Search

In this paper, we shall solve a time-fractional nonlinear Schrödinger equation by using the quintic non-polynomial spline and the L1 formula. The unconditional stability, unique solvability and convergence of our numerical scheme are proved by the Fourier method. It is shown that our method is sixth order accurate in the spatial dimension and $(2-\gamma )$ ( 2 − γ ) th order accurate in the temporal dimension, where γ is the fractional order. The efficiency of the proposed numerical scheme is further illustrated by numerical experiments, meanwhile the simulation results indicate better performance over previous work in the literature.

Global asymptotical stability of the positive equilibrium (PE) of a dynamical system is one of the research focus in theoretical studies of both continuous and discrete bio-mathematical models. In this paper, we shall establish the global asymptotical stability of the PE of a discrete Logistic competitive model in certain planar region. Indeed, sufficient conditions, dependent only on the parameters of the model, are obtained to ensure the global asymptotical stability of the PE in this region. The parameter region that corresponds to these sufficient conditions can be illustrated graphically and several examples of such regions are presented. Our approach to establish the global asymptotical stability of the PE involves proving the global attractivity of the PE in the planar region concerned and a key process here is the derivation of the maxima of the related functions in the planar region.

In this paper, a mid-knot cubic non-polynomial spline is applied to obtain the numerical solution of a system of second-order boundary value problems. The numerical method is proved to be uniquely solvable and it is of second-order accuracy. We further include three examples to illustrate the accuracy of our method and to compare with other methods in the literature. Published version

We study two discrete predator-prey models in patchy environment, one without dispersal corridors and one with dispersal corridors. Dispersal corridors are passes that allow the migration of species from one patch to another and their existence may influence the permanence of the model. We will offer sufficient conditions to guarantee the permanence of the two predator-prey models. By comparing the two permanence criteria, we discuss the effects of dispersal corridors on the permanence of the predator-prey model. It is found that the dispersion of the prey from one patch to another is helpful to the permanence of the prey if the population growth of the prey is density dependent; however, this dispersion of the prey could be disadvantageous or advantageous to the permanence of the predator. Five numerical examples are presented to confirm the theoretical results obtained and to illustrate the effects of dispersal corridors on the permanence of the predator-prey model. Published version

We are concerned with the existence of homoclinic solutions for the following second order nonautonomous singular Hamiltonian systemsu¨+atWuu=0, (HS) where-∞

This paper presents a linearization theorem for the impulsive differential equations when the linear system has ordinary dichotomy. We prove that when the linear impulsive system has ordinary dichotomy, the nonlinear systemx˙(t)=A(t)x(t)+f(t,x),t≠tk,Δx(tk)=A~(tk)x(tk)+f~(tk,x),k∈ℤ, is topologically conjugated tox˙(t)=A(t)x(t),t≠tk,Δx(tk)=A~(tk)x(tk),k∈ℤ, whereΔx(tk)=x(tk+)-x(tk-),x(tk-)=x(tk), represents the jump of the solutionx(t)att=tk. Finally, two examples are given to show the feasibility of our results.

Based on Mawhin's coincidence degree theory, sufficient conditions are obtained for the existence of at leasttwopositive periodic solutions for a plant-hare model with toxin-determined functional response (nonmonotone). Some new technique is used in this paper, because standard arguments in the literature are not applicable.

This paper considers the topological classification of non-autonomous dynamic equations on time scales. In this paper we show, by a counterexample, that the trivial solutions of two topologically conjugated systems may not have the same uniform stability. This is contrary to the expectation that two topologically conjugated systems should have the same topological structure and asymptotic behaviors. To counter this mismatch in expectation, we propose a new definition of strong topological conjugacy that guarantees the same topological structure, and in particular the same uniform stability, for the corresponding solutions of two strongly topologically conjugated systems. Based on the new definition, a new version of the generalized Hartman–Grobman theorem is developed. We also include some examples to illustrate the feasibility and effectiveness of the new generalized Hartman–Grobman theorem.

In this paper, we first prove some perturbation results on the ranges of maximal monotone operators, one of which is then used to show that the non-linear elltic equation involving the generalized -Laplacian operator with Neumann boundary conditions has a unique solution in . This unique solution is shown to be the zero point of a suitably defined non-linear m-accretive mapping. Finally, two kinds of iterative sequences are constructed and proved to converge strongly and weakly to the unique solution, respectively. Some new techniques of constructing appropriate operators and decomposing the equations are employed, which extend and complement some of the previous work.

In this paper, we establish two multi-dimensional discrete Halanay-type inequalities. As an application, we consider a malaria transmission model with two delays and employ the two discrete Halanay-type inequalities to establish the global (exponential) asymptotical stability of the disease free equilibrium of the model. It is further shown that the disease free equilibrium of the delayed model is globally asymptotically stable when the basic reproduction number of the model is less than 1. Published version

In this paper we initiate the oscillation theory for fractional differential equations. Oscillation criteria are obtained for a class of nonlinear fractional differential equations of the form $$D_a^q x + f_1 (t,x) = v(t) + f_2 (t,x),\mathop {\lim }\limits_{t \to a} J_a^{1 - q} x(t) = b_1 $$ , where Daq denotes the Riemann-Liouville differential operator of order q, 0 < q ≤ 1. The results are also stated when the Riemann-Liouville differential operator is replaced by Caputo’s differential operator.

By using some results of pseudo-monotone operator, we discuss the existence and uniqueness of solution for a non-linear Dirichlet boundary value problem involving the p-Laplacian operator. An iterative scheme that strongly converges to the solution will also be proposed.

We consider the following system of Urysohn integral equations By using fixed point theorems and fixed point index theory, criteria are established for the existence of single, double and multiple solutions of the system that are of constant signs, i.e., for each 1 ≤ i ≤ n, u i (t) ≥ 0 or u i (t) ≤ 0 for t ∈ [0,1]. We also consider the eigenvalue problem where criteria are derived such that for λ in some interval, the existence of a constant-sign solution is guaranteed.

We consider the following differential equation u ( n ) ( t ) = F ( t , u ( t ) ) , t ∈ [ 0 , 1 ] , together with the three-point focal type boundary conditions u ( j ) ( 0 ) = 0 , 0 ≤ j ≤ n − 3 , u ( n − 2 ) ( p ) = 0 , u ( n − 1 ) ( 1 ) = 0 , where 1 2 p 1 . By using two different fixed-point theorems, we offer criteria for the existence of three positive solutions of this problem. Examples are also included to illustrate the results obtained.

We consider the following system of difference equations where T ≥ N>0. By using fixed point theorems and fixed point index theory, criteria are developed to guarantee the existence of single, double and multiple solutions of the system that are of constant signs, i.e. for each 1 ≤ i ≤ n, u i (k) ≥ 0 or u i (k) ≤ 0 for k ∈ {0,1,…,T}. We also consider the eigenvalue problem and establish criteria for the existence of a constant-sign solution when λ is in some interval.

Some new oscillation theorems for higher-order nonlinear functional differential equations of the form >0, are established.

We consider the following model that describes the dynamics of epidemics in homogeneous/heterogeneous populations as well as the spreading of multiple inter-related infectious diseases: u i ( k ) = ∑ l = k - τ i k - 1 g i ( k , l ) f i ( l , u 1 ( l ) , u 2 ( l ) , … , u n ( l ) ) , k ∈ Z , 1 ⩽ i ⩽ n . Our aim is to establish criteria such that the above system has one or multiple constant-sign periodic solutions ( u 1 , u 2 , … , u n ) , i.e., for each 1 ⩽ i ⩽ n , u i is periodic and θ i u i ⩾ 0 where θ i ∈ { 1 ,- 1 } is fixed. Examples are also included to illustrate the results obtained.

There are two contributions in this paper. The first is that the abstract result for the existence of the unique solution of certain nonlinear parabolic equation is obtained by using the properties of H-monotone operators, consequently, the proof is simplified compared to the corresponding discussions in the literature. The second is that the connections between resolvent of H-monotone operators and solutions of nonlinear parabolic equations are shown, and this strengthens the importance of H-monotone operators, which have already attracted the attention of mathematicians because of the connections with practical problems. Published version

We consider the following system of generalized right focal boundary value problems u i ‴ ( t ) = f i ( t , u 1 ( ϕ 1 ( t ) ) , u 2 ( ϕ 2 ( t ) ) , … , u n ( ϕ n ( t ) ) ) , t ∈ [ a , b ] , u i ( a ) = u i ′ ( t ∗ ) = 0 , ξ u i ( b ) + δ u i ″ ( b ) = 0 , 1 ⩽ i ⩽ n , where 1 2 ( a + b ) t ∗ b , ξ ⩾ 0 , δ > 0 and ϕ i , 1 ⩽ i ⩽ n are deviating arguments. By using different fixed point theorems, we develop several criteria for the existence of three solutions of the system which are of fixed sign on the interval [ a , b ] , i.e., for each 1 ⩽ i ⩽ n , θ i u i ( t ) ⩾ 0 for all t ∈ [ a , b ] and fixed θ i ∈ { 1 , − 1 } . Examples are also included to illustrate the results obtained.

We consider the following system of third-order three-point generalized right focal boundary value problems: u i ‴ ( t ) = f i ( t , u 1 ( t ) , u 2 ( t ) , … , u n ( t ) ) , t ∈ [ a , b ] , u i ( a ) = u i ′ ( t i ) = 0 , γ i u i ( b ) + δ i u i ″ ( b ) = 0 , where i = 1 , 2 , … , n , γ i ⩾ 0 and δ i > 0 . Using a nonlinear alternative of Leray–Schauder type and Krasnosel'skii's fixed point theorem, we establish the existence of one or more constant-sign solutions for the system.

We consider the following system of difference equations where μ>0, T≥N>m≥0, the function f may take negative values and f( \cdot ,u_{1},u_{2}, \dddot ,u_{n}) may be singular at u_{j} = 0, j \in \{ 1,2, \dddot ,n \} . Our aim is to establish criteria such that the above system has a constant-sign solution. To illustrate the generality of the results obtained, application to a well known boundary value problem is included. We also extend the above problem to that on {\rm \shade N = \{ 0,1, \dddot \} }

We consider the following system of differential equations u i ( m i ) ( t ) + P i ( t , u 1 ( t ) , u 2 ( t ) , … , u n ( t ) ) = 0 , t ∈ [ 0 , 1 ] , 1 ⩽ i ⩽ n together with Sturm–Liouville boundary conditions u i ( j ) ( 0 ) = 0 , 0 ⩽ j ⩽ m i − 3 , ξ u i ( m i − 2 ) ( 0 ) − η u i ( m i − 1 ) ( 0 ) = 0 , ω u i ( m i − 2 ) ( 1 ) + δ u i ( m i − 1 ) ( 1 ) = 0 , 1 ⩽ i ⩽ n , where m i ⩾ 2 for each 1 ⩽ i ⩽ n , η ⩾ 0 , δ ⩾ 0 , η + ξ > 0 , δ + ω > 0 , and ξ ω + ξ δ + η ω > 0 . By using two different fixed point theorems, we offer criteria for the existence of three solutions of the system which are of fixed signs on the interval [ 0 , 1 ] . Examples are also included to illustrate the results obtained.

We consider the discrete system Δmui(k)=Pi(k,u1(k),u2(k),…,un(k)),k∈{0,1,…,N},1⩽i⩽n, together with Hermite boundary conditions, Δjui(kν)=0,j=0,…,mν−1,ν=1,…,r,1⩽i⩽n, where r⩾2, mν⩾1 for ν=1,…,r, ∑ν=1rmν=m, and kν's are integers such that 0=k1

The modelling of the spread of infectious disease is discussed for time t in the real (R), discrete (Z) and time scale (T) domains. We shall offer criteria for the existence of a nontrivial and nonnegative periodic solution for the model in all the three domains. These criteria can be implemented numerically and an algorithm is given.

We study the existence and uniqueness of positive solutions for a class of singularm-point boundary value problems of second order differential equations on a measure chain. A sharper sufficient condition for the existence and uniqueness ofCrd⁡1[0,T]positive solutions as well asCrd⁡1[0,T]positive solutions is obtained by the technique of lower and upper solutions and the maximal principle theorem.

This paper gives a version of Hartman-Grobman theorem for the impulsive differential equations. We assume that the linear impulsive system has a nonuniform exponential dichotomy. Under some suitable conditions, we proved that the nonlinear impulsive system is topologically conjugated to its linear system. Indeed, we do construct the topologically equivalent function (the transformation). Moreover, the method to prove the topological conjugacy is quite different from those in previous works (e.g., see Barreira and Valls, 2006). Published version

Nonuniform exponential dichotomy has been investigated extensively. The essential condition of these previous results is based on the assumption that the nonlinear term satisfies|f(t,x)|≤μe−ε|t|. However, this condition is very restricted. There are few functions satisfying|f(t,x)|≤μe−ε|t|. In some sense, this assumption is not reasonable enough. More suitable assumption should be|f(t,x)|≤μ. To the best of the authors' knowledge, there is no paper considering the existence and uniqueness of solution to the perturbed nonautonomous system with a relatively conservative assumption|f(t,x)|≤μ. In this paper, we prove that if the nonlinear term is bounded, the perturbed nonautonomous system with nonuniform exponential dichotomy has a unique solution. The technique employed to prove Theorem 4 is the highlight of this paper.

The main purpose of this paper is to study the periodicity and global asymptotic stability of a generalized Lotka-Volterra’s competition system with delays. Some sufficient conditions are established for the existence and stability of periodic solution of such nonlinear differential equations. The approaches are based on Mawhin’s coincidence degree theory, matrix spectral theory, and Lyapunov functional. Published version

In this paper we offer the best possible error estimates for discrete Abel-Gontscharoff interpolation. As an application, these error bounds are used to provide tests for the right disfocality as well as disconjugacy for higher order difference equations.

We consider the following boundary value problem: , where is fixed. With the aid of fixed point theorems for operators on a cone, existence criteria are developed for multiple (at least three) positive solutions of the boundary value problem. As as application, we establish the existence of radial solutions of certain partial difference equation. Several examples are also included to dwell upon the importance of the results obtained.

We consider the following boundary value problem:[formula][formula]where 1 ≤ p ≤ n − 1 is fixed. Using a fixed point theorem for operators on a cone, we develop criteria for the existence of two positive solutions of the boundary value problem. In addition, upper and lower bounds for these positive solutions are established for special cases. We also include several examples to dwell upon the importance of the results obtained.

We consider the following boundary value problem where 1≤p≤n−1 is fixed. Criteria are developed for the existence of two positive solutions of the boundary value problem. For special cases we also establish upper and lower bounds for these positive solutions. Several examples are included to dwell upon the importance of the results obtained.

We develop criteria for the nonexistence of eventually positive (negative) and nondecreasing (nonincreasing) solutions of the partial difference equation ∇ m ∇ n y(m, n) + P(m, n, y(m + k, n + l)) = Q(m, n, y(m + k, n + l)) and ∇ m ∇ n y(m, n) + ∑ i = 1 τ P i (m, n, y(m + k i , n + l i )) = ∑ i = 1 τ Q i (m, n, y(m + k i , n + l i )). Several examples which dwell upon the importance of our results are also included.

We present a survey on the existence of periodic solutions of singular differential equations. In particular, we pay our attention to singular scalar differential equations, singular damped differential equations, singular impulsive differential equations, and singular differential systems. Published version

Dynamical aspects of initial/boundary value problems for ordinary differential equations have become a rapidly growing area of research in the theory of differential equations and dynamical systems and have gathered substantial research interests during the last decades. The attractiveness of this field not only is derived from theoretical interests but also is motivated by the insights that such dynamical aspects could reveal in several phenomena observed in applied sciences. The current special issue places its emphasis on the study of the dynamical aspects of initial/boundary value problems for ordinary differential equations. Call for papers has been carefully prepared by the guest editors and posted on the journals web page, which has attracted many researchers to submit their contribution on wide topics such as oscillation theory, delay differential equation, impulsive differential equation, multipoint boundary value problems, stochastic mutualism system, chaotic system, homoclinic solutions, Hamiltonian systems, stability and bifurcation, exponential extinction, singular elliptic problem, nonuniform exponential contraction and dichotomy. All manuscripts submitted to this special issue went through a thorough peer-refereeing process. Based on the reviewers’ reports, we collect twenty-five original research articles by more than fifty active international researchers in differential equations and from different countries such as Korea, China, Malaysia, Singapore, Czech Republic, Turkey, Slovenia, India, and USA. Besides, one survey on recent results for the existence of singular periodic problems is also contained. It is certainly impossible to provide in this short editorial note a more comprehensive description for all articles in this special issue. However, the team of the guest editors believes that the results included reflect some recent trends in research and outline new ideas for future studies of dynamical aspects of initial/boundary value problems for ordinary differential equations.

This paper is concerned with the existence and uniqueness of solutions for boundary value problems with p-Laplacian delay differential equations on the half-line. The existence of solutions is derived from the Schauder fixed point theorem, whereas the uniqueness of solution is established by the Banach contraction principle. As an application, an example is given to demonstrate the main results. MSC: 34K10; 34B18; 34B40

In this paper some simple inequalities are used to offer sufficient conditions for the oscillation of all solutions of the differential equation[formula]where 0 p / q with p , q odd integers, or p even and q odd integers. Several examples which dwell upon the importance of our results are also included.

We consider the following complementary Lidstone boundary value problem (−1) m y (2m+1) (t )= λF(t,y(t),y � (t)), t ∈ (0,1) y(0) =0 , y (2k−1) (0) = y (2k−1) (1) =0 , 1≤ k ≤ m where l > 0. The values of l are characterized so that the boundary value problem has a positive solution. Moreover, we derive explicit intervals of l such that for any l in the interval, the existence of a positive solution of the boundary value problem is guaranteed. Some examples are also included to illustrate the results obtained. Note that the nonlinear term F depends on y’ and this derivative dependence is seldom investigated in the literature. AMS Subject Classification: 34B15.

This paper gives an application of Mawhin’s coincidence degree and matrix spectral theory to a predator-prey model withM-predators andN-preys. The method is different from that used in the previous work. Some new sufficient conditions are obtained for the existence and global asymptotic stability of the periodic solution. The existence and stability conditions are given in terms of spectral radius of explicit matrices which are much different from the conditions given by the algebraic inequalities. Finally, an example is given to show the feasibility of our results.

Three-point boundary value problems of second-order differential equation with ap-Laplacian on finite and infinite intervals are investigated in this paper. By using a new continuation theorem, sufficient conditions are given, under the resonance conditions, to guarantee the existence of solutions to such boundary value problems with the nonlinear term involving in the first-order derivative explicitly.