This paper studies channel consisting of a manufacturer and two retailers. As a basis for comparison, the first, multiagent Stackelberg model has been structured based on perfect rationality. Further, fairness preference theory will be embedded in marketing channel multiagent Stackelberg model, and the results show that if the retailers have a jealous fairness preference, the manufacturer will reduce the wholesale price, retailers will increase the effort level, product sales will be increased, and the total channel utility and manufacturers’ utility will be pareto improvement, but the pareto improvement of retailers’ utility is associated with the interval of jealousy fairness preference coefficient. If the retailers have a sympathetic fairness preference, the manufacturer increases wholesale price, retailers reduce the effort level, and the total channel utility, manufacturer’s utility, and retailers’ utility are less than that of the no fairness preference utility.

We discuss the existence of positive solutions to a class of fractional boundary value problem with changing sign nonlinearity and advanced arguments $x(\mathrm{0})={\mathrm{x}}^{\mathrm{\prime }}(\mathrm{0})=\mathrm{0},$ $x(\mathrm{1})=\beta x(\eta )+\lambda [x],\mathrm{}\mathrm{}\mathrm{}\mathrm{}\beta >\mathrm{0},\; and\mathrm{}\mathrm{}\mathrm{}\mathrm{}\eta \in (\mathrm{0,1}),$ where $D^{\alpha }$ is the standard Riemann-Liouville derivative, $f:[\mathrm{0},\mathrm{\infty })\to [\mathrm{0},\mathrm{\infty })$ is continuous, $f(\mathrm{0})>\mathrm{0}$,$\mathrm{}\mathrm{}\mathrm{\hspace{0.17em}h\hspace{0.17em}}:[\mathrm{0,1}]\to (-\mathrm{\infty },+\mathrm{\infty })$, and $\mathrm{}\mathrm{}a(t)$ is the advanced argument. Our analysis relies on a nonlinear alternative of Leray-Schauder type. An example is given to illustrate our results.

We consider positive solutions and optimal control problem for a second order impulsive differential equation with mixed monotone terms. Firstly, by using a fixed point theorem of mixed monotone operator, we study positive solutions of the boundary value problem for impulsive differential equations with mixed monotone terms, and sufficient conditions for existence and uniqueness of positive solutions will be established. Also, we study positive solutions of the initial value problem for our system. Moreover, we investigate the control problem of positive solutions to our equations, and then, we prove the existence of an optimal control and its stability. In addition, related examples will be given for illustrations.

In the present paper, we consider an incompressible magnetohydrodynamic flow of two-dimensional upper-convected Maxwell fluid over a porous stretching plate with suction and injection. The nonlinear partial differential equations are reduced to an ordinary differential equation by the similarity transformations and taking into account the boundary layer approximations. This equation is solved approximately by means of the optimal homotopy asymptotic method (OHAM). This approach is highly efficient and it controls the convergence of the approximate solutions. Different approximations to the solution are given, showing the exceptionally good agreement between the analytical and numerical solutions of the nonlinear problem. OHAM is very efficient in practice, ensuring a very rapid convergence of the solutions after only one iteration even though it does not need small or large parameters in the governing equation.

In flotation processes, concentrate grade is the key production index but is difficult to be measured online. The mechanism models reflect the basic tendency of concentrate grade changes but cannot provide adequate prediction precision. The data-driven models based on froth image features provide accurate prediction within well-sampled space but rely heavily on sample data with less generalization capability. So, a hybrid intelligent model combining the two kinds of model is proposed in this paper. Since the information of image features is enormous, and the relationship between image features and concentrate grade is nonlinear, a B-spline partial least squares (BS-PLS) method is adopted to construct the data-driven model for concentrate grade prediction. In order to gain better generalization capability and prediction accuracy, information entropy is introduced to integrate the mechanism model and the BS-PLS model together and modify the model output online through an output deviation compensation strategy. Moreover, a slide window scheme is employed to update the hybrid model in order to improve its adaptability. The industrial practical data testing results show that the performance of the hybrid model is better than either of the two single models and it satisfies the accuracy and stability requirements in industrial applications.

We study the transient flow of a Newtonian fluid in rectangular microchannels taking into account boundary slip. An exact solution is derived by using the separation of variables in space and Fourier series expansion in time. It is found that, for different forms of driving pressure field, the effect of boundary slip on the flow behavior is qualitatively different. If the pressure gradient is constant, the flow rate is almost linearly proportional to the slip parameter $\mathcal{l}$ when $\mathcal{l}$ is large; if the pressure gradient is in a waveform, as the slip parameter $\mathcal{l}$ increases, the amplitude of the flow rate increases until approaching a constant value when $\mathcal{l}$ becomes sufficiently large.

We give the analytical solution and the series expansion solution of a class of singularly perturbed partial differential equation (SPPDE) by combining traditional perturbation method (PM) and reproducing kernel method (RKM). The numerical example is studied to demonstrate the accuracy of the present method. Results obtained by the method indicate the method is simple and effective.

Recent research confirms that slip of a fluid on the solid surface occurs at micrometer scale. Slip on solid surface may cause the change of interior material deformation which consequently leads to the change of velocity profile and stress field. This paper concerns the time periodic electroosmotic flow in a channel with slip boundary driven by an alternating electric field, which arises from the study of particle manipulation and separation such as flow pumping and mixing enhancement. Although exact solutions to various flow problems of electroosmotic flows under the no-slip condition have been obtained, exact solutions for problems under slip boundary conditions have seldom been addressed. In this paper, an exact solution is derived for the time periodic electroosmotic flow in two-dimensional straight channels under slip boundary conditions.

We consider the optimal tracking problem for a divergent-type parabolic PDE system, which can be used to model the spatial-temporal evolution of the magnetic diffusion process in a tokamak plasma. With the feedforward trajectories generated by numerical optimization, we derive a linear system to describe the error evolution. The weak variation approach is then extended to cylindrical coordinates to obtain a Riccati-type PDE for feedback kernel synthesis. Then, a finite difference method is used to give numerical solutions for the Riccati-type PDE. The obtained feedback kernel is finally used to simulate the closed-loop system. The simulation results demonstrate that the proposed tracking method can attenuate the tracking error effectively.

We deal with the existence and uniqueness of positive solutions to a class of nonlinear parabolic partial differential equations, by using some fixed point theorems for mixed monotone operators with perturbation.

The weak Galerkin finite element method is combined with the method of characteristics to treat the convection-diffusion problems on the triangular mesh. The optimal order error estimates in $H^1$ and $L^2$ norms are derived for the corresponding characteristics weak Galerkin finite element procedure. Numerical tests are performed and reported.

A Lotka-Volterra competition model with nonlinear boundary conditions is considered. First, by using upper and lower solutions method for nonlinear boundary problems, we investigate the existence of positive solutions in weak competition case. Next, we prove that $-d_1\mathrm{\Delta }u=u(a_1-b_{1}u-c_{1}v)$, $x\in \mathrm{\Omega }$; $-d_2\mathrm{\Delta }v=v(a_2-b_{2}u-c_{2}v)$, $x\in \mathrm{\Omega }$; $\partial u/\partial \nu +f(u)=0$, $x\in \partial \mathrm{\Omega }$; $\partial v/\partial \nu +g(v)=0$, $x\in \partial \mathrm{\Omega }$, has no positive solution when one of the diffusion coefficients is sufficiently large.

We consider a predator-prey system with Beddington-DeAngelis functional response and delays, in which not only the stage structure on prey but also the delay due to digestion is considered. First, we give a sufficient and necessary condition for the existence of a unique positive equilibrium by analyzing the corresponding locations of a hyperbolic curve and a line. Then, by constructing an appropriate Lyapunov function, we prove that the positive equilibrium is locally asymptotically stable under a sufficient condition. Finally, by using comparison theorem and the $\omega$-limit set theory, we study the global asymptotic stability of the boundary equilibrium and the positive equilibrium, respectively. Also, we obtain a sufficient condition to assure the global asymptotic stability.

We establish the existence of positive solutions to a class of singular nonlocal fractional order differential system depending on two parameters. Our methods are based on Schauder’s fixed point theorem.

We consider a fully discrete $H^1$-Galerkin mixed finite element approximation of one nonlinear integrodifferential model which often arises in mathematical modeling of the process of a magnetic field penetrating into a substance. We adopt the Crank-Nicolson discretization for time derivative. Optimal order a priori error estimates for the unknown function in $L^2$ and $H^1$ norm and its gradient function in $L^2$ norm are presented. A numerical example is given to verify the theoretical results.

We study the existence of positive and monotone solution to the boundary value problem $u^{\prime \prime }(t)+f(t,u(t))=0$, $0⩽\mathrm{t}⩽1$, $u(0)=\xi u(1)$, $u\mathrm{\text{'}}(1)=\eta u\mathrm{\text{'}}(0)$, where $\xi ,\eta \in (\mathrm{0,1})\cup (1,\mathrm{\infty })$. The main tool is the fixed point theorem of cone expansion and compression of functional type by Avery, Henderson, and O’Regan. Finally, four examples are provided to demonstrate the availability of our main results.

Some delta-nabla type maximum principles for second-order dynamic equations on time scales are proved. By using these maximum principles, the uniqueness theorems of the solutions, the approximation theorems of the solutions, the existence theorem, and construction techniques of the lower and upper solutions for second-order linear and nonlinear initial value problems and boundary value problems on time scales are proved, the oscillation of second-order mixed delat-nabla differential equations is discussed and, some maximum principles for second order mixed forward and backward difference dynamic system are proved.

This paper investigates the stabilization problem for impulsive switched systems with time delays. First, exponential stability criteria of the delayed impulsive switched systems are established by use of the Lyapunov-Krasovskii functional method. Based on these results, sufficient conditions for the existence of a guaranteed cost control are also given. Subject to these sufficient conditions, the closed-loop impulsive switched system under the guaranteed cost control law will be exponentially stable with a guaranteed cost value.

We are concerned with the uniqueness of solutions for a class of p-Laplacian fractional order nonlinear systems with nonlocal boundary conditions. Based on some properties of the p-Laplacian operator, the criterion of uniqueness for solutions is established.

The approach of Kruzkov’s device of doubling the variables is applied to establish the local stability of strong solutions for a nonlinear partial differential equation in the space $L^1(R)$ by assuming that the initial value only lies in the space $L^1(R)\cap L^{\infty }(R)$.

We consider a class of coupled systems with damping terms. By using multiplier method and the estimation techniques of the energy, we show that even if the kernel function is nonincreasing and integrable without additional conditions, the energy of the system decays also to zero in a good rate.

We study a nonlinear elliptic problem defined in a bounded domain involving biharmonic operator together with an asymptotically linear term. We establish at least three nontrivial solutions using the topological degree theory and the critical groups.

We study the existence of positive solution for the eigenvalue problem of semipositone fractional order differential equation with multipoint boundary conditions by using known Krasnosel'skii's fixed point theorem. Some sufficient conditions that guarantee the existence of at least one positive solution for eigenvalues $\lambda >0$ sufficiently small and $\lambda >0$ sufficiently large are established.

The well-posedness of global strong solutions for a nonlinear partial differential equation including the Novikov equation is established provided that its initial value $v_{\mathrm{0}}(x)$ satisfies a sign condition and $v_{\mathrm{0}}(x)\in H^s(R)$ with $s>\mathrm{3}/\mathrm{2}$. If the initial value $v_{\mathrm{0}}(x)\in H^s(R)(\mathrm{1}\le s\le \mathrm{3}/\mathrm{2})$ and the mean function of $(\mathrm{1}-{\partial }_x^{\mathrm{2}})v_{\mathrm{0}}(x)$ satisfies the sign condition, it is proved that there exists at least one global weak solution to the equation in the space $v(t,x)\in L^{\mathrm{2}}([\mathrm{0},+\mathrm{\infty }),H^s(R))$ in the sense of distribution and $v_x\in L^{\mathrm{\infty }}([\mathrm{0},+\mathrm{\infty })\times R)$.

We discuss the existence and uniqueness of positive solutions for the following fractional switched system: (${Dc}_{0+}^{\alpha }u(t)+f_{\sigma (t)}(t,u(t))+{\mathrm{g}}_{\sigma (t)}(t,u(t))=\mathrm{0},\mathrm{}\mathrm{t}\in J=[\mathrm{0,1}]$); $\mathrm{(u}(\mathrm{0})=u^{\prime \prime }(\mathrm{0})=\mathrm{0},u(\mathrm{1})={\int }_0^{1}u(s)\hspace{0.17em}d\mathrm{s)}$, where ${Dc}_{0+}^{\alpha }$ is the Caputo fractional derivative with , $\sigma (t):J\to \{\mathrm{1,2},\dots ,N\}$ is a piecewise constant function depending on $t$, and ${ℝ}^{+}=[0,+\mathrm{\infty })$,$f_i,g_i\in C[J\times {ℝ}^{+},{ℝ}^{+}$], $i=\mathrm{1,2},\dots ,N$. Our results are based on a fixed point theorem of a sum operator and contraction mapping principle. Furthermore, two examples are also given to illustrate the results.

We study the boundary stabilization of a semilinear wave equation with variable coefficients under the time-varying and nonlinear feedback. By the Riemannian geometry methods, we obtain the stability results of the system under suitable assumptions of the bound of the time-varying term and the nonlinearity of the nonlinear term.

A four-point coupled boundary value problem of fractional differential equations is studied. Based on Mawhin’s coincidence degree theory, some existence theorems are obtained in the case of resonance.

By constructing some general type conditions and using fixed point theorem of cone, this paper investigates the existence of at least one and at least two positive solutions for systems of nonlinear higher order differential equations with integral boundary conditions. As application, some examples are given.

We present a two-grid finite element scheme for the approximation of a second-order nonlinear hyperbolic equation in two space dimensions. In the two-grid scheme, the full nonlinear problem is solved only on a coarse grid of size $H$. The nonlinearities are expanded about the coarse grid solution on the fine gird of size $h$. The resulting linear system is solved on the fine grid. Some a priori error estimates are derived with the $H^1$-norm $O(h+H^2)$ for the two-grid finite element method. Compared with the standard finite element method, the two-grid method achieves asymptotically same order as long as the mesh sizes satisfy $h=O(H^2)$.

We study the pricing of American options in an incomplete market in which the dynamics of the underlying risky asset is driven by a jump diffusion process with stochastic volatility. By employing a risk-minimization criterion, we obtain the Radon-Nikodym derivative for the minimal martingale measure and consequently a linear complementarity problem (LCP) for American option price. An iterative method is then established to solve the LCP problem for American put option price. Our numerical results show that the model and numerical scheme are robust in capturing the feature of incomplete finance market, particularly the influence of market volatility on the price of American options.

Now dealing with power system became the most important arts. In this paper we report the dynamic analysis of a fractional-order power system with parameter $Q_1$. To the best know of our knowledge, that was the first report about bifurcation analysis of the fractional order power system. So first we discuss the dynamic analysis with different fractional order and different parameters. Furthermore we will establish its numerical simulations which are provided to demonstrate the feasibility and efficacy of our analysis.

We consider the two-sided first-exit problem for a jump process having jumps with rational Laplace transform. We derive the joint distribution of the first passage time to two-sided barriers and the value of process at the first passage time. As applications, we present explicit expressions of the dividend formulae for barrier strategy and threshold strategy.