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In this article, we take a time-space fractional convection-diffusion problem with a nonlinear reaction term on a finite domain. We use the L1 operator to discretize the Caputo fractional derivative and the weighted shifted Grünwald difference(WSGD) method to approximate the Riesz fractional derivative. Furthermore, we apply the Crank Nicolson difference scheme with weighted shifted Grünwald-Letnikov and obtain that the numerical method is unconditionally stable and convergent with the accuracy of O(τ2-α + h2), where α ∈ (0,1]. For finding the numerical solution of the nonlinear system of equation, we apply the fixed iteration method. At the end, numerical simulations are treated to verify the effectiveness and consistency of the proposed method.

In this paper, we study the one dimensional Schrödinger equation with Liouville–Weyl fractional derivatives $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} {}_tD_\infty ^\alpha ({}_{-\infty }D_t^\alpha u(t))+V(t)u(t)=f(t,u(t)), \ \ \ \ t\in {\mathbb {R}},\\ u \in {H^\alpha }(\mathbb {R}), \end{array} \right. \end{aligned}$$ tD∞α(-∞Dtαu(t))+V(t)u(t)=f(t,u(t)),t∈R,u∈Hα(R), For the case when f(t, u) is superlinear at infinity, by applying Ekeland Variational Principle and Mountain Pass Theorem, we obtain the existence of at least two nontrivial solutions. For the case of asymptotically linear at infinity, we prove that the above problem has one nontrivial solution under certain assumptions. Some recent results are extended and improved.

A nonlocal delayed reaction-diffusive two-species model with Dirichlet boundary condition and general functional response is investigated in this paper. Based on the Lyapunov–Schmidt reduction, the existence, bifurcation direction and stability of Hopf bifurcating periodic orbits near the positive spatially nonhomogeneous steady-state solution are obtained, where the time delay is taken as the bifurcation parameter. Moreover, the general results are applied to a diffusive Lotka–Volterra type food-limited population model with nonlocal delay effect, and it is found that diffusion and nonlocal delay can also affect the other dynamic behavior of the system by numerical experiments. [ABSTRACT FROM AUTHOR]

In this note, we point out that Theorems 5.1 and 6.3 in Biswas et al. (2015) are not true. We present a new result and discuss the existence and local stability of the equilibria for the delay model. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a class of impulsive differential equations with Dirichlet boundary conditions are considered. Multiplicity results are obtained by critical point theory. Recent results in the literature are generalized and significantly improved. [Copyright &y& Elsevier]

In this paper, by using the theory of abstract continuous theorem of k-set contractive operator and some analysis techniques, we obtain some sufficient conditions for the existence of positive periodic solutions for a neutral population model.

A class of nonlinear delay difference equations are considered and some sufficient conditions for global attractivity of solutions of the equation are obtained. It is shown that the stability properties, both local and global, of the equilibrium of the delay equation can be derived from those of an associated nondelay equation.

In this paper, we study the asymptotic behavior of solutions for a class of nonlinear difference equations and give the results about the solutions of the equations convergent to a constant. Our results generalize all conclusions obtained in [1].

In this paper, we study the nonlinear coupled system of equations with fractional integral boundary conditions involving the Caputo fractional derivative of orders θ1 and θ2 and Riemann–Liouville derivative of orders ϱ1 and ϱ2 with the p-Laplacian operator, where n−1<θ1,θ2,ϱ1,ϱ2≤n, and n≥3. With the help of two Green’s functions Gϱ1w,ℑ,Gϱ2w,ℑ, the considered coupled system is changed to an integral system. Since topological degree theory is more applicable in nonlinear dynamical problems, the existence and uniqueness of the suggested coupled system are treated using this technique, and we find appropriate conditions for positive solutions to the proposed problem. Moreover, necessary conditions are highlighted for the Hyer–Ulam stability of the solution for the specified fractional differential problems. To confirm the theoretical analysis, we provide an example at the end.

The major concern of this article is to propose the notion of picture fuzzy rough sets (PFRSs) over two different universes which depend on δ,ζ,ϑ-cut of picture fuzzy relation ℛ on two different universes (i.e., by combining picture fuzzy sets (PFSs) with rough sets (RSs)). Then, we discuss several interesting properties and related results on the PFRSs. Furthermore, we define some notions related to PFRSs such as (Type-I/Type-II) graded PFRSs, the degree α and β with respect to ℛδ,ζ,ϑ on PFRSs, and (Type-I/Type-II) generalized PFRSs based on the degree α and β with respect to ℛδ,ζ,ϑ and investigate the basic properties of above notions. Finally, an approach based on the rough picture fuzzy approximation operators on two different universes in decision-making problem is introduced, and we give an example to show the validity of this approach.

A class of nonlinear difference systems is considered in this paper. By exploring the relationship between this system and a correspondent first-order difference system, some permanence results are obtained.