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Theorem 3.1 If the system (1.2) satisfies one of the conditions of Theorem 2.1, the origin is a center. Furthermore, the phase diagram near the origin is symmetric w.r.t. a straight line. | Figure 3.1 is the phase diagram of the system (1.2), if its parameters satisfy \( {a}_{1} = {b}_{1} = \) \( {b}_{2} = - {a}_{3} = - {b}_{3} = 1/2,{a}_{2} = 7/4 \) for the condition \( \left( {\mathrm{A}}_{11}\right) \) in Theorem 2.1. In this case, the system takes the form\n\n\[ \dot{x} = - y + 1/2{x}^{2} + 7/{4xy} - ... | Yes |
Theorem 4.1 Assume the SGCDRE admits a solution \( {P}_{i}\left( t\right) \in {\mathbb{R}}^{n \times n} \) on \( t \in \left\lbrack {0, T}\right\rbrack \), and the finite horizon LQ optimal control problem (2.1), (2.4) is well-posed. Then, the optimal control in the admissible class \( {\mathcal{U}}_{\mathrm{{ad}}} \) ... | Proof The proof is immediate from Lemma 4.3. | No |
Theorem 3.4 Let \( \left\{ {\varepsilon }_{k}\right\} \) be a positive sequence which is convergent to zero. Suppose that \( {z}_{k} \) is a stationary point of the problem (2.11) with \( \varepsilon = {\varepsilon }_{k} \) . If \( \bar{z} \) is an accumulation point of the sequence \( \left\{ {z}_{k}\right\} \) such t... | Proof To prove the above result, we only need to prove that \( {\eta }_{i}^{G * } = 0,{\eta }_{i}^{H * } \geq 0 \) for \( i \in {I}_{00}\left( \bar{z}\right) \) . Obviously, Assumption (*) implies that \( \operatorname{supp}\left( {\bar{\gamma }}^{ * }\right) \cap {I}_{00}\left( \bar{z}\right) = \varnothing \) and \( \... | Yes |
Theorem 1.1 Consider the system (1.5) with \( \left| \varepsilon \right| \) small enough. Using the first order Melnikov function, the maximal number of limit cycles which bifurcate from the periodic annulus of the origin of system (1.5) \( {\left. \right| }_{\varepsilon = 0} \) is \( n \) . Moreover, the corresponding... | ## 2. Proof of Theorem 1.1\n\nFrom Theorem 2.2 in [9] and Remark 2.3 in [12], we know that the first order Melnikov function \( M\left( h\right) \) of system (1.5) has the following form\n\n\[ M\left( h\right) = {\int }_{\overset{⏜}{{A}_{2}{A}_{1}}}{g}_{1}\left( {x, y}\right) \mathrm{d}x - {f}_{1}\left( {x, y}\right) \... | Yes |
Lemma 2.2 For \( h \in \left( {0, + \infty }\right) \) ,\n\n\[ M\left( h\right) = {h\psi }\left( h\right) + \sqrt{h}{\varphi }_{n}\left( \sqrt{h}\right) \]\n\nwhere \( \psi \left( h\right) \) is a polynomial in \( h \) of degree \( \left\lbrack \frac{n - 1}{2}\right\rbrack \) and \( {\varphi }_{n}\left( u\right) \) is ... | Proof From Lemma 2.1, we have\n\n\[ M\left( h\right) = \sqrt{h}{\varphi }_{n}\left( \sqrt{h}\right) + \mathop{\sum }\limits_{{k = 1}}^{m}\left\lbrack {{\alpha }_{k}\left( h\right) {I}_{0,0}^{k}\left( h\right) + {\beta }_{k}\left( h\right) {I}_{1,0}^{k}\left( h\right) + {\gamma }_{k}\left( h\right) {I}_{0,1}^{k}\left( h... | Yes |
Corollary 3 When \( m \neq n \), a hypothesis test of level \( \alpha \) for the null hypothesis \( p = q \) has the acceptance region\n\n\[ \n{\operatorname{MMD}}_{b}\left( {\mathcal{F}, X, Y}\right) < \sqrt{\frac{K\left( {m + n}\right) }{mn}}\left( {1 + \sqrt{2\log {\alpha }^{-1}}}\right) .\n\]\n\n(3.1) | Proof When \( p = q \) and \( m \neq n \), we get\n\n\[ \n{\mathrm{E}}_{X, Y}\left\lbrack {{\operatorname{MMD}}_{b}\left( {\mathcal{F}, X, Y}\right) }\right\rbrack = {\mathrm{E}}_{X, Y}\left\lbrack {\mathop{\sup }\limits_{{f \in \mathcal{F}}}\left\lbrack {\frac{1}{m}\mathop{\sum }\limits_{{i = 1}}^{m}f\left( {x}_{i}\ri... | Yes |
Lemma 2.1 Suppose (V) and (K) hold. Let \( k \mathrel{\text{:=}} \frac{\alpha }{p - 1} \) . Then there exist \( \delta > 0 \) and \( C \geq 1 \) such that for all \( {u}_{0} \in {L}_{k}^{\infty } \) satisfying \( {\begin{Vmatrix}{u}_{0}\end{Vmatrix}}_{\infty, k} \leq \delta \), the corresponding solution \( u \) of (1.... | Proof Without loss of generality, assume \( u \geq 0 \) . Let \( U\left( x\right) \mathrel{\text{:=}} \frac{1}{{\left( A + {\left| x\right| }^{2}\right) }^{\frac{k}{2}}} \) for \( A > 0 \), then\n\n\[ {\Delta U} = \mathop{\sum }\limits_{{i = 1}}^{n}\frac{{\partial }^{2}U}{\partial {x}_{i}^{2}} = \mathop{\sum }\limits_{... | Yes |
Corollary 2.1 Under the assumptions of Lemma 2.1, if \( {u}_{0} \in {L}_{k,0}^{\infty } \), then \( \mathop{\lim }\limits_{{t \rightarrow \infty }}\parallel u\left( t\right) {\parallel }_{\infty, k} = \) 0. | Proof (2.2) also implies that \( u \leq \frac{{C}^{\prime }\delta }{1 + {\left| x\right| }^{k}} \) . Then for all \( x \) and \( t > 0 \), taking \( 0 < \delta < \) \( {\left( \frac{{a}_{1}}{2{C}^{\prime \prime }}\right) }^{\frac{1}{p - 1}} \), we have\n\n\[ \n{u}_{t} - {\Delta u} + \frac{1}{2}V\left( x\right) u = K\le... | Yes |
Lemma 4.1 Let \( {u}_{0} \in {L}^{\infty } \cap {H}^{1} \) . If \( T\left( {u}_{0}\right) = \infty \), then for each \( k \geq 0 \), the \( \omega \) -limit set \( {\omega }_{k}\left( {u}_{0}\right) \) consists of equilibria. | Proof Assume \( \widetilde{v} \in {\omega }_{k}\left( {u}_{0}\right) \), then there exists \( {t}_{j} \rightarrow \infty \) such that \( u\left( {t}_{j}\right) \rightarrow \widetilde{v} \) in \( {L}_{k}^{\infty } \) . Fixing \( t > 0 \) and by the continuity of semigroup, we have\n\n\[ u\left( {t + {t}_{j}}\right) = S\... | Yes |
Lemma 5.2(Uniform a priori estimate for global solutions) If \( T\left( {u}_{0}\right) = \infty \), then\n\n\[ \mathop{\sup }\limits_{{t \geq 0}}\parallel u\left( t\right) {\parallel }_{\infty } \leq C\left( {{\begin{Vmatrix}{u}_{0}\end{Vmatrix}}_{\infty } + {\begin{Vmatrix}{u}_{0}\end{Vmatrix}}_{{H}^{1}}}\right) \]\n\... | Proof By the standard local theory, there exists \( \tau = \tau \left( {\begin{Vmatrix}{u}_{0}\end{Vmatrix}}_{\infty }\right) > 0 \) such that\n\n\[ {\begin{Vmatrix}u\left( t;{u}_{0}\right) \end{Vmatrix}}_{\infty } \leq {\begin{Vmatrix}{u}_{0}\end{Vmatrix}}_{\infty } + 1,\;0 \leq t \leq \tau . \]\n\n(5.4)\n\nArguing by... | No |
Lemma 3.1 Assume that the conditions \( \mathrm{H}\left( \xi \right) ,\mathrm{H}\left( \beta \right) ,\left( {\mathrm{H}}_{1}\right) \) and \( \left( {\mathrm{H}}_{2}\right) \) hold. Then \( u = 0 \) is a local minimum of \( \mathcal{J} \) and \( {\mathcal{J}}_{ \pm } \) for any \( \lambda \leq 0 \) . | Proof From the conditions \( \left( {\mathrm{H}}_{1}\right) \) and \( \left( {\mathrm{H}}_{2}\right) \), for given \( \epsilon > 0 \), there exists \( {A}_{1} > 0 \) such that\n\n\[ \left| {F\left( {x, u}\right) }\right| \leq \frac{p\left( x\right) + \epsilon }{2}{\left| u\right| }^{2} + {A}_{1}{\left| u\right| }^{\gam... | Yes |
Lemma 3.3 Assume that \( \lambda \leq 0 \), and the conditions \( H\left( \xi \right), H\left( \beta \right) ,\left( {\mathrm{H}}_{1}\right) ,\left( {\mathrm{H}}_{2}\right) ,\left( {\mathrm{H}}_{3}\right) \) with \( l = + \infty \) and \( \left( {\mathrm{H}}_{4}\right) \) hold. If \( {u}_{ \pm } \) are the isolated non... | Proof Let\n\n\[{\mathcal{J}}_{s}\left( u\right) = s{\mathcal{J}}_{ + }\left( u\right) + \left( {1 - s}\right) \mathcal{J}\left( u\right), s \in \left\lbrack {0,1}\right\rbrack ,\]\n\nthen \( {\mathcal{J}}_{s} \) satisfies the condition \( {\left( \mathrm{C}\right) }_{c} \) by Lemma 3.2, and \( {u}_{ + } \) is a critica... | Yes |
Lemma 3.6 Under the conditions \( \mathrm{H}\left( \xi \right) ,\mathrm{H}\left( \beta \right) ,\left( {\mathrm{H}}_{1}\right) ,\left( {\mathrm{H}}_{2}\right) ,\left( {\mathrm{H}}_{3}\right) \) with \( l = + \infty \) and \( \left( {\mathrm{H}}_{4}\right) \) , the functionals \( \mathcal{J} \) and \( {\mathcal{J}}_{ \p... | Proof We only sketchily give the proof of \( {\mathcal{J}}_{ + } \), the cases of \( \mathcal{J} \) and \( {\mathcal{J}}_{ - } \) are similar. This proof is essentially equal to our the previous section of the proof of Lemma 3.2 and the last section of the proof of Proposition 3 of [9]. Hence, we omit it here. | No |
Theorem 3.7 If the conditions \( \mathrm{H}\left( \xi \right) ,\mathrm{H}\left( \beta \right) ,\left( {\mathrm{H}}_{1}\right) \) and \( \left( {\mathrm{H}}_{2}\right) \) hold, then there exists \( {\lambda }^{ * } > 0 \) such that for all \( 0 < \lambda < {\lambda }^{ * } \) we can find \( \rho > 0 \) for which we have... | Proof By the conditions \( \left( {\mathrm{H}}_{1}\right) \) and \( \left( {\mathrm{H}}_{2}\right) \), given \( \epsilon > 0 \), there exists \( {A}_{3} > 0 \) such that\n\n\[ \n{f}_{ + }\left( {x, u}\right) \leq \frac{p\left( x\right) + \epsilon }{2}{\left( {u}^{ + }\right) }^{2} + {A}_{3}{\left( {u}^{ + }\right) }^{\... | Yes |
Theorem 4.1 Equations (4.9),(4.11) indeed admit a unique solution \( {\phi }^{ * }\left( t\right) = \left( {{\phi }_{1}^{ * }\left( t\right) ,{\phi }_{2}^{ * }\left( t\right) }\right. \) , \( \left. {{\phi }_{3}^{ * }\left( t\right) ,{\phi }_{4}^{ * }\left( t\right) ,{\phi }_{5}^{ * }\left( t\right) }\right) ,{u}^{ * }... | Proof It is easy to verify that \( {W}_{1}\left( {\beta }_{1}\right) ,{W}_{2}\left( {\beta }_{2}\right) \) defined by (4.12) is an increasing function of the variable \( {\beta }_{1},{\beta }_{2} \), respectively. Moreover, \( {W}_{1}\left( 0\right) = - \left( {{r}_{1} - {r}_{0} + \left( {\lambda + {\lambda }_{1}}\righ... | Yes |
Theorem 4.2 Let \( W\left( {\phi }_{6}\right) = \frac{\gamma {h}^{P}}{{\vartheta }_{6}}{\phi }_{6}\ln {\phi }_{6} + {h}^{P}{\phi }_{6} - \frac{\delta }{\varsigma } \), then \( W\left( {\phi }_{6}\right) \) has a unique positive root \( {\phi }_{6} \) . | Proof Since \( {W}^{\prime }\left( {\phi }_{6}\right) = \frac{\gamma {h}^{P}}{{\vartheta }_{6}}\left( {\ln {\phi }_{6} + 1}\right) + {h}^{P} \), it is easy to verify \( W\left( {\phi }_{6}\right) \) is a decreasing function on \( \left( {0,{\mathrm{e}}^{-\frac{\gamma + {\vartheta }_{6}}{\gamma }}}\right) \) and increas... | Yes |
Theorem 4.3 If there exists a function and a control policy \( \left( {{\alpha }^{ * }\left( t\right) ,{\beta }_{1}^{ * }\left( t\right) ,{\beta }_{2}^{ * }\left( t\right) ,{\beta }_{3}^{ * }\left( t\right) }\right) \) , which satisfy the HJB equation (4.1), then \( M\left( {t, x, l, z}\right) \) is the corresponding v... | Proof From [11], the above theorem will hold if \( \left( {{\alpha }^{ * }\left( t\right) ,{\beta }_{1}^{ * }\left( t\right) ,{\beta }_{2}^{ * }\left( t\right) ,{\beta }_{2}^{ * }\left( t\right) }\right) \) and the corresponding candidate value function \( M\left( {t, x, l, z}\right) \) has the following three properti... | No |
Theorem 3.1 Consider Eq.(1.1), \( a\left( t\right), b\left( t\right) \) and \( c\left( t\right) \) are all \( \omega \) -periodic continuous functions on \( \mathbb{R} \) . Suppose that the following conditions hold:\n\n\( \left( {\mathrm{H}}_{1}\right) a\left( t\right) > 0 \)\n\n\( \left( {\mathrm{H}}_{2}\right) b\lef... | Proof 1) We prove that Eq.(1.1) has two positive \( \omega \) -periodic continuous solutions \( {\gamma }_{1}\left( t\right) \) and \( {\gamma }_{2}\left( t\right) \).\n\nDefine a set as follows:\n\n\[ B = \left\{ {\varphi \left( t\right) \in C\left( {\mathbb{R},\mathbb{R}}\right) \left| {\;\varphi \left( {t + \omega }... | Yes |
Theorem 3.2 Under the conditions of Theorem 3.1, Eq.(1.1) has exactly two \( \omega \) -periodic continuous solutions: \( {\gamma }_{1}\left( t\right) \) and \( {\gamma }_{2}\left( t\right) \) . | Proof The proof of the existence of \( {\gamma }_{1}\left( t\right) \) and \( {\gamma }_{2}\left( t\right) \) is given in Theorem 3.1. Now, we prove that Eq.(1.1) has exactly two \( \omega \) -periodic continuous solutions: \( {\gamma }_{1}\left( t\right) \) and \( {\gamma }_{2}\left( t\right) \) . We know if \( x\left... | Yes |
Corollary 3.1 Consider the following Bernoulli's equation:\n\n\[ \frac{\mathrm{d}x}{\mathrm{\;d}t} = a\left( t\right) {x}^{2} + b\left( t\right) x \] \n\n\( \left( {3.47}\right) \) \n\nand \( a\left( t\right), b\left( t\right) \) are both \( \omega \) -periodic continuous functions on \( \mathbb{R} \) . Suppose that th... | 1) One \( \omega \) -periodic continuous solution is \( {\gamma }_{1}\left( t\right) = 0 \), and \( {\gamma }_{1}\left( t\right) \) is attractive if given initial value on \( {D}_{1} = \left\{ {x\left( {t}_{0}\right) \mid x\left( {t}_{0}\right) < \frac{1}{\zeta \left( {t}_{0}\right) }}\right\} \), and unstable if given... | Yes |
Theorem 3.3 Consider Eq.(1.1), \( a\left( t\right), b\left( t\right) \) and \( c\left( t\right) \) are all \( \omega \) -periodic continuous functions on \( \mathbb{R} \) . Suppose that the following conditions hold:\n\n\( \left( {\mathrm{H}}_{7}\right) a\left( t\right) < 0 \)\n\n\( \left( {\mathrm{H}}_{8}\right) b\lef... | Proof Consider the following equation:\n\n\[ \frac{\mathrm{d}x}{\mathrm{\;d}t} = \widehat{a}\left( t\right) {x}^{2} + \widehat{b}\left( t\right) x + \widehat{c}\left( t\right) ,\]\n\n(3.48)\n\nhere \( \widehat{a}\left( t\right) = - a\left( t\right) > 0,\widehat{b}\left( t\right) = - b\left( t\right) < 0,\widehat{c}\lef... | Yes |
Theorem 3.4 Under the conditions of Theorem 3.3, Eq.(1.1) has exactly two \( \omega \) -periodic continuous solutions: \( {\gamma }_{1}\left( t\right) \) and \( {\gamma }_{2}\left( t\right) \) . | Proof Theorem 3.4 also follows by applying Theorem 3.2 to the equation (3.48). | No |
Corollary 3.2 Consider Bernoulli’s Eq.(3.47), and \( a\left( t\right), b\left( t\right) \) are both \( \omega \) -periodic continuous functions on \( \mathbb{R} \) . Suppose that the following conditions hold:\n\n\( \left( {\mathrm{H}}_{11}\right) a\left( t\right) < 0 \)\n\n\( \left( {\mathrm{H}}_{12}\right) b\left( t\... | 1) One \( \omega \) -periodic continuous solution is \( {\gamma }_{1}\left( t\right) = 0 \), and \( {\gamma }_{1}\left( t\right) \) is unstable on \( \mathbb{R} \) ;\n\n2) Another \( \omega \) -periodic continuous solution is\n\n\[ \n{\gamma }_{2}\left( t\right) = - \frac{1}{{\int }_{-\infty }^{t}{\mathrm{e}}^{-{\int }... | Yes |
Consider the following equation:\n\n\\[ \n\\frac{\\mathrm{d}x}{\\mathrm{\\;d}t} = \\left( {2 + \\sin t}\\right) {x}^{2} + \\left( {\\sin t - 8}\\right) x + 2 - \\cos t. \n\\]\n\n(4.1) | It is easy to calculate that \\( {a}_{M} = 3,{a}_{L} = 1,{b}_{M} = - 7,{b}_{L} = - 9,{c}_{M} = 3,{c}_{L} = 1 \\), and\n\n\\[ \n{b}_{M}^{2} - 4{a}_{M}{c}_{M} = {13} > 0. \n\\]\n\nClearly, the conditions \\( \\left( {\\mathrm{H}}_{1}\\right) - \\left( {\\mathrm{H}}_{4}\\right) \\) of Theorem 3.1 are satisfied. It follows... | Yes |
Consider the following equation:\n\n\[ \frac{\mathrm{d}x}{\mathrm{\;d}t} = \left( {-2 + \sin t}\right) {x}^{2} + \left( {\sin t + 8}\right) x - 2 + \cos t. \] | It is easy to calculate that \( {a}_{M} = - 1,{a}_{L} = - 3,{b}_{M} = 9,{b}_{L} = 7,{c}_{M} = - 1,{c}_{L} = - 3 \), and\n\n\[ {b}_{L}^{2} - 4{a}_{L}{c}_{L} = {13} > 0. \]\n\nClearly, the conditions \( \left( {\mathrm{H}}_{7}\right) - \left( {\mathrm{H}}_{10}\right) \) of Theorem 3.3 are satisfied. It follows from Theor... | Yes |
Lemma 2.4 If \( \psi \notin {\delta }_{2} \), then \( {\lambda }_{\varphi ,\omega }^{ \circ } \) contains \( {\ell }_{1} \) . | Proof For any \( \varepsilon > 0 \), by \( \psi \notin {\delta }_{2} \) there is a sequence \( \left\{ {u}_{i}\right\} \) of positive numbers such that \( \varphi \left( {{2}^{2}{u}_{1}}\right) \omega \left( 1\right) < \frac{\varepsilon }{2}, \n\n\[ \n\varphi \left( {{2}^{i + 1}{u}_{i}}\right) \leq \left( {{2}^{i + 1} ... | Yes |
Lemma 2.5 If \( \omega \) is not regular, then \( {\lambda }_{\varphi ,\omega }^{ \circ } \) contains \( {\ell }_{\infty }^{n} \) uniformly. | Proof Since \( \omega \) is not regular, we see that for any \( n \) and any \( \varepsilon \in \left( {0,1}\right) \), there exists a nature number \( \alpha > 1 \) such that\n\n\[\n\mathop{\sum }\limits_{{t = 1}}^{{n\alpha }}\omega \left( t\right) \leq \left( {1 + \varepsilon }\right) \mathop{\sum }\limits_{{t = 1}}^... | Yes |
Theorem 4.1 Orlicz-Lorentz sequence space \( {\lambda }_{\varphi ,\omega }^{ \circ } \) is uniformly non-square if and only if\n\n(a) \( \omega \) is regular,\n\n(b) \( \varphi ,\psi \in {\delta }_{2} \) . | Proof (Necessity) By Lemmas 2.3-2.4, we can see \( \varphi \in {\delta }_{2} \) and \( \psi \in {\delta }_{2} \) . Suppose \( \omega \) is not regular, then we can find an infinite sequence \( \left\{ {t}_{n}\right\} \subset \mathbb{N} \) such that\n\n\[ \frac{S\left( {2{t}_{n}}\right) }{S\left( {t}_{n}\right) } \right... | Yes |
Proposition 3.2 Let \( {H}_{0} \) be a Hamiltonian function of the form \( {H}_{0} = N\left( p\right) + \epsilon \widetilde{H}\left( {p, q}\right) \) with \( N,\widetilde{H} \in {\mathfrak{A}}_{{p}^{ * },\varrho } \) . Denote\n\n\[ \n{\overrightarrow{\lambda }}_{0} = \left( {{\partial }_{{p}_{1}}N\left( {p}^{ * }\right... | Proof Step 1 Let \( \lambda = \nabla N\left( {p}^{ * }\right) \) be the unperturbed frequency. By the classic KAM theorem, there exists a symplectic differmorphism \( \Phi : {B}_{{p}_{ * },\frac{1}{2}\rho } \rightarrow {B}_{{p}_{ * },\frac{1}{2}\rho } \) such that\n\n\[ \nH \circ \Phi = z + \lambda \cdot \left( {p - {p... | Yes |
Theorem 2.1 Under the validity of the first-order condition in (1.2), and for intermediate sequences \( k \), i.e., sequences of integer values \( k = k\left( n\right) ,1 \leq k < n \), such that\n\n\[ k = k\left( n\right) \rightarrow \infty ,\;k/n \rightarrow 0,\;\text{ as }\;n \rightarrow \infty ,\]\n\nthe \( {N}_{\a... | Proof Denote \( {K}_{\alpha }\left( \frac{i}{k + 1}\right) \mathrel{\text{:=}} {K}_{\alpha } \) and by Lemma 2.1, we have\n\n\[ {U}_{i} \approx \gamma {E}_{i} + A\left( {n/k}\right) {\left( \frac{i}{k + 1}\right) }^{-\rho }{E}_{i} + {o}_{p}\left( {A\left( {n/k}\right) }\right) \]\n\n\[ = \gamma {E}_{i} + A\left( {n/k}\... | Yes |
Lemma 3.1 Suppose Assumption G holds. Then there exist constants \( {M}_{p},{M}_{y},{M}_{m} > 0 \) , such that\n\n\[ \begin{Vmatrix}{p}_{k}\end{Vmatrix} \leq {M}_{p},\;\begin{Vmatrix}{y}_{k + 1}\end{Vmatrix} \leq {M}_{y},\;\left| {{m}_{k}\left( \alpha \right) }\right| \leq {M}_{m}\alpha \]\n\nfor all \( k \) and \( \al... | Proof From (G1) we have that the right-hand side of (2.3) is uniformly bounded. Additionally, Assumptions (G2), (G3) and (G4) guarantee that the inverse of the matrix in (2.3) exists and is uniformly bounded for all \( k \) . Consequently, the solution of \( \left( {2.3}\right) \left( {\left( {p}_{k},{y}_{k + 1}\right)... | Yes |
Lemma 3.2 Suppose Assumption (G1) holds. Then there exist constants \( {C}_{\theta },{C}_{\omega } > 0 \) such that for all \( k \) and \( \alpha \in (0,1\rbrack \)\n\n\[ \left| {\theta \left( {{x}_{k} + \alpha {p}_{k}}\right) - \left( {1 - \alpha }\right) \theta \left( {x}_{k}\right) }\right| \leq {C}_{\theta }{\alpha... | Proof The proof of (3.4a) can be found in Lemma 3.3 from [14].\n\nNext, we give the proof of (3.4b). From the second order Taylor expansions,\n\n\[ \left| {\omega \left( {{x}_{k} + \alpha {p}_{k}}\right) - \omega \left( {x}_{k}\right) - {m}_{k}\left( \alpha \right) }\right| \]\n\n\[ \overset{\left( {2.9}\right) }{ = }\... | No |
Lemma 3.3 Suppose Assumption G holds. If \( \left\{ {x}_{{k}_{i}}\right\} \) is a subsequence of iterates for which \( \chi \left( {x}_{{k}_{i}}\right) \geq {\epsilon }_{1} \) with constants \( {\epsilon }_{1} > 0 \) independent of \( i \) . Then there exists \( {\epsilon }_{2} > 0 \) independent of \( i \), we have\n\... | Proof If \( \theta \left( {x}_{{k}_{i}}\right) = 0 \), but \( \chi \left( {x}_{{k}_{i}}\right) > 0 \), Algorithm I would not terminated. From (2.3) and (3.3),\n\n\[ {m}_{{k}_{i}}\left( \alpha \right) /\alpha = {\left( {g}_{{k}_{i}} - {A}_{k}{y}_{{k}_{i} + 1}\right) }^{\mathrm{T}}{H}_{{k}_{i}}{p}_{{k}_{i}} = - {\begin{V... | Yes |
Lemma 3.4 Suppose Assumption G holds. Then the trial point \( {x}_{k}\left( {\alpha }_{k, l}\right) \) could not be rejected by \( {x}_{k} \) if \( {\alpha }_{k, l} \) is sufficiently small. | Proof There are two cases.\n\nCase \( 1\;\left( {\theta \left( {x}_{k}\right) > 0}\right) \) . Following directly from the second order Taylor expansion of \( \theta \left( x\right) \) , we have\n\n\[ \theta \left( {{x}_{k}\left( {\alpha }_{k, l}\right) }\right) - \theta \left( {x}_{k}\right) + \mu \left( {\alpha }_{k,... | Yes |
Theorem 3.1 Suppose Assumption G holds. Then\n\n\[ \mathop{\lim }\limits_{{k \rightarrow \infty }}\theta \left( {x}_{k}\right) = 0 \] | Proof The proof is similar to Lemma 8 in [15]. | No |
Lemma 3.7 Suppose Assumption G holds. Let \( \left\{ {x}_{{k}_{i}}\right\} \) be a subsequence. There exists a certain constant \( \bar{\alpha } > 0 \) such that for all \( {k}_{i} \) and \( \alpha \leq \bar{\alpha } \)\n\n\[ \omega \left( {{x}_{{k}_{i}} + \alpha {p}_{{k}_{i}}}\right) - \omega \left( {x}_{{k}_{i}}\righ... | Proof Let \( {M}_{p} \) and \( {C}_{\omega } \) be the constants from Lemmas 3.1 and 3.2. It then follows for all \( \alpha \leq \bar{\alpha } \) with \( \bar{\alpha } \mathrel{\text{:=}} \frac{\left( {1 - {\eta }_{\omega }}\right) {\epsilon }_{2}}{{C}_{\omega }{M}_{p}^{2}} \) and from (3.5) that\n\n\[ \omega \left( {{... | Yes |
Lemma 3.8 Suppose that Assumption G holds and that the filter is augmented only a finite number of times, i.e., \( \left| \mathcal{Z}\right| < \infty \) . Then\n\n\[ \mathop{\lim }\limits_{{k \rightarrow \infty }}\chi \left( {x}_{k}\right) = 0 \] | The proof of Lemma 3.8 is based on Lemma 8 in [4] by using the new filter pair. | No |
Lemma 3.9 Suppose Assumption G holds. Let \( \left\{ {x}_{{k}_{i}}\right\} \) be a subsequence and \( {m}_{{k}_{i}}\left( \alpha \right) \leq \) \( - \alpha {\epsilon }_{2} \) for a constant \( {\epsilon }_{2} > 0 \) independent of \( {k}_{i} \) and for all \( \alpha \in (0,1\rbrack \) . Then there exist constants \( {... | The proof of Lemma 3.9 is similar to Lemma 9 in [4]. So, it is omitted. | No |
Lemma 3.10 Suppose Assumption G holds. Let \( \left\{ {x}_{{k}_{i}}\right\} \) be a subsequence of \( \left\{ {x}_{k}\right\} \) with \( \chi \left( {x}_{{k}_{i}}\right) \geq \epsilon \) for a constant \( \epsilon > 0 \) independent of \( {k}_{i} \) . Then there exists \( K \in \mathbb{N} \) such that for all \( {k}_{i... | Lemma 3.10 can be proved by using the idea of Lemma 10 in [4]. | No |
Theorem 3.2 Suppose Assumption G holds. Then\n\n\\[ \n\\mathop{\\lim }\\limits_{{k \\rightarrow \\infty }}\\theta \\left( {x}_{k}\\right) = 0 \n\\]\n\n\\( \\left( {{3.11}\\mathrm{a}}\\right) \\)\n\nand\n\n\\[ \n\\mathop{\\liminf }\\limits_{{k \\rightarrow \\infty }}\\chi \\left( {x}_{k}\\right) = 0 \n\\]\n\n(3.11b)\n\n... | Proof (3.11a) follows from Theorem 3.1. In the case of the filter is augmented only a finite number of times, (3.11b) has been proved by Lemma 3.8. Otherwise, there exists a subsequence \\( \\left\\{ {x}_{{k}_{i}}\\right\\} \\) such that \\( {k}_{i} \\in \\mathcal{Z} \\) for all \\( i \\) . Now suppose that \\( \\lim \... | Yes |
Theorem 2.1 For all \( f \in {L}_{{p}_{2}\left( \cdot \right) ,{q}_{2}}, g \in {L}_{p\left( \cdot \right), q},0 < {p}^{ + },{p}_{2}^{ + } < \infty ,0 < q,{q}_{2} \leq \infty \), with \( \frac{1}{{p}_{1}} = \frac{1}{p} + \frac{1}{{p}_{2}},\frac{1}{{q}_{1}} = \frac{1}{q} + \frac{1}{{q}_{2}} \), we have | \[ \parallel {fg}{\parallel }_{{L}_{{p}_{1}\left( \cdot \right) ,{q}_{1}}} \leq c\parallel f{\parallel }_{{L}_{{p}_{2}\left( \cdot \right) ,{q}_{2}}}\parallel g{\parallel }_{{L}_{p\left( \cdot \right), q}}. \] | Yes |
Theorem 3.1 Let \( p\\left( \\cdot \\right) ,{p}_{2}\\left( \\cdot \\right) \\in P\\left( \\Omega \\right) \) satisfy the condition (2.4), \( 0 < q,{q}_{2} < \\infty, v \\in \) \( {V}_{p\\left( \\cdot \\right), q} \) with \( \\frac{1}{{p}_{1}\\left( \\cdot \\right) } = \\frac{1}{p\\left( \\cdot \\right) } + \\frac{1}{{... | Proof Using the pointwise estimation\n\n\[ s\\left( {{T}_{v}\\left( f\\right) }\\right) \\leq M\\left( v\\right) s\\left( f\\right) \]\n\n(3.1)\n\nby Hölder's inequality in Theorem 2.1, we have\n\n\[ {\\begin{Vmatrix}s\\left( {T}_{v}\\left( f\\right) \\right) \\end{Vmatrix}}_{{p}_{1}\\left( \\cdot \\right) ,{q}_{1}} \\... | Yes |
Lemma 4. \( {\mathbf{1}}^{\left\lbrack {17}\right\rbrack } \) Let \( p\left( \cdot \right) \in P\left( \Omega \right) ,0 < q \leq \infty ,0 < \theta < 1 \) and \( \frac{1}{p\left( \cdot \right) } = \frac{1 - \theta }{\widehat{p}\left( \cdot \right) } \) . Then | \[ {\left( {H}_{\widehat{p}\left( \cdot \right) }^{s},{H}_{\infty }^{s}\right) }_{\theta, q} = {H}_{p\left( \cdot \right), q}^{s}. \] | Yes |
Theorem 4.1 Let \( {p}_{1}\left( \cdot \right) ,{p}_{2}\left( \cdot \right) \in P\left( \Omega \right) ,0 < {p}_{1}\left( \cdot \right) < {p}_{2}\left( \cdot \right) < \infty \) and \( 0 < q < \infty \) . Suppose that \( f = {\left\{ {f}_{n}\right\} }_{n \geq 0} \in {H}_{{p}_{1}\left( \cdot \right), q}^{s} \), one of i... | Proof From the definition of \( {v}_{j}^{-1} \), it is easy to see that the process \( {v}^{-1} = {\left\{ {v}_{j}^{-1}\right\} }_{j \geq 1} \) is adapted to \( {\left\{ {\mathcal{F}}_{j}\right\} }_{j \geq 1} \) and \( {v}_{j}^{-1} \leq 1 \) for every \( j \geq 1 \) . Then \( g = {\left\{ {g}_{n}\right\} }_{n \geq 0} \... | Yes |
Theorem 4.2 Let \( {p}_{1}\left( \cdot \right) ,{p}_{2}\left( \cdot \right) \in P\left( \Omega \right) ,0 < {p}_{1}\left( \cdot \right) < {p}_{2}\left( \cdot \right) < \infty \) and \( 0 < {q}_{1} < {q}_{2} < \infty \) . Suppose that \( f = {\left\{ {f}_{n}\right\} }_{n \geq 0} \in {H}_{{p}_{1}\left( \cdot \right) ,{q}... | \[ {g}_{n} = \mathop{\sum }\limits_{{k = 0}}^{n}{v}_{k - 1}^{-1}d{f}_{k},\;n \geq 0, \] where \( {v}_{j}^{-1} \mathrel{\text{:=}} \min \left\{ {\mathrm{E}\left( {{s}_{j + 1}{\left( {f}^{0}\right) }^{-\beta } \mid {\mathcal{F}}_{j}}\right) ,1}\right\} \) for any \( j \geq - 1,{f}^{0} \) is given by (4.2) and \( \beta = ... | Yes |
For the MARE (1.1), if \( K \) in (1.2) is a nonsingular M-matrix or an irreducible singular M-matrix and the parameters \( \alpha ,\beta \) satisfy\n\n\[ \alpha \geq \max \left\{ {a}_{ii}\right\} ,\;\beta \geq \max \left\{ {d}_{ii}\right\} \]\n\n(3.5)\n\nthen the sequence \( \left\{ {X}_{k}\right\} \) generated by (3.... | Proof We prove (3.6) by induction. When \( k = 0 \), it is clear that \( 0 = {X}_{0} \leq S \) . Thus, (3.6) holds true for \( k = 0 \) .\n\nSuppose that (3.6) holds true for \( k = l \) . Then we have\n\n\[ \left( {{\beta I} + A - {X}_{l}C}\right) \left( {{X}_{l + 1} - S}\right) \left( {{\alpha I} + D - C{X}_{l}}\righ... | Yes |
Theorem 3.1 For the MARE (1.1), if \( K \) in (1.2) is a nonsingular M-matrix or an irreducible singular M-matrix and the parameters \( \alpha ,\beta \) satisfy (3.5), then the sequence \( \left\{ {X}_{k}\right\} \) generated by the new iteration method (3.4) is well defined, monotonically increasing, and converges to ... | Proof We have shown in Lemma 3.1 and Lemma 3.2 that \( \left\{ {X}_{k}\right\} \) is nonnegative, monotonically increasing and is bounded from above. Thus there is a nonnegative matrix \( {S}^{ * } \) such that \( \mathop{\lim }\limits_{{k \rightarrow \infty }}{X}_{k} = {S}^{ * } \) . From Lemma 3.1, we have \( {S}^{ *... | Yes |
Example 4.1 Consider the MARE (1.1) with\n\n\[ A = - {10}{E}_{n \times n} + {180.002}{I}_{n},\;B = {0.001}{E}_{n \times m}, \]\n\n\[ C = {B}^{\mathrm{T}},\;D = {0.018}{I}_{m}, \]\n\nwhere \( {E}_{m \times n} \) is the \( m \times n \) matrix with all ones and \( {I}_{m} \) is the identity matrix of size \( m \) with \(... | Tab. 4.1 Numerical results of Example 4.1\n\n<table><thead><tr><th>Method</th><th>IT</th><th>CPU</th><th>RES</th></tr></thead><tr><td>Newton</td><td>3</td><td>0.0022</td><td>\( {7.4339}\mathrm{e} \) - \( {08} \)</td></tr><tr><td>FP3</td><td>8</td><td>0.0043</td><td>\( {4.8065}\mathrm{e} \) - \( {07} \)</td></tr><tr><td... | No |
Example 4.2 Consider the MARE (1.1) with\n\n\[ A = \left( \begin{matrix} 3 & - 1 & & \\ & 3 & \ddots & \\ & & \ddots & - 1 \\ - 1 & & & 3 \end{matrix}\right) \in {\mathbb{R}}^{n \times n},\;B = 2{I}_{n},\;C = {20}{I}_{n},\;D = {10A}. \] | This example is from [12], where the corresponding \( K \) is an irreducible singular M-matrix. For different sizes of \( n \), the numerical results are summarized in Tab. 4.2.\n\nTab. 4.2 Numerical results of Example 4.2\n\n<table><thead><tr><th>\( n \)</th><th>Method</th><th>IT</th><th>CPU</th><th>RES</th></tr></the... | No |
Theorem 1.1 Suppose \( u \) satisfies \( {u}_{xx} + {x}^{2\sigma }{u}_{yy} = f \) in \( \Omega \) and \( u\left( {0, y}\right) = \varphi \left( y\right), u\left( {\kappa, y}\right) = \phi \left( y\right) \), then the following holds. | 1) Let \( 0 < \alpha < 1 \) when \( k = 0,1 \) . If \( f\left( {x, y}\right) \in {L}^{\infty }\left( \Omega \right) ,\varphi \left( y}\right) \in {C}_{ * }^{k,\alpha }\left( \mathbb{R}\right) \) and \( \psi \left( y}\right) \in {C}^{k,\alpha }\left( \mathbb{R}\right) \) , then \( u \in {C}_{ * }^{k,\alpha }\left( \bar{... | Yes |
Lemma 3.1 Suppose \( \Omega \) is a bounded domain in \( {\mathbb{R}}_{ + }^{2} \), and \( \mathcal{L}u \leq 0 \) in \( \Omega, u \geq 0 \) on \( \partial \Omega \) , then \( u \geq 0 \) in \( \Omega \) . | It is easily seen by applying the usual methods for the elliptic equations, so we omit the proof. | No |
Lemma 3.2 Let \( Q = \\left( {0,1}\\right) \\times \\left( {-2,2}\\right) \) . Suppose\n\n\[ \n{u}_{xx} + {x}^{2\\sigma }{u}_{yy} = 0,\\;\\left( {x, y}\\right) \\in Q = \\left( {0,1}\\right) \\times \\left( {-2,2}\\right) \n\]\n\nwith \( u\\left( {0, y}\\right) = 0 \), and \( \\left| u\\right| \\leq 1,\\left( {x, y}\\r... | Proof By Lemma 3.1 we know\n\n\[ \n\\left| u\\right| \\leq 1,\\;\\left( {x, y}\\right) \\in Q. \n\]\n\nWe consider the function\n\n\[ \nv\\left( {x, y}\\right) = - {x}^{2} + {y}^{2} + {2x} \n\]\n\nthen\n\n\[ \n{v}_{xx} + {x}^{2\\sigma }{v}_{yy} = - 2 + 2{x}^{2\\sigma } \\leq 0,\\;\\left( {x, y}\\right) \\in {Q}_{1}^{ +... | Yes |
Lemma 3.3 Let\n\n\\[ \n{u}_{xx} + {x}^{2\\sigma }{u}_{yy} = f,\\;\\left( {x, y}\\right) \\in {Q}_{1}^{ + },\n\\]\n\n(3.1)\n\nand \\( u\\left( {0, y}\\right) = \\varphi \\left( y\\right), y \\in \\left( {-1,1}\\right) \\) . There exist constants \\( 0 < r < \\frac{1}{2} \\) and \\( \\delta \\) small such that if\n\n\\[\... | Proof Let \\( v\\left( {x, y}\\right) \\) satisfy\n\n\\[\n\\left\\{ \\begin{array}{ll} {v}_{xx} + {x}^{2\\sigma }{v}_{yy} = 0, & \\left( {x, y}\\right) \\in {Q}_{1}^{ + }, \\\\ v\\left( {0, y}\\right) = 0, & y \\in \\left( {-1,1}\\right) , \\\\ v = u, & \\left( {x, y}\\right) \\in \\partial {Q}_{1}^{ + } \\smallsetminu... | Yes |
Lemma 3.4 Suppose \( u \) is a weak solution of\n\n\[ \n{u}_{xx} + {x}^{2\sigma }{u}_{yy} = 0,\;\left( {x, y}\right) \in {Q}_{1}^{ + }, \]\n\nwith the boundary condition\n\n\[ \nu\left( {0, y}\right) = 0,\;y \in \left( {-1,1}\right) \]\n\nand \( \left| {u\left( {x, y}\right) }\right| \leq 1,\left( {x, y}\right) \in {Q}... | This lemma can be proved by the odd extension and Lemma 2 in [10]. | No |
Lemma 3.5 Suppose\n\n\\[ \n{u}_{xx} + {x}^{2\\sigma }{u}_{yy} = f,\\;\\left( {x, y}\\right) \\in {Q}_{1}^{ + },\n\\]\n\nand \\( u\\left( {0, y}\\right) = \\varphi \\left( y\\right) ,\\left| u\\right| \\leq 1,\\left( {x, y}\\right) \\in \\partial {Q}_{1}^{ + } \\) . Then, for any \\( 0 < \\alpha < 1 \\), there exist con... | Proof Let \\( v\\left( {x, y}\\right) \\) satisfy\n\n\\[ \n\\left\\{ \\begin{array}{ll} {v}_{xx} + {x}^{2\\sigma }{v}_{yy} = 0, & \\left( {x, y}\\right) \\in {Q}_{1}^{ + }, \\\\ v\\left( {0, y}\\right) = 0, & y \\in \\left( {-1,1}\\right) , \\\\ v = u, & \\left( {x, y}\\right) \\in \\partial {Q}_{1}^{ + } \\smallsetmin... | Yes |
Lemma 4.1 Let \( A \geq 0 \) and \( 0 < \delta \leq 1 \) . Suppose\n\n\[ \n\mathcal{L}u = f,\;\left( {x, y}\right) \in {Q}_{\frac{1}{2}}^{ + }, \n\]\n\nwith\n\n\[ \nu\left( {0, y}\right) = \varphi \left( y\right) ,\;y \in \left( {-{\left( \frac{1}{2}\right) }^{1 + \sigma },{\left( \frac{1}{2}\right) }^{1 + \sigma }}\ri... | Proof Let\n\n\[ \nv\left( {x, y}\right) = \frac{1 + A}{2}\delta \left( {1 - 4{x}^{2}}\right) . \n\]\n\nThen we have\n\n\[ \nv\left( {x, y}\right) \geq u\left( {x, y}\right) ,\;\left( {x, y}\right) \in \partial {Q}_{\frac{1}{2}}^{ + } \n\]\n\nand\n\n\[ \n\mathcal{L}v = - 4\left( {1 + A}\right) \delta - 4\left( {1 + A}\r... | Yes |
Lemma 4.2 Suppose\n\n\\[ \n\\mathcal{L}u = f,\\;\\left( {x, y}\\right) \\in {Q}_{1}^{ + },\n\\]\n\nwith \\( u\\left( {0, y}\\right) = \\varphi \\left( y\\right) \\), and \\( \\left| u\\right| \\leq 1,\\left( {x, y}\\right) \\in \\partial {Q}_{1}^{ + } \\) . Then, for any \\( 0 < \\alpha < 1 \\), there exist constants \... | Proof Let \\( v\\left( {x, y}\\right) \\) satisfy\n\n\\[ \n\\left\\{ \\begin{array}{ll} {v}_{xx} + {x}^{2\\sigma }{v}_{yy} = 0, & \\left( {x, y}\\right) \\in {Q}_{1}^{ + }, \\\\ v\\left( {0, y}\\right) = 0, & y \\in \\left( {-1,1}\\right) , \\\\ v = u, & \\left( {x, y}\\right) \\in \\partial {Q}_{1}^{ + } \\smallsetmin... | Yes |
Theorem 4.1 Let \( l > \sigma - 1,0 < \alpha < 1,\varphi \left( y\right) \) be \( {C}_{ * }^{1,\alpha } \) at \( y = 0 \) . Suppose\n\n\[ \left\{ \begin{array}{ll} \mathcal{L}u = f, & \left( {x, y}\right) \in {Q}_{{r}_{0}}^{ + }, \\ u\left( {0, y}\right) = \varphi \left( y\right) , & y \in \left( {-{r}_{0}^{1 + \sigma ... | Proof We will modify the proof of Theorem 3.1 to obtain this theorem. Since \( \varphi \left( y\right) \) is \( {C}_{ * }^{1,\alpha } \) at \( y = 0 \), we have\n\n\[ \left| {\varphi \left( y\right) - p\left( y\right) }\right| \leq {\left\lbrack \varphi \right\rbrack }_{{C}_{ * }^{1,\alpha }\left( 0\right) }{y}^{\frac{... | Yes |
Theorem 2.1 Let \( \mathcal{A} = W + \mathrm{i}T \in {\mathbb{C}}^{n \times n} \) be a non-Hermitian but symmetric matrix \( \left( {\mathcal{A} \neq {\mathcal{A}}^{ * },\mathcal{A} = {\mathcal{A}}^{\mathrm{T}}}\right) \) with \( W, T \in {\mathbb{R}}^{n \times n} \) are both symmetric, and \( W \) being positive defin... | Proof (i) By (2.4) and direct calculations, we have\n\n\[ \n\rho \left( {T}_{\alpha ,\beta }\right) = \rho \left( {\mathrm{i}{\widehat{M}}_{\alpha ,\beta }\left( {{\beta I} - \alpha {W}^{-1}T}\right) }\right) \n\]\n\n\[ \n\leq {\begin{Vmatrix}\mathrm{i}{\widehat{M}}_{\alpha ,\beta }\left( \beta I - \alpha {W}^{-1}T\rig... | Yes |
Corollary 2.1 Assume that the conditions of Theorem 2.1 are satisfied, then the optimal relation between two parameters \( \alpha ,\beta \) that minimizes the upper bound \( {\delta }_{\alpha ,\beta } \) of the spectral radius \( \rho \left( {T}_{\alpha ,\beta }\right) \) is given by\n\n\[ \alpha = \beta \frac{1 - {\la... | Proof By introducing \( \tau = \alpha /\beta \), and\n\n\[ g\left( \tau \right) = \frac{1 - \tau {\lambda }_{\min }}{\tau + {\lambda }_{\min }},\;h\left( \tau \right) = \frac{\tau {\lambda }_{\max } - 1}{\tau + {\lambda }_{\max }}, \]\n\nwe have\n\n\[ {g}^{\prime }\left( \tau \right) = - \frac{1 + {\lambda }_{\min }^{2... | Yes |
Theorem 2.2 Let \( \mathcal{A} = W + \mathrm{i}T \in {\mathbb{C}}^{n \times n} \) be a non-Hermitian but symmetric matrix \( \left( {\mathcal{A} \neq {\mathcal{A}}^{ * },\mathcal{A} = {\mathcal{A}}^{\mathrm{T}}}\right) \) with \( W, T \in {\mathbb{R}}^{n \times n} \) be both symmetric, and \( W \) being positive defini... | Proof Let \( \lambda \) be an eigenvalue of the matrix \( {T}_{\alpha ,\beta } \) and \( x \) the corresponding eigenvector, i.e., \( {M}_{\alpha ,\beta }^{-1}{N}_{\alpha ,\beta }x = {\lambda x} \), or equivalently, \( \lambda \left( {{\alpha W} + {\beta T}}\right) x = \mathrm{i}\left( {{\beta W} - {\alpha T}}\right) x... | Yes |
Theorem 2.3 Let \( \mathcal{A} = W + \mathrm{i}T \in {\mathbb{C}}^{n \times n} \) be a non-Hermitian but symmetric matrix \( \left( {\mathcal{A} \neq {\mathcal{A}}^{ * },\mathcal{A} = {\mathcal{A}}^{\mathrm{T}}}\right) \) with \( W, T \in {\mathbb{R}}^{n \times n} \) be both symmetric, and \( W \) being positive defini... | Proof Let \( \lambda \) be an eigenvalue of the matrix \( {M}_{\alpha ,\beta }^{-1}\mathcal{A} \) and \( x \) the corresponding eigenvector with \( \parallel x{\parallel }_{2} = 1 \) . It is known that\n\n\[ \left( {\alpha - \beta \mathrm{i}}\right) \left( {W + \mathrm{i}T}\right) x = \lambda \left( {{\alpha W} + {\bet... | Yes |
Lemma 2.1 Systems (1.2) and (1.3) have a unique global positive solution on \( t > - \tau \) for any initial data given above, respectively. | Remark 2.1 The proof is very standard and is omitted here. Readers may refer to [17]. | No |
Theorem 3.3 If Assumption 3.2 holds, then the equilibrium state of (3.5) is stable in probability, i.e., the equilibrium state of (1.3) is stable in probability. | Proof Define two same functionals \( {V}_{1},{V}_{2} \) as before, using Itô’s formula and computing \( {LV} \) along (3.6), we have\n\n\[ L\left( {{V}_{1} + {V}_{2}}\right) = 2\left( {{y}_{1} - \frac{{a}_{12}{x}_{1}^{ * }}{1 - {\tau }_{1}^{\prime }}{\int }_{t - {\tau }_{1}\left( t\right) }^{t}{y}_{1}\left( s\right) \m... | Yes |
Theorem 2.1 Let Assumptions 1-3 hold. Then, for small enough \( \varepsilon > 0 \) there exists a \( {\lambda }^{ * }\left( \varepsilon \right) \) with \( {\lambda }^{ * }\left( \varepsilon \right) = {\lambda }_{1}\varepsilon + \mathcal{O}\left( {\varepsilon }^{3/2}\right) \) such that when \( \lambda = {\lambda }^{ * ... | ## 3. Proof of Theorem 2.1\n\nTo prove Theorem 2.1 we need the following lemma whose proof can be found in [24].\n\nLemma 3.1 \( {}^{\left | No |
Lemma 3.2 Assume \( \theta \in \left( {0, H}\right), H \in \left( {\frac{1}{2},1}\right) \), and let \( {\xi }_{t} \) be defined in (3.1). Then, as \( t \rightarrow T \) :\n\n1) if \( 0 < \theta < \frac{1}{2} \), then\n\n\[ \n{\left( T - t\right) }^{1 - {2\theta }}{\int }_{0}^{t}{\left( T - u\right) }^{{2\theta } - 2}{... | Proof of Theorem 3.1 By the formula (2.6), we obtain that, for any \( t \in \lbrack 0, T) \) ,\n\n\[ \n\frac{1}{2}{\left( T - t\right) }^{{2\theta } - 1}{\xi }_{t}^{2} = \frac{1 - {2\theta }}{2}{\int }_{0}^{t}{\left( T - u\right) }^{{2\theta } - 2}{\xi }_{u}^{2}\mathrm{\;d}u + {\int }_{0}^{t}{\left( T - u\right) }^{{2\... | Yes |
1) If \( \theta \in \left( {0,1 - H}\right) \) then, as \( t \rightarrow T \) ,\n\n\[ \left( {F,{\left( T - t\right) }^{1 - H - \theta }{\int }_{0}^{t}{\left( T - u\right) }^{\theta - 1}\mathrm{\;d}{Y}_{u}^{\left( 1\right) }}\right) \overset{\text{ law }}{ \rightarrow }\left( {F,\sqrt{{\sigma }_{1}}N}\right) . \] | Proof of Theorem 3.2 1) Assume that \( \theta \in \left( {0,1 - H}\right) \) . By Lemma 3.4, we have\n\n\[ {\left( T - t\right) }^{\theta - H}\left( {\theta - {\widehat{\theta }}_{t}}\right) = \frac{{\left( T - t\right) }^{1 - H - \theta }{\eta }_{t}}{{\left( T - t\right) }^{1 - {2\theta }}{\int }_{0}^{t}{\left( T - u\... | Yes |
Theorem 2.1 The Pitman's measure of closeness of preliminary test estimator related to unrestricted estimator are as follows:\n\n(i) If \( {F}_{\alpha } \leq \frac{m\left( {p + \delta - \gamma }\right) }{p\left( {m + \gamma }\right) } \), then\n\n\[ \operatorname{PC}\left( {{\widehat{\sigma }}^{2PT},{\widehat{\sigma }}... | Proof By direct computation, we know that\n\n\[ L\left( {{\widehat{\sigma }}^{2PT},{\sigma }^{2}}\right) = {\left( \frac{{\widehat{\sigma }}^{2PT}}{{\sigma }^{2}} - 1\right) }^{2} = I\left( {F \geq {F}_{\alpha }}\right) {\left( \frac{{u}_{2}}{m + \gamma } - 1\right) }^{2} + I\left( {F < {F}_{\alpha }}\right) {\left( \f... | Yes |
Example 1 Symmetry reduction and solution of the system (2.1) which admits the GCS (2.2). | We get solutions in exponential form by integrating the system (2.2)\n\n\[ u = {\phi }_{1}\left( t\right) + {\phi }_{2}\left( t\right) {\mathrm{e}}^{{b}_{1}x},\;v = {\psi }_{1}\left( t\right) + {\psi }_{2}\left( t\right) {\mathrm{e}}^{{b}_{1}x}. \]\n\n(3.1)\n\nThe system (2.1) can be reduced to the following system of ... | Yes |
The system (2.9) can be rewritten as\n\n\[ \n{u}_{t} = {\left\lbrack \left( d + av\right) u\right\rbrack }_{xx} + u\left( {{a}_{11} - {4a}{c}^{2}v}\right) ,\;{v}_{t} = {\left\lbrack \left( d + bu\right) v\right\rbrack }_{xx} + v\left( {{a}_{21} - {4b}{c}^{2}u}\right) .\n\] \n\n(3.3) | Since \( a > 0, b > 0, d > 0,{a}_{11} > 0,{a}_{21}{c}^{2} > 0 \), the system (3.3) is competition model. The form of GCSs is\n\n\[ \n{\eta }_{1} = {u}_{xxx} - {c}^{2}{u}_{x},\;{\eta }_{2} = {v}_{xxx} - {c}^{2}{v}_{x}.\n\] \n\n(3.4)\n\nWe obtain the exponential form solutions by integrating the system (3.4)\n\n\[ \nu = ... | Yes |
Theorem 3.1(Weak duality theorem) Let \( x \in M \) be feasible for \( \left( \mathcal{P}\right) \) and \( \left( {\lambda ,\rho }\right) (\lambda \in \) \( {T}^{ * },\rho \geq 0) \) be a feasible point for \( \left( \mathcal{D}\right) \), then\n\n\[ \operatorname{val}\left( \mathcal{D}\right) \leq \operatorname{val}\l... | Proof As \( \mathcal{L}\left( {x,\lambda ,\rho }\right) \leq \psi \left( {x,0}\right) \), it follows that\n\n\[ \mathop{\inf }\limits_{{x \in M}}\mathcal{L}\left( {x,\lambda ,\rho }\right) \leq \psi \left( {x,0}\right) \]\n\ntherefore\n\n\[ \mathop{\inf }\limits_{{x \in M}}\mathcal{L}\left( {x,\lambda ,\rho }\right) \l... | Yes |
Corollary 3.2 Suppose the condition (Q) holds, and \( \mu \) is lower semi-continuous at 0 . then \( \operatorname{val}\left( \mathcal{P}\right) = \operatorname{val}\left( \mathcal{D}\right) \) . | Proof According to the condition (Q), there exist real constants \( c \) and \( b \), such that \( \mu \left( z\right) \geq c - {b\sigma }\left( z\right) ,\forall z \in T. \)\n\nLet \( \lambda \in {T}^{ * } \), then\n\n\[ \mu \left( z\right) \geq c + \langle \lambda, z\rangle - \parallel \lambda {\parallel }_{1}\parall... | Yes |
Example 3.1 Consider the following problem:\n\n\\[ \n\\text{(}\\mathcal{P}\\text{)}\\min f\\left( x\\right) = {x}_{1}^{2} - {x}_{2}^{2} + 2{x}_{2} \n\\]\n\n\\[ \n\\text{s.t}\\;g\\left( {x,\\omega }\\right) = {x}_{1} + \\left( {1 - \\omega }\\right) {x}_{2} \\leq 0\\text{,}\n\\]\n\n(3.9)\n\n\\[ \n\\omega \\in \\left\\lb... | As \\( \\mathcal{L}\\left( {x,\\lambda ,\\rho }\\right) \\leq \\psi \\left( {x,0}\\right) \\), it follows that\n\n\\[ \n\\mathop{\\inf }\\limits_{{x \\in M}}\\mathcal{L}\\left( {x,\\lambda ,\\rho }\\right) \\leq \\psi \\left( {x,0}\\right) \n\\]\n\ntherefore\n\n\\[ \n\\mathop{\\inf }\\limits_{{x \\in M}}\\mathcal{L}\\l... | Yes |
We consider the case of the LAD loss function and the error term with a standard normal distribution. The estimated results are shown in Tab. 1. | From Tab. 1 we find that FR+LLA method performs best in the case of ultra-high dimensional data. The RSIS+Lasso method performs better for the estimation error and the prediction error. However, considering the ability to correctly exclude unimportant variables and to misidentify important variables, the FR+LLA method ... | Yes |
Lemma 2.2 Let \( {x}^{ * } \in {X}^{ * } \) . Then, the sequences \( \left\{ {x}_{k}\right\} \) and \( \left\{ {z}_{k}\right\} \) generated by Algorithm SGM satisfy\n\n\[ \n{\begin{Vmatrix}{x}_{k + 1} - {x}^{ * }\end{Vmatrix}}^{2} \leq {\begin{Vmatrix}{x}_{k} - {x}^{ * }\end{Vmatrix}}^{2} - {\sigma }^{2}{\begin{Vmatrix... | Proof See the proof of Lemma 3.2 in [8]. | No |
Lemma 3.1 Suppose that Assumption (A2) holds. Then, there exists a constant \( \gamma > 0 \) such that the stepsize \( {\alpha }_{k} \) defined in Step 3 of Algorithm SGM satisfies\n\n\[ \n{\alpha }_{k} \geq \min \left\{ {1,\gamma \frac{{\begin{Vmatrix}{F}_{k}\end{Vmatrix}}^{2}}{{\begin{Vmatrix}{d}_{k}\end{Vmatrix}}^{2... | Proof See the proof of Lemma 3.4 in [8]. | No |
Lemma 3.2 Let \( \\left\\{ {x}_{k}\\right\\} \) be an infinite sequence generated by Algorithm SGM. If Assumption (A2) holds, then we have\n\n\[ \n\\mathop{\\lim }\\limits_{{k \\rightarrow + \\infty }}\\inf \\begin{Vmatrix}{F}_{k}\\end{Vmatrix} = 0 \n\]\n\nFurthermore, the sequence \( \\left\\{ {x}_{k}\\right\\} \) con... | Proof See the proof of Theorem 3.5 in [8]. | No |
Theorem 3.1 Suppose that Assumptions (A1) and (A2) hold. Then, there exists a constant \( \tau > 0 \) such that\n\n\[ \operatorname{dist}\left( {{x}_{k},{X}^{ * }}\right) \leq \frac{\tau }{\sqrt{k}} \]\n\n(3.7) | Proof Let \( {\bar{x}}_{k} \in {X}^{ * } \) be the closest point to \( {x}_{k} \), i.e.,\n\n\[ \begin{Vmatrix}{{x}_{k} - {\bar{x}}_{k}}\end{Vmatrix} = \operatorname{dist}\left( {{x}_{k},{X}^{ * }}\right) \]\n\n(3.8)\n\nThen, it follows from (2.6) and (3.8) that\n\n\[ {\operatorname{dist}}^{2}\left( {{x}_{k + 1},{X}^{ *... | Yes |
Theorem 3.2 Suppose that Assumption (A2) holds. If the mapping \( F \) is strongly monotone with modulus \( \mu > 0 \), i.e., \n\n\[ \n\langle F\left( x\right) - F\left( y\right), x - y\rangle \geq \mu \parallel x - y{\parallel }^{2},\forall x, y \in X. \n\] \n\nthen \n\n\[ \n\begin{Vmatrix}{{x}_{k} - \bar{x}}\end{Vmat... | Proof By the Cauchy-Schwarz inequality and the strong monotonicity of \( F \), we obtain \n\n\[ \n\begin{Vmatrix}{F}_{k}\end{Vmatrix} = \begin{Vmatrix}{{F}_{k} - F\left( \bar{x}\right) }\end{Vmatrix} \geq \mu \begin{Vmatrix}{{x}_{k} - \bar{x}}\end{Vmatrix}, \n\] \n\n(3.16) \n\nwhich, together (3.10) and (3.11), implies... | Yes |
Lemma 4.1 Let \( \left\{ {x}_{k}\right\} \) and \( \left\{ {z}_{k}\right\} \) be the two sequences generated by UAF. If \( {x}^{ * } \in {X}^{ * } \) , then we have the following conclusions:\n\n1) If the stepsize \( {\alpha }_{k} \) is performed by using the line search schemes (4.4),(4.5) and (4.9), respectively, the... | Proof From the monotonicity of \( F \) and \( {x}^{ * } \in {X}^{ * } \), it follows that\n\n\[ \nF{\left( {z}_{k}\right) }^{\mathrm{T}}\left( {{z}_{k} - {x}^{ * }}\right) = {\left( F\left( {z}_{k}\right) - F\left( {x}^{ * }\right) \right) }^{\mathrm{T}}\left( {{z}_{k} - {x}^{ * }}\right) \geq 0,\forall k.\n\]\n\n(4.13... | No |
Lemma 4.2 Suppose that Assumption (A2) holds. Then, there exists a constant \( {c}_{10} > \) 0 such that the stepsize \( {\alpha }_{k} \) defined in Step 3 of UAF satisfies\n\n\[{\alpha }_{k} \geq \min \left\{ {\tau ,{c}_{10}\frac{{\begin{Vmatrix}{F}_{k}\end{Vmatrix}}^{2}}{{\begin{Vmatrix}{d}_{k}\end{Vmatrix}}^{2}}}\ri... | Proof The proof is similar to that of Lemma 2 in [12], and we therefore omit it. | No |
Theorem 4.1 Let \( \\left\\{ {x}_{k}\\right\\} \) be an infinite sequence generated by UAF. If the stepsize \( {\\alpha }_{k} \) is performed by using the line search schemes (4.4), (4.5) and (4.9), respectively, then we have the following conclusions:\n\n(I) Suppose that Assumption (A2) holds. Then \( \\mathop{\\lim }... | Proof Using Lemma 4.11) and Lemma 4.2, we can prove this theorem in a similar way as Lemma 3.2, Theorem 3.1 and Theorem 3.2. | No |
Theorem 4.2 Let \( \left\{ {x}_{k}\right\} \) be an infinite sequence generated by UAF. If the stepsize \( {\alpha }_{k} \) is performed by using the line search schemes (4.3) and (4.7), respectively, then we have the following conclusions:\n\n(I) Suppose that Assumption (A2) holds. Then \( \mathop{\lim }\limits_{{k \r... | Proof Using Lemma 4.12) and Lemma 4.2, we can prove this theorem in a similar way as Theorem 3.4, Theorem 3.6 and Theorem 3.8 in [7]. | No |
The elements of function \( F \) are given by\n\n\( {F}_{i}\left( x\right) = \ln \left( {{x}_{i} + 1}\right) - \frac{{x}_{i}}{n}, i = 1,2,\ldots, n \), and \( X = \left\{ {x \in {\mathbb{R}}^{n} \mid {x}_{i} \geq 0, i = 1,2,\ldots, n}\right\} \) . | Throughout the experiments, the direction \( {d}_{k} \) is defined as \( {}^{\left\lbrack 1\right\rbrack } \) :\n\n\[ \n{d}_{k} = \left\{ \begin{array}{ll} - {F}_{k}, & k = 0 \\ - {\theta }_{k}{F}_{k}, & k \geq 1 \end{array}\right. \]\n\n(5.1)\n\nwhere\n\n\[ \n{\theta }_{k} = \left\{ \begin{array}{ll} 1, & k = 0 \\ \fr... | Yes |
Lemma 2.2 There exist \( m > 0 \) and \( C > 0 \) such that the solution of (1.5) fulfills\n\n\[{\int }_{\Omega }u\left( {\cdot, t}\right) \leq m\text{ for all }t \in \left( {0,{T}_{\max }}\right) ,\]\n\n(2.2)\n\nand\n\n\[{\int }_{t}^{t + \tau }{\int }_{\Omega }{u}^{\alpha } \leq C\text{ for all }t \in \left( {0,{T}_{\... | Proof The proof is analogous to Lemma 2.2 in [14]. To avoid repetition, we omit it. | No |
Lemma 2.3 Suppose that \( \left( {u, v, w}\right) \) is a classical solution to (1.5). Then for each \( p \in \lbrack 1,\infty ) \), we have\n\n\[ \n{\int }_{\Omega }{v}^{p}\left( {\cdot, t}\right) \leq {\int }_{\Omega }{v}_{0}^{p}\text{ for all }t \in \left( {0,{T}_{\max }}\right) .\n\]\n\n(2.4)\n\nIn particular, when... | Proof Multiplying the second equation in (3.5) by \( {v}^{p - 1} \), integrating by parts, we obtain\n\n\[ \n\frac{1}{p}\frac{\mathrm{d}}{\mathrm{d}t}{\int }_{\Omega }{v}^{p} = {\int }_{\Omega }{v}^{p - 1}{\Delta v} - {\int }_{\Omega }{v}^{p}w\n\]\n\n\[ \n= - \left( {p - 1}\right) {\int }_{\Omega }{v}^{p - 2}{\left| \n... | Yes |
Lemma 2.4 There exists \( C > 0 \) such that the solution of (1.5) satisfies\n\n\[ \mathop{\sup }\limits_{{t \in \left( {0,{T}_{\max }}\right) }}\parallel w\left( {\cdot, t}\right) {\parallel }_{{L}^{\alpha }\left( \Omega \right) } \leq C. \] | Proof Multiplying the third equation of (1.5) by \( {w}^{\alpha - 1} \) and using Young’s inequality, we obtain that\n\n\[ \frac{1}{\alpha }\frac{\mathrm{d}}{\mathrm{d}t}{\int }_{\Omega }{w}^{\alpha } = {\int }_{\Omega }{w}^{\alpha - 1}u - \delta {\int }_{\Omega }{w}^{\alpha } \]\n\n\[ \leq \frac{\delta }{2}{\int }_{\O... | Yes |
Lemma 3.1 Let \( \delta > 0 \) and \( \alpha \geq 2 \) . Assume that\n\n\[ \mu > {\left( 2\epsilon \right) }^{\frac{1}{\alpha - 1}},\;0 < \epsilon < {2\delta }.\]\n\nThen the global solution of (3.1) has the property that there exists a constant \( \eta > 0 \) such that the nonnegative functions \( E \) and \( F \) def... | Proof The nonnegative of \( E\left( t\right) \) are similar to the recent works \( {}^{\left\lbrack {10},{16} - {17}\right\rbrack } \), so we omit it. In accordance with strong maximal principle and \( {U}_{0} > 0 \), it is readily yielded that \( U \) is positive in \( \bar{\Omega } \times \left( {0,\infty }\right) \)... | Yes |
Theorem 2.1 Let \( \Phi \in \mathcal{G},0 < q < \infty \) and \( \omega \) be a nonnegative random variable on \( \left( {\Omega ,\mathcal{F}, P}\right) \) . If the martingale \( f = {\left( {f}_{n}\right) }_{n \geq 0} \) is in \( {H}_{\Phi, q,\omega }^{\sigma } \), then there exists a sequence \( {\left( {a}^{k}\right... | Proof Assume \( f \in {H}_{\Phi, q,\omega }^{\sigma } \) . Let us consider the stopping times for all \( k \in \mathbb{Z} \) , \n\n\[ \n{\tau }_{k} = \inf \left\{ {n \in \mathbb{N} : {\sigma }_{n + 1}\left( f\right) > {2}^{k}}\right\} ,\;\inf \varnothing = \infty . \n\] \n\nFor each stopping time \( {\tau }_{k} \), we ... | Yes |
Theorem 2.2 Let \( \Phi \in \mathcal{G},0 < q < \infty \) and \( \omega \) be a nonnegative random variable on \( \left( {\Omega ,\mathcal{F}, P}\right) \) . If the martingale \( f = {\left( {f}_{n}\right) }_{n \geq 0} \) is in \( {\mathcal{Q}}_{\Phi, q,\omega } \), then there exists a sequence \( {\left( {a}^{k}\right... | Proof The proof is similar to the one of Theorem 2.1, so we only give it in sketch. If \( f = {\left( {f}_{n}\right) }_{n \geq 0} \in {\mathcal{Q}}_{\Phi, q,\omega } \), the stopping times \( {\tau }_{k} \) are defined by\n\n\[ \n{\tau }_{k} = \inf \left\{ {n \in \mathbb{N} : {\lambda }_{n} > {2}^{k}}\right\} ,\;\inf \... | Yes |
Theorem 2.3 Let \( \Phi \in \mathcal{G},0 < q < \infty \) and \( \omega \) be a nonnegative random variable on \( \left( {\Omega ,\mathcal{F}, P}\right) \) . If the martingale \( f = {\left( {f}_{n}\right) }_{n \geq 0} \) is in \( {\mathcal{D}}_{\Phi, q,\omega } \), then there exists a sequence \( {\left( {a}^{k}\right... | Proof The only difference in the proof is that \( {\lambda }_{n} = \mathop{\sum }\limits_{{k \in \mathbb{Z}}}{\mu }_{k}{\begin{Vmatrix}S\left( {a}^{k}\right) \end{Vmatrix}}_{\infty }{\chi }_{\left\{ {\tau }_{k} \leq n\right\} } \) is replaced by \( {\lambda }_{n} = \mathop{\sum }\limits_{{k \in \mathbb{Z}}}{\mu }_{k}{\... | No |
Lemma 4.1 Let \( \Phi \in \mathcal{G} \cap \Delta \cap \nabla ,0 < {r}_{1},{r}_{2}, p, q \leq \infty \) such that \( \frac{1}{r} = \frac{1}{{r}_{1}} + \frac{1}{{r}_{2}} \) . If \( f \in {L}_{{\Phi }^{\left( {r}_{1}\right) }}, g \in {L}_{{\Phi }^{\left( {r}_{2}\right) }} \), then \( {fg} \in {L}_{{\Phi }^{\left( r\right... | The proof is similar to Page 64 in [7] and we omit it. | No |
Theorem 4.1 Let \( \Phi \in \mathcal{G} \cap \Delta \cap \nabla ,0 < {q}_{0},{q}_{1} < 1,0 < {r}_{1},{r}_{2} \leq \infty ,\omega \) is nonnegative random variable and\n\n\[ \n\frac{1}{r} = \frac{1}{{r}_{1}} + \frac{1}{{r}_{2}},\;\frac{1}{q} = \frac{1 - \theta }{{q}_{0}} + \frac{\theta }{{q}_{1}},0 < \theta < 1.\n\]\n\n... | Proof Let \( f \in {H}_{{\Phi }^{\left( {r}_{2}\right) }, q,\omega }^{\sigma } \) . There exists a sequence \( {\left( {a}^{k}\right) }_{k \in \mathbb{Z}} \) of \( \left( {1,\Phi ,\infty ,\omega }\right) \) atoms and a sequence \( {\left( {\mu }_{k}\right) }_{k \in \mathbb{Z}} \in {\ell }_{q} \) of real numbers such th... | Yes |
Lemma 2.3 Let \( L \) be defined by (2.2), then\n\n\[ \operatorname{Ker}L = \left\{ {u \in X \mid u\left( t\right) = c,\forall t \in \left\lbrack {0, T}\right\rbrack, c \in {\mathbb{R}}^{2}}\right\} ,\n\]\n\n\[ \operatorname{Im}L = \left\{ {y \in Y \mid \left( \begin{matrix} {\int }_{0}^{T}{\left( T - s\right) }^{\alph... | Proof Obviously, by the Caputo fractional derivative and Riemann-Liouville fractional integral, we can see that (2.4) holds.\n\nIf \( y \in \operatorname{Im}L \), then there exists \( u \in \operatorname{dom}L \) such that \( y = {Lu} \) . That is, \( {y}_{1}\left( t\right) = {D}_{{0}^{ + }}^{\alpha }{u}_{1}\left( t\ri... | Yes |
Lemma 2.4 Let \( L \) be defined by (2.2), then \( L \) is a Fredholm operator of index zero, and the linear continuous projector operators \( P : X \rightarrow X \) and \( Q : Y \rightarrow Y \) can be defined as\n\n\[ \n{Pu}\left( t\right) = \left( \begin{matrix} {t}^{1 - \alpha }{u}_{1}\left( t\right) { \mid }_{t = ... | Proof Obviously, \( \operatorname{Im}P = \operatorname{Ker}L \) and \( {P}^{2}u = {Pu} \) . It follows from \( u = \left( {u - {Pu}}\right) + {Pu} \) that \( X = \operatorname{Ker}P + \operatorname{Ker}L \) . By simple calculation, we can get that \( \operatorname{Ker}P \cap \operatorname{Ker}L = \{ 0\} \) . Then we ge... | Yes |
Theorem 3.2 Suppose that the condition \( \left( {\mathrm{H}}_{2}\right) \) holds. Further, assume that the following condition holds.\n\n\( \left( {\mathrm{H}}_{3}\right) \) there exists a nonnegative number \( r \geq 0 \) such that\n\n\[ \mathop{\lim }\limits_{{\left| u\right| + \left| v\right| \rightarrow + \infty }... | Proof Note that \( {\left\lbrack \frac{4{T}^{\beta }r}{\Gamma \left( {\beta + 1}\right) }\left( 1 + \frac{{4}^{\nu - 1}{T}^{\alpha + \nu - 2}}{{\left( \Gamma \left( \alpha + 1\right) \right) }^{\nu - 1}}\right) \right\rbrack }^{\frac{1}{{P}_{m} - 1}} < 1 \), then there is a constant \( \varepsilon > 0 \) such that \( {... | Yes |
Consider the following mixed fractional periodic boundary value problem with \( p\left( t\right) \) -Laplacian operator: | By Theorem 3.1, the problem (3.11) has at least one solution. | No |
Lemma 2.6 The functions \( {g}_{1} \) and \( {g}_{2} \) given by (2.10) have the properties:\n\n(i) \( {g}_{1}\left( {t, s}\right) \leq \frac{B}{\Gamma \left( \alpha \right) }{t}^{\alpha - 1} \) for all \( t, s \in \left\lbrack {0,1}\right\rbrack \) ;\n\n(ii) \( {g}_{2}\left( {t, s}\right) \leq \frac{{t}^{\alpha - q - ... | Proof (i) From (2.10), we have\n\n\[ {g}_{1}\left( {t, s}\right) \leq \frac{1}{\Gamma \left( \alpha \right) }{t}^{\alpha - 1}\left\lbrack {B{\left( 1 - s\right) }^{\alpha - p - 1} - \left( {B - 1}\right) {\left( 1 - s\right) }^{\alpha - q}}\right\rbrack \]\n\n\[ \leq \frac{1}{\Gamma \left( \alpha \right) }{t}^{\alpha -... | Yes |
Lemma 2.7 \( G\left( {t, s}\right) \) is continuous on \( \left\lbrack {0,1}\right\rbrack \times \left\lbrack {0,1}\right\rbrack \), and satisfies\n\n(i) \( G\left( {t, s}\right) \leq D{t}^{\alpha - 1} \) for all \( t, s \in \left\lbrack {0,1}\right\rbrack \), where \( D \) is given by (2.5);\n\n(ii) \( G\left( {t, s}\... | Proof By definition of the function \( G\left( {t, s}\right) \), we know that for all \( t, s \in \left\lbrack {0,1}\right\rbrack \), the function \( G\left( {t, s}\right) \) is continuous.\n\n(i) By Lemma 2.6, we obtain for all \( t, s \in \left\lbrack {0,1}\right\rbrack \),\n\n\[ G\left( {t, s}\right) = {g}_{1}\left(... | Yes |
Theorem 3.1 \( T : P \rightarrow P \) is completely continuous. | Proof \( T : P \rightarrow P \) is continuous in view of nonnegative and continuity of \( G\left( {t, s}\right) \) and \( f \) . Next, we shall show that \( T \) is compact in \( P \) . Suppose \( \Omega \subset P \) is bounded, then, \( \exists M > 0 \) , \( \forall x \in \Omega \), s.t. \( \parallel x\parallel \leq M... | Yes |
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