
Citation: Qian Cao, Xin Long. New convergence on inertial neural networks with time-varying delays and continuously distributed delays[J]. AIMS Mathematics, 2020, 5(6): 5955-5968. doi: 10.3934/math.2020381
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Inertial neural networks system is a class of second order delay differential equations proposed by Babcock and Westervelt [1]. It is established by introducing an inertia term into the multi-directional associative memory neural networks, and is widely used in the fields of optimization, associative memory, image processing, psychophysics, and adaptive pattern recognition [1]. Therefore, it is of great significance to study the dynamic behaviors (such as stability [2,3,4,5,6], dissipation [7,8,9], Hopf bifurcation [10,11,12], Lagrange stability [13,14,15], synchronization [16,17,18,19,20], etc.) of the system in the application of inertial neural networks. It is worth noting that the dynamics analysis on inertial neural networks is usually to convert them into a first-order differential system by reducing order variable substitution under the assumption that the activation functions are bounded [21,22,23,24]. In particular, the periodicity, stability and convergence for the inertial neural networks systems have been established in [25,26,27,28,29,30] by using the reduced order method. However, this method needs to introduce some new parameters, which will raise the dimension in the inertial neural networks system. This will increase huge amounts of computation and it is difficult to achieve in practice [3,4,20]. Therefore, the authors of [3,4,20] have developed some non-reduced order methods to establish the stability and synchronization conditions of inertial neural networks with constant or time-varying delays, respectively.
Because there are many parallel paths with a series of different axon sizes and lengths in the neural networks, it is necessary to introduce continuous distributed delays to describe the transmission of neuron signals. In recent years, a large number of literatures have studied the dynamic behaviors of inertial neural networks with unbounded distributed delays [31,32,33,34,35,36,37]. In particular, the author of literature [36] studied the global convergence of inertial neural networks with continuous distributed delays by using a non-reduced order method:
x″i(t)=−ai(t)x′i(t)−bi(t)xi(t)+n∑j=1cij(t)˜Pj(xj(t))+n∑j=1hij(t)∫+∞0Kij(u)˜Rj(xj(t−u))du+Ji(t),i∈S:={1,2,⋯,n}, | (1.1) |
and
xi(s)=φi(s), x′i(s)=ψi(s), −∞≤s≤0, φi, ψi∈BC((−∞,0],R), | (1.2) |
where BC((−∞,0],R) is the set of all continuous and bounded functions from (−∞,0] to R, x(t)=(x1(t), x2(t),⋯,xn(t)) is the state vector, x″(t) is called an inertial term of (1.1), the time-varying connection weights cij, hij:R→R and ai, bi:R→(0,+∞) are bounded and continuous functions, the delay kernel Kij:[0, +∞)→R is a continuous function, the external input Ji(t) and the activation function ˜Pj and ˜Rj are continuous, and i, j∈S.
Unfortunately, in the initial values (1.2) which also adopted in [24,35], the assumption that
x′i(s)=ψi(s), −∞≤s≤0, ψi∈BC((−∞, 0],R), |
is incorrect. In fact, in the system (1.1), the transmission term ai(t)x′i(t) is not affected by the delays. Combined with the theory of the delay differential equation, we can see that in the initial problem (1.2), it is not necessary to assume that x′i(t) is bounded and continuous on (−∞, 0], which leaves room for further improvement.
On the other hand, the dynamic characteristics of the inertial neural networks are usually affected by time-varying delays and distributed delays. Therefore, it is especially significant to study the following inertial neural networks system with bounded time-varying delays and unbounded continuously distributed delays:
x″i(t)=−ai(t)x′i(t)−bi(t)xi(t)+n∑j=1cij(t)˜Pj(xj(t))+n∑j=1dij(t)˜Qj(xj(t−τij(t)))+n∑j=1hij(t)∫+∞0Kij(u)˜Rj(xj(t−u))du+Ji(t),i∈S, | (1.3) |
where Ji,cij,dij,hij:R→R, ai, bi:R→(0,+∞) and τij:R→R+ are bounded and continuous functions, the delay kernel Kij∈C([0,+∞),R) is a continuous function, the activation functions ˜Pi, ˜Qi, ˜Ri are continuous, and i,j∈S.
As is well known that the Lyapunov function structure of unbounded time-delay systems is more complex than bounded time-delay systems, therefore, the stability of the former is more difficult to establish than the latter. Especially, there are few studies on the dynamic behaviors of inertial neural networks with bounded time-varying delays and unbounded distributed delays. So far, we only find that the authors of [38] have discussed the existence and exponential stability of the periodic solution of system (1.3) with periodic input functions. However, to the best of our knowledge, there has not yet been research work on the global convergence analysis for the system (1.3) by utilizing a non-reduce method.
Regarding the above discussions, in this manuscript, the initial value (1.2) is modified to
xi(s)=φi(s), x′i(0)=ψi, −∞≤s≤0, φi∈BC((−∞,0],R), ψi∈R, | (1.4) |
and the global convergence criterion of the inertial neural networks (1.3) is established by applying a non-reduce method. In a nutshell, the contributions of this paper can be summarized as follows. 1) Without adopting the periodicity on the input functions, a class of inertial neural networks with bounded time-varying delays and unbounded continuously distributed delays are proposed; 2) Under some appropriate assumptions, a non-reduce approach is developed to show that all solutions and their derivatives in the proposed model are convergent to the zero vector; 3) The initial value conditions (1.2) in [24,35,36] are relaxed into a broader situation; 4) Numerical results including comparisons are presented to verify the obtained theoretical results.
The structure of this paper is as follows. The global convergence of the solutions and their derivatives of system (1.3) is established in section 2. Then, section 3 presents a concrete example with the numeric simulations to show the feasibility of the main results. Section 4 contains a conclusion of this paper and the further research of the topic.
In this section, we use the following Barbarat's lemma to prove the global convergence of the system (1.3).
Lemma 2.1 [36] If g(t) is uniformly continuous on interval [0, +∞), and ∫+∞0g(s)ds exists and is bounded, then limt→+∞g(t)=0.
Assumptions:
(G1) there exist three constants LPj, LQj and LRj such that
|˜Pj(u)|≤LPj|u|, |˜Qj(u)|≤LQj|u|, |˜Rj(u)|≤LRj|u|, for all u∈R, j∈S. |
(G2) for i,j∈S, |Kij(t)| is integrable on [0,+∞).
(G3) u(t)=maxi∈S|Ji(t)|, U(t)=∫t0u(s)ds is bounded on [0,+∞).
(G4) for i,j∈S, a′i(t), b′i(t) and (|cij(t)|LPj+|dij(t)|LQj+|hij(t)|LRj∫+∞0|Kij(u)|du)′ are bounded and continuous on [0,+∞).
(G5) there exist constants βi>0 and αi≥0,γi≥0 satisfying
supt∈[0,+∞)Di(t)<0, inft∈[0,+∞){4Di(t)Ei(t)−F2i(t)}>0, ∀t∈R, i∈S, | (2.1) |
where
{Di(t)=αiγi−ai(t)α2i+12α2in∑j=1(|cij(t)|LPj+|dij(t)|LQj+|hij(t)|LRj∫+∞0|Kij(u)|du),Ei(t)=−bi(t)αiγi+12n∑j=1(|cij(t)|LPj+|dij(t)|LQj+|hij(t)|LRj∫+∞0|Kij(u)|du)|αiγi| +12n∑j=1α2j(|cji(t)|LPi+d+jiLQi11−˙τ+ji+h+jiLRi∫+∞0|Kji(u)|du) +12n∑j=1(|cji(t)|LPi+d+jiLQi11−˙τ+ji+h+jiLRi∫+∞0|Kji(u)|du)|αjγj|,Fi(t)=βi+γ2i−ai(t)αiγi−bi(t)α2i,˙τ+ij=supt∈[0, +∞)τ′ij(t), d+ij=supt∈[0, +∞)|dij(t)|, h+ij=supt∈[0, +∞)|hij(t)|, i,j∈S. |
(G6) for i, j∈S, τij is continuously differentiable, and τ′ij(t)=˙τij<1 for all t∈R.
Remark 2.1. Combining (G1), (G2) and the basic theory on functional differential equation with infinite delay in [40], one can show that all solutions of the initial value problem (1.3) and (1.4) exist on [0, +∞).
Remark 2.2. In this paper, (G3) means that the input functions are absolutely integrable on [0, +∞), and (G4), which is related the delay kernels, implies the specific effect of time delay functions on the convergence of system (1.3).
Theorem 2.1 Assume that (G1)–(G6) hold. Let x(t)=(x1(t),x2(t),⋯,xn(t)) be a solution of the initial value problem (1.3) and (1.4). Then limt→+∞xi(t)=0, limt→+∞x′i(t)=0.
Proof. According to (G3), (G5) and the boundedness of coefficients in (1.3), one can choose two positive constants ρ and ϱ such that
−ρ=maxi∈Ssupt∈[0,+∞)e−U(t)Di(t), −ϱ=maxi∈Ssupt∈[0,+∞)e−U(t)[Ei(t)−(Fi(t))24Di(t)]. | (2.2) |
Let
W(t)=e−U(t){12n∑i=1βix2i(t)+12n∑i=1(αix′i(t)+γixi(t))2+12n∑i=1n∑j=1(α2id+ij+|αiγi|d+ij)LQj11−˙τ+ij∫tt−τij(t)x2j(s)ds+12n∑i=1n∑j=1(α2ih+ij+|αiγi|h+ij)LRj∫+∞0|Kij(u)|∫tt−ux2j(s)dsdu+12n∑i=1α2i}. |
It follows from (G2), (G3), (G6) and (1.3) that
W′(t)=−u(t)W(t)+e−U(t){n∑i=1(βi+γ2i)xi(t)x′i(t)+n∑i=1(α2ix′i(t)+αiγixi(t))×[−ai(t)x′i(t)−bi(t)xi(t)+n∑j=1cij(t)˜Pj(xj(t))+n∑j=1dij(t)˜Qj(xj(t−τij(t)))+n∑j=1hij(t)∫+∞0Kij(u)˜Rj(xj(t−u))du+Ji(t)]+n∑i=1αiγi(x′i(t))2+12n∑i=1n∑j=1(α2id+ij+|αiγi|d+ij)LQj11−˙τ+ijx2j(t)−12n∑i=1n∑j=1(α2id+ij+|αiγi|d+ij)LQj11−˙τ+ijx2j(t−τij(t))(1−τ′ij(t))+12n∑i=1n∑j=1(α2ih+ij+|αiγi|h+ij)LRj∫+∞0|Kij(u)|dux2j(t)−12n∑i=1n∑j=1(α2ih+ij+|αiγi|h+ij)LRj∫+∞0|Kij(u)|x2j(t−u)du≤e−U(t){−u(t)12n∑i=1[(αix′i(t)+γixi(t))2−2αi|αix′i(t)+γixi(t)|+α2i]+n∑i=1(βi+γ2i−ai(t)αiγi−bi(t)α2i)xi(t)x′i(t)+n∑i=1(αiγi−ai(t)α2i)(x′i(t))2−n∑i=1bi(t)αiγix2i(t)+12n∑i=1n∑j=1(α2id+ij+|αiγi|d+ij)LQj11−˙τ+ijx2j(t)−12n∑i=1n∑j=1(α2id+ij+|αiγi|d+ij)LQjx2j(t−τij(t))+12n∑i=1n∑j=1(α2ih+ij+|αiγi|h+ij)LRj∫+∞0|Kij(u)|dux2j(t)−12n∑i=1n∑j=1(α2ih+ij+|αiγi|h+ij)LRj∫+∞0|Kij(u)|x2j(t−u)du+n∑i=1n∑j=1(α2i|x′i(t)|+|αiγi||xi(t)|)|cij(t)||˜Pj(xj(t))|+n∑i=1n∑j=1(α2i|x′i(t)|+|αiγi||xi(t)|)|dij(t)||˜Qj(xj(t−τij(t)))|+n∑i=1n∑j=1(α2i|x′i(t)|+|αiγi||xi(t)|)|hij(t)|∫+∞0|Kij(u)||˜Rj(xj(t−u))|du}≤e−U(t){n∑i=1(βi+γ2i−ai(t)αiγi−bi(t)α2i)xi(t)x′i(t)+n∑i=1(αiγi−ai(t)α2i)(x′i(t))2+n∑i=1[−bi(t)αiγi+12n∑j=1(α2jd+ji+|αjγj|d+ji)LQi11−˙τ+ji+12n∑j=1(α2jh+ji+|αjγj|h+ji)LRi∫+∞0|Kji(u)|du]x2i(t)−12n∑i=1n∑j=1(α2id+ij+|αiγi|d+ij)LQjx2j(t−τij(t))−12n∑i=1n∑j=1(α2ih+ij+|αiγi|h+ij)LRj∫+∞0|Kij(u)|x2j(t−u)du+n∑i=1n∑j=1(α2i|x′i(t)|+|αiγi||xi(t)|)|cij(t)||˜Pj(xj(t))|+n∑i=1n∑j=1(α2i|x′i(t)|+|αiγi||xi(t)|)|dij(t)||˜Qj(xj(t−τij(t)))|+n∑i=1n∑j=1(α2i|x′i(t)|+|αiγi||xi(t)|)|hij(t)|∫+∞0|Kij(u)||˜Rj(xj(t−u))|du}. (2.3) |
The assumption (G1) and the fact that uv≤12(u2+v2)(u,v∈R) entail that
n∑i=1n∑j=1(α2i|x′i(t)|+|αiγi||xi(t)|)|cij(t)||˜Pj(xj(t))|≤12n∑i=1n∑j=1α2i|cij(t)|LPj(x′i(t))2+12n∑i=1n∑j=1(|αiγi||cij(t)|LPj+α2j|cji(t)|LPi+|αjγj||cji(t)|LPi)x2i(t), |
n∑i=1n∑j=1(α2i|x′i(t)|+|αiγi||xi(t)|)|dij(t)||˜Qj(xj(t−τij(t)))|≤12n∑i=1n∑j=1α2i|dij(t)|LQj(x′i(t))2+12n∑i=1n∑j=1|αiγi||dij(t)|LQjx2i(t)+12n∑i=1n∑j=1|(α2i|dij(t)|LQj+|αiγi||dij(t)|LQj)x2j(t−τij(t)), |
and
n∑i=1n∑j=1(α2i|x′i(t)|+|αiγi||xi(t)|)|hij(t)|∫+∞0|Kij(u)||˜Rj(xj(t−u))|du≤12n∑i=1n∑j=1α2i|hij(t)|LRj∫+∞0|Kij(u)|du(x′i(t))2+12n∑i=1n∑j=1|αiγi||hij(t)|LRj∫+∞0|Kij(u)|dux2i(t)+12n∑i=1n∑j=1(α2i|hij(t)|LRj+|αiγi||hij(t)|LRj)∫+∞0|Kij(u)|x2j(t−u)du, |
which, together with (G4), (2.2) and (2.3), give
W′(t)≤e−U(t){n∑i=1(βi+γ2i−ai(t)αiγi−bi(t)α2i)xi(t)x′i(t)+n∑i=1[αiγi−ai(t)α2i+12α2in∑j=1(|cij(t)|LPj+|dij(t)|LQj+|hij(t)|LRj∫+∞0|Kij(u)|du)](x′i(t))2+n∑i=1[−bi(t)αiγi+12n∑j=1(|cij(t)|LPj+|dij(t)|LQj+|hij(t)|LRj∫+∞0|Kij(u)|du)|αiγi|+12n∑j=1α2j(|cji(t)|LPi+d+jiLQi11−˙τ+ji+h+jiLRi∫+∞0|Kji(u)|du)+12n∑j=1(|cji(t)|LPi+d+jiLQi11−˙τ+ji+h+jiLRi∫+∞0|Kji(u)|du)|αjγj|]x2i(t)} =e−U(t){n∑i=1Di(t)(x′i(t)+Fi(t)2Di(t)xi(t))2+n∑i=1(Ei(t)−(Fi(t))24Di(t))x2i(t)}≤−ρn∑i=1(x′i(t)+Fi(t)2Di(t)xi(t))2−ϱn∑i=1x2i(t)≤0, ∀t∈[0,+∞). | (2.4) |
This implies that W(t)≤W(0) for all t∈[0,+∞), and
12n∑i=1βix2i(t)+12n∑i=1(αix′i(t)+γixi(t))2≤+∞, t∈[0,+∞). |
Since αi|x′i(t)|≤|αix′i(t)+γixi(t)|+|γixi(t)|, it follows that x′i(t) and xi(t) are uniformly bounded on [0, +∞) for all i∈S. According to the continuity of right-hand side functions in (1.3), it is easy to see that x″i(t) is also uniformly bounded on [0, +∞) for all i∈S, which combining with (G4) lead that n∑i=1(x′i(t)+Fi(t)2Di(t)xi(t))2 and n∑i=1x2i(t) are uniformly continuous on [0, +∞).
In addition, (2.4) entails that
n∑i=1(x′i(t)+Fi(t)2Di(t)xi(t))2≤−1ρW′(t), n∑i=1x2i(t)≤−1ϱW′(t), ∀t≤0, |
and
limt→∞∫t0n∑i=1(x′i(s)+Fi(s)2Di(s)xi(s))2ds≤W(0)ρ, limt→∞∫t0n∑i=1x2i(s)ds≤W(0)ϱ, |
which, together with Lemma 2.1, lead to
limt→∞xi(t)=0, limt→∞(x′i(t)+Fi(t)2Di(t)xi(t))=0, limt→∞x′i(t)=0, i∈S. |
The proof is complete.
Remark 2.3. Obviously, system (1.1) is a special case of system (1.3) when dij=0, i,j∈S, and the restrictions on initial value condition (1.4) are weaker than those ones in (1.2), hence all the results in [30] can be derived from theorem 2.1. Moreover, the global Lipschitz conditions on the activation functions were crucial in [3,20,31] where the convergence on the state vector of inertial neural networks system was considered. However, in this paper, the global Lipschitz conditions have been abandoned and the global convergence on the inertial neural networks system with bounded time-varying delays and unbounded continuously distributed delays has been established. This implies that Theorem 2.1 generalizes and complements the main results of [3,20,30,31].
Example 3.1. Regard the following inertial neural networks with mixed delays:
{x″1(t)=−(4.56+sin2t)x′1(t)−(13.58+sin2t)x1(t)+1.21(sint)˜P1(x1(t)) +1.51(cost)˜P2(x2(t))−1.42(sin2t)˜Q1(x1(t−0.2sin2t)) +1.95(cos2t)˜Q2(x2(t−0.2sin2t)) −1.83(sin2t)∫+∞0sin(2u)1+u2˜R1(x1(t−u))du +0.71(cos2t)∫+∞0sin(2u)1+u2˜R2(x2(t−u))du+20sin4te−t2,x″2(t)=−(4.71+sin2t)x′2(t)−(14.45+sin2t)x2(t)−0.83(sint)˜P1(x1(t)) −1.47(cost)˜P2(x2(t))−1.52(sin2t)˜Q1(x1(t−0.2sin2t)) +0.95(cos2t)˜Q2(x2(t−0.2sin2t)) −3.51(sin2t)∫+∞0 cos(2u)1+u2˜R1(x1(t−u))du +1.17(cos2t)∫+∞0 cos(2u)1+u2˜R2(x2(t−u))du+30sin4te−t2, | (3.1) |
where ˜P1(u)=˜Q1(u)=˜R1(u)=0.25(|u+1|−|u−1|), ˜P2(u)=˜Q2(u)=˜R2(u)=0.5u(sin3u). Take αi=γi=1, β1=18.14, β2=19.16, LPi=LQi=LKi=0.5, i=1,2, we gain Di(t)<0, 4Di(t)Ei(t)>(Fi(t))2, i=1,2, t∈R. By Theorem 2.1, we can conclude that all solutions of (3.1) and the derivatives are convergent to the zero vector, respectively. Simulations in Figures 1 and 2 reflect that the theoretical convergence is in sympathy with the numerically observed behavior. Here, it can be seen from the moving trend of the trajectories in Figures 1, 2 that xi(t) and x′i(t) are convergent to 0 as t→+∞.
Remark 3.1. It should be pointed out that ˜P2(u)=˜Q2(u)=˜R2(u)=0.5u(sin3u) does not satisfy the global Lipschitz condition, and d11, d12, d21,d22≠0, then all results in the references [24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70] can not be straightly applied to show that every solution and its derivative convergent to the zero vector in system (3.1). Moreover, as far as the authors know, the convergence on inertial neural networks with bounded time-varying delays and unbounded continuously distributed delays without applying the reduced-order method has not been touched in the previous literature. Consequently, the main results established in this paper are essentially new and complement some existing ones.
In this paper, applying differential inequality techniques coupled with Lyapunov function method instead of the reduced order method, we study the global convergence on inertial neural networks with bounded time-varying delays and unbounded continuously distributed delays. Some sufficient assertions have been established to guarantee that every solution and its derivative of the addressed model are convergent to the zero vector. It should be mentioned that the method applied in this paper provides a possible approach to study the topic on dynamical behaviours of other inertial neural networks model with bounded time-varying delays and unbounded continuously distributed delays.
The authors would like to express the sincere appreciation to the editor and reviewers for their helpful comments in improving the presentation and quality of the paper. This work was supported by the Postgraduate Scientific Research Innovation Project of Hunan Province (No. CX20200892).
We confirm that we have no conflict of interest.
[1] |
K. Babcock, R. Westervelt, Stability and dynamics of simple electronic neural networks with added inertia, Phys. D, 23 (1986), 464-469. doi: 10.1016/0167-2789(86)90152-1
![]() |
[2] | Y. Zhou, X. Wan, C. Huang, et al. Finite-time stochastic synchronization of dynamic networks with nonlinear coupling strength via quantized intermittent control, Appl. Math. Comput., 376 (2020), 125157. |
[3] |
C. Huang, B. Liu, New studies on dynamic analysis of inertial neural networks involving nonreduced order method, Neurocomputing, 325 (2019), 283-287. doi: 10.1016/j.neucom.2018.09.065
![]() |
[4] |
C. Huang, Exponential stability of inertial neural networks involving proportional delays and nonreduced order method, J. Exp. Theor. Artif. Intell., 32 (2020), 133-146. doi: 10.1080/0952813X.2019.1635654
![]() |
[5] |
Z. Cai, J. Huang, L. Huang, Periodic orbit analysis for the delayed Filippov system, Proc. Am. Math. Soc., 146 (2018), 4667-4682. doi: 10.1090/proc/13883
![]() |
[6] |
C. Huang, R. Su, J. Cao, et al. Asymptotically stable high-order neutral cellular neural networks with proportional delays and D operators, Math. Comput. Simulation, 171 (2020), 127-135. doi: 10.1016/j.matcom.2019.06.001
![]() |
[7] | X. Yang, S. Wen, Z. Liu, et al. Dynamic properties of foreign exchange complex network, Mathematics, 7 (2019), 832. |
[8] |
C. Huang, X. Yang, J. Cao, Stability analysis of Nicholson's blowflies equation with two different delays, Math. Comput. Simulation, 171 (2020), 201-206. doi: 10.1016/j.matcom.2019.09.023
![]() |
[9] |
C. Huang, J. Wang, L. Huang, Asymptotically almost periodicity of delayed Nicholson-type system involving patch structure, Electron. J. Differ. Equ., 2020 (2020), 1-17. doi: 10.1186/s13662-019-2438-0
![]() |
[10] | H. Zhao, X. Yu, L. Wang, Bifurcation and control in an inertial two-neuron system with time delays, Int. J. Bifurcat. Chaos, 22 (2012), 1250036. |
[11] |
C. Huang, X. Long, J. Cao, Stability of anti-periodic recurrent neural networks with multiproportional delays, Math. Methods Appl. Sci., 43 (2020), 6093-6102. doi: 10.1002/mma.6350
![]() |
[12] |
C. Huang, L. Yang, J. Cao, Asymptotic behavior for a class of population dynamics, AIMS Math., 5 (2020), 3378-3390. doi: 10.3934/math.2020218
![]() |
[13] |
Q. Cao, X. Guo, Anti-periodic dynamics on high-order inertial Hopfield neural networks involving time-varying delays, AIMS Math., 5 (2020), 5402-5421. doi: 10.3934/math.2020347
![]() |
[14] | C. Huang, S.Wen, M. Li, et al. An empirical evaluation of the influential nodes for stock market network: Chinese A-shares case, Finance Res. Lett., (2020), 101517. |
[15] |
L. Duan, L. Huang, Z. Guo, et al. Periodic attractor for reaction-diffusion high-order hopfield neural networks with time-varying delays, Comput. Math. Appl., 73 (2017), 233-245. doi: 10.1016/j.camwa.2016.11.010
![]() |
[16] |
S. Wen, Y. Tan, M. Li, et al. Analysis of global remittance based on complex networks, Front. Phys., 8 (2020), 1-9. doi: 10.3389/fphy.2020.00001
![]() |
[17] |
T. Chen, L. Huang, P. Yu, et al. Bifurcation of limit cycles at infinity in piecewise polynomial systems, Nonlinear Anal. Real World Appl., 41 (2018), 82-106. doi: 10.1016/j.nonrwa.2017.10.003
![]() |
[18] |
J. Wang, X. Chen, L. Huang, The number and stability of limit cycles for planar piecewise linear systems of node-saddle type, J. Math. Anal. Appl., 469 (2019), 405-427. doi: 10.1016/j.jmaa.2018.09.024
![]() |
[19] | Z. Ye, C. Hu, L. He, et al. The dynamic time-frequency relationship between international oil prices and investor sentiment in China: A wavelet coherence analysis, Energy J., 41 (2020). DOI: 10.5547/01956574.41.5.fwen. |
[20] |
X. Li, X. Li, C. Hu, Some new results on stability and synchronization for delayed inertial neural networks based on non-reduced order method, Neural Networks, 96 (2017), 91-100. doi: 10.1016/j.neunet.2017.09.009
![]() |
[21] | C. Huang, H. Kuang, X. Chen, et al. An LMI approach for dynamics of switched cellular neural networks with mixed delays, Abstr. Appl. Anal., 2013 (2013), 1-8. |
[22] |
Y. Ke, C. Miao, Anti-periodic solutions of inertial neural networks with time delays, Neural Process. Lett., 45 (2017), 523-538. doi: 10.1007/s11063-016-9540-z
![]() |
[23] |
C. Xu, Q. Zhang, Existence and global exponential stability of anti-periodic solutions for BAM neural networks with inertial term and delay, Neurocomputing, 153 (2015), 108-116. doi: 10.1016/j.neucom.2014.11.047
![]() |
[24] |
L. Wang, M. Ge, J. Hu, et al. Global stability and stabilization for inertial memristive neural networks with unbounded distributed delays, Nonlinear Dyn., 95 (2019), 943-955. doi: 10.1007/s11071-018-4606-2
![]() |
[25] | L. Wang, H. He, Z. Zeng, Global synchronization of fuzzy memristive neural networks with discrete and distributed delays, IEEE Trans. Fuzzy Syst., (2019), 1-12. DOI: 10.1109/TFUZZ.2019.2930032. |
[26] | L. Wang, Z. Zeng, M. Ge, A disturbance rejection framework for finite-time and fixed-time stabilization of delayed memristive neural networks, IEEE Transactions on Systems, Man, and Cybernetics, (2019), 1-11. DOI: 10.1109/TSMC.2018.2888867. |
[27] | J. Zhao, J. Liu, L. Fang, Anti-periodic boundary value problems of second-order functional differential equations, Bull. Malays. Math. Sci. Soc., 37 (2014), 311-320. |
[28] |
C. Aouiti, I. B. Gharbia, Dynamics behavior for second-order neutral Clifford differential equations: Inertial neural networks with mixed delays, Comput. Appl. Math., 39 (2020), 1-31. doi: 10.1007/s40314-019-0964-8
![]() |
[29] | C. Aouiti, E. A. Assali, Effect of fuzziness on the stability of inertial neural networks with mixed delay via non-reduced-order method, Int. J. Comput. Math. Comput. Syst. Th., 4 (2019), 151-170, 30. C. Huang, C. Peng, X. Chen, et al. Dynamics analysis of a class of delayed economic model, Abstr. |
[30] | Appl. Anal., 2013 (2013), 1-12. |
[31] |
L. Wang, Z. Zeng, M. Ge, et al. Global stabilization analysis of inertial memristive recurrent neural networks with discrete and distributed delays, Neural Networks, 105 (2018), 65-74. doi: 10.1016/j.neunet.2018.04.014
![]() |
[32] |
C. Huang, B. Liu, X. Tian, Global convergence on asymptotically almost periodic SICNNs with nonlinear decay functions, Neural Process. Lett., 49 (2019), 625-641. doi: 10.1007/s11063-018-9835-3
![]() |
[33] |
K. Zhu, Y. Xie, F. Zhou, Pullback attractors for a damped semilinear wave equation with delays, Acta Math. Sin. (Engl. Ser.), 34 (2018), 1131-1150. doi: 10.1007/s10114-018-7420-3
![]() |
[34] |
J. Wang, C. Huang, L. Huang, Discontinuity-induced limit cycles in a general planar piecewise linear system of saddle-focus type, Nonlinear Anal. Hybrid Syst., 33 (2019), 162-178. doi: 10.1016/j.nahs.2019.03.004
![]() |
[35] |
J. Zhang, C. Huang, Dynamics analysis on a class of delayed neural networks involving inertial terms, Adv. Differ. Equ., 2020 (2020), 1-12. doi: 10.1186/s13662-019-2438-0
![]() |
[36] |
Y. Xu, Convergence on non-autonomous inertial neural networks with unbounded distributed delays, J. Exp. Theor. Artif. Intell., 32 (2020), 503-513. doi: 10.1080/0952813X.2019.1652941
![]() |
[37] | C. Huang, H. Yang, J. Cao, Weighted pseudo almost periodicity of multi-proportional delayed shunting inhibitory cellular neural networks with D operator, Discrete Contin. Dyn. Syst. Ser. S, (2018), 1-14. DOI:10.3934/dcdss.2020372. |
[38] | L. Yao, Q. Cao, Anti-periodicity on high-order inertial Hopfield neural networks involving mixed delays, J. Inequal. Appl., (2020), Available from: https://doi.org/10.1186/s13660-020-02444-3. |
[39] |
J. Li, J. Ying, D. Xie, On the analysis and application of an ion size-modified Poisson-Boltzmann equation, Nonlinear Anal. Real World Appl., 47 (2019), 188-203. doi: 10.1016/j.nonrwa.2018.10.011
![]() |
[40] | Y. Hino, S. Murakami, T. Naito, Functional differential equations with infinite delay, In Lecture in math-ematics Berlin: Springer, 1991. |
[41] | Q. Cao, G. Wang, C. Qian, New results on global exponential stability for a periodic Nicholson's blowflies model involving time-varying delays, Adv. Differ. Equ., 2020 (2020), 43. |
[42] | H. Hu, T. Yi, X. Zou, On spatial-temporal dynamics of Fisher-KPP equation with a shifting environment, Proc. Am. Math. Soc., 148 (2020), 213-221. |
[43] | X. Long, S. Gong, New results on stability of Nicholson's blowflies equation with multiple pairs of time-varying delays, Appl. Math. Lett., 100 (2020), 106027. |
[44] |
C. Huang, H. Zhang, L. Huang, Almost periodicity analysis for a delayed Nicholson's blowflies model with nonlinear density-dependent mortality term, Commun. Pure Appl. Anal., 18 (2019), 3337-3349. doi: 10.3934/cpaa.2019150
![]() |
[45] | C. Huang, H. Zhang, J. Cao, et al. Stability and Hopf bifurcation of a delayed prey-predator model with disease in the predator, Int. J. Bifurcation Chaos, 29 (2019), 1950091. |
[46] |
H. Zhang, Global Large Smooth Solutions for 3-D Hall-magnetohydrodynamics, Discrete Contin. Dyn. Syst., 39 (2019), 6669-6682. doi: 10.3934/dcds.2019290
![]() |
[47] | X. Zhang, H. Hu, Convergence in a system of critical neutral functional differential equations, Appl. Math. Lett., 107 (2020), 106385. |
[48] | C. Qian, New periodic stability for a class of Nicholson's blowflies models with multiple different delays, Int. J. Control, (2020), 1-13. DOI: 10.1080/00207179.2020.1766118. |
[49] |
C. Huang, X. Long, L. Huang, et al. Stability of almost periodic Nicholson's blowflies model involving patch structure and mortality terms, Can. Math. Bull., 63 (2020), 405-422. doi: 10.4153/S0008439519000511
![]() |
[50] | Y. Xu, Q. Cao, X. Guo, Stability on a patch structure Nicholson's blowflies system involving distinctive delays, Appl. Math. Lett., 105 (2020), 106340, |
[51] |
H. Hu, X. Yuan, L. Huang, et al. Global dynamics of an SIRS model with demographics and transfer from infectious to susceptible on heterogeneous networks, Math. Biosci. Eng., 16 (2019), 5729-5749. doi: 10.3934/mbe.2019286
![]() |
[52] |
Y. Tan, Dynamics analysis of Mackey-Glass model with two variable delays, Math. Biosci. Eng., 17 (2020), 4513-4526. doi: 10.3934/mbe.2020249
![]() |
[53] |
L. Li, Q. Jin, B. Yao, Regularity of fuzzy convergence spaces, Open Math., 16 (2018), 1455-1465. doi: 10.1515/math-2018-0118
![]() |
[54] |
C. Huang, L. Liu, Boundedness of multilinear singular integral operator with non-smooth kernels and mean oscillation, Quaest. Math., 40 (2017), 295-312. doi: 10.2989/16073606.2017.1287136
![]() |
[55] | C. Huang, J. Cao, F. Wen, et al. Stability analysis of SIR model with distributed delay on complex networks, PLoS One, 11 (2016), e0158813. |
[56] |
X. Li, Y. Liu, J. Wu, Flocking and pattern motion in a modified cucker-smale model, Bull. Korean Math. Soc., 53 (2016), 1327-1339. doi: 10.4134/BKMS.b150629
![]() |
[57] |
Y. Xie, Q. Li, K. Zhu, Attractors for nonclassical diffusion equations with arbitrary polynomial growth nonlinearity, Nonlinear Anal. Real World Appl., 31 (2016), 23-37. doi: 10.1016/j.nonrwa.2016.01.004
![]() |
[58] |
F. Wang, P. Wang, Z. Yao, Approximate controllability of fractional partial differential equation, Adv. Differ. Equ., 2015 (2015), 1-10. doi: 10.1186/s13662-014-0331-4
![]() |
[59] |
Y. Liu, J. Wu, Multiple solutions of ordinary differential systems with min-max terms and applications to the fuzzy differential equations, Adv. Differ. Equ., 2015 (2015), 1-13. doi: 10.1186/s13662-014-0331-4
![]() |
[60] |
L. Yan, J. Liu, Z. Luo, Existence and multiplicity of solutions for second-order impulsive differential equations on the half-line, Adv. Differ. Equ., 2013 (2013), 1-12, doi: 10.1186/1687-1847-2013-1
![]() |
[61] |
Y. Liu, J. Wu, Fixed point theorems in piecewise continuous function spaces and applications to some nonlinear problems, Math. Methods Appl. Sci., 37 (2014), 508-517. doi: 10.1002/mma.2809
![]() |
[62] |
X. Li, Z. Liu, J. Li, Existence and controllability for nonlinear fractional control systems with damping in Hilbert spaces, Acta Math. Sci., 39 (2019), 229-242. doi: 10.1007/s10473-019-0118-5
![]() |
[63] |
C. Qian, Y. Hu, Novel stability criteria on nonlinear density-dependent mortality Nicholson's blowflies systems in asymptotically almost periodic environments, J. Inequal. Appl., 2020 (2020), 1-18. doi: 10.1186/s13660-019-2265-6
![]() |
[64] |
S. Zhou, Y. Jiang, Finite volume methods for N-dimensional time fractional Fokker-Planck equations, Bull. Malays. Math. Sci. Soc., 42 (2019), 3167-3186. doi: 10.1007/s40840-018-0652-7
![]() |
[65] |
Y. Tan, C. Huang, B. Sun, et al. Dynamics of a class of delayed reaction-diffusion systems with Neumann boundary condition, J. Math. Anal. Appl., 458 (2018), 1115-1130. doi: 10.1016/j.jmaa.2017.09.045
![]() |
[66] |
F. Liu, L. Feng, A. Vo, et al. Unstructured-mesh Galerkin finite element method for the twodimensional multi-term time-space fractional Bloch-Torrey equations on irregular convex domains, Comput. Math. Appl., 78 (2019), 1637-1650. doi: 10.1016/j.camwa.2019.01.007
![]() |
[67] | Q. Jin, L. Li, G. Lang, p-regularity and p-regular modification in T-convergence spaces, Mathematics, 7 (2019), 370. |
[68] |
L. Huang, Endomorphisms and cores of quadratic forms graphs in odd characteristic, Finite Fields Appl, 55 (2019), 284-304. doi: 10.1016/j.ffa.2018.10.006
![]() |
[69] | L. Huang, B. Lv, K. Wang, Erdos-Ko-Rado theorem, Grassmann graphs and p(s)-Kneser graphs for vector spaces over a residue class ring, J. Combin. Theory Ser. A, 164 (2019), 125-158. |
[70] |
Y. Li, M. Vuorinen, Q. Zhou, Characterizations of John spaces, Monatsh. Math., 188 (2019), 547-559. doi: 10.1007/s00605-018-1231-6
![]() |
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