In this paper, we study the asymptotic behavior of the solutions of a new class of difference equations $$x_{n+1}=ax_{n-l}+bx_{n-k}+f ( x_{n-l},x_{n-k} ), $$ where l and k are nonnegative integers, a and b are nonnegative real numbers, the initial values $x_{-s}, x_{-s+1},\ldots, x_{0}$ are positive real numbers, $s=\max\{l,k\}$ , and $f (u,v ): ( 0,\infty ) ^{2}\rightarrow ( 0...
A singular biological economic model which considers a prey-predator system with time delay and stage structure is proposed in this paper. The local stability at the equilibrium point and the dynamic behavior of the model are studied. Local stability analysis of the model without time delay reveals that there is a phenomenon of singularity-induced bifurcation due to the economic...
In this paper, we get a new form equivalent integral equation for a class of evolution equations of fractional order with nonlocal conditions on the half-line. With the aid of it, the uniqueness of the mild solution is obtained by the Banach contraction theorem. Also, we present the existence and uniqueness theorem of positive mild solutions by the monotone iterative method...
This paper is concerned with a non-zero sum differential game problem of an anticipated forward-backward stochastic differential delayed equation under partial information. We establish a maximum principle and a verification theorem for the Nash equilibrium point by virtue of the duality and convex variation approach. We study a linear-quadratic system under partial information...
The main purpose of this study is to investigate a fractional discontinuous Sturm-Liouville problem with transmission conditions. We shall consider a fractional boundary value problem involving an operator with two parts. It is shown that the eigenvalues and corresponding eigenfunctions of the main problem coincide with the eigenvalues and corresponding eigenfunctions of the...
A new nonlinear partial differential system called two-mode higher-order Boussinesq-Burgers system is established. We aim to use the simplified bilinear method to find the necessary constraint conditions that guarantee the existence of both regular and singular multiple soliton solutions of the model. To study the correctness of the obtained results, we use the hyperbolic-tangent...
We present a closed-form formula for the general solution to the difference equation $$x_{n+k}-q_{n}x_{n}=f_{n},\quad n\in \mathbb {N}_{0}, $$ where $k\in \mathbb {N}$ , $(q_{n})_{n\in \mathbb {N}_{0}}$ , $(f_{n})_{n\in \mathbb {N}_{0}}\subset \mathbb {C}$ , in the case $q_{n}=q$ , $n\in \mathbb {N}_{0}$ , $q\in \mathbb {C}\setminus\{0\}$ . Using the formula, we show the...
We formulate a stochastic SIS epidemic model with vaccination by introducing a Lévy noise and regime switching into the epidemic model. First, we prove that the stochastic model admits a unique global positive solution. Moreover, we study the asymptotic behavior of the stochastic regime switching SIS model with vaccination driven by Lévy noise.
In this paper, we study a Lotka-Volterra prey-predator system with feedback control. We establish sufficient conditions under which a unique positive equilibrium is globally stable. Further, we show that a suitable feedback control on predator species can make prey species that is on the brink of extinction become globally stable, but under the conditions of small feedback...
We consider adaptive compensation for infinite number of actuator failures in the tracking control of uncertain nonlinear systems. We construct an adaptive controller by combining the common Lyapunov function approach and the structural characteristic of neural networks. The proposed control strategy is feasible under the presupposition that the systems have a nonstrict-feedback...
The square iterative roots for strictly monotonic and upper semicontinuous functions with one set-valued point were fully described in (Li et al. in Publ. Math. (Debr.) 75:203-220, 2009). As a continuation, we study both strictly monotonic and nonmonotonic multifunctions. We present sufficient and necessary conditions under which those multifunctions have nth iterative roots...
A strongly coupled cooperative parabolic system, which describes fecally-orally epidemic model with cross-diffusion in a heterogeneous environment, was formulated and analyzed. The basic reproduction number R 0 D $R_{0} ^{D}$ , which serves as a threshold parameter that predicts whether the coexistence will exist or not, is introduced by the next infection operator and the...
We present a fractional-order model of two-species facultative mutualism with harvesting. We investigate the stability of the equilibrium points of the model by using the linearization method for noncoexistence of equilibrium points and the Lyapunov direct method for the positive coexistence of an equilibrium point. In addition, we obtain sufficient conditions to ensure the local...
In this paper, a reaction-diffusion trimolecular biochemical model with autocatalysis and fractional-order derivative is proposed. We establish the existence and uniqueness of a positive solution to this system in a Besov space. Besides, for this system, we obtain stability, Hopf and Turing bifurcations and spatial patterns. These dynamic behaviors of this system are slightly...
Many numerical methods have been developed for nonlinear fractional integro-differential Volterra’s population model (FVPG). In these methods, to approximate a function on a particular interval, only a restricted number of points have been employed. In this research, we show that it is possible to use the fuzzy transform method (F-transform) to tackle with FVPG. It makes the F...
The square iterative roots for strictly monotonic and upper semicontinuous functions with one set-valued point were fully described in (Li et al. in Publ. Math. (Debr.) 75:203-220, 2009). As a continuation, we study both strictly monotonic and nonmonotonic multifunctions. We present sufficient and necessary conditions under which those multifunctions have nth iterative roots...
We discuss the calculation of discriminating kernel for the discrete-time dynamic game and continuous-time dynamic game (namely differential game) using the viability kernel and reachable set. For the discrete-time dynamic game, we give an approximation of the viability kernel by the maximal reachable set. Then, based on the relationship between viability and discriminating...
In this paper, we consider the nonergodic Ornstein-Uhlenbeck process X 0 = 0 , d X t = θ X t d t + d B t a , b , t ≥ 0 , $$ X_{0}=0, \quad\quad dX_{t}=\theta X_{t} \,dt+dB_{t}^{a,b},\quad t\geq0, $$ driven by the weighted fractional Brownian motion B t a , b $B_{t}^{a,b}$ with parameter a and b. Our goal is to estimate the unknown parameter θ > 0 $\theta>0$ based on the discrete...
We begin by proving a local existence result for a fractional Caputo nonlocal thermistor problem. Then additional existence and continuation theorems are obtained, ensuring global existence of solutions.
We investigate the collective dynamics of multi-quasi-synchronization of coupled fractional-order neural networks with delays. Using the pinning impulsive strategy, we design a novel controller to pin the coupled networks to realize the multi-quasi-synchronization. Based on the comparison principle and mathematical analysis, we derive some novel criteria of the multi-quasi...