This paper is concerned with the blow-up solutions of the critical Gross-Pitaevskii equation, which models the Bose-Einstein condensate. The existence and qualitative properties of the minimal blow-up solutions are obtained.

The aim of this paper is to extend the work of Sun et al. (2012) to a more general case for a wider range of function classes of $f$ and $g$. Our results include the case of the previous work, which are essential improvement of the work of Sun et al. (2012), especially.

We study the boundary value problems of second-order singular differential equations. At first, we reduce the BVPs to initial value problems of first-order singular integrodifferential equations and then we employ the quasilinearization method in studying the IVPs and obtain two monotone iterative sequences, which converge uniformly and quadratically to the unique solution of the IVPs. Finally, we get the similar result for the given BVPs.

We study the equity premium and option pricing under jump-diffusion model with stochastic volatility based on the model in Zhang et al. 2012. We obtain the pricing kernel which acts like the physical and risk-neutral densities and the moments in the economy. Moreover, the exact expression of option valuation is derived by the Fourier transformation method. We also discuss the relationship of central moments between the physical measure and the risk-neutral measure. Our numerical results show that our model is more realistic than the previous model.

Using the Fountain theorem and a version of the Local Linking theorem, we obtain some existence and multiplicity results for a class of fourth-order elliptic equations.

We study the slip flow of fluids driven by the combined effect of electrical force and pressure gradient. The underlying boundary value problem is solved through the use of Fourier series expansion in time and Bessel function in space. The exact solutions and numerical investigations show that the slip length and electrical field parameters have significant effects on the velocity profile. By varying these system parameters, one can achieve smooth velocity profiles or wave form profiles with different wave amplitude and frequency. This opens the way for optimizing the flow by choosing the slip length, the electrical field, and electrolyte solutions.

The Cauchy problem for the Boussinesq equation in multidimensions is investigated. We prove the asymptotic behavior of the global solutions provided that the initial data are suitably small. Moreover, our global solutions can be approximated by the solutions to the corresponding linear equation as time tends to infinity when the dimension of space $n\ge \mathrm{3}$.

This paper provides sufficient conditions for the existence and uniqueness of positive solutions to a singular differential system with integral boundary value. The emphasis here is that the boundary conditions are coupled and this is where the main novelty of this work lies. By mixed monotone method, the existence and uniqueness results of the problem are established. An example is given to demonstrate the main results.

This paper is concerned with the boundedness and attractiveness of nonlinear switched delay systems whose subsystems have different equilibria. Some sufficient conditions which can guarantee the system’s boundedness are obtained. In addition, we work out the region where the solution will remain and furthermore the relationship between the initial function and the bounded region. Based on the new concept of attractor with switching laws, we show that the nonlinear switched delay system is attractive and then obtain the attractive region.

This work is concerned with a mixed boundary value problem for the semilinear parabolic equation with a memory term and generalized Lewis functions which depends on both spacial variable and time. Under suitable conditions, we prove the existence and uniqueness of global solutions and the energy functional decaying exponentially or polynomially to zero as the time goes to infinity by introducing brief Lyapunov function and precise priori estimates.

A rigorous mathematical characterization for early-stage spatial and temporal patterns formation in a Leslie-Gower predator-prey model with cross diffusion is investigated. Given any general perturbation near an unstable constant equilibrium, we prove that its nonlinear evolution is dominated by the corresponding linear dynamics along a fixed finite number of the fastest growing modes.

We obtain sufficient conditions for the existence of unbounded solutions of the following nonlinear differential equation ${({\phi }_p(x^{\prime }))}^{\prime }+(p-1)[\alpha {\phi }_p(x^{+})-\beta {\phi }_p(x^{-})]=(p-1)f(t,x,x^{\prime })$, where ${\phi }_p(u)={|u|}^{p-2}u,p>1,x^{+}=\text{max}\{x,0\},x^{-}=\text{max}\{-x,0\},\alpha ,\beta$ are positive constants, and $f$ is continuous, bounded, and $T$-periodic in $t$ for some $T>0$.

It is intended to present some extensions of the famous Altman and Rothe types fixed-point theorems. The inequality conditions are relaxed to the $\alpha$-positive-homogeneous operator. Some new fixed-point theorems are obtained with the help of the theory of topological degree.

A nonlinear Schrödinger equation with a higher-order dispersive term describing the propagation of ultrashort femtosecond pulses in optical fibres is considered and is transformed into a second-order nonlinear ordinary differential equation. We investigate the exact travelling wave solutions of the nonlinear Schrödinger equation using three methods, namely, the auxiliary equation method, the first integral method, and the direct integral method. As a result, Jacobi elliptic function solution, hyperbolic function solution, trigonometric function solution, and rational solution with parameters are obtained successfully. When the parameters are taken as special values, the two known solitary wave solution and periodic wave solution are derived from the solutions obtained. The aim of the paper is to compare the efficiency of the three methods.

By using the cone theory and the Banach contraction mapping principle, the existence and uniqueness results are established for nonlinear higher-order differential equation boundary value problems with sign-changing Green’s function. The theorems obtained are very general and complement previous known results.

We obtain some new common fixed point theorems satisfying a weak contractive condition in the framework of partially ordered metric spaces. The main result generalizes and extends some known results given by some authors in the literature.

The long time behavior of solutions of the nonautonomous three-components reversible Gray-Scott system defined on the entire space ${ℝ}^n$ is studied when the external forcing terms are unbounded in a phase space. The existence of a pullback global attractor for the equation is established in ${\left[L^2\left({ℝ}^n\right)\right]}^3$ and ${\left[H^1\left({ℝ}^n\right)\right]}^3$, respectively. The pullback asymptotic compactness of solutions is proved by using uniform estimates on the tails of solutions on unbounded domains.

We study the following Schrödinger-Poisson system: $-\mathrm{\Delta }u+V(x)u+\varphi u=|u{|}^{p-\mathrm{1}}u$, $-\mathrm{\Delta }\varphi =u^{\mathrm{2}}$, ${\lim }_{|x|\to +\infty }\varphi (x)=\mathrm{0}$, where $u,\varphi :{ℝ}^{\mathrm{3}}\to ℝ$ are positive radial functions, $p\in (\mathrm{1},+\mathrm{\infty })$, $x=(x_{\mathrm{1}},x_{\mathrm{2}},x_{\mathrm{3}})\in {ℝ}^{\mathrm{3}}$, and $V(x)$ is allowed to take two different forms including $V(x)=\mathrm{1}/(x_{\mathrm{1}}^{\mathrm{2}}+x_{\mathrm{2}}^{\mathrm{2}}+x_{\mathrm{3}}^{\mathrm{2}}{)}^{\alpha /\mathrm{2}}$ and $V(x)=\mathrm{1}/(x_{\mathrm{1}}^{\mathrm{2}}+x_{\mathrm{2}}^{\mathrm{2}}{)}^{\alpha /\mathrm{2}}$ with $\alpha >\mathrm{0}$. Two theorems for nonexistence of nontrivial solutions are established, giving two regions on the $\alpha -p$ plane where the system has no nontrivial solutions.

We study the existence and uniqueness of the positive solution for the fractional differential system involving the Riemann-Stieltjes integral boundary conditions $-{\mathcal{D}}_t^{\alpha }x(t)=f(t,y(t))$, $-{\mathcal{D}}_t^{\beta }y(t)=g(t,x(t))$, $t\in (\mathrm{0,1})$, $x(\mathrm{0})=y(\mathrm{0})=\mathrm{0}$, $x(\mathrm{1})={\int }_{\mathrm{0}}^{\mathrm{1}}‍x(s)dA(s)$, and $y(\mathrm{1})={\int }_{\mathrm{0}}^{\mathrm{1}}‍y(s)dB(s)$, where , $\beta \le \mathrm{2}$, and ${\mathcal{D}}_t^{\alpha }$ and ${\mathcal{D}}_t^{\beta }$ are the standard Riemann-Liouville derivatives, $A$ and $B$ are functions of bounded variation, and ${\int }_{\mathrm{0}}^{\mathrm{1}}\mathrm{‍}{\mathcal{D}}_t^{\beta }x(s)dA(s)$ and ${\int }_{\mathrm{0}}^{\mathrm{1}}\mathrm{‍}{\mathcal{D}}_t^{\beta }y(s)dB(s)$ denote the Riemann-Stieltjes integral. Our results are based on a generalized fixed point theorem for weakly contractive mappings in partially ordered sets.

We investigate a compound Poisson infinite factor diffusion model which describes the relationship between the infinite-dimension random risk resource and the corresponding stochastic process. We derive the no-arbitrage condition on the drift of instantaneous forward rates in the compound model and study the impact of random jump on the price of the zero-coupon bond.

This paper is concerned with the existence and uniqueness of positive solutions for a Volterra nonlinear fractional system of integral equations. Our analysis relies on a fixed point theorem of a sum operator. The conditions for the existence and uniqueness of a positive solution to the system are established. Moreover, an iterative scheme is constructed for approximating the solution. The case of quadratic system of fractional integral equations is also considered.

A mathematical technique based on an auxiliary equation and the symbolic computation system Matlab is employed to investigate a generalized KdV-mKdV equation which possesses high-order nonlinear terms. Some new solutions including the Jacobi elliptic function solutions, the degenerated soliton-like solutions, and the triangle function solutions to the equation are obtained.

The ($G^{\prime }/G$)-expansion method and the symbolic computation system Mathematica are employed to investigate the coupled Schrödinger-Boussinesq equations. The hyperbolic function solutions, trigonometric function solutions, and rational function solutions to the equations are obtained. The decaying properties of several solutions are analyzed.

In this paper, by using a fixed point theorem, we investigate the existence of a positive solution to the singular fractional boundary value problem ${D_{C}}_{0+}^{\alpha }u+f(t,u,{D_{C}}_{0+}^{\nu }u,{D_{C}}_{0+}^{\mu }u)+g(t,u,{D_{C}}_{0+}^{\nu }u,{D_{C}}_{0+}^{\mu }u)=0$, $u(0)={u}^{\prime }(0)={u}^{\prime \prime }(0)={u}^{\prime \prime \prime }(0)=0$, where $3<\alpha <4$, $0<\nu <1$, $1<\mu <2$, ${D_{C}}_{0+}^{\alpha }$ is Caputo fractional derivative, $f(t,x,y,z)$ is singular at the value 0 of its arguments $x,y,z$, and $g(t,x,y,z)$ satisfies the Lipschitz condition.

For a nonlinear generalization of the Camassa-Holm equation, we investigate the dynamic properties of solutions for the equation under the assumption that the initial value $u_0(x)$ lies in the space $H^1(R)$. A one-sided upper bound estimate on the first-order spatial derivative, $L^p$ bound estimate, and a space-time higher-norm estimate for the solutions are obtained.

The existence and uniqueness of local strong solutions for a nonlinear equation are investigated in the Sobolev space $C([\mathrm{0},T);H^s(R))\hspace{0.17em}\cap C^{\mathrm{1}}([\mathrm{0},T);H^{s-\mathrm{1}}(R))$ provided that the initial value lies in $H^s(R)$ with $s>\mathrm{3}/\mathrm{2}$. Meanwhile, we prove the existence of global weak solutions in $L^{\mathrm{\infty }}([\mathrm{0},\mathrm{\infty });L^{\mathrm{2}}(R))$ for the equation.