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Lemma 3 Assume that \( x\left( t\right) \) is an eventually positive solution of (1) and satisfies the property (I) of Lemma 2. Then there exists a \( {\left. {t}_{1} \in \left\lbrack {t}_{0},\infty \right) \right) }_{T} \) such that\n\n\[ \n{z}^{\Delta }\left( t\right) \geq {\left( \frac{B\left( {{t}_{1}, t}\right) }{... | Proof According to (1), there exists a \( {\left. {t}_{1} \in \left\lbrack {t}_{0},\infty \right) \right) }_{T} \) and a \( t \in \left\lbrack {{t}_{1},\infty }\right) \) such that\n\n\[ \n{z}^{\Delta }\left( t\right) > 0,\;{\left( a\left( t\right) {\left( {z}^{\Delta }\left( t\right) \right) }^{\alpha }\right) }^{\Del... | Yes |
Assume that \( x\left( t\right) \) is an eventually positive solution of (1) and satifies the property (I) of Lemma 2. If\n\n\[ \n{\int }_{{t}_{0}}^{\infty }\delta \left( t\right) q\left( \mathrm{t}\right) {\Delta t} = \infty \n\]\n\nthen there exists a \( {\left. {t}_{1} \in \left\lbrack {t}_{0},\infty \right) \right)... | Proof Define \( Z\left( t\right) = z\left( t\right) - t{z}^{\Delta }\left( t\right) \) . Thus\n\n\[ \n{Z}^{\Delta }\left( t\right) = {z}^{\Delta }\left( t\right) - \left( {{z}^{\Delta }\left( t\right) + \sigma \left( t\right) {z}^{\Delta \Delta }\left( t\right) }\right) = - \sigma \left( t\right) {z}^{\Delta \Delta }\l... | Yes |
Lemma 5 Assume that \( x\left( t\right) \) is an eventually positive solution of (1) and satisfies the property (II) of Lemma 2. If\n\n\[{\int }_{{t}_{0}}^{\infty }{\left( \frac{1}{a\left( u\right) }{\int }_{u}^{\infty }{\left( \frac{1}{b\left( v\right) }{\int }_{v}^{\infty }q\left( s\right) \Delta s\right) }^{\frac{1}... | Proof It is clear \( \mathop{\lim }\limits_{{t \rightarrow \infty }}z\left( t\right) = l \geq 0 \) . Next we claim that \( l = 0 \) . Otherwise, there exists a \( {t}_{1} \in \lbrack {t}_{0},\infty {)}_{T} \) such that \( z\left( t\right) \geq l > 0 \) for all \( t \in {\left\lbrack {t}_{1},\infty \right) }_{T} \) . By... | Yes |
Theorem 2 Assume that \( \left( {\mathrm{A}}_{3}\right) ,\left( 2\right) ,\left( {13}\right) ,\left( {16}\right) \) and \( {\alpha \beta } \geq 1,\gamma \geq 1 \) hold. Furthermore, assume that there exists a positive function \( r \in {C}_{rd}^{1}\left( {{\left\lbrack {t}_{0},\infty \right) }_{T},{R}_{ + }}\right) \),... | Proof Suppose that \( x\left( t\right) \) is a nonoscillatory solution of (1). Without loss of generality, there exists a \( {\left. {t}_{1} \in \left\lbrack {t}_{0},\infty \right) \right) }_{T} \) such that \( x\left( t\right) > 0, x\left( {\tau \left( t\right) }\right) > 0, x\left( {\delta \left( t\right) }\right) > ... | Yes |
Theorem 3 Assume that \( \left( {\mathrm{A}}_{3}\right) ,\left( 2\right) ,\left( {13}\right) ,\left( {16}\right) \) and \( {\alpha \beta } \geq 1,\gamma \geq 1 \) hold. Suppose that there exists a function \( H \) with property \( P \), a function \( h \in {C}_{rd}^{1}\left( {{D}_{0}, R}\right) \), and a function \( \e... | Proof Suppose that \( x\left( t\right) \) is a nonoscillatory solution of (1). Without loss of generality, there exists a \( {t}_{1} \in {\left\lbrack {t}_{0},\infty \right) }_{T} \) such that \( x\left( t\right) > 0, x\left( {\tau \left( t\right) }\right) > 0, x\left( {\delta \left( t\right) }\right) > 0 \) on \( {\le... | Yes |
Theorem 4 Assume that \( \left( {\mathrm{A}}_{3}\right) ,\left( 2\right) ,\left( {13}\right) ,\left( {16}\right) \) and \( {\alpha \beta } \leq 1 \) hold. If\n\n\[ \mathop{\limsup }\limits_{{t \rightarrow \infty }}\left( {{t}^{\alpha \beta }{\left( \frac{B\left( {{t}_{1}, t}\right) }{a\left( t\right) }\right) }^{\frac{... | Proof Suppose that \( x\left( t\right) \) is a nonoscillation solution of (1). Without loss of generality, there exists a \( {\left. {t}_{1} \in \left\lbrack {t}_{0},\infty \right) \right) }_{T} \) such that \( x\left( t\right) > 0, x\left( {\tau \left( t\right) }\right) > 0, x\left( {\delta \left( t\right) }\right) > ... | Yes |
Consider the third-order nonlinear equation on time scales\n\n\[ \n{\left( {\left( {\left( {t}^{\frac{1}{3}}{\left( {\left( \frac{1}{2}x\left( t\right) \right) }^{\Delta }\right) }^{\frac{1}{3}}\right) }^{\Delta }\right) }^{\Delta }\right) }^{\Delta } + {\left( {\int }_{0}^{t}{s}^{2}\Delta s\right) }^{\frac{4}{5}}{x}^{... | It is clear that (2), (13) and the (16) hold. By Theorem 2, it remains to prove (23) holds. Let \( m = 1, r\left( t\right) = 1 \) . We have\n\n\[ \n\mathop{\limsup }\limits_{{t \rightarrow \infty }}\frac{1}{t}{\int }_{{t}_{1}}^{t}\left( {c\left( {t - s}\right) q\left( s\right) - \frac{0}{\left( {t - s}\right) {\left( \... | Yes |
Lemma 1 Suppose \( {f}_{k} \) is monotonically decreasing and bounded below. If for all \( k \) , \[ {h}_{k + 1} \leq \left( {\beta - {\gamma }^{2}}\right) {h}_{k} + \gamma \left( {{f}_{k} - {f}_{k + 1}}\right) \text{ or }{f}_{k} - {f}_{k + 1} \geq \gamma {h}_{k + 1} \] (9) holds, then \( {h}_{k} \rightarrow 0 \) . | Proof Suppose \( K \) is a sufficient large positive integer. If \( {h}_{k + 1} \leq \left( {\beta - {\gamma }^{2}}\right) {h}_{k} + \gamma \left( {{f}_{k} - {f}_{k + 1}}\right) \) holds for all \( k > K \), then two conditions follows. 1) If \( {f}_{k} - {f}_{k + 1} \geq \gamma {h}_{k + 1} \) holds on an infinite subs... | Yes |
Lemma 2 Suppose assumptions A1-A3 hold, \( \left\{ {x}_{k}\right\} \) is an infinite sequence of iterative points accepted to \( {\mathcal{F}}_{k} \) produced by Algorithm 1, then\n\n\[ \mathop{\lim }\limits_{{k \rightarrow \infty }}h\left( {x}_{k}\right) = 0 \] | Proof Here we use the methodology of ([14], Lemma 3.3). If the conclusion is not true, there must exist an infinite subsequence \( \left\{ {x}_{{k}_{i}}\right\} \subset \left\{ {x}_{k}\right\} \) and a constant \( \bar{\epsilon } > 0 \) such that\n\n\[ h\left( {x}_{{k}_{i}}\right) > \bar{\epsilon } \]\n\n(10)\n\nholds.... | Yes |
Lemma 3 Suppose assumptions A1-A3 hold, and let \( {d}_{k} \) be a feasible point of \( {QP} \) subproblem (3). It then follows that\n\n\[ \Delta {f}_{k} = f\left( {x}_{k}\right) - f\left( {x}_{k + 1}\right) \geq {\Delta \phi }\left( {x}_{k}\right) - n{\rho }^{2}M, \]\n\n\[ \left| {{c}_{i}\left( {{x}_{k} + {d}_{k}}\rig... | Proof By the definition of \( h\left( x\right) \) and ([9], Lemma 3), the conclusion follows. | No |
Lemma 4 Suppose assumptions A1-A3 hold, then the inner iteration terminates finitely. | Proof If \( {x}_{k} \) is a KKT point of problem (1), then \( {d}_{k} = 0 \) and the algorithm terminates. Otherwise, if the inner iteration does not terminate finitely, then \( \rho \rightarrow 0 \) by the algorithm mechanism. We consider two cases.\n\n(a) \( {h}_{k} > 0 \) . For all \( {d}_{k} \) such that \( {\begin... | Yes |
Lemma 2 (Cauchy integral formula) Let \( D \) be a bounded domain with piecewise smooth boundary. If \( f\left( z\right) \) is analytic on \( D \), and extends smoothly to the boundary of \( D \), then | \[ f\left( z\right) = \frac{1}{2\pi i}{\int }_{\partial D}\frac{f\left( \zeta \right) }{\zeta - z}\mathrm{\;d}\zeta ,\;z \in D. \] | Yes |
Lemma 1 Let \( u = u\left( {x, t}\right) \) be a smooth solution to the problem (1), if for some \( B > 0,\operatorname{supp}u\left( t\right) \subseteq \left\lbrack {-B, B}\right\rbrack \), then for all \( \lambda ,\theta \in \mathbb{R} \), we have\n\n\[ \left| {F\left( {u\left( t\right) }\right) \left( {\lambda + {i\t... | Proof Using the Cauchy-Schwarz inequality and \( {L}^{2}\left( \mathbb{R}\right) \) -norm conservation law, we may prove this lemma like Carvajal and Panthee in [8]. | No |
Lemma 2 Let \( u = u\left( {x, t}\right) \in C\left( {\left\lbrack {0, T}\right\rbrack ,{H}^{s}\left( \mathbb{R}\right) }\right), s \geq 4 \), be a solution to the problem (1), \( B \) as in the Theorem 1, then we have\n\n\[ \rho \left( \lambda \right) \lesssim \frac{\sqrt{B}M}{1 + {\lambda }^{4}}. \] | Proof Applying the Cauchy-Schwarz inequality and \( {L}^{2}\left( \mathbb{R}\right) \) -norm conservation law, we may verify this lemma like Carvajal and Panthee in [8]. | No |
Lemma 5 If \( \rho \left( \lambda \right) \) is a function defined in (2), then there exist constants \( {a}_{0} > \) \( 0,{a}_{1} > 0,{a}_{2} > 0 \) such that\n\n\[ g\left( \lambda \right) = {\int }_{\left| \xi \right| \geq \left| \lambda \right| }{\xi }^{2}\rho \left( {\lambda - \xi }\right) \mathrm{d}\xi \leq {a}_{0... | Proof Since \( g \) is an even function, we suppose \( \lambda > 0 \) . By Lemma 2, it follows that\n\n\[ g\left( \lambda \right) = {\int }_{\lambda }^{+\infty }{\xi }^{2}\rho \left( {\lambda - \xi }\right) \mathrm{d}\xi + {\int }_{-\infty }^{-\lambda }{\xi }^{2}\rho \left( {\lambda - \xi }\right) \mathrm{d}\xi \]\n\n\... | Yes |
Lemma 1 Suppose \( \mu > 0,0 < \alpha < 1 \), Model 1 and Model 2 are proper, continuous and coercive. Then the minimizers exist. | Proof We only prove the coercive property of Model 1. Obviously, we have\n\n\[ \sqrt{{\left| {x}_{1}\right| }^{2} + {\left| {x}_{2}\right| }^{2}} \leq \left| {x}_{1}\right| + \left| {x}_{2}\right| \]\n\nthen\n\n\[ {P}_{1}\left( f\right) \geq \left( {1 - \alpha }\right) \mu \left( \left( {{\begin{Vmatrix}{D}_{x}f\end{Vm... | Yes |
Lemma 2 For all \( f,{f}^{s} \), we have\n\n\[ \n{P}_{1}\left( f\right) \leq {Q}_{1}\left( {f,{f}^{s}}\right) ,\;{P}_{1}\left( f\right) = {Q}_{1}\left( {f, f}\right) .\n\]\n\nIf the sequence \( \left\{ {f}^{s}\right\} \) is generated by (10), then\n\n\[ \n{P}_{1}\left( {f}^{s}\right) - {P}_{1}\left( {f}^{s + 1}\right) ... | Proof In fact, since \( \parallel {Df}{\parallel }_{2,1} - \left\langle {{Df},{q}^{s}}\right\rangle \) is convex on \( f \), then we can get the minimum is 0 . After simple calculation, we have \( - \parallel {Df}{\parallel }_{2,1} \leq - \left\langle {{D}_{f},{q}^{s}}\right\rangle \) . Then, \( {P}_{1}\left( f\right) ... | Yes |
Theorem 1 Under the assumptions in Lemma 1, any non-zero limit point of \( \left\{ {f}^{s}\right\} \) generated by (10) is a stationary point of \( {P}_{1}\left( f\right) \) . | Proof Firstly, \( {Q}_{1}\left( {f,{f}^{s}}\right) \) is convex on \( f \) . Secondly, under the assumption \( \frac{0}{0} = 1 \) , it is easy to know that the directional derivative \( {\left. {Q}_{1f}{}^{\prime }\left( f,{f}^{s};d\right) \right| }_{f = {f}^{s}} \) (only with respect to \( f \) ) equals to \( {P}_{1f}... | Yes |
Theorem 2 The pricing formula of a defaultable bond with constant recovery under a hybrid model is given by\n\n\[ B\\left( {V, r,\\lambda, t}\\right) = P\\left( {r, t}\\right) \\left( {\\left\\lbrack {{2N}\\left( {{d}_{1}\\left( t\\right) }\\right) - 1}\\right\\rbrack \\left( {1 - {R}_{0}}\\right) {\\mathrm{e}}^{{a}_{3... | Proof It is well known that the stochastic recovery model reduces to a constant recovery model when setting \( {R}_{1} = 0 \) in (6). In this case of the constant recovery, \( {A}_{3}\\left( t\\right) = 0,{A}_{5}\\left( t\\right) = 0 \), and \( {A}_{6}\\left( t\\right) = 0 \) . Moreover, \( {A}_{1}\\left( t\\right) \) ... | Yes |
Lemma 3 (i) \( \operatorname{Dim}\operatorname{Ker}{F}_{\left( u, v\right) }\left( {\bar{b},\theta ,0}\right) = 1 \) . Moreover, \( \operatorname{Ker}{F}_{\left( u, v\right) }\left( {\bar{b},\theta ,0}\right) = {\left( {\Phi }_{1},{\Phi }_{2}\right) }^{T} \) with \( {\Phi }_{1} < 0,{\Phi }_{2} > 0 \) in \( \Omega \) an... | Proof By (5) and (6), the linearization of \( F\left( {b, u, v}\right) = 0 \) with respect to \( \left( {u, v}\right) = \) \( \left( {\theta ,0}\right) \) can be written as\n\n\[ - {d}_{1}{\Delta u} - {au} + {2\theta u} + \frac{\theta v}{m + \theta } = 0,\;x \in \Omega ,\]\n\n\[ - {d}_{2}{\Delta v} - {bv} = 0,\;x \in \... | Yes |
Theorem 1 The trivial solution \( \left( {0,0}\right) \) of system \( \left( 2\right) \) is unstable. | Proof Using (3), we can immediately obtain the following characteristic equation at \( \left( {0,0}\right) \) for the system \( \left( 2\right) \) :\n\n\[ \left( {\lambda + {\mu }_{k}{d}_{2} + \beta }\right) \left( {\lambda + {\mu }_{k}{d}_{3} - r}\right) = 0, \]\n\nwhich implies that one of characteristic roots is \( ... | Yes |
Theorem 2 Assume that \( r < {b}_{3}\beta /{a}_{2} \), then \( \left( {0, r/{b}_{3}}\right) \) is locally stable for the system (2). | Proof Using (3), we can obtain the following characteristic equation at \( \left( {0, r/{b}_{3}}\right) \) for the system (2):\n\n\[\n\left( {\lambda + {\mu }_{k}{d}_{2} + \beta - {a}_{2}r/{b}_{3}}\right) \left( {\lambda + {\mu }_{k}{d}_{3} + r}\right) = 0.\n\]\n\nUsing the condition \( r < {b}_{3}\beta /{a}_{2} \), we... | Yes |
Theorem 3 Assume that \( r > {b}_{3}\beta /{a}_{2} \) and \( {b}_{2}{b}_{3} > {a}_{2}{a}_{4} \), then \( \left( {{u}_{2}^{ * },{u}_{3}^{ * }}\right) \) is locally stable for the system (2). | Proof When \( {\tau }_{2} = {\tau }_{3} = 0 \), the characteristics equation (3) can be rewritten as\n\n\[ \n{\lambda }^{2} + \left( {p + q}\right) \lambda + {pq} + {a}_{2}{a}_{4}{u}_{2}^{ * }{u}_{3}^{ * } = 0, \n\]\n\n(4)\n\nwhere\n\n\[ \np = {\mu }_{k}{d}_{2} + \beta - {a}_{2}{u}_{3}^{ * } + 2{b}_{2}{u}_{2}^{ * },\;q... | Yes |
Theorem 6 The trivial solution \( \left( {0,0,0}\right) \) of system \( \left( 1\right) \) is unstable. | Proof The characteristic equation of system (1) at \( \left( {0,0,0}\right) \) is\n\n\[ \left| \begin{matrix} \lambda + {\mu }_{k}{d}_{1} + \alpha & 0 & 0 \\ 0 & \lambda + {\mu }_{k}{d}_{2} + \beta & 0 \\ 0 & 0 & \lambda + {\mu }_{k}{d}_{3} - r \end{matrix}\right| = 0, \]\n\nthat is\n\n\[ \left( {\lambda + {\mu }_{k}{d... | Yes |
Theorem 1 (Weak duality) Let \( \left( {\bar{x},\bar{y}}\right) \) and \( \left( {{x}^{\prime },{y}^{\prime },{z}^{\prime },{y}^{ * },{z}^{ * }}\right) \) be feasible points for (SOP) and (MWD), respectively. Suppose that \( \left( {F \times G}\right) \) is \( A \) -invex at \( \left( {{x}^{\prime },\left( {{y}^{\prime... | Proof We proceed by contradiction. Suppose that \( \bar{y} - {y}^{\prime } \in - \operatorname{int}\left( D\right) \) . Since \( {y}^{ * } \in {D}^{ * } \smallsetminus \left\{ {0}_{{Y}^{ * }}\right\} \), we have\n\n\[ {y}^{*T}\left( {\bar{y} - {y}^{\prime }}\right) < 0 \]\n\n(3)\n\nAnd because \( \left( {\bar{x},\bar{y... | Yes |
Theorem 2 (Strong duality) Let \( \left( {\bar{x},\bar{y}}\right) \in \operatorname{graph}\left( F\right) \) be a weak minimizer of (SOP). Suppose that for some \( \left( {{y}^{ * },{z}^{ * }}\right) \in \left( {{D}^{ * } \smallsetminus \left\{ {0}_{{Y}^{ * }}\right\} }\right) \times {E}^{ * } \) and \( \bar{z} \in G\l... | Proof Since equations (7) and (8) hold, it is obviously that \( \left( {\bar{x},\bar{y},\bar{z},{y}^{ * },{z}^{ * }}\right) \) is a feasible solution for (MWD). Afterwards, we will prove that\n\n\[ \n\left( {{W}_{1} - \bar{y}}\right) \cap \operatorname{int}\left( D\right) = \varnothing .\n\]\n\nIn fact, assume that the... | Yes |
Theorem 3 (Converse duality) Let \( \left( {{x}^{\prime },{y}^{\prime },{z}^{\prime },{y}^{ * },{z}^{ * }}\right) \) be a weak maximizer of the problem (MWD) and \( {z}^{\prime } \in G\left( {x}^{\prime }\right) \cap \left( {-E}\right) \) . Suppose that \( \left( {F \times G}\right) \) is \( A \) -invex at \( \left( {{... | Proof Firstly, it is clearly that \( \left( {{x}^{\prime },{y}^{\prime }}\right) \) is a feasible solution of the problem (SOP). Next, assuming that \( \left( {{x}^{\prime },{y}^{\prime }}\right) \) is not a weak minimizer of the problem (SOP), then\n\n\[
\left( {F\left( \Omega \right) - {y}^{\prime }}\right) \cap \lef... | Yes |
Theorem 4 (Weak duality) Let \( \left( {\bar{x},\bar{y}}\right) \) and \( \left( {{x}^{\prime },{y}^{\prime },{z}^{\prime },{y}^{ * },{z}^{ * }}\right) \) be feasible points for (SOP) and (WD), respectively. Suppose that \( \left( {F \times G}\right) \) is \( A \) -invex at \( \left( {{x}^{\prime },\left( {{y}^{\prime ... | Proof Suppose that\n\n\[ \bar{y} - {y}^{\prime } - {z}^{*T}{z}^{\prime } \cdot {d}_{0} \in - \operatorname{int}\left( D\right) . \]\n\nSince \( G\left( \bar{x}\right) \cap \left( {-E}\right) \neq \varnothing \), let \( \bar{z} \in G\left( \bar{x}\right) \cap \left( {-E}\right) \), then, \( {z}^{*T}\bar{z} \leq 0 \) . H... | Yes |
Theorem 5 (Strong duality) Let \( \left( {\bar{x},\bar{y}}\right) \in \operatorname{graph}\left( F\right) \) and \( \bar{z} \in G\left( \bar{x}\right) \cap \left( {-E}\right) \) . Suppose that \( \left( {\bar{x},\bar{y}}\right) \) is a weak minimizer of (SOP) and for some \( \left( {{y}^{ * },{z}^{ * }}\right) \in \lef... | Proof Since equations (7) and (8) are fulfilled, it is obviously that \( \left( {\bar{x},\bar{y},\bar{z},{y}^{ * },{z}^{ * }}\right) \) is a feasible solution for (WD). Next, we show that\n\n\[ \left( {{W}_{2} - \bar{y} - {z}^{*T}\bar{z} \cdot {d}_{0}}\right) \cap \operatorname{int}\left( D\right) = \varnothing . \]\n\... | Yes |
Theorem 6 (Converse duality) Let \( \left( {{x}^{\prime },{y}^{\prime },{z}^{\prime },{y}^{ * },{z}^{ * }}\right) \) be a weak maximizer of the problem (WD) with \( {z}^{\prime } \in G\left( {x}^{\prime }\right) \cap \left( {-E}\right) \) and \( {z}^{*T}{z}^{\prime } = 0 \) . Suppose that \( \left( {F \times G}\right) ... | Proof It is clearly that \( \left( {{x}^{\prime },{y}^{\prime }}\right) \) is a feasible solution of the problem (SOP). Let \( \left( {{x}^{\prime },{y}^{\prime }}\right) \) is not a weak minimizer of the problem (SOP), then\n\n\[ \left( {F\left( \Omega \right) - {y}^{\prime }}\right) \cap \left( {-\operatorname{int}\l... | Yes |
Proposition 1 Under Assumption 1, for any \( h \in \mathcal{H} \) measurable with respect to \( {\mathcal{F}}_{n} \), and any \( {h}^{ * } \in \mathcal{H},\eta \in N \), | \[ \mathbb{E}\left\lbrack {\frac{1}{m}\mathop{\sum }\limits_{{t = n + 1}}^{{n + m}}\mathbb{E}\left\lbrack {F\left( {h;{z}_{t}}\right) - F\left( {{h}^{ * };{z}_{t}}\right) \mid {\mathcal{F}}_{n}}\right\rbrack }\right\rbrack \]\n\[ \leq E\left\lbrack {f\left( h\right) }\right\rbrack - f\left( {h}^{ * }\right) + {2GR\alph... | Yes |
Proposition 2 Under Assumption 1 and Assumption 2, denote the sequence of outputs of the online algorithm by \( {h}_{t} \) . Then for any \( \eta \in N \) , | \[ \mathop{\sum }\limits_{{t = 1}}^{n}\left\lbrack {f\left( {h}_{t}\right) - f\left( {h}^{ * }\right) }\right\rbrack \leq {\mathfrak{C}}_{n} + {G\eta }\mathop{\sum }\limits_{{t = 1}}^{n}{\lambda }_{t} + {2\eta GR} \] \[ + \mathop{\sum }\limits_{{t = 1}}^{n}\left\lbrack {f\left( {h}_{t}\right) - F\left( {{h}_{t};{z}_{t ... | Yes |
Lemma 1 Consider the situation of Assumption 1, and additionally assume that \( h,\omega \in \mathcal{H} \) be measurable with respect to the \( \sigma \) -field \( {\mathcal{F}}_{t} \) . Then for any \( \eta \in N \) , | Proof Recalling that \( f\left( h\right) = {\mathbb{E}}_{\Pi }\left\lbrack {F\left( {h;z}\right) }\right\rbrack \), the definition of the underlying measure \( \mu \) and the densities \( \pi \) and \( \rho \) in (2), then we have\n\n\[ \mathbb{E}\left\lbrack {F\left( {h;{z}_{t + \eta }}\right) - F\left( {\omega ;{z}_{... | Yes |
Lemma 3.1 Suppose that the infinite sequence \( \left\{ {x}_{k}\right\} \) is generated by Algorithm \( {3.1},{d}_{k} \) is the solution of equation (4) at the \( k \) th iteration, the merit function \( \varphi \left( x\right) \) decreases in the direction \( {d}_{k} \) at \( {x}_{k} \in {\mathbb{R}}^{n} \) . | Proof The statement is obvious. Firstly, we have\n\n\[ \n{\varphi }^{\prime }\left( {{x}_{k};{d}_{k}}\right) = \mathop{\sum }\limits_{{i = 1}}^{n}{g}_{i}\left( {x}_{k}\right) {H}_{i}^{\prime }\left( {{x}_{k};{d}_{k}}\right) \]\n\n\[ \n\leq {f}_{{j}_{1}}^{1}{\left( \nabla {f}_{{j}_{1}}^{1}\right) }^{\mathrm{T}}{d}_{k} +... | Yes |
Lemma 3.2 Suppose that \( \left\{ {x}_{k}\right\} \) is generated by Algorithm 3.1, there exists a smallest integer \( m \in \lbrack 0, + \infty ) \) such that\n\n\[ \varphi \left( {{x}_{k} + {\beta }^{m}{d}_{k}}\right) - \varphi \left( {x}_{k}\right) \leq \sigma {\beta }^{m}{H}_{k}^{\mathrm{T}}{V}_{k}{d}_{k}. \]\n | Proof We prove the statement by contradiction. Without loss of generality, we assume that the following inequality holds when \( m \rightarrow \infty \) ,\n\n\[ \varphi \left( {{x}_{k} + {\beta }^{m}{d}_{k}}\right) - \varphi \left( {x}_{k}\right) > \sigma {\beta }^{m}{H}_{k}^{\mathrm{T}}{V}_{k}{d}_{k}. \]\n\n(9)\n\nThe... | Yes |
Theorem 3.2 Let Assumption 3.1 and Assumption 3.2 hold, if \( \left\{ {x}_{k}\right\} \) is any infinite sequence generated by Algorithm 3.1 and one of the accumulation points of \( \left\{ {x}_{k}\right\} \) is an isolated solution of VCP, then the entire sequence \( \left\{ {x}_{k}\right\} \) converges to the accumul... | Proof We start with proving the existence of at least one accumulation of \( \left\{ {x}_{k}\right\} \) . Since Algorithm 3.1 is decent, \( \left\{ {x}_{k}\right\} \) is bounded for any initial point \( {x}_{0} \), there exists at least one convergent subsequence of \( \left\{ {x}_{k}\right\} \) . For convenience, deno... | Yes |
Problem 4.1 (Kojima-Shindo problem) Consider a nonlinear complementarity problem, where \( {f}_{1}\left( x\right) = x \in {\mathbb{R}}^{n} \), \[ {f}_{2}\left( x\right) = \left( \begin{matrix} 3{x}_{1}^{2} + 2{x}_{1}{x}_{2} + 2{x}_{2}^{2} + {x}_{3} + 3{x}_{4} - 6 \\ 2{x}_{1}^{2} + {x}_{1} + {x}_{2}^{2} + {10}{x}_{3} + ... | There exists a nondegenerate solution \( {x}^{ * } = {\left( 1,0,3,0\right) }^{\mathrm{T}} \) and a degenerate solution \( {x}^{* * } = \) \( {\left( \frac{\sqrt{6}}{2},0,0,\frac{1}{2}\right) }^{\mathrm{T}} \) in this problem. | Yes |
Lemma 1 Supposed that \( {x}^{ * } \) is an equilibrium point of the system (9) with \( {\mu }_{\left( P,{\mathbb{R}}^{n}\right) }\left( H\right) < 0 \), then \( {x}^{ * } \) is globally exponentially stable, and for any solution \( x\left( t\right) \) initiated from \( {x}_{0} = x\left( {t}_{0}\right) \), we have\n\n\... | Proof In the review of \( H\left( {x}^{ * }\right) = 0 \) and Definition 2, we get\n\n\[ \n\frac{\mathrm{d}{\begin{Vmatrix}x\left( t\right) - {x}^{ * }\end{Vmatrix}}_{P}^{2}}{\mathrm{\;d}t} = 2\left\langle {x\left( t\right) - {x}^{ * },{PH}\left( {x\left( t\right) }\right) }\right\rangle \n\]\n\n\[ \n\leq 2{\mu }_{\lef... | Yes |
Theorem 1 Under Assumptions (H1) and (H2), if\n\n\[ \Delta = {I}_{N} \otimes \widehat{M} + c{\left( \left( A - {A}_{d}\right) \otimes P\Gamma \widehat{L}\right) }^{s} + c\left( {{A}_{d} - K}\right) \otimes {\left( P\Gamma L\right) }^{s} < 0, \]\n\nthen the systems (6) and (4) are globally exponentially synchronized, wh... | Proof Let\n\n\[ e\left( t\right) = {\left( {e}_{1}^{T}\left( t\right) ,{e}_{2}^{T}\left( t\right) ,\cdots ,{e}_{N}^{T}\left( t\right) \right) }^{T}, \]\n\n\[ H\left( {e\left( t\right) }\right) = {\left( {H}_{1}\left( e\left( t\right) \right) ,{H}_{2}\left( e\left( t\right) \right) ,\cdots ,{H}_{N}\left( e\left( t\right... | Yes |
Theorem 3 In case of \( G\left( x\right) = x \), if Assumption (H1) holds, and\n\n\[ \Delta = {I}_{N} \otimes M + c{\left( \left( A - K\right) \otimes P\Gamma \right) }^{s} < 0, \]\n\nthen the systems (6) and (4) are globally exponentially synchronized with the controllers being \( {u}_{i} = - c{k}_{i}\left( {{x}_{i}\l... | Proof Define\n\n\[ e\left( t\right) = {\left( {e}_{1}^{T}\left( t\right) ,{e}_{2}^{T}\left( t\right) ,\cdots ,{e}_{N}^{T}\left( t\right) \right) }^{T},\;H\left( {e\left( t\right) }\right) = {\left( {H}_{1}\left( e\left( t\right) \right) ,{H}_{2}\left( e\left( t\right) \right) ,\cdots ,{H}_{N}\left( e\left( t\right) \ri... | No |
Corollary 1 Assume that (H1) and (H2) are satisfied, and \( \Gamma \) is a positive definite matrix. If \( \Delta < 0 \) in Theorem 1, then\n\n\[ \rho - {c\delta }\left( {{\eta }_{i} + {k}_{i}}\right) < 0,\;0 \leq i \leq r,\;\rho - {c\delta }{\eta }_{i} < 0,\;r + 1 \leq i \leq N, \]\n\n(19)\n\nwhere \( \rho ,\delta \) ... | Analogy to the discussion in [16], the pinning controllers should be give priority to the nodes that have low degree by (19) when \( c > 0 \) is given. | No |
Theorem 1 If the controller \( U\left( t\right) \) is designed as the form in (7),(8) and (12), and all the eigenvalues \( {\lambda }_{1},{\lambda }_{2},\cdots ,{\lambda }_{n} \) of \( B + \Lambda \) satisfy\n\n\[ \left| {\arg \left( {\lambda }_{i}\right) }\right| > \frac{\alpha \pi }{2},\;i = 1,2,\cdots, n, \]\n\nthen... | Proof Submitting systems (7), (8) and (12) into system (6), the error system can be written as follows\n\n\[ {D}^{\alpha }e\left( t\right) = {D}^{\alpha }\left( {y\left( t\right) - {Cx}\left( {t - \tau }\right) }\right) = {D}^{\alpha }y\left( t\right) - C{D}^{\alpha }x\left( {t - \tau }\right) \]\n\n\[ = {By} + G\left(... | Yes |
Lemma 2 There exists a \( M > 0 \) such that for any solution of system (2) with initial conditions (4) and (5), there must be a \( T > 0 \) such that \( {S}_{i}\left( t\right) \leq M \) and \( {I}_{i}\left( t\right) \leq M \) for \( i = 1,2 \) and \( t \geq T \) . | Proof For \( {N}_{1}\left( t\right) = {S}_{1}\left( t\right) + {I}_{1}\left( t\right) ,{N}_{2}\left( t\right) = {S}_{2}\left( t\right) + {I}_{2}\left( t\right) \), we have\n\n\[ \left\{ \begin{array}{l} \frac{d{N}_{1}\left( t\right) }{dt} = {A}_{1} - \left( {{d}_{1} + {a}_{1}}\right) {N}_{1}\left( t\right) + {a}_{2}{N}... | Yes |
Proposition 1 The following inequality\n\n\[ \n{R}_{0} > \max \left\{ {\frac{{\beta }_{1}}{{d}_{1} + {\delta }_{1} + {a}_{1}},\frac{{\beta }_{2}}{{d}_{2} + {\delta }_{2} + {a}_{2}}}\right\} \n\]\n\nholds. | Proof From (7), we have\n\n\[ \n{R}_{0} > \frac{{\beta }_{1}}{2\left( {{d}_{1} + {\delta }_{1} + {a}_{1}}\right) } + \frac{{\beta }_{2}}{2\left( {{d}_{2} + {\delta }_{2} + {a}_{2}}\right) } + \frac{1}{2}\left| {\frac{{\beta }_{1}}{{d}_{1} + {\delta }_{1} + {a}_{1}} - \frac{{\beta }_{2}}{{d}_{2} + {\delta }_{2} + {a}_{2... | Yes |
Theorem 2 Let \( {R}_{0} > 1 \), then there is a positive constant \( \eta \) such that every solution \( \left( {{S}_{1}\left( t\right) ,{I}_{1}\left( t\right) ,{S}_{2}\left( t\right) ,{I}_{2}\left( t\right) }\right) \) of system (2) with initial conditions (4) and (5), \( {\phi }_{1}\left( 0\right) + \) \( {\phi }_{2... | From system (2), we obtain\n\n\[ \frac{d{S}_{i}}{dt} \geq {A}_{i} - \left( {{d}_{i} + {\beta }_{i} + {a}_{i}}\right) {S}_{i},\;i = 1,2, \] | No |
Corollary 1 Let \( {R}_{0} > 1 \), then system (2) is permanent for every solution \( \left( {{S}_{1}\left( t\right) }\right. \) , \( \left. {{I}_{1}\left( t\right) ,{S}_{2}\left( t\right) ,{I}_{2}\left( t\right) }\right) \) of system (2) with | In 3.1, we know that the limiting equations for \( {I}_{1}\left( t\right) \) and \( {I}_{2}\left( t\right) \) are as follows \[ \left\{ \begin{array}{l} \frac{d{I}_{1}\left( t\right) }{dt} = \left\lbrack {{\beta }_{1} - \left( {{d}_{1} + {\delta }_{1} + {a}_{1}}\right) }\right\rbrack {I}_{1}\left( t\right) - \frac{{\be... | Yes |
Theorem 5 If \( {R}_{0} > 1 \) ,(2) has a unique endemic equilibrium \( {E}^{ * } = \left( {{S}_{1}^{ * },{I}_{1}^{ * },{S}_{2}^{ * },{I}_{2}^{ * }}\right) \) , which is globally attractive in\n\n\[ \sum = \left\{ {\Phi = \left( {{\varphi }_{1},{\phi }_{1},{\varphi }_{2},{\phi }_{2}}\right) \in C\left( {\left\lbrack {-... | Proof Since \( {N}_{i}\left( t\right) = {S}_{i}\left( t\right) + {I}_{i}\left( t\right) \), then \( {S}_{i}^{ * } + {I}_{i}^{ * } = {N}_{i}^{ * }, i = 1,2 \), and \( {S}_{i}^{ * } = {N}_{i}^{ * } - {I}_{i}^{ * } \) . According to the theory of asymptotic autonomous systems \( {}^{\left\lbrack {19}\right\rbrack } \) and... | Yes |
Theorem 1 When \( b \geq \varepsilon {K}^{2}\left( {b + {2qK}}\right) \), system (1) has no immune control equilibrium in \( D \) as \( {\mathcal{R}}_{0} \leq 1 \), and a unique simple immune control equilibrium \( {E}_{1}^{ * }\left( {{y}_{1}^{ * },{z}_{1}^{ * }}\right) \) in \( D \) as \( {\mathcal{R}}_{0} > 1 \) . | Proof Direct calculation gives \( {F}^{\prime }\left( y\right) = \left\lbrack {\left( {b + {2qy}}\right) \varepsilon {y}^{2} - b}\right\rbrack /{y}^{2} \) . Note that function \( \left( {b + {2qy}}\right) \varepsilon {y}^{2} - b \) is increasing with respect to \( y \), then, when \( b \geq \varepsilon {K}^{2}\left( {b... | Yes |
Theorem 3.1 Suppose that (H1) holds and \( {bc} \leq 1 \) . Then:\n\n(i) If \( \beta \in \left( {0,{\beta }_{* * }}\right) \), then \( \left( {{\theta }_{{d}_{1}, m},0}\right) \) is globally asymptotically stable for all \( {d}_{1},{d}_{2} > 0 \) ;\n\n(ii) If \( \beta \in \left( {{\beta }^{* * },\infty }\right) \), the... | Proof We consider the case (i). We can claim that \( \left( {{\theta }_{{d}_{1}, m},0}\right) \) is linearly stable. That is\n\n\[{d}_{2}{\Delta \psi } + \left( {{\beta a} - b{\theta }_{{d}_{1}, m}}\right) \psi + {\mu }_{1}\psi = 0,\;{\partial }_{v}\psi = 0,\;\text{ on }\partial \Omega .\]\n\nWe only need to show that ... | Yes |
Theorem 3.2 Suppose that (H1) holds. Then there exist two unique positive numbers \( {\beta }_{{d}_{1},{d}_{2}}^{U},{\beta }_{{d}_{1},{d}_{2}}^{V} \in \left( {{\beta }_{* * },{\beta }^{* * }}\right) \) for any \( {d}_{1},{d}_{2} > 0 \) such that:\n\n(i) If \( \beta \in \left( {0,{\beta }_{{d}_{1},{d}_{2}}^{U}}\right) \... | To establish Theorem 3.2, we only need to prove that the following two lemmas. In fact, we can get more information about of two numbers \( {\beta }_{{d}_{1},{d}_{2}}^{U},{\beta }_{{d}_{1},{d}_{2}}^{V} \) . More precisely, we have: | No |
Lemma 3.1 Suppose that (H1) holds. Then there exists a unique number \( {\beta }_{{d}_{1},{d}_{2}}^{U} \) for any \( {d}_{1},{d}_{2} > 0 \) such that\n\n\[ \n{\mu }_{1}\left( {{d}_{2},{\beta }_{{d}_{1},{d}_{2}}^{U}a - b{\theta }_{{d}_{1}, m}}\right) = 0. \n\]\n\n(9)\n\nIn particular, we can get:\n\n(i) If \( \beta \in ... | Proof For (i), we only need to show that there exists a unique \( {\beta }_{{d}_{1},{d}_{2}}^{U} > 0 \) such that\n\n\[ \n\left\{ \begin{array}{lll} {\mu }_{1}\left( {{d}_{2},{\beta a} - b{\theta }_{{d}_{1}, m}}\right) > 0, & \text{ if } & \beta \in \left( {0,{\beta }_{{d}_{1},{d}_{2}}^{U}}\right) , \\ {\mu }_{1}\left(... | Yes |
Lemma 3.2 Suppose that (H1) holds. Then there exists a unique number \( {\beta }_{{d}_{1},{d}_{2}}^{V} \) for any \( {d}_{1},{d}_{2} > 0 \) such that\n\n\[ \n{\mu }_{1}\left( {{d}_{1}, m - c{\beta }_{{d}_{1},{d}_{2}}^{V}a}\right) = 0.\n\]\n\n(11)\n\nIn particular, we can get:\n\n(i) If \( \beta \in \left( {0,{\beta }_{... | Proof It is clear that \( {\mu }_{1}\left( {{d}_{1}, m - {c\beta a}}\right) < 0 \) for \( \beta \ll 1 \) and \( {\mu }_{1}\left( {{d}_{1}, m - {c\beta a}}\right) > 0 \) for \( \beta \gg 1 \) for any \( {d}_{1},{d}_{2} > 0 \) . Moreover, by Lemma 2.2, we see that \( {\mu }_{1}\left( {{d}_{1}, m - {c\beta a}}\right) \) i... | Yes |
Theorem 3.3 Suppose that (H1) holds and we set\n\n\[ \nL \mathrel{\text{:=}} \frac{\mathop{\inf }\limits_{\bar{\Omega }}m}{\mathop{\sup }\limits_{\bar{\Omega }}m}. \]\n\nThen we have:\n\n(i) If \( 0 < {bc} < L \), then \( {\beta }^{ * } < {\beta }_{ * } \), and both \( \left( {{\theta }_{{d}_{1}, m},0}\right) \) and \(... | Proof For (i), we can claim that \( \left( {{\theta }_{{d}_{1}, m},0}\right) \) is unstable for \( \beta > {\beta }^{ * } \) . We only need to show that \( {\mu }_{1}\left( {{d}_{2},{\beta a} - b{\theta }_{{d}_{1}, m}}\right) < 0 \) . Easily\n\n\[ \n{\beta a} - b{\theta }_{{d}_{1}, m} > {\beta a} - b\mathop{\sup }\limi... | Yes |
Lemma 3.3 Suppose that (H1) holds and \( L < {bc} \leq 1 \) . (I) If \( L < {bc} < \frac{\bar{m}}{{\theta }_{{d}_{1}, m}},\;\text{ then }\frac{b}{a}\overline{{\theta }_{{d}_{1}, m}} < \frac{1}{c}\frac{\bar{m}}{a}. | Proof It is clear that \( {\beta }_{ * } > {\beta }_{* * } \) since \( {bc} < 1 \) . Hence, \( \left( {{\theta }_{{d}_{1}, m},0}\right) \) is linearly stable for any \( \beta \in \left( {{\beta }_{ * },\frac{b}{a}\overline{{\theta }_{{d}_{1}, m}}}\right) ,\;{d}_{2} > {\check{d}}_{2} = \frac{1}{{\lambda }_{1}\left( {{\b... | Yes |
Theorem 3.4 Suppose that (H1) holds and for any \[ \beta \in \left( {\frac{b}{a}\mathop{\inf }\limits_{\bar{\Omega }}{\theta }_{{d}_{1}, m},\frac{b}{a}\overline{{\theta }_{{d}_{1}, m}}}\right) ,\;\text{ with }{bc} < \frac{\bar{m}}{\overline{{\theta }_{{d}_{1}, m}}}. \] Then there exist small \( {c}^{ * } > 0 \) and lar... | Proof First, we claim that \( \left( {{\theta }_{{d}_{1}, m},0}\right) \) is linearly stable. We only need to show that \( {\mu }_{1}\left( {{d}_{2},{\beta a} - b{\theta }_{{d}_{1}, m}}\right) > 0 \) . Clearly \[ {\int }_{\Omega }\left( {{\beta a} - b{\theta }_{{d}_{1}, m}}\right) < 0,\;{\beta a} - b{\theta }_{{d}_{1},... | Yes |
Theorem 3.7 Suppose that (H1) holds and for all \( {d}_{1},{d}_{2} \) small with\n\n\[ \n{bc} < \frac{\mathop{\inf }\limits_{\bar{\Omega }}m}{\mathop{\inf }\limits_{\bar{\Omega }}{\theta }_{{d}_{1}, m}}.\n\]\n\nThen for any \( \beta \in \left( {\frac{b}{a}\mathop{\inf }\limits_{\Omega }{\theta }_{{d}_{1}, m},{\beta }_{... | Proof We claim that if both \( {d}_{1} \) and \( {d}_{2} \) are small, then \( \left( {{\theta }_{{d}_{1}, m},0}\right) \) is unstable. We only need to show that \( {\mu }_{1}\left( {{d}_{2},{\beta a} - b{\theta }_{{d}_{1}, m}}\right) < 0 \) . Since \( m > 0 \) on \( \bar{\Omega } \), we have \( {\theta }_{{d}_{1}, m} ... | Yes |
Theorem 1 Let \( \left\{ {{\mathbf{e}}_{1},{\mathbf{e}}_{2},\cdots ,{\mathbf{e}}_{k + 1},\cdots }\right\} \) be the tracking error sequence generated by the CDOILC algorithm (4)-(8), then \( {\begin{Vmatrix}{\mathbf{e}}_{k + 1}\end{Vmatrix}}^{2} \leq {\begin{Vmatrix}{\mathbf{e}}_{k}\end{Vmatrix}}^{2} \) . | Proof It is easy to draw the conclusion from the fact as follows\n\n\[ \n{\begin{Vmatrix}{\mathbf{e}}_{k + 1}\end{Vmatrix}}^{2} - {\begin{Vmatrix}{\mathbf{e}}_{k}\end{Vmatrix}}^{2} \n\]\n\n\[ \n= {\begin{Vmatrix}{\mathbf{e}}_{k} - {\alpha }_{k}\mathbf{H}{\mathbf{P}}_{k}\end{Vmatrix}}^{2} - {\begin{Vmatrix}{\mathbf{e}}_... | Yes |
Example 1 Consider a linear discrete time-invariant SISO system \( {}^{\left\lbrack {12}\right\rbrack } \) as follows\n\n\[ G\left( z\right) = \frac{{0.02771z} - {0.02713}}{{z}^{2} - {1.9580z} + {0.9589}}. \]\n\n(12)\n\nThe operation discrete time interval is set as \( \{ 0,1,2,\cdots ,{499}\} \) . without loss of gene... | Comparative tracking performances of the proposed CDOILC with that of NOILC and POILC are shown in Figure 1, where the solid curve is the iteration-wise tracking error curve produced by the proposed CDOILC algorithm, the dot profile is made by the NOILC scheme and the dash-dot one is generated by the POILC scheme, resp... | Yes |
Lemma 4 Let \( {Q}_{3} - e \) be the graph obtained from the hypercube \( {Q}_{3} \) by removing an edge. Then \( {Q}_{3} - e \) has an eigenvalue 1 with the eigenvector \( \mathbf{x} = {\left( {x}_{1},{x}_{2},\cdots ,{x}_{8}\right) }^{T} \) shown in Figure 3. | Proof Recall that the characteristic equation is \( A\left( {{Q}_{3} - e}\right) \mathbf{x} = \lambda \mathbf{x} \), which is equivalent to \( \lambda {x}_{i} = \mathop{\sum }\limits_{{j \sim i}}{x}_{j} \) . It is straightforward to check that the equations \( {x}_{1} = {x}_{2} + {x}_{3} + {x}_{7},\cdots ,{x}_{8} = {x}... | Yes |
Lemma 5 Let \( X \) be a graph with maximum degree 3 . If \( X \) has \( m \) disjoint \( {Q}_{3} - e \) as its induced subgraphs, then it has eigenvalue 1 with multiplicity at least \( m \) . | Proof We construct \( m \) eigenvectors \( \mathbf{x} = {\left( {x}_{1},{x}_{2},\cdots ,{x}_{n}\right) }^{T} \) for eigenvalue 1 in graph \( X \) . Take any induced subgraph \( {Q}_{3} - e \), assign the coordinates of \( \mathbf{x} \) corresponding to vertices of the \( {Q}_{3} - e \) the values as shown in Figure 3. ... | No |
Theorem 1 For any graph \( X \), the number of subgraphs \( {P}_{k}\left( {4 \leq k \leq 6}\right) \) contained in \( X \) (not necessarily induced) can be enumerated by the following formulas:\n\n(i)\n\n\[ \n{N}_{X}\left( {P}_{4}\right) = \mathop{\sum }\limits_{{{v}_{i}{v}_{j} \in E\left( X\right) }}d\left( {{v}_{i}{v... | Proof We only prove part (iii) in detail. For parts (i) and (ii), the proof is similar and is omitted. Let \( {v}_{g}{v}_{h}{v}_{i}{v}_{j}{v}_{k}{v}_{l} \) be any path \( {P}_{6} \) . Then \( {v}_{h}{v}_{i}{v}_{j}{v}_{k} \) is called the central \( {P}_{4} \) of the path \( {P}_{6} \) . It is easy to see that a path \(... | No |
For any graph \( X \), the number of subgraph \( {T}_{1,1,2} \) contained in \( X \) (not necessary induced) can be enumerated by the following formula\n\n\[ \n{N}_{X}\left( {T}_{1,1,2}\right) = \mathop{\sum }\limits_{{{v}_{i} \in V\left( X\right), d\left( {v}_{i}\right) > 2}}{\widetilde{d}}_{2}\left( {v}_{i}\right) - ... | Proof For any subgraph \( {K}_{1,3} \) of graph \( X \), let \( {v}_{i} \) be the vertex of degree 3 and \( {v}_{j},{v}_{k},{v}_{l} \) be its three neighbors. Then we construct a \( {T}_{1,1,2} \) from the \( {K}_{1,3} \) by the following procedure. Pick a neighbor of \( {v}_{i} \) arbitrarily, say \( {v}_{j} \) . Exte... | Yes |
Lemma 6 Let \( G \cong {P}_{k}▱{P}_{2} \) and \( H \) be a graph generalized cospectral with \( G \) . Then \( H \) is a bipartite graph with \( k - 1 \) quadrangles and has an identical degree sequence as that of \( G \) . | Proof It is obvious that \( G \) is a bipartite graph with \( k - 1 \) quadrangles. The cospectrality of \( G \) and \( H \) implies that \( H \) is also bipartite, moreover, Lemma 3(vii) gives that \( H \) has \( k - 1 \) quadrangles. From Lemma 3(ii),(vi), we see that \( \left| {E\left( G\right) }\right| = \left| {E\... | Yes |
Lemma 7 Let \( G \cong {P}_{k}▱{P}_{2} \) and \( H \) be a graph generalized cospectral with \( G \) . Then \( G \) and \( H \) have the same number of paths \( {P}_{5} \) as their subgraph (not necessary induced). | Proof By Lemma 2(iv), for any graph \( X \), the number of walks of length 4 determined by the following equation\n\n\[ \n{N}_{4}\left( X\right) = 2{N}_{X}\left( {P}_{2}\right) + 8{N}_{X}\left( {P}_{3}\right) + 4{N}_{X}\left( {P}_{4}\right) + 2{N}_{X}\left( {P}_{5}\right) \n\]\n\n\[ \n+ 6{N}_{X}\left( {K}_{1,3}\right) ... | Yes |
Lemma 8 Let \( G \cong {P}_{k}▱{P}_{2}\left( {k \geq 5}\right) \) and \( H \) be a graph generalized cospectral with \( G \) . Then \( G \) and \( H \) have the same number of \( {T}_{1,1,2} \) as their subgraphs (not necessarily induced). | Proof Since \( {N}_{G}\left( {C}_{3}^{1}\right) = {N}_{H}\left( {C}_{3}^{1}\right) = 0 \), Theorem 3 gives that \( H\left\lbrack {V}^{ * }\right\rbrack \cong 2{P}_{2} \), and so does the subgraph induced by the vertices of degree 2 of \( G \) . By (4), it is straightforward to verify that \( {N}_{G}\left( {T}_{1,1,2}\r... | Yes |
Lemma 9 Let \( G \cong {P}_{k}▱{P}_{2} \) and \( H \) be a graph generalized cospectral with \( G \) . Then \( {N}_{G}\left( {P}_{6}\right) + 2{N}_{G}\left( {C}_{4}^{1}\right) = {N}_{H}\left( {P}_{6}\right) + 2{N}_{H}\left( {C}_{4}^{1}\right) \) . | Proof Recall that \( G \) and \( H \) are bipartite, so \( G \) and \( H \) have no \( {C}_{3},{C}_{3}^{1},{C}_{3}^{2},{C}_{3}^{3} \) , \( {C}_{4}^{2},{C}_{5} \) as their subgraphs. Lemma 3(v) gives that generalized cospectral graphs have the same number of walks of length \( k \) . Take \( k = 5, X \cong G \) or \( H ... | Yes |
Corollary 3 Let \( G \cong {P}_{k}▱{P}_{2} \) and \( H \) be a graph generalized cospectral with \( G \) . Then \[ \mathop{\sum }\limits_{{{v}_{h}{v}_{i}{v}_{j}{v}_{k} \in {\mathbb{P}}_{4}\left( G\right) }}d\left( {{v}_{h}{v}_{i}{v}_{j}{v}_{k}}\right) = \mathop{\sum }\limits_{{{v}_{h}{v}_{i}{v}_{j}{v}_{k} \in {\mathbb{... | Proof Since \( G \) and \( H \) are both bipartite, they have no \( {C}_{3}^{2},{C}_{5} \) and \( {C}_{4}^{2} \) as their subgraphs. Lemma 3(vii) implies that \( G \) and \( H \) have \( k - 1 \) cycle \( {C}_{4} \) . It follows our result by substituting (3) into Lemma 9. | No |
Lemma 10 Let \( G \cong {P}_{k}▱{P}_{2}\left( {k \geq 5}\right) \) . Then \[ \mathop{\sum }\limits_{{{v}_{h}{v}_{i}{v}_{j}{v}_{k} \in {\mathbb{P}}_{4}\left( G\right) }}d\left( {{v}_{h}{v}_{i}{v}_{j}{v}_{k}}\right) = {48k} - {134}. \] | Proof Let \( \Gamma \) be a graph with the same degree sequence as the grid graph \( {P}_{k}▱{P}_{2} \) , in which the subgraph induced by the four vertices of degree 2 is isomorphic to \( 2{P}_{2} \) . Suppose that \( {v}_{h}{v}_{i}{v}_{j}{v}_{k} \) is any \( {P}_{4} \) in \( \Gamma \), then we classify the \( {P}_{4}... | Yes |
Proposition 1 Let \( B \) be a bipartite graph with maximum degree 3 . Suppose that \( v \in V\left( B\right) \) and \( q\left( v\right) \) is the number of quadrangles in which \( v \) lies in, then \( q\left( v\right) = \) \( 0,1,2,3,4,6 \) . Moreover, | (i) \( \;q\left( v\right) = 6 \) if and only if \( B \cong {K}_{3,3} \ | No |
Lemma 12 Part (i) of Theorem 4 does not occur. | Proof Assume that the two paths \( {P}_{4} \) between \( {P}_{2}^{a} \) and \( {P}_{2}^{b} \) are \( {P}_{4}^{a} = {v}_{1}{v}_{5}{v}_{6}{v}_{3},{P}_{4}^{b} = \) \( {v}_{2}{v}_{7}{v}_{8}{v}_{4} \) . We conclude that each of these eight vertices \( {v}_{i} \) for \( i = 1,2,\cdots ,8 \) lies in at most in one quadrangle,... | Yes |
Theorem 5 If \( k ≢ 2\left( {\;\operatorname{mod}\;3}\right) \), then the grid graph \( {P}_{k}▱{P}_{2} \) is determined by its generalized spectrum. | Proof Let \( H \) be a graph generalized cospectral with the grid graph \( {P}_{k}▱{P}_{2} \) . Since \( k ≢ 2\left( {\;\operatorname{mod}\;3}\right) \), Corollary 1 gives that \( {P}_{k}▱{P}_{2} \) has no zero eigenvalues, and so does \( H \) . Thus \( H \) has neither \( {K}_{2,3} \) nor \( {K}_{3,3} - e \) as its in... | No |
Theorem 1 For any solution \( \left( {{x}_{1}\left( t\right) ,{x}_{2}\left( t\right) ,{x}_{3}\left( t\right) }\right) \) of system (1) with \( t \) large enough, there exists a constant \( M > 0 \), such that \( {x}_{1}\left( t\right) < M,{x}_{2}\left( t\right) < M \) and \( {x}_{3}\left( t\right) < M \) . | Proof Let \( X\left( t\right) = \left( {{x}_{1}\left( t\right) ,{x}_{2}\left( t\right) ,{x}_{3}\left( t\right) }\right) \) be a solution of system (1) with initial value \( \left( {{x}_{10},{x}_{20},{x}_{30}}\right) \) . Define \( \omega \left( {t, X\left( t\right) }\right) = {a}_{1}{x}_{1}\left( t\right) + {a}_{2}{x}_... | Yes |
Corollary 1 If \( {\mu }_{i} = 0\left( {i = 1,2}\right) \) and \( l = 1 \) . The solution \( \left( {0,0,{x}_{3}^{ * }\left( t\right) }\right) \) is globally asymptotically stable if | \[ {p}^{m} > \max \left\{ {\frac{{d}_{i}m{r}_{3}{B}_{1}{r}_{i}T}{{c}_{i}{A}_{1}}, i = 1,2}\right\} ,\] where \[ {A}_{1} = \left( {1 - \exp \left( {-{r}_{3}{mT}}\right) }\right) ,\;{B}_{1} = {\left( 1 - \exp \left( -{r}_{3}T\right) \right) }^{m}. \] | Yes |
Corollary 2 If \( p = 0 \) and \( l = 1 \) . The solution \( \left( {0,0,{x}_{3}^{ * }\left( t\right) }\right) \) is locally stable if that | \[ T < \min \left\{ {\frac{1}{{r}_{1}}\ln \frac{1}{1 - {\mu }_{1}},\frac{1}{{r}_{2}}\ln \frac{1}{1 - {\mu }_{2}}}\right\} \] | Yes |
Theorem 3 System (1) is permanent if the following conditions (H1), (H2) hold. | Proof Let \( X\left( t\right) = \left( {{x}_{1}\left( t\right) ,{x}_{2}\left( t\right) ,{x}_{3}\left( t\right) }\right) \) be any solution of system (1) with \( X\left( 0\right) > \) 0 . From Theorem 1, we know that exists a constant \( M > 0 \) such that \( {x}_{1}\left( t\right) < \) \( M,{x}_{2}\left( t\right) < M,{... | Yes |
Corollary 4 If \( p = 0 \) and \( l = 1 \) . System (1) is permanent if \( {k}_{1} > {h}_{1}{k}_{2},{k}_{2} > {h}_{2}{k}_{1} \) and | \[ T > \max \left\{ {\frac{1}{\left( {r}_{1} - \frac{{r}_{1}{h}_{1}{k}_{2}}{{k}_{1}}\right) }\ln \frac{1}{1 - {\mu }_{1}},\frac{1}{\left( {r}_{2} - \frac{{r}_{2}{h}_{2}{k}_{1}}{{k}_{2}}\right) }\ln \frac{1}{1 - {\mu }_{2}}}\right\} . \] | Yes |
Lemma 3 Suppose that the unique optimal solution of (6) is \( {\lambda }_{j}^{k}, j \in {J}^{k} \), then the optimal solution of (2) is\n\n\[ \n{y}_{k + 1} = {x}_{k} - {\mu }_{k}\mathop{\sum }\limits_{{j \in {J}^{k}}}{\lambda }_{j}^{k}{g}_{j} \n\] | Proof The Lagrange function of (5) is\n\n\[ \nL\left( {w, x,\lambda }\right) = w + \frac{1}{2{\mu }_{k}}{\begin{Vmatrix}x - {x}_{k}\end{Vmatrix}}^{2} + \mathop{\sum }\limits_{{j \in {J}^{k}}}{\lambda }_{j}\left\lbrack {f\left( {x}_{k}\right) + {g}_{j}^{T}\left( {x - {x}_{k}}\right) - {\alpha }_{j}^{k} - w}\right\rbrack... | Yes |
Lemma 4 Let \( {\left\{ {\lambda }_{j}^{k}\right\} }_{j \in {J}^{k}} \) be the optimal solution of (6). Then, we have:\n\n(i) \( {s}_{k} \in \partial {\widehat{f}}_{k}\left( {y}_{k + 1}\right) \) ;\n\n(ii) \( {\widehat{f}}_{k}\left( {y}_{k + 1}\right) = f\left( {x}_{k}\right) - {\mu }_{k}{\begin{Vmatrix}{s}_{k}\end{Vma... | Proof (i) By the optimality condition of (2), we get\n\n\[ 0 \in \partial {\widehat{f}}_{k}\left( {y}_{k + 1}\right) + \frac{1}{{\mu }_{k}}\left( {{y}_{k + 1} - {x}_{k}}\right) \]\n\nFrom Lemma \( 3,{y}_{k + 1} = {x}_{k} - {\mu }_{k}{s}_{k} \), so \( {s}_{k} \in \partial {\widehat{f}}_{k}\left( {y}_{k + 1}\right) \).\n... | Yes |
Lemma 5 For \( {f}_{k}^{a} \) defined by (9), we have\n\n\[ \n{f}_{k}^{a}\left( x\right) = f\left( {x}_{k}\right) + {s}_{k}^{T}\left( {x - {x}_{k}}\right) - {e}_{k},\;{f}_{k}^{a}\left( x\right) \leq {\widehat{f}}_{k}\left( x\right) \leq f\left( x\right) .\n\] | Proof By the proof process of Lemma 4 (iv), it is straightforward. | No |
Lemma 6 Consider Algorithm 1, denote \( \bar{f} = \mathop{\min }\limits_{{x \in {R}^{n}}}f\left( x\right) > - \infty \), then\n\n\[ \mathop{\sum }\limits_{l}{v}_{k\left( l\right) } \leq \frac{f\left( {x}_{1}\right) - \bar{f}}{{m}_{1}} \] | Proof As \( l \) is the index of descent step, \( {x}_{k\left( l\right) + 1} = {y}_{k + 1} \) . From Step \( 3,\forall l, f\left( {y}_{k + 1}\right) \leq \) \( f\left( {x}_{k\left( l\right) }\right) - {m}_{1}{v}_{k\left( l\right) } \) . Summing up over \( l \), we have\n\n\[ \mathop{\sum }\limits_{l}{v}_{k\left( l\righ... | Yes |
Lemma 7 Suppose the stabilized center point generated by the last serious step is \( {x}_{{k}_{0}} \) and \( {\left\{ {y}_{k}\right\} }_{k \geq {k}_{0}} \) are trial points. Then, for all \( k \geq {k}_{0} \) and \( x \in {R}^{n} \), we have\n\n\[ f\left( {x}_{{k}_{0}}\right) - {v}_{k} + \frac{1}{2{\mu }_{k}}{\begin{Vm... | Proof Using the suppose and the definition of \( {v}_{k} \), we have\n\n\[ {v}_{k} = f\left( {x}_{{k}_{0}}\right) - {\widehat{f}}_{k}\left( {y}_{k + 1}\right) - \frac{1}{2{\mu }_{k}}{\begin{Vmatrix}{y}_{k + 1} - {x}_{{k}_{0}}\end{Vmatrix}}^{2}, \]\n\nfor \( k \geq {k}_{0} \) . Taking the expression of \( {v}_{k} \) int... | Yes |
Consider the following nonsmooth convex optimization problem\n\n\[ \mathop{\min }\limits_{{x \in {R}^{4}}}f\left( x\right) = \mathop{\max }\limits_{{1 \leq i \leq 4}}{f}_{i}\left( x\right) \]\n\nwhere\n\n\[ {f}_{1}\left( x\right) = {x}_{1}^{2} + {x}_{2}^{2} + 2{x}_{3}^{2} + {x}_{4}^{2} - 5{x}_{1} - 5{x}_{2} - {21}{x}_{... | Let \( {x}_{0} = \left( {0,0,0,0}\right) ,{\mu }_{0} = 1/\begin{Vmatrix}{g}^{0}\end{Vmatrix} \) . As is known, \( {x}^{ * } = \left( {0,1,2, - 1}\right) ,{f}^{ * } = - {44} \) . | Yes |
Consider the following nonsmooth convex optimization problem\n\n\\[ \n\\mathop{\\operatorname{minimize}}\\limits_{{x \\in {R}^{2}}}f\\left( x\\right) = \\mathop{\\max }\\limits_{{1 \\leq i \\leq 3}}{f}_{i}\\left( x\\right) \n\\]\n\nwhere\n\n\\[ \n{f}_{1}\\left( x\\right) = 5{x}_{1} + {x}_{2},\\;{f}_{2}\\left( x\\right)... | Let \\( {x}_{0} = \\left( {1,1}\\right) ,{\\mu }_{0} = 1 \\) . As is known, its optimal solution is \\( {x}^{ * } = \\left( {0, - 3}\\right) \\) with \\( f\\left( {x}^{ * }\\right) = - 3 \\) . | Yes |
Lemma 1 Let \( f : {\mathcal{N}}_{ + } \rightarrow \mathcal{C} \) satisfy\n\n\[ f\left( {{u}^{2} + 2{v}^{2}}\right) = {f}^{2}\left( u\right) + 2{f}^{2}\left( v\right) \]\n\nfor all \( u, v \in {\mathcal{N}}_{ + }, f\left( 1\right) = a \) and \( f\left( 2\right) = b \) . Then we have\n\n\[ {f}^{2}\left( 3\right) = 9{a}^... | Proof Since \( f\left( 1\right) = a, f\left( 2\right) = b \) and \( 3 = {1}^{2} + 2 \times {1}^{2} \), then we have \( f\left( 3\right) = 3{a}^{2} \) and thus (10) holds. We note that\n\n\[ {27} = {3}^{2} + 2 \times {3}^{2} = {5}^{2} + 2 \times {1}^{2},\;{33} = {1}^{2} + 2 \times {4}^{2} = {5}^{2} + 2 \times {2}^{2}, \... | Yes |
Lemma 3 Let \( f : {\mathcal{N}}_{ + } \rightarrow \mathcal{C} \) satisfy\n\n\[ f\left( {{u}^{2} + 3{v}^{2}}\right) = {f}^{2}\left( u\right) + 3{f}^{2}\left( v\right) \]\n\nfor all \( u, v \in {\mathcal{N}}_{ + }, f\left( 1\right) = a \) and \( f\left( 2\right) = b \) . Then we have\n\n\[ {f}^{2}\left( 3\right) = \frac... | Proof Since \( f\left( 1\right) = a, f\left( 2\right) = b \) and\n\n\[ {28} = {5}^{2} + 3 \times {1}^{2} = {1}^{2} + 3 \times {3}^{2} = {4}^{2} + 3 \times {2}^{2},\;{52} = {2}^{2} + 3 \times {4}^{2} = {5}^{2} + 3 \times {3}^{2}, \]\n\nwe have\n\n\[ \left\{ \begin{array}{l} {f}^{2}\left( 5\right) = 3{f}^{2}\left( 3\righ... | Yes |
Lemma 5 Let \( f : {\mathcal{N}}_{ + } \rightarrow \mathcal{C} \) satisfy\n\n\[ f\left( {{u}^{2} + 4{v}^{2}}\right) = {f}^{2}\left( u\right) + 4{f}^{2}\left( v\right) \]\n\nfor all \( u, v \in {\mathcal{N}}_{ + }, f\left( 1\right) = a \) and \( f\left( 2\right) = b \). Then (22)-(26) and (34) hold and\n\n\[ {f}^{2}\lef... | Proof Since\n\n\[ f\left( 1\right) = a,\;f\left( 2\right) = b,\;{20} = {4}^{2} + 4 \times {1}^{2} = {2}^{2} + 4 \times {2}^{2}, \]\n\n\[ {65} = {7}^{2} + 4 \times {2}^{2} = {1}^{2} + 4 \times {4}^{2},\;{68} = {8}^{2} + 4 \times {1}^{2} = {2}^{2} + 4 \times {4}^{2}, \]\n\nwe have\n\n\[ {f}^{2}\left( 4\right) = 5{f}^{2}\... | Yes |
Lemma 6 \( f\left( 1\right) \in \left\{ {0,1, - 1,\frac{1}{5}, - \frac{1}{5}}\right\} \) . | Proof Firstly, by \( f\left( 1\right) = a \) and \( f\left( 5\right) = {f}^{2}\left( 1\right) + 4{f}^{2}\left( 1\right) = 5{f}^{2}\left( 1\right) = 5{a}^{2} \), we have\n\n\[ \n{f}^{2}\left( 5\right) = {25}{a}^{4} \n\]\n\n(37)\n\nSimilarly, by \( f\left( 2\right) = b \) and \( f\left( {20}\right) = {f}^{2}\left( 2\righ... | Yes |
Lemma 7 Let \( f : {\mathcal{N}}_{ + } \rightarrow \mathcal{C} \) satisfy\n\n\[ f\left( {{u}^{2} + 5{v}^{2}}\right) = {f}^{2}\left( u\right) + 5{f}^{2}\left( v\right) \]\n\nfor all \( u, v \in {\mathcal{N}}_{ + }, f\left( 1\right) = a \) and \( f\left( 2\right) = b \) . Then (22)-(26),(34),(42) and (43) hold:\n\n\[ {f}... | Proof Since\n\n\[ f\left( 1\right) = a,\;f\left( 2\right) = b,\;{21} = {4}^{2} + 5 \times {1}^{2} = {1}^{2} + 5 \times {2}^{2}, \]\n\n\[ {84} = {8}^{2} + 5 \times {2}^{2} = {2}^{2} + 5 \times {4}^{2},\;{69} = {7}^{2} + 5 \times {2}^{2} = {8}^{2} + 5 \times {1}^{2}, \]\n\n\[ {129} = {7}^{2} + 5 \times {4}^{2} = {2}^{2} ... | Yes |
Lemma 1 Let \( A \) be \( n \times n \) Hermitian matrices and \( B \) be a principal submatrix of \( A \) of order \( m \) . If \( {\lambda }_{1}\left( A\right) \geq {\lambda }_{2}\left( A\right) \geq \cdots \geq {\lambda }_{n}\left( A\right) \) is the eigenvalues of \( A \) and \( {\mu }_{1}\left( B\right) \geq \) \(... | \[ {\lambda }_{n - m + i}\left( A\right) \leq {\mu }_{i}\left( B\right) \leq {\lambda }_{i}\left( A\right) ,\;i = 1,2,\cdots, m. \] | Yes |
Lemma 2 Let \( G \) be a simple connected graph of order \( n \) with distance eigenvalues \( {\lambda }_{1}\left( {D\left( G\right) }\right) \geq {\lambda }_{2}\left( {D\left( G\right) }\right) \geq \cdots \geq {\lambda }_{n}\left( {D\left( G\right) }\right), H \) is a induced subgraph of \( G \) of order \( m \) with... | \[ {\lambda }_{n - m + i}\left( {D\left( G\right) }\right) \leq {\mu }_{i}\left( {D\left( H\right) }\right) \leq {\lambda }_{i}\left( {D\left( G\right) }\right) ,\;i = 1,2,\cdots, m. \] | Yes |
Lemma 3 Let \( T \) be a tree of order \( n \) and \( D\left( T\right) \) be the distance matrix of \( T \) . If \( {\lambda }_{n - 1}\left( {D\left( T\right) }\right) \in \left\lbrack {-{2.4295},0}\right\rbrack \), then \( T \) is \( {T}^{ * } \) -free, where \( {T}^{ * } = \left\{ {{P}_{7},{T}_{1},{T}_{2},{T}_{3}}\ri... | Proof By contradiction, we assume that \( T \) contains one of \( {T}^{ * } \) as a induced subgraph, then \( D\left( T\right) \) contains one of \( D\left( {P}_{7}\right) \) and \( D\left( {T}_{i}\right) \) for \( i = 1,2,3 \) as a principal submatrix. By a simple calculation,\n\n\[ \n{\lambda }_{6}\left( {D\left( {T}... | Yes |
Theorem 1 If \( T \) is a tree of order \( n \) and \( D\left( T\right) \) be the distance matrix of \( T \), then \( {\lambda }_{n - 1}\left( {D\left( T\right) }\right) \in \left\lbrack {-{2.4295},0}\right\rbrack \) if and only if \( T \in \left\{ {{P}_{6},{T}_{4}\left( {a, b}\right) ,{T}_{5}\left( {a, b}\right) }\rig... | Proof If \( T \in \left\{ {{P}_{6},{T}_{4}\left( {a, b}\right) ,{T}_{5}\left( {a, b}\right) }\right\} \), then by Lemma 4, the result holds. For converse, by Lemma 3, we get that \( T \) is \( {T}^{ * } \) -free, then \( d\left( T\right) \leq 5 \) . If \( d\left( T\right) = 5 \), then \( T \cong {P}_{6} \), if \( d\lef... | Yes |
Lemma 5 Let \( G \) be a unicyclic graph of order \( n \) and \( D\left( G\right) \) be the distance matrix of \( G \) . If \( {\lambda }_{n - 1}\left( {D\left( G\right) }\right) \in \left\lbrack {-2,0}\right\rbrack \), then \( G \) is \( {G}^{ * } \) -free, where \( {G}^{ * } = \left\{ {{P}_{6},{T}_{3},{G}_{1},{G}_{2}... | Proof By Lemmas 3 and 4, \( G \) is \( \left\{ {{P}_{6},{T}_{3}}\right\} \) -free. By contradiction, we may assume that \( G \) contains one of \( \left\{ {{G}_{1},{G}_{2},{G}_{3},{G}_{4},{G}_{5}}\right\} \) as an induced subgraph, then \( D\left( G\right) \) must contains one of \( D\left( {G}_{i}\right) \) for \( i =... | Yes |
Lemma 6 If \( G \in \left\{ {{G}_{6},{G}_{7},{G}_{8},{G}_{9},{G}_{10}}\right\} \) is a unicyclic graph of order \( n \), then\n\n\[ \n{\lambda }_{n - 1}\left( {D\left( G\right) }\right) \in \left\lbrack {-2,0}\right\rbrack \text{.} \n\] | Proof By a straightforward calculation, we have\n\n\[ \n\Phi \left\lbrack {D\left( {G}_{6}\right), t}\right\rbrack = {\left( -1\right) }^{a + b + 1}{\left( t + 2\right) }^{a + b - 2}\left\lbrack {{t}^{5} - 2\left( {a + b - 2}\right) {t}^{4} - \left( {{13a} + {13b} + {5ab} - 1}\right) {t}^{3}}\right. \n\]\n\n\[ \n- 2\le... | Yes |
Theorem 2 If \( G \) is a unicyclic graph of order \( n \) and \( D\left( G\right) \) be the distance matrix of \( G \), then \( {\lambda }_{n - 1}\left( {D\left( G\right) }\right) \in \left\lbrack {-2,0}\right\rbrack \) if and only if \( G \in \left\{ {{G}_{6},{G}_{7},{G}_{8},{G}_{9},{G}_{10}}\right\} \) . | Proof The sufficient part is obvious by Lemma 6. So we only need to prove the necessity. By Lemma \( 5,{P}_{6} \) and \( {G}_{4} \) are forbidden subgraphs of \( G \), then \( d\left( G\right) \leq 4 \) and \( g\left( G\right) \leq 4 \) . Since \( G \) is \( \left\{ {{T}_{3},{G}_{1},{G}_{2},{G}_{3},{G}_{5}}\right\} \) ... | Yes |
Theorem 3 If \( G \) is a bicyclic graph of order \( n \) and \( D\left( G\right) \) be the distance matrix of \( G \), then \( {\lambda }_{n - 1}\left( {D\left( G\right) }\right) \in \left\lbrack {-2,0}\right\rbrack \) if and only if\n\n\[ G \in \left\{ {{G}_{j},{G}_{k},{G}_{l},{G}_{m},{G}_{n},{G}_{o},{G}_{p},{G}_{q},... | Proof The sufficient part is obvious by Lemma 8. So we only need to prove the necessity. By Lemma \( 7, G \) is \( {G}^{ * } \) -free and \( {G}^{* * } \) -free, then \( d\left( G\right) \leq 4 \) and \( g\left( G\right) \leq 4 \) . Thus we discuss the following two cases.\n\nCase \( {1G} \) contains an \( \infty \) -g... | Yes |
Lemma 3 Let \( C \) be a closed convex nonempty subset of \( H \) . Suppose that \( T \) : \( C \rightarrow C \) is a quasi-nonexpansive mapping. Then \( F\left( T\right) \) is a convex closed subset of \( C \) . | Proof we prove first that \( F\left( T\right) \) is closed. Let \( \left\{ {p}_{n}\right\} \subset F\left( T\right) \) with \( {p}_{n} \rightarrow p\left( {n \rightarrow \infty }\right) \) , we prove that \( p \in F\left( T\right) \) . Since \( T \) is quasi-nonexpansive, one has \( \begin{Vmatrix}{{Tp} - {p}_{n}}\end{... | Yes |
Theorem 2 Let \( C \) be a closed convex nonempty subset of \( H \) . Suppose that \( T : C \rightarrow C \) is a closed quasi-nonexpansive mapping and \( S : C \rightarrow C \) is a L-Lipschitz and quasi-nonexpansive mapping such that \( F\left( T\right) \cap F\left( S\right) \neq \varnothing \) . Assume that \( \left... | Proof By Lemma 3 and the assumption that \( F\left( T\right) \cap F\left( S\right) \neq \varnothing \), we know that \( {P}_{F\left( T\right) \cap F\left( S\right) }{x}_{0} \) is well defined for every \( {x}_{0} \in C \) . It follows from the construction of \( {C}_{n} \) and Lemma 2 that \( {C}_{n} \) is closed and c... | No |
Corollary 3 Let \( C \) be a closed convex nonempty subset of \( H \) . Suppose that \( {T}_{1} : C \rightarrow C \) is a closed quasi-nonexpansive mapping and \( \left\{ {{T}_{k} : C \rightarrow C, k = 2,\cdots, N}\right\} \) are \( N - 1{L}_{k} \) -Lipschitz and quasi-nonexpansive mappings such that \( \mathop{\bigca... | The proof of Corollary 3 is similar to Theorem 2 and omitted here. | No |
Proposition 5 Assume \( f = {\left\{ {f}_{n}\right\} }_{n = 0}^{{N}_{1} - 1}, h = {\left\{ {h}_{n}\right\} }_{n = 0}^{{N}_{2} - 1} \), and \( {N}_{1} \neq {N}_{2} \) . Without loss of generality, assume that \( {N}_{1} > {N}_{2} \) . Let\n\n\[ \widetilde{h} = {\left\lbrack {h}_{0},{h}_{1},\cdots ,{h}_{{N}_{2} - 1},\ove... | Proof The convolution matrix \( \widetilde{H} \in {R}^{\left( {2{N}_{1} - 1}\right) \times {N}_{1}} \) generated by \( \widetilde{h} \) and \( f \) is\n\n\[ \widetilde{H} = \left( \begin{matrix} H \\ 0 \end{matrix}\right) \]\n\nwhere \( H \in {R}^{\left( {{N}_{1} + {N}_{2} - 1}\right) \times {N}_{1}} \) is the convolut... | Yes |
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