\left( \frac{{t}^{\beta }}{1 - \beta }\right) }^{1/2},\varphi \left( t\right) = {\left( \frac{{t}^{\beta }}{1 - \be... | Yes |
Lemma 3.5 Let \( {\varphi }^{n} \geq 0, n = 1,2,\cdots, N,{\varphi }^{0} = 0,\mu \) is a positive constant and there exist positive constants \( {\lambda }_{1},{\lambda }_{2} \) such that \( 0 < {\lambda }_{1} \leq \lambda \left( \mathbf{x}\right) \leq {\lambda }_{2} \), which satisfies \( {\widetilde{B}}_{0}\lambda \l... | Proof The result (3.3) can be proved by mathematical induction. Obviously, the case \( n = 1 \) is trivial. Suppose the result (3.3) is true for all \( n = 1,2,\cdots, k \), then we need to prove that they hold also for \( n = k + 1\left( {0 \leq k \leq N - 1}\right) \) . By means of (2.3) and \( - \mathop{\sum }\limit... | Yes |
Theorem 5.1 Assume that \( {u}^{n},{U}^{n} \) be solutions of (2.1) and (2.5) at \( t = {t}_{n} \), respectively. If \( u,{u}_{t} \in {H}^{2}\left( \omega \right) ,{u}_{tt} \in {L}^{2}\left( \omega \right) \), we get\n\n\[ \n{\begin{Vmatrix}{u}^{n} - {U}^{n}\end{Vmatrix}}_{0} \leq C{h}^{2}\left( {{\begin{Vmatrix}{u}_{t... | Proof By (2.1) and (2.5), we obtain the following error equation:\n\n\[ \n\left( {{\widetilde{P}}_{\alpha ,{\alpha }_{1},{\alpha }_{2},\cdots ,{\alpha }_{r}}\left( {\widetilde{D}}_{t}\right) \left( {{u}^{n} - {U}^{n}}\right) ,{v}_{h}}\right) + \left( {\omega \left( \mathbf{x}\right) \nabla \left( {{u}^{n} - {U}^{n}}\ri... | Yes |
Lemma 2.1 It holds that for \( t > 0,0 < \lambda < 1 \)\n\n1) \( {g}_{\lambda }\left( t\right) = 1,{g}_{\lambda }^{\prime }\left( t\right) = 0 \) if \( 0 < t < \frac{1}{\lambda } \) ;\n\n2) \( {g}_{\lambda }^{\prime }\left( t\right) t \leq {g}_{\lambda }\left( t\right) \leq \frac{{c}_{\lambda }}{t} \), where \( {c}_{\l... | Proof The lemma can be obtained by direct calculation. | No |
Lemma 2.3 Assume the condition (V) holds, then the imbedding \( {W}_{V}^{1, p}\left( {\mathbb{R}}^{N}\right) \hookrightarrow \) \( {L}^{q}\left( {\mathbb{R}}^{N}\right) \left( {p \leq q < {p}^{ * }}\right) \) is compact. | Proof This lemma can be proved by almost the same way as that of Lemma 5.1 in [22]. So we omit it. | No |
Lemma 3.1 It holds that for \( u, v \in {W}_{V}^{1, p}\left( {\mathbb{R}}^{N}\right) \)\n\n1) \( \parallel {DJ}\left( u\right) - {DJ}\left( v\right) \parallel \leq c\parallel u - v{\parallel }_{{W}_{V}^{1, p}\left( {\mathbb{R}}^{N}\right) }^{p - 1} \) for \( 1 < p < 2 \), \n\n\( \parallel {DJ}\left( u\right) - {DJ}\lef... | Proof 1) Let \( \varphi \in {W}_{V}^{1, p}\left( {\mathbb{R}}^{N}\right) \) . We have\n\n\[ \langle {DJ}\left( u\right) - {DJ}\left( v\right) ,\varphi \rangle \]\n\n\[ = {\int }_{{\mathbb{R}}^{N}}\left( {{\left| \nabla u\right| }^{p - 2}\nabla u - {\left| \nabla v\right| }^{p - 2}\nabla v}\right) \nabla \varphi \mathrm... | Yes |
Lemma 3.5 For \( 0 < \lambda < 1 \), there exists \( {\delta }_{0} > 0 \) such that for \( 0 < \delta < {\delta }_{0}, A\left( {\partial P}\right) \subset \) \( P, A\left( {\partial Q}\right) \subset Q. \) | Proof We only prove \( A\left( {\partial P}\right) \subset P \) . For \( u \in \partial P \), let \( v = {Au} \) . Taking \( \varphi = {v}^{ + } \) in the equation (3.2), we have\n\n\[ \n{\begin{Vmatrix}{v}^{ + }\end{Vmatrix}}_{{L}^{qr}\left( {\mathbb{R}}^{N}\right) }^{p} \leq c{\begin{Vmatrix}{v}^{ + }\end{Vmatrix}}_{... | Yes |
Lemma 3.6 There exist \( {\delta }_{0} \) and \( {c}^{ * } = c\left( \delta \right) \) such that \( {\Gamma }_{\lambda }\left( u\right) \geq {c}^{ * } \), for \( u \in \partial P \cap \partial Q \) . | Proof For \( u \in \partial P \cap \partial Q \), we have\n\n\[ \n{\Gamma }_{\lambda }\left( u\right) \geq I\left( u\right) = \frac{1}{p}{\int }_{{\mathbb{R}}^{N}}\left( {{\left| \nabla u\right| }^{p} + V\left( x\right) {\left| u\right| }^{p}}\right) \mathrm{d}x - \frac{1}{2q}{\int }_{{\mathbb{R}}^{N}}{\int }_{{\mathbb... | Yes |
Theorem 3.1 Assume (V) holds. Then the functional \( {\Gamma }_{\lambda },\lambda \in \left( {0,1}\right) \) has infinitely many sign-changing critical points, the corresponding critical values are defined as\n\n\[ \n{c}_{j} = \mathop{\inf }\limits_{{E \in {\Gamma }_{j}}}\mathop{\sup }\limits_{{u \in E \smallsetminus O... | Proof 1) All the assumptions are fulfilled and it’s easy to prove that \( \left\{ {c}_{j}\right\} \) is increasing and \( {E}_{j} = {\varphi }_{j + 1}\left( {B}_{j + 1}\right) \in {\Gamma }_{j} \) . For \( t \in {B}_{j + 1}, u = {\varphi }_{j + 1}\left( t\right) \), we have \( {g}_{\lambda }\left( {{\psi }^{\frac{1}{2}... | Yes |
Theorem 5.1 Let \( \left\{ {{u}^{n},{\overrightarrow{p}}^{n}}\right\} \) and \( \left\{ {{U}^{n},{\overrightarrow{P}}^{n}}\right\} \) be solutions of (3.2) and (5.2), respectinely. Suppose that \( u \in {H}^{2}\left( \Omega \right) ,{u}_{t} \in {H}^{3}\left( \Omega \right) ,\overrightarrow{p},\overrightarrow{{p}_{t}} \... | Proof \( \;\forall {v}^{h} \in {V}^{h},{\overrightarrow{w}}^{h} \in {\overrightarrow{W}}^{h} \), by (3.2) and (5.2), we gain the following error equations\n\n\[ \n\left( {\nabla {\bar{\partial }}_{t}{\xi }^{n},\nabla {v}^{h}}\right) = \left( {{\overrightarrow{\theta }}^{n - \frac{1}{2}},\nabla {v}^{h}}\right) + \left( ... | No |
Proposition 2.1 The function \( F \) defined by (1.3) is finite everywhere, convex and differentiable; its gradient is \( g\left( x\right) = \nabla F\left( x\right) = \frac{x - p\left( x\right) }{\lambda } \). Furthermore, there holds for all \( x \) and \( y \) in \( {\mathbb{R}}^{n} \): | \[ \parallel g\left( x\right) - g\left( y\right) \parallel \leq \frac{\parallel x - y\parallel }{\lambda }. \] | Yes |
Lemma 2.1 Let \( {p}^{\alpha }\left( {x,\varepsilon }\right) \) be a vector satisfying (2.3). If \( {F}^{\alpha }\left( {x,\varepsilon }\right) \) and \( {g}^{\alpha }\left( {x,\varepsilon }\right) \) are defined by (2.4) and (2.5), respectively, then we get\n\n(i) \( F\left( x\right) \leq {F}^{\alpha }\left( {x,\varep... | (2.8)\n\nThe above lemma shows that the approximations \( {F}^{\alpha }\left( {x,\varepsilon }\right) \) and \( {g}^{\alpha }\left( {x,\varepsilon }\right) \) may be arbitrarily close to \( F\left( x\right) \) and \( g\left( x\right) \), respectively, if the parameter \( \varepsilon \) is small enough. | Yes |
Lemma 4.1 Algorithm 3.1 is well defined, i.e., at the \( k \) -th iteration of the algorithm, the stepsize \( {\alpha }_{k} \) can be determined finitely in Step 3. | Proof It suffices to show that at the \( k \) -th iteration, there exists an \( {\widetilde{\alpha }}_{k} > 0 \) such that\n\n\[ \n{F}^{\alpha }\left( {{x}_{k} + \alpha {d}_{k},{\varepsilon }_{k + 1}}\right) \leq {F}^{\alpha }\left( {{x}_{k},{\varepsilon }_{k}}\right) - \sigma {\alpha }^{2}{\begin{Vmatrix}{d}_{k}\end{V... | Yes |
Lemma 4.2 Let \( \left\{ {x}_{k}\right\} \) be the sequence generated by Algorithm 3.1. Then we have \( {x}_{k} \in {\mathcal{L}}_{0} \) for all \( k \) . Furthermore, if Assumption A holds, then\n\n\[ \mathop{\sum }\limits_{{k = 0}}^{{+\infty }}{\alpha }_{k}^{2}{\begin{Vmatrix}{d}_{k}\end{Vmatrix}}^{2} < \infty \] | Proof we prove the first assertion by induction. It is clear that \( {x}_{0} \in {\mathcal{L}}_{0} \) . Assume that \( {x}_{k} \in {\mathcal{L}}_{0} \) . Then, it follows from (2.6) and the line search scheme (3.1) that\n\n\[ F\left( {x}_{k + 1}\right) \leq {F}^{\alpha }\left( {{x}_{k + 1},{\varepsilon }_{k + 1}}\right... | Yes |
Lemma 4.3 Suppose that \( {\varepsilon }_{k} = O\left( {{\alpha }_{k}^{2}{\begin{Vmatrix}{d}_{k}\end{Vmatrix}}^{2}}\right) \) . Then there exist positive constants \( {c}_{1},{c}_{2} \) and a positive integer \( {k}_{0} > 0 \) such that\n\n\[ \frac{{z}_{k}^{\mathrm{T}}{s}_{k}}{{\begin{Vmatrix}{s}_{k}\end{Vmatrix}}^{2}}... | Proof By (2.12) and (2.13), we have that\n\n\[ \frac{{z}_{k}^{\mathrm{T}}{s}_{k}}{{\begin{Vmatrix}{s}_{k}\end{Vmatrix}}^{2}} \geq \frac{t{\begin{Vmatrix}{s}_{k}\end{Vmatrix}}^{2}}{{\begin{Vmatrix}{s}_{k}\end{Vmatrix}}^{2}} = t \]\n\n(4.10)\n\nand\n\n\[ \begin{Vmatrix}{z}_{k}\end{Vmatrix} \leq \begin{Vmatrix}{y}_{k}\end... | Yes |
Lemma 4.4 Suppose that Assumption A holds. If \( {\varepsilon }_{k} = O\left( {{\alpha }_{k}^{2}{\begin{Vmatrix}{d}_{k}\end{Vmatrix}}^{2}}\right) \), then there exists an integer \( {k}_{1} > 0 \) such that\n\n\[ \frac{1}{n{c}_{2}}\begin{Vmatrix}{{g}^{\alpha }\left( {{x}_{k},{\varepsilon }_{k}}\right) }\end{Vmatrix} \l... | Proof By using the formulas (2.10) and (2.11), we can easily compute the trace of matrices \( {\widetilde{B}}_{k} \) and \( {\widetilde{H}}_{k} \), respectively, as follows\n\n\[ \operatorname{trace}\left( {\widetilde{B}}_{k}\right) = n\frac{{\begin{Vmatrix}{z}_{k - 1}\end{Vmatrix}}^{2}}{{z}_{k - 1}^{\mathrm{T}}{s}_{k ... | Yes |
Theorem 4.1 Let \( \\left\\{ {x}_{k}\\right\\} \) be an infinite sequence generated by Algorithm 3.1. Suppose that Assumption A holds. If \( {\\varepsilon }_{k} = O\\left( {{\\alpha }_{k}^{2}{\\begin{Vmatrix}{d}_{k}\\end{Vmatrix}}^{2}}\\right) \), then\n\n\[ \n\\mathop{\\lim }\\limits_{{k \\rightarrow + \\infty }}\\beg... | Proof From (4.21), it follows that\n\n\[ \n{F}^{\\alpha }\\left( {{x}_{k},{\\varepsilon }_{k}}\\right) - {F}^{\\alpha }\\left( {{x}_{k + 1},{\\varepsilon }_{k + 1}}\\right) + {\\varepsilon }_{k} \\geq \\eta {\\begin{Vmatrix}{g}^{\\alpha }\\left( {x}_{k},{\\varepsilon }_{k}\\right) \\end{Vmatrix}}^{2},\\forall k \\geq {... | Yes |
Theorem 3.1 Let \( g \) satisfy \( \left( {\mathrm{G}}_{1}\right) - \left( {\mathrm{G}}_{3}\right), F\left( {t,{u}_{t}}\right) \) subject to assumptions \( \left( {\mathrm{F}}_{1}\right) - \left( {\mathrm{F}}_{4}\right), h \in \) \( {L}_{\text{loc }}^{2}\left( {\mathbb{R};H}\right) \) and satisfy \( \left( {1.12}\right... | For the proof of Theorem 3.1, we can refer to [8-9, 20] for details. | No |
Lemma 3.1 Let \( g \) satisfy \( \left( {\mathrm{G}}_{1}\right) - \left( {\mathrm{G}}_{3}\right), F\left( {t,{u}_{t}}\right) \) subject to assumptions \( \left( {\mathrm{F}}_{1}\right) - \left( {\mathrm{F}}_{4}\right), h \in \) \( {L}_{\text{loc }}^{2}\left( {\mathbb{R};H}\right) \) and satisfy (1.12). Then the solutio... | Proof Taking the scalar product in \( H \) of (1.1) with \( v = {\partial }_{t}u + {\sigma u}\left( {\sigma > 0}\right) \), we find that\n\n\[ \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\left( {{\left| v\right| }^{2} + \parallel {\Delta u}{\parallel }^{2}}\right) + \sigma \parallel {\Delta u}{\parallel }^{2} + \left( {\n... | Yes |
Theorem 3.2 Let assumptions of Lemma 3.1 be in force. Then the family \( {\widehat{D}}_{1} = \) \( \left\{ {{D}_{1}\left( t\right) ;t \in \mathbb{R}}\right\} \) with \( {D}_{1}\left( t\right) = {\widehat{B}}_{{C}_{\mathcal{H}}}\left( {0,\rho \left( t\right) }\right) \), the closed ball in \( {C}_{\mathcal{H}} \) of cen... | Proof That \( {\widehat{D}}_{1} \) is pullback \( {\mathcal{D}}_{{\alpha }_{1}} \) -absorbing set for the problem (1.1) is an immediate consequence of (3.1) in Lemma 3.1.\n\nThanks to (3.14), we have \( {\mathrm{e}}^{{\alpha }_{1}t}{\rho }^{2}\left( t\right) \rightarrow 0 \), as \( t \rightarrow - \infty \) . Then \( {... | Yes |
Theorem 4.1 Let \( t = \left( {{x}_{{m}_{1} + 1},{x}_{{m}_{1} + 2},\cdots ,{x}_{{m}_{2}}}\right) \) be Type-I doubly censored samples from the Topp-Leone distribution (2.1), if the prior distribution of \( \theta \) is given by (4.2), the following conclusions can be obtained.\n\n(i) Under the squared error loss functi... | Proof (i)Under the squared loss function, the Bayesian estimation of \( \theta \) is the mean value of its posterior distribution. So we have\n\n\[ \n{\widehat{\theta }}_{B1} = {\int }_{0}^{\infty }{\theta \pi }\left( {\theta \mid t}\right) \mathrm{d}\theta \n\]\n\n\[ \n= {B}^{-1}\mathop{\sum }\limits_{{j = 0}}^{{n - {... | Yes |
Theorem 4.2 Let \( t = \left( {{x}_{{m}_{1} + 1},{x}_{{m}_{1} + 2},\cdots ,{x}_{{m}_{2}}}\right) \) be Type-I doubly censored samples from the Topp-Leone distribution (2.1), if the prior distribution of \( \theta \) is given by (4.2), we can obtain the following conclusions:\n\n(i) Under the squared error loss function... | Proof (i) Under the squared loss function, the Bayesian estimation of \( R\left( x\right) \) is\n\n\[ \n{\widehat{R}}_{B1}\left( x\right) = {\int }_{0}^{\infty }R\left( x\right) \pi \left( {\theta \mid t}\right) \mathrm{d}\theta\n\]\n\n\[ \n= 1 - {B}^{-1}\mathop{\sum }\limits_{{j = 0}}^{{n - {m}_{2}}}{\left( -1\right) ... | Yes |
Theorem 2.1 Let Assumptions 1.1,1.2 hold and \( \alpha \geq \beta \) . The constant \( {\rho }_{1} = \rho \left( {\alpha ,\beta }\right) \) is defined in Lemma 2.1. If\n\n\[ \frac{a}{1 + {\rho }_{1}} \land b > K \]\n\n\( \left( {2.2}\right) \)\n\nthere exists a unique global solution to Eq.(1.1), denoted by \( x\left( ... | Proof Let \( y = \mathbf{0} \), by (1.5), so we have \( {\left| g\left( x\right) \right| }^{2} \leq K\left( {1 + {\left| x\right| }^{\beta }}\right) {\left| x\right| }^{2} \) . This, together with (1.4), yields\n\n\[ 2{x}^{\mathrm{T}}f\left( x\right) + {\left| g\left( x\right) \right| }^{2} \leq - a{\left| x\right| }^{... | Yes |
Lemma 2.2 For a fixed \( \theta \in \lbrack 0,1/2) \) and \( h \in \left( {0,1}\right) \), let \( {\left\{ {X}_{k}\right\} }_{k \geq 0} \) be defined by (2.5). Then, for any initial data with \( {X}_{0} \neq 0 \), we have \( \mathbb{P}\left( {\left| {X}_{k}\right| \geq \frac{4{B}^{k}}{\sqrt{h}},\forall k \geq 1}\right)... | Proof First, let us deduce that for \( k \geq 1 \) , \[ \left| {X}_{k}\right| \geq \frac{4{B}^{k}}{\sqrt{h}}\text{ and }\left| {\Delta {w}_{k}}\right| \leq {B}^{k} \Rightarrow \left| {X}_{k + 1}\right| \geq \frac{4{B}^{k + 1}}{\sqrt{h}}. \] (2.6) By (2.5), we have \[ \left| {X}_{k + 1}\right| + {\theta h}\left| {X}_{k ... | Yes |
For any \( 0 < h < {h}^{ * } \) and \( \theta \in (1/2,1\rbrack \), let\n\n\[ \n{I}_{1} = {\left( \frac{1 - \theta }{\theta }\right) }^{2}\text{ and }{I}_{2} = \frac{K + \frac{3}{2}n{K}^{2}h - b\left( {1 - \theta }\right) }{b\theta }, \n\n\[ \n{I}_{3} = \left\lbrack {1 - a\left( {1 - \theta }\right) h + {Kh}\left( {1 +... | Proof If \( \theta = 1 \), by \( h < {h}^{ * } \) and the condition (2.2), we have \( {I}_{1} = 0 \) and \( 0 < {I}_{2},{I}_{3} < 1 \). \n\nIf \( \theta \in \left( {1/2,1}\right) \), we have \( 0 < {I}_{1} < 1 \) and \( {I}_{2},{I}_{3} < 1 \). \n\nThus, for any \( \theta \in (1/2,1\rbrack \), we have \( 0 < {I}_{1} \ve... | Yes |
\[ \left\{ \begin{array}{l} \mathrm{d}x\left( t\right) = \left\lbrack {-\frac{3}{2}x\left( t\right) + \frac{1}{2}{x}^{3}\left( t\right) - 2{x}^{5}\left( t\right) }\right\rbrack \mathrm{d}t + \left\lbrack {x\left( t\right) + \frac{1}{2}{x}^{2}\left( t\right) }\right\rbrack \mathrm{d}w\left( t\right) ,\;t > 0, \\ x\left(... | Let \( f\left( x\right) = - \frac{3}{2}x + \frac{1}{2}{x}^{3} - 2{x}^{5} \), and \( g\left( x\right) = x + \frac{1}{2}{x}^{2} \) . Eq. (1.1) can be rewritten as Eq. (1.1). Clearly, \( f \) and \( g \) satisfy Assumption 1.1.\n\nBy Lemma 2.1, we can compute that\n\n\[ {2xf}\left( x\right) = - 3{x}^{2} + {x}^{4} - 4{x}^{... | Yes |
Consider the following matrices:\n\n\[ \n{A}_{1} = \left\lbrack \begin{matrix} 4 & - 1 & 0 & 0 \\ - 3 & 3 & - 1 & - 1 \\ - 1 & - {0.1} & 3 & - 3 \\ 0 & 0 & - 2 & 4 \end{matrix}\right\rbrack \text{ and }{A}_{2} = \left\lbrack \begin{matrix} 3 & - 1 & 0 & 0 \\ - 3 & 2 & 0 & 0 \\ 0 & - 3 & 2 & 0 \\ 0 & 0 & - 2 & 2 \end{ma... | Obviously, \( {A}_{1} \) is a \( Z \) -matrix, so \( {B}^{ + } = {A}_{1} \) . Let \( {B}^{ + } = \left\lbrack {b}_{ij}\right\rbrack \) . Then \( \left\lbrack {\left| {b}_{11}\right| - {r}_{1}^{2}\left( {B}^{ + }\right) }\right\rbrack \left| {b}_{22}\right| = \) \( {12} > 5 = \left| {b}_{12}\right| {r}_{2}\left( {B}^{ +... | Yes |
Lemma 3.1 If \( \left( {u, v}\right) \in E \smallsetminus \{ \left( {0,0}\right) \} \) is a weak solution of (EQ). Then \( \left( {u, v}\right) \in \mathrm{N} \) . | Proof The proof is following from the definition of \( \mathrm{N} \) . | No |
Lemma 3.2 Assume that \( \left( {\mathrm{v}}_{1}\right) - \left( {\mathrm{v}}_{3}\right) \) and \( \left( {\mathrm{f}}_{1}\right) - \left( {\mathrm{f}}_{5}\right) \) hold. Then\n\n1) \( \mathrm{N} \) is a \( {C}^{1} \) manifold;\n\n2) For any \( \left( {u, v}\right) \in \mathrm{N} \), there exists \( \rho > 0 \) such t... | Proof (1) It follows from \( \left( {\mathrm{f}}_{5}\right) \) that for all \( t \neq 0 \)\n\n\[{f}^{\prime }\left( t\right) {t}^{2} - f\left( t\right) t > 0,{g}^{\prime }\left( t\right) {t}^{2} - g\left( t\right) t > 0.\]\n\n(3.2)\n\nFor \( \left( {u, v}\right) \in \mathrm{N} \) we get\n\n\[ \parallel \left( {u, v}\ri... | Yes |
Lemma 3.3 Assume that \( \left( {\mathrm{v}}_{1}\right) - \left( {\mathrm{v}}_{3}\right) \) and \( \left( {\mathrm{f}}_{1}\right) - \left( {\mathrm{f}}_{5}\right) \) hold. For any \( \left( {u, v}\right) \in E \smallsetminus \{ \left( {0,0}\right) \} \) , there is a unique \( {t}_{0} > 0 \) such that \( \left( {{t}_{0}... | Proof It follows from the assumption \( \left( {\mathrm{f}}_{4}\right) \) that\n\n\[ \mathop{\lim }\limits_{{\left| t\right| \rightarrow + \infty }}\frac{F\left( t\right) }{{t}^{2}} = + \infty ,\mathop{\lim }\limits_{{\left| t\right| \rightarrow + \infty }}\frac{G\left( t\right) }{{t}^{2}} = + \infty . \]\n\n(3.6)\n\nF... | Yes |
Lemma 3.4 Suppose that \( \left( {\mathrm{f}}_{5}\right) \) holds. Then\n\n\[ \n{c}_{N} > 0\text{.} \n\] | Proof Define \( H\left( t\right) = f\left( t\right) t - {2F}\left( t\right) \) and \( K\left( t\right) = g\left( t\right) t - {2G}\left( t\right) \) . We claim that\n\n\[ \nH\left( t\right) > 0, K\left( t\right) > 0,\;\text{ for any }t \in \mathbb{R} \smallsetminus \{ 0\} . \n\]\n\n(3.8)\n\nIn fact, it follows from the... | Yes |
Lemma 3.6 Assume that \( \left( {\mathrm{v}}_{1}\right) - \left( {\mathrm{v}}_{3}\right) \) and \( \left( {\mathrm{f}}_{1}\right) - \left( {\mathrm{f}}_{6}\right) \) hold. Then there exists a nontrivial ground state solution for (EQ). | Proof It follows from \( \left( {\mathrm{f}}_{6}\right) \) that, for all \( \left( {u, v}\right) \in E \)\n\n\[ I\left( {\left| u\right| ,\left| v\right| }\right) \leq I\left( {u, v}\right) \]\n\nLet \( \left( {{u}_{0},{v}_{0}}\right) \in \mathrm{N} \) be the ground state obtained in Lemma 3.5. In light of Lemma 3.3, t... | Yes |
Lemma 4.1 Assume that \( \left( {\mathrm{f}}_{3}\right) \) holds. Then there exists \( {\mu }_{0} > 0 \) such that \( {c}_{N} < \frac{2}{N}{S}^{\frac{N}{4}} \) for all \( \mu > {\mu }_{0} \) . | Proof Let \( \left( {u, v}\right) \in E \) such that \( u, v \geq 0 \) and \( u, v ≢ 0 \) . It follows from Lemma 3.3 that there exists \( {t}_{0} > 0 \) such that \( \left( {{t}_{0}u,{t}_{0}v}\right) \in \mathrm{N} \) . Then we have\n\n\[ \parallel \left( {u, v}\right) {\parallel }^{2} - 2{\int }_{{\mathbb{R}}^{N}}\la... | Yes |
Lemma 4.3 For \( \left( {u, v}\right) \in E \smallsetminus \{ \left( {0,0}\right) \} \), there exists \( \widetilde{t} > 0 \) such that\n\n\[ T\left( {\widetilde{t}u,\widetilde{t}v}\right) = \mathop{\max }\limits_{{t \geq 0}}T\left( {{tu},{tv}}\right) ,\;\left\langle {{T}^{\prime }\left( {\widetilde{t}u,\widetilde{t}v}... | Proof The method is inspired by [18]. For the reader's convenience, we sketch the proof here. By a similar way as in the proof of Lemma 2.3, there exists \( \widetilde{t} > 0 \) such that (4.5) holds. Next we show that (4.6) holds.\n\nCase 1 One of \( u, v \) is zero. Without loss of generality, we may assume that \( u... | Yes |
Lemma 4.5 If \( p = q = {2}_{ * } \) . Assume that \( \left( {\mathrm{v}}_{1}\right) - \left( {\mathrm{v}}_{3}\right) ,\left( {\mathrm{f}}_{1}\right) - \left( {\mathrm{f}}_{3}\right) \) and \( \left( {\mathrm{f}}_{6}\right) - \left( {\mathrm{f}}_{7}\right) \) hold. Then there exists \( {\mu }_{0} > 0 \) such that the s... | Proof In analogous way to the proof of Theorem 1.1. Let \( \left\{ \left( {{u}_{n},{v}_{n}}\right) \right\} \subset \mathrm{N} \) be the minimizing sequence satisfying (3.9). Following from Lemma 4.2, passing to a subsequence, we way assume that \( \left( {{u}_{n},{v}_{n}}\right) \rightharpoonup \left( {{u}_{0},{v}_{0}... | No |
Lemma 4.6 Under the condition \( p = q = {2}_{ * } \), assume that \( \left( {\mathrm{v}}_{1}\right) - \left( {\mathrm{v}}_{3}\right) ,\left( {\mathrm{f}}_{1}\right) - \left( {\mathrm{f}}_{3}\right) \) and \( \left( {\mathrm{f}}_{6}\right) - \left( {\mathrm{f}}_{7}\right) \) hold. Then there exists \( {\widetilde{\mu }... | Proof The proof is similar to Lemmas 3.6 and 3.7. Here is omitted. | No |
Lemma 5.1 Let \( \left( {u, v}\right) \in E \) be a weak solution of \( {\left( \mathrm{{EQ}}\right) }_{ * } \), then \( \left( {u, v}\right) \) satisfies the following identity\n\n\[ \n\frac{N - 4}{2}\left( {{\int }_{{\mathbb{R}}^{N}}{\left| \Delta u\right| }^{2}\mathrm{\;d}x + {\int }_{{\mathbb{R}}^{N}}{\left| \Delta... | Proof The proof is standard (see e.g. Lemma 3.1 in [4]), so we omit it here. | No |
Lemma 5.2 Let \( {V}_{i}\left( x\right) \in {C}^{1}\left( {{\mathbb{R}}^{N},\mathbb{R}}\right) \left( {i = 1,2}\right) \) is nonnegative and \( \lambda \left( x\right) \in {C}^{1}\left( {{\mathbb{R}}^{N},\mathbb{R}}\right) \) . Assume that \( \left( {\mathrm{v}}_{4}\right) - \left( {\mathrm{v}}_{5}\right) \) holds. If ... | Proof Since \( \left( {u, v}\right) \) is a positive solution for the problem \( {\left( \mathrm{{EQ}}\right) }_{ * } \), one has\n\n\[ \n{\int }_{{\mathbb{R}}^{N}}\left( {{\left| \Delta u\right| }^{2} + {\left| \Delta v\right| }^{2}}\right) \mathrm{d}x + {\int }_{{\mathbb{R}}^{N}}\left( {{\left| \nabla u\right| }^{2} ... | Yes |
Corollary 3.1 Suppose that the condition (iv) of Theorem 3.1 is replaced by \( {\left( \mathrm{{iv}}\right) }^{\prime } \) \( T \in {\mathfrak{T}}^{\prime } \), while the other conditions remain the same. Then there exists at least one solution to problem (1.5). | Proof It follows from Remark 2.2 that (iv)' implies (iv), so the conclusion follows as an easy consequence of Theorem 3.1. | Yes |
Proposition 3.1 \( {\Xi }_{1} \subset {\Xi }_{2} \) . | Proof Denote by \( \Upsilon \) the subset of \( A \) defined as follows:\n\n\[ \Upsilon \mathrel{\text{:=}} \left\{ {x \in A : {Px}{ \preccurlyeq }_{1}{Qx}, R\left( {Tx}\right) { \preccurlyeq }_{2}S\left( {Tx}\right) }\right\} .\n\]\n\nLet \( z \in {\Xi }_{1} \) . Then we have \( {F}_{z,{Tz}}\left( t\right) = {F}_{A, B... | Yes |
Theorem 1.1 Suppose \( \beta \geq 3,{u}_{0} \in {H}^{1}\left( {\mathbb{R}}^{3}\right) \) with \( \nabla \cdot {u}_{0} = 0 \) and \( {\theta }_{0} \in {L}^{2}\left( {\mathbb{R}}^{3}\right) \) . Then the system (1.1) exists a unique global strong solution satisfying for any given \( T > 0 \)\n\n\[ u \in {L}^{\infty }\lef... | ## 2. Proof of Theorem 1.1\n\nThis section is devoted to proving Theorem 1.1. We start with the following a priori estimates, which is important for the existence part of Theorem 1.1.\n\nProposition 2.1 Suppose | No |
Lemma 2.2 \( {}^{\left\lbrack {14}\right\rbrack } \) Aussume that \( h \in C\left( {0,1}\right) \cap {L}^{1}\left( {0,1}\right) \) is such that \( {D}_{{0}^{ + }}^{\alpha }h \in C\left( {0,1}\right) \cap {L}^{1}\left( {0,1}\right) \) | then\n\n\[
{I}_{{0}^{ + }}^{\alpha }{D}_{{0}^{ + }}^{\alpha }h\left( t\right) = h\left( t\right) + {c}_{1}{t}^{\alpha - 1} + {c}_{2}{t}^{\alpha - 2} + \cdots + {c}_{n}{t}^{\alpha - n},
\]
\n\nwhere \( {c}_{i} \in \mathbb{R}, i = 1,2,\ldots, n, n = \left\lbrack \alpha \right\rbrack + 1 \) . | Yes |
Lemma 2.4 \( K\left( s\right) > 0 \) for all \( s \in \left\lbrack {0,1}\right\rbrack \), and \( K\left( s\right) \) is nondecreasing on \( \left\lbrack {0,1}\right\rbrack \) . | Proof From Lemma 2.3 and hypothesis made by (1.3), we have\n\n\[ K\left( 0\right) = L = 1 - \mathop{\sum }\limits_{{i = 1}}^{\infty }{\eta }_{i}{\xi }_{i}^{\alpha - \beta - 1} > 0. \]\n\n\[ {K}^{\prime }\left( s\right) = - \left( {\alpha - \beta - 1}\right) \mathop{\sum }\limits_{{s < {\xi }_{i}}}{\eta }_{i}{\left( \fr... | Yes |
Theorem 3.1 Denote\n\n\\[ \n{N}_{1} = {\\left( {\\int }_{0}^{1}G\\left( 1, s\\right) \\cdot {\\varphi }_{q}\\left( {\\int }_{0}^{s}{b}_{1}\\left( \\tau \\right) \\mathrm{d}\\tau \\right) \\mathrm{d}s\\right) }^{-1},{N}_{2} = {\\left( {\\int }_{0}^{1}G\\left( 1, s\\right) \\cdot {\\varphi }_{\\widetilde{q}}\\left( {\\in... | Proof For \\( \\left( {x, y}\\right) \\in \\overline{{P}^{2}\\left( {\\mu, r}\\right) } \\), by using \\( \\left( {\\mathrm{H}}_{1}\\right) \\) and Lemma 2.5, we get\n\n\\[\n\\mu \\left( {\\mathcal{T}\\left( {x, y}\\right) }\\right) = \\mathop{\\max }\\limits_{{t \\in \\left\\lbrack {0,1}\\right\\rbrack }}\\left| {{\\m... | Yes |
Lemma 2.2 For any \( {\mu }_{1} \neq {\mu }_{2} \in {\mathbb{R}}_{ + } \), we have\n\n\[ \left| {\phi \left( {{\mu }_{1}, t}\right) - \phi \left( {{\mu }_{2}, t}\right) }\right| \leq 2\left| {{\mu }_{1} - {\mu }_{2}}\right| \] | Proof Without loss of generality, we assume that \( 0 \leq {\mu }_{1} < {\mu }_{2} \) .\n\n1) If \( t < 0 \), then \( \left| {\phi \left( {{\mu }_{1}, t}\right) - \phi \left( {{\mu }_{2}, t}\right) }\right| = 0 \), then conclusion is obvious.\n\n2) If \( t > {\mu }_{2} \), then\n\n\[ \left| {\phi \left( {{\mu }_{1}, t}... | Yes |
Lemma 2.4 Function \( H\left( {\mu, x, y}\right) \) is continuously differentiable on \( {\mathbb{R}}_{+ + } \times {\mathbb{R}}^{n} \times {\mathbb{R}}^{n} \) and | \[ {H}^{\prime }\left( {\mu, x, y}\right) = \left\lbrack \begin{matrix} 1 & 0 & 0 \\ 0 & - {F}^{\prime }\left( x\right) & I \\ {d}_{\mu } & {D}_{x} & {Dy} \end{matrix}\right\rbrack ,\] where \( I \) denotes the \( n \times n \) identity matrix and \[ {d}_{\mu } \mathrel{\text{:=}} {\left( {\phi }_{\mu }^{\prime }\left(... | Yes |
Lemma 3.1 Let \( H\left( z\right) \mathrel{\text{:=}} H\left( {\mu, x, y}\right) \) be defined by (1.3). For any \( z \mathrel{\text{:=}} \left( {\mu, x, y}\right) \in \) \( {\mathbb{R}}_{+ + } \times {\mathbb{R}}^{2n} \), define the level set\n\n\[ \n{L}_{\mu }\left( {z}^{0}\right) \mathrel{\text{:=}} \left\{ {\left( ... | Proof By Lemma 2.2, for all \( \left( {x, y}\right) \in {L}_{\mu }\left( {{z}^{0},{\mu }_{1},{\mu }_{2}}\right) \), we have\n\n\[ \n\parallel G\left( {0, x, y}\right) {\parallel }_{1} \leq \parallel G\left( {\mu, x, y}\right) - G\left( {0, x, y}\right) {\parallel }_{1} + \parallel G\left( {\mu, x, y}\right) {\parallel ... | Yes |
Lemma 4.2 Let function \( \Phi \) and \( H \) be defined by (1.4) and (1.3), respectively. Then\n\n(a) \( \Phi \left( {\cdot ,\cdot , \cdot }\right) \) is strongly semismooth on \( {\mathbb{R}}_{ + } \times {\mathbb{R}}^{2n} \) .\n\n(b) If \( {F}^{\prime }\left( x\right) \) is Lipschitz continuous on \( {\mathbb{R}}^{n... | Proof It is not difficult to show that \( c - d,{\left( c - d\right) }^{2} \) is a strongly semismooth for all \( \left( {c, d}\right) \in {\mathbb{R}}^{2} \) . By recalling the definition of \( \Phi \) and the fact that the composition of strongly semismooth functions is strongly semismooth, we obtain immediately that... | Yes |
Lemma 2.1 Assume that the conditions (2.1)-(2.4) hold. Set\n\n\[ \n{\alpha }_{1} = \frac{1}{2}\frac{{\partial }^{2}f}{\partial {x}^{2}}\left( {0,0,0}\right) ,\;{\alpha }_{2} = \frac{{\partial }^{2}f}{\partial x\partial y}\left( {0,0,0}\right) ,\;{\alpha }_{3} = \frac{{\partial }^{3}f}{\partial {v}^{3}}\left( {0,0,0}\ri... | Proof Introduce a new variable \( \widehat{x} \) satisfying\n\n\[ \nx = \widehat{x} - \frac{{\alpha }_{2}}{2{\alpha }_{1}}y\n\]\n\nUsing (2.1)-(2.3), system (1.2) is transformed into the following form\n\n\[ \n{\widehat{x}}^{\prime } = {\alpha }_{1}{\widehat{x}}^{2} - \frac{{\alpha }_{3}}{{48}{\alpha }_{1}^{3}}{y}^{3} ... | Yes |
A small neighborhood of \( {q}_{0} \) is mapped diffeomorphically onto a neighborhood of \( {\Pi }_{2}\left( {q}_{0}\right) \) and\n\n\[ \n{\Pi }_{2}\left( {q}_{0}\right) = \left( {{\delta }^{-1/2},{\sigma }^{1/3} + O\left( {r}_{2}^{2/3}\right) ,{r}_{2}}\right) .\n\] | Proof By transformation \( \widehat{{y}_{2}} = {y}_{2} - {\sigma }^{1/3} \), system (2.14) becomes\n\n\[ \n{x}_{2}^{\prime } = {x}_{2}^{2} - 3{\sigma }^{\frac{2}{3}}{\widehat{y}}_{2} + O\left( {{\widehat{y}}_{2}^{2},{\widehat{y}}_{2}^{3},{r}_{2}}\right) ,\;{\widehat{y}}_{2}^{\prime } = {r}_{2}\left( {-1 + O\left( {r}_{... | Yes |
Proposition 2.2 System (2.15) has the following properties for \( \rho > 0 \) and \( \delta > 0 \) sufficiently small:\n\n1) The linearization of system (2.15) at \( {p}_{a} \) (respectively, \( {p}_{r} \) ) has the following real eigenvalues: \( {\lambda }_{1} = - 2 \) (respectively, \( {\lambda }_{1} = 2 \) ) with th... | Proof The first assertion follows from simple computations. Assertions 2)-5) follow from the first assertion and the standard theory of center manifold \( {}^{\left\lbrack 5\right\rbrack } \) . | No |
Proposition 2.3 For sufficiently small \( \rho ,\delta \) and \( {\beta }_{1} \), the transition map \( {\Pi }_{1} \) has the following properties\n\n1) \( \Pi \left( {R}_{1}\right) \) is a wedge-like region in \( {\Delta }_{1}^{\text{out }} \) and \( {\Pi }_{1}^{-1}\left( {R}_{2}\right) \) is also a wedge-like region ... | Proof From the second and third equations of (2.15), we have\n\n\[ \n- \frac{\mathrm{d}{r}_{1}}{\mathrm{\;d}{\varepsilon }_{1}} = \frac{{r}_{1}}{6{\varepsilon }_{1}}\left( {1 + O\left( {r}_{1}\right) }\right) \n\]\n\nIntegrating the above equation, we obtain\n\n\[ \n\frac{1}{{r}_{1}} = {c}_{0}{\varepsilon }_{1}^{\frac{... | Yes |
Proposition 2.4 The transition map \( {\Pi }_{3} \) defined by the flow of system (2.19) has the form \[ {\Pi }_{3}\left( {{r}_{3},{y}_{3},\delta }\right) = \left( \begin{matrix} \rho \\ {\Pi }_{32}\left( {{r}_{3},{y}_{3},\delta }\right) \\ {\left( \frac{{r}_{3}}{\rho }\right) }^{6}\delta \end{matrix}\right) \] where \... | Proof Fixed \( \left( {{r}_{3},{y}_{3},{\varepsilon }_{3}}\right) \in {\Delta }_{3}^{\text{in }} \), we consider a solution \( \left( {r, y,\varepsilon }\right) \left( t\right) \) of system (2.20) satisfying \( \left( {r, y,\varepsilon }\right) \left( 0\right) = \left( {{r}_{3},{y}_{3},{\varepsilon }_{3}}\right) \) and... | Yes |
Proposition 2.5 System (2.30) has the following properties:\n\n1) There exists a unique orbit \( {\gamma }_{2} \) which can be parametrized as \( \left( {{x}_{2},{y}_{2}\left( {x}_{2}\right) }\right) ,{x}_{2} \in \mathbb{R} \) , where\n\n\[ \n{y}_{2}\left( {x}_{2}\right) = {x}_{2}^{2/3} + o\left( \frac{1}{{x}_{2}}\righ... | Proof Equation \( \mathrm{d}{x}_{2}/\mathrm{d}{y}_{2} = {y}_{2}^{3} - {x}_{2}^{2} \) is a special Riccati equation. Its general solution can be expressed as follows\n\n\[ \n{x}_{2}\left( {y}_{2}\right) = \frac{{y}_{2}^{3/2}\left( {c\operatorname{BesselI}\left( {-\frac{4}{5},\frac{2}{5}{y}_{2}^{5/2}}\right) - \operatorn... | Yes |
Proposition 2.7 The transition map \( {\Pi }_{3} \) formed by the flow of system (2.32) has the form \[ {\Pi }_{3}\left( {{r}_{3},{y}_{3},\delta }\right) = \left( \begin{matrix} \rho \\ {\Pi }_{32}\left( {{r}_{3},{y}_{3},\delta }\right) \\ {\left( \frac{{r}_{3}}{\rho }\right) }^{5}\delta \end{matrix}\right) \] where \[... | Proof \( \operatorname{Fix}\left( {{r}_{3},{\widehat{y}}_{3},{\varepsilon }_{3}}\right) \in {\Delta }_{3}^{\text{in }} \) . We consider a solution \( \left( {r,\widehat{y},\varepsilon }\right) \left( t\right) \) of system (2.35) satisfying \( \left( {r,\widehat{y},\varepsilon }\right) \left( 0\right) = \left( {{r}_{3},... | Yes |
Theorem 2.1 Model (1.2) admits a unique solution \( \mathbf{X}\left( t\right) \) for any initial value \( \mathbf{X}\left( 0\right) \) , and solution will remain in \( {\mathbb{R}}_{ + }^{3} \) with probability one. | Proof We construct a non-negative \( {C}^{2} \) -function \( {W}_{1} = S - 1 - \ln S + V - 1 - \ln V + I - 1 - \ln I \) by the similar approaches in [17-19], then the generalized Itô’s formula implies that\n\n\[ \mathrm{d}{W}_{1} = \mathcal{L}{W}_{1}\mathrm{\;d}t + \left( {S - 1}\right) {\sigma }_{1}\left( r\right) \ma... | No |
Theorem 2.2 If \( {R}_{0}^{s} > 1 \), then the model (1.2) admits a unique stationary distribution \( \nu \left( \cdot \right) \), which has the ergodic property, where\n\n\[ \n{R}_{0}^{s} = \frac{{\left\lbrack \mathop{\sum }\limits_{{k = 1}}^{N}{\pi }_{k}{q}_{1}\left( k\right) \right\rbrack }^{3}}{\mathop{\sum }\limit... | Proof Firstly, the assumption \( {\gamma }_{ij} > 0 \) for \( i \neq j \) implies that condition (H1) in Lemma 2.1 is satisfied. Secondly, we consider the bounded open subset \( D = \left( {\frac{1}{d}, d}\right) \times \left( {\frac{1}{d}, d}\right) \times \left( {\frac{1}{d}, d}\right) \subset {\mathbb{R}}_{ + }^{3} ... | Yes |
Theorem 3.1 If the following condition holds\n\n\[ \n{R}_{0}^{e} = \frac{\mathop{\sum }\limits_{{k = 1}}^{N}{\pi }_{k}\left( {{\widehat{a}}_{3}\beta \left( k\right) + {\widehat{a}}_{1}\eta \left( k\right) }\right) }{{\widehat{a}}_{1}{\widehat{a}}_{3}\mathop{\sum }\limits_{{k = 1}}^{N}{\pi }_{k}{q}_{3}\left( k\right) } ... | Proof Define \( {W}_{7} = \ln I \), applying the generalized Itô’s formula to \( {W}_{7} \), we obtain\n\n\[ \n\mathrm{d}{W}_{7} = \mathcal{L}{W}_{7}\mathrm{\;d}t + {\sigma }_{3}\left( {r\left( t\right) }\right) \mathrm{d}{B}_{3}\left( t\right) , \]\n\nwhere\n\n\[ \n\mathcal{L}{W}_{7} = \frac{\beta \left( {r\left( t\ri... | Yes |
Example 4.1 We choose (I) in Tab. 4.1 with \( {\sigma }_{1} = {0.5},{\sigma }_{2} = {0.5},{\sigma }_{3} = {0.5} \) for \( k = 1 \) and \( {\sigma }_{1} = {0.3},{\sigma }_{2} = {0.3},{\sigma }_{3} = {0.3} \) for \( k = 2 \), we thus derive \( {R}_{0}^{s} \approx {1.0553} > 1 \), so the condition of Theorem 2.2 is satisf... | Fig. 4.1 shows that the susceptible, the vaccinated and the infected are persistent in the mean for a long run, the corresponding frequencies with probability density functions are demonstrated in Fig. 4.2. | No |
The extinction is discussed by choosing (II) in Tab. 4.1 with \( {\sigma }_{1} = {\sigma }_{2} = \) \( {\sigma }_{3} = {0.5} \) for \( k = 1 \) and \( {\sigma }_{1} = {\sigma }_{2} = {\sigma }_{3} = {0.3} \) for \( k = 2 \). It is easy to check that the condition of Theorem 3.1 is satisfied, that is, \( {R}_{0}^{e} \ap... | The simulations in Fig. 4.3 show that the increasing of the intensities for the white noises accelerates the extinction of infectious diseases. | No |
Theorem 2.1 Let \( f : {\mathbb{R}}^{n} \rightarrow \mathbb{R} \) be a continuously differentiable function. For a given iterative point \( {x}_{k} \) and a search direction \( {d}_{k} \) at \( {x}_{k} \) . If \( {g}_{k}^{\mathrm{T}}{d}_{k} < 0 \), then there is always a step-size \( {\alpha }_{k} \) in the set \( \lef... | Proof We now prove the conclusion by the inductive method. For \( n = 0 \) ,(2.1) can be written\n\n\[ f\left( {{x}_{0} + {\alpha }_{0}{d}_{0}}\right) \leq {\widehat{C}}_{l\left( 0\right) } + {\delta }_{0}{\alpha }_{0}{g}_{0}^{\mathrm{T}}{d}_{0} \]\n\n(2.7)\n\nSince \( {\widehat{C}}_{l\left( 0\right) } = f\left( {x}_{0... | Yes |
Theorem 3.1 Suppose that \( \\left\\{ {x}_{k}\\right\\} \) is the iterative sequence generated by Algorithm 2.1, and there is \( {g}_{k}^{\\mathrm{T}}{d}_{k} < 0 \) for \( k \\geq 0 \), then \( \\forall l \\geq 1 \) ,\n\n\\[ \n\\mathop{\\max }\\limits_{{1 \\leq j \\leq M}}f\\left( {x}_{{Ml} + j}\\right) \\leq \\mathop{... | Proof In order to prove (3.1), we first prove that the following inequality holds.\n\n\\[ \nf\\left( {x}_{{Ml} + j}\\right) \\leq \\mathop{\\max }\\limits_{{1 \\leq j \\leq M}}{C}_{M\\left( {l - 1}\\right) + j} + {\\alpha }_{{Ml} + j - 1}{\\delta }_{{Ml} + j - 1}{g}_{{Ml} + j - 1}^{\\mathrm{T}}{d}_{{Ml} + j - 1}.\n\\]\... | Yes |
Lemma 3.1 Suppose that Assumptions 3.2 holds. If \( {\alpha }_{k} \) is the iterative step-size generated by Algorithm 2.1, then under the conditions of (2.4) and (2.5) for \( k \geq 0 \) ,\n\n\[ \n{\alpha }_{k} \geq \frac{\rho {c}_{1}\left( {1 - {\delta }_{\max }}\right) }{L{c}_{2}} > 0.\n\]\n\n(3.6) | Proof For the obtained step-size \( {\alpha }_{k} \), if \( \frac{{\alpha }_{k}}{\rho } \) does not satisfy (2.1), then\n\n\[ \nf\left( {{x}_{k} + \frac{{\alpha }_{k}}{\rho }{d}_{k}}\right) > {\widehat{C}}_{l\left( k\right) } + {\delta }_{k}\frac{{\alpha }_{k}}{\rho }{g}_{k}^{\mathrm{T}}{d}_{k}\n\]\n\nThus\n\n\[ \n{f}_... | Yes |
Theorem 3.2 Suppose that \( \left\{ {x}_{k}\right\} \) is the iterative sequence generated by Algorithm 2.1. Under the conditions of Assumptions 3.1, 3.2 and (2.4) and (2.5), it holds that\n\n\[ \mathop{\lim }\limits_{{k \rightarrow \infty }}\begin{Vmatrix}{g}_{k}\end{Vmatrix} = 0 \] | Proof Firstly, let us prove that there exists a constant \( \gamma \) such that\n\n\[ \begin{Vmatrix}{g}_{k + 1}\end{Vmatrix} \leq \gamma \begin{Vmatrix}{g}_{k}\end{Vmatrix} \]\n\n(3.10)\n\nFrom Algorithm 2.1, we get that \( {\alpha }_{k} \leq 1 \) . By (1.2) and (2.5), we have that\n\n\[ \begin{Vmatrix}{{x}_{k + 1} - ... | Yes |
Theorem 3.3 Suppose that \( \left\{ {x}_{k}\right\} \) is the iterative sequence generated by Algorithm 2.1. Under Assumptions 3.1, 3.2 and 3.3, it holds that \[ \mathop{\lim }\limits_{{k \rightarrow \infty }}\inf \begin{Vmatrix}{g}_{k}\end{Vmatrix} = 0 \] | Proof Assuming that (3.15) does not true. Then, for each \( k \), there exists a constant \( \beta > 0 \) such that \[ \begin{Vmatrix}{g}_{k}\end{Vmatrix} \geq \beta \text{.} \] By Assumption 3.3, we can get \[ \left| {{g}_{k}^{\mathrm{T}}{d}_{k}}\right| \geq {c}_{1}{\beta }^{2} \] Since (2.4) holds under the condition... | Yes |
Lemma 2.1 Suppose that \( \left( {\mathrm{F}}_{1}\right) \) and \( \left( {\mathrm{F}}_{4}\right) \) hold. Then there exists \( {d}_{2} > 0 \), such that \( \left| {F\left( {x, t}\right) }\right| \leq {d}_{2}G\left( \left| t\right| \right) {t}^{2} \) for all \( x \in {\mathbb{R}}^{3} \) and \( \left| t\right| \geq {t}_... | Proof Take\n\[ h\left( s\right) = F\left( {x,{st}}\right) ,\forall s \geq \frac{{t}_{\infty }}{\left| t\right| }, t \in \mathbb{R}, x \in {\mathbb{R}}^{3}. \]\n\n(2.1)\n\nCombining (2.1) with \( \left( {\mathrm{F}}_{1}\right) \), we have\n\n\[ {h}^{\prime }\left( s\right) = f\left( {x,{st}}\right) t \leq \frac{1}{s}\le... | Yes |
Lemma 2.2 Suppose that \( \\left( \\mathrm{V}\\right) ,\\left( {\\mathrm{F}}_{1}\\right) ,\\left( {\\mathrm{F}}_{3}\\right) \) and \( \\left( {\\mathrm{F}}_{4}\\right) \) hold. Then, for any \( \\varepsilon > 0 \) and \( x \\in {\\mathbb{R}}^{3}, t \\in \\mathbb{R} \), there exists \( {d}_{\\varepsilon } > 0 \), such t... | Proof From Lemma 2.1 and \( \\left( {\\mathrm{F}}_{4}\\right) \), there exists \( {d}_{3} > 0 \) such that\n\n\[F\\left( {x, t}\\right) \\leq {d}_{3}{t}^{2},\\text{ for }\\left| t\\right| \\geq {t}_{\\infty }.\]\n\n(2.9)\n\nIt follows from \( \\left( {\\mathrm{F}}_{1}\\right) \) that\n\n\[\\left| {f\\left( {x, t}\\righ... | Yes |
In the special case \( \Omega = {\mathbb{R}}^{n} \) in Theorem 1.1, we have by (1.2) that | \[ {\int }_{{\mathbb{R}}^{n}}\frac{{\left| x\right| }^{2}}{2}{\left| {u}_{t}\right| }^{2}\mathrm{\;d}x + {\int }_{{\mathbb{R}}^{n}}\frac{{\left| x\right| }^{2}}{2}\frac{\mathrm{d}}{\mathrm{d}t}\left( {\frac{{\left| \nabla u\right| }^{2}}{2} - F\left( {x, u}\right) }\right) \mathrm{d}x = \frac{n - 2}{2}{\int }_{{\mathbb... | Yes |
Let \( u\left( {x, t}\right) \) solve the following semilinear parabolic equation \[ \left\{ \begin{array}{ll} {u}_{t} - {\Delta u} = \frac{u}{{\left| x\right| }^{2}}, & x \in \Omega, t > 0, \\ u\left( {x, t}\right) = 0, & x \in \partial \Omega, t > 0, \\ u\left( {x,0}\right) = {u}_{0}, & x \in \Omega , \end{array}\rig... | Proof By Theorem 1.1, we derive that \[ {\int }_{\partial \Omega }\frac{{\left| \nabla u\right| }^{2}}{2}\left( {x \cdot \nu }\right) \mathrm{d}S = {\int }_{\Omega }\frac{{\left| x\right| }^{2}}{2}{\left| {u}_{t}\right| }^{2}\mathrm{\;d}x + {\int }_{\Omega }\frac{{\left| x\right| }^{2}}{2}\frac{\mathrm{d}}{\mathrm{d}t}... | Yes |
Example 3.2 Let \( u\left( {x, t}\right) \) solve the following semilinear parabolic equation\n\n\[ \left\{ \begin{array}{ll} {u}_{t} - {\Delta u} = \left( {1 - {\left| u\right| }^{2}}\right) u, & x \in {\mathbb{R}}^{n} \times \left( {0,\infty }\right) , \\ u = 0, & \left| x\right| \rightarrow \infty . \end{array}\righ... | Proof By Corollary 1.1, one has \n\n\[ {\int }_{{\mathbb{R}}^{n}}\frac{{\left| x\right| }^{2}}{2}{\left| {u}_{t}\right| }^{2}\mathrm{\;d}x + {\int }_{{\mathbb{R}}^{n}}\frac{{\left| x\right| }^{2}}{2}\frac{\mathrm{d}}{\mathrm{d}t}\left( {\frac{{\left| \nabla u\right| }^{2}}{2} - \frac{{\left( 1 - {\left| u\right| }^{2}\... | Yes |
Example 3.3 Let \( \left( {u\left( {x, t}\right), v\left( {x, t}\right) }\right) \) solve the following semilinear parabolic systems\n\n\[ \left\{ \begin{array}{ll} {u}_{t} - {\Delta u} = \frac{2u}{1 + {u}^{2} + {v}^{2}}, & x \in {\mathbb{R}}^{n} \times \left( {0,\infty }\right) , \\ {v}_{t} - {\Delta v} = \frac{2v}{1 ... | Proof By Corollary 1.1, we have\n\n\[ {\int }_{{\mathbb{R}}^{n}}\frac{{\left| x\right| }^{2}}{2}{\left| {u}_{t}\right| }^{2}\mathrm{\;d}x + {\int }_{{\mathbb{R}}^{n}}\frac{{\left| x\right| }^{2}}{2}\frac{\mathrm{d}}{\mathrm{d}t}\left( {\frac{{\left| \nabla u\right| }^{2}}{2} - \ln \left( {1 + {u}^{2} + {v}^{2}}\right) ... | Yes |
Theorem 2.1 An incomplete Boolean control networks with delay of order \( \mu \) has the following algebraic form: \( x\left( {t + 1}\right) = {L}_{1}^{w}u\left( t\right) x\left( t\right) x\left( {t - 1}\right) x\left( {t - 2}\right) \cdots x\left( {t - \mu + 1}\right) \), where \( {L}_{1}^{w} = \left\lbrack \begin{arr... | We only need to prove that the incomplete system is equivalent to the state trajectory with control state avoidance set \( S \) . If \( u\left( t\right) x\left( x\right) = {\delta }_{{2}^{m + n}}^{i} \notin W \), then \[ x\left( {t + 1}\right) = {L}_{1}u\left( t\right) x\left( t\right) x\left( {t - 1}\right) \cdots x\l... | Yes |
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