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...

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.

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 , n ∈ N 0 , $$x_{n+k}-q_{n}x_{n}=f_{n},\quad n\in \mathbb {N}_{0}, $$ where k ∈ N $k\in \mathbb {N}$ , ( q n ) n ∈ N 0 $(q_{n})_{n\in \mathbb {N}_{0}}$ , ( f n ) n ∈ N 0 ⊂ C $(f_{n})_{n\in \mathbb {N}_{0}}\subset \mathbb {C}$ , in the case q n = q $q_{n}=q$ , n ∈ N 0 $n\in...

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...

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.

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 , n ∈ N 0 , $$x_{n+k}-q_{n}x_{n}=f_{n},\quad n\in \mathbb {N}_{0}, $$ where k ∈ N $k\in \mathbb {N}$ , ( q n ) n ∈ N 0 $(q_{n})_{n\in \mathbb {N}_{0}}$ , ( f n ) n ∈ N 0 ⊂ C $(f_{n})_{n\in \mathbb {N}_{0}}\subset \mathbb {C}$ , in the case q n = q $q_{n}=q$ , n ∈ N 0 $n\in...

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...

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.

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 , n ∈ N 0 , $$x_{n+k}-q_{n}x_{n}=f_{n},\quad n\in \mathbb {N}_{0}, $$ where k ∈ N $k\in \mathbb {N}$ , ( q n ) n ∈ N 0 $(q_{n})_{n\in \mathbb {N}_{0}}$ , ( f n ) n ∈ N 0 ⊂ C $(f_{n})_{n\in \mathbb {N}_{0}}\subset \mathbb {C}$ , in the case q n = q $q_{n}=q$ , n ∈ N 0 $n\in...

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...

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...

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...

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...