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Abstract: The problem of a classification of robust homoclinic cycles in low-dimensional spaces has been frequently asked in recent years. In this paper, we resume the results in and and we solve the problem in in the case of orientation-preserving group actions. [Copyright &y& Elsevier]

Abstract: This paper deals with the existence of multiple positive solutions for the one-dimensional p-Laplacian subject to one of the following boundary conditions: or where , . By means of a fixed point theorem due to Avery and Peterson, sufficient conditions are obtained that guarantee the existence of at least three positive solutions. The interesting point is the nonlinear term f is involved with the first-order derivative explicitly. [Copyright &y& Elsevier]

In this paper, in the setting of sequential spaces, we consider the value functions vS and vI defined by vS(x)=supy∈K(x)f(x,y) and vI(x)=infy∈K(x)f(x,y), and we introduce sufficient conditions on the value functions and on the data, weaker than upper semicontinuity, which guarantee the existence of maximum for such functions. Various examples illustrate the results and the connections with previous results. [Copyright &y& Elsevier]

In the case of oscillatory potentials, we establish an oscillation theorem for the forced sublinear differential equation , . No restriction is imposed on the forcing term to be the nth derivative of an oscillatory function. In particular, we show that all solutions of the equation , , are oscillatory for all if . This provides an analogue of a result of Nasr [Proc. Amer. Math. Soc. 126 (1998) 123] for the forced superlinear equation and answers a question raised in an earlier paper [J.S.W. Wong, SIAM J. Math. Anal. 19 (1988) 673]. [Copyright &y& Elsevier]

In this article, we construct a general series for 1/π. We indicate that Ramanujan''s 1/π-series are all special cases of this general series and we end the paper with a new class of 1/π-series. Our work is motivated by series recently discovered by Takeshi Sato. [Copyright &y& Elsevier]

This paper is concerned with a class of quasilinear parabolic equations with singularity and arbitrary degeneracy. The existence and uniqueness of generalized solutions to a kind of boundary value problem is established. [Copyright &y& Elsevier]

In this paper, by means of a new twin fixed-point theorem in a cone, the existence of at least two positive solutions of m-point boundary value problem for second order dynamic equations on time scales is considered. [Copyright &y& Elsevier]

In this paper, Schaefer''s fixed point theorem and nonlinear alternative of Leray–Schauder type are used to investigate the existence of solutions for second order boundary value problem for impulsive dynamic equations on time scales. [Copyright &y& Elsevier]

In this paper, we consider a kind of neutral functional differential equation as follows: x(t)−cx(t−τ)′=gt,xt−μ(t)+e(t) in the critical case &z.sfnc;c&z.sfnc;=1. By employing the topological degree theory and some analysis techniques, we obtain some new results on the existence of periodic solutions. [Copyright &y& Elsevier]

In this paper, we study existence of solutions of first order dynamic inclusions on time scales with general boundary conditions. Both the ∇-derivative and Δ-derivative cases are considered. Examples are presented to illustrate that the Δ-derivative case needs more restrictive assumptions. [Copyright &y& Elsevier]

This paper is concerned with a cross-diffusion system arising in a prey–predator population model. The main purpose is to discuss the stability analysis for coexistence steady-state solutions obtained by Kuto and Yamada (J. Differential Equations, to appear). We will give some criteria on the stability of these coexistence steady states. Furthermore, we show that the Hopf bifurcation phenomenon occurs on the steady-state solution branch under some conditions. [Copyright &y& Elsevier]

In this paper, we consider boundary value problems for nonlinear differential equations on the semi-axis (0,∞) and also on the whole axis (−∞,∞), under the assumption that the left-hand side being a second order linear differential expression belongs to the Weyl limit-circle case. The boundary value problems are considered in the Hilbert spaces L2(0,∞) and L2(−∞,∞), and include boundary conditions at infinity. The existence and uniqueness results for solutions of the considered boundary value problems are established. [Copyright &y& Elsevier]

This paper is concerned with the nonoscillatory problems of odd-dimensional systems of linear retarded functional differential equations. Based upon the corresponding characteristic equations, we get some criteria for nonoscillations by utilizing the matrix measures. [Copyright &y& Elsevier]

In this paper, a pair of nondifferentiable multiobjective programming problems is first formulated, where each of the objective functions contains a support function of a compact convex set in Rn. For a differentiable function h :Rn×Rn→R, we introduce the definitions of the higher-order F-convexity (F-pseudo-convexity, F-quasi-convexity) of function f :Rn→R with respect to h. When F and h are taken certain appropriate transformations, all known other generalized invexity, such as η-invexity, type I invexity and higher-order type I invexity, can be put into the category of the higher-order F-invex functions. Under these the higher-order F-convexity assumptions, we prove the higher-order weak, higher-order strong and higher-order converse duality theorems related to a properly efficient solution. [Copyright &y& Elsevier]

We consider the nth order nonlinear neutral differential equation of the form x(t)+∫lower limit a, upper limit b p(t,μ)xτ(t,μ) dμ(n)+λ∫lower limit c, upper limit d q(t,ξ)fxσ(t,ξ) dξ=h(t). In this paper, we obtain sufficient conditions which ensure that all solutions are either oscillatory or limt→∞&z.sfnc;x(t)&z.sfnc;=∞ or else liminft→∞&z.sfnc;x(t)&z.sfnc;=0. [Copyright &y& Elsevier]

This paper deals with nonautonomous Lie´nard-type systems. Sufficient conditions are given for the zero solution of the systems to be globally asymptotically stable. The main result is proved by means of phase plane analysis with a Liapunov function. Examples are included to contrast our theorem with results which were presented by Hatvani and Cantarelli. Some global phase portraits are also attached. [Copyright &y& Elsevier]

In this paper, we consider the existence of at least three positive solutions for the 2nth order m-point boundary value problem y(2n)(t)=ft,y(t),y″(t),…,y(2(n−1))(t), 0&les;t&les;1,y(2i)(0)=0, y(2i)(1)=∑lower limit j=1, upper limit m−2 kijy(2i)(ξj), 0&les;i&les;n−1, where kij>0 (i=0,1,…,n−1; j=1,2,…,m−2), 0<ξ1<ξ2<⋯<ξm−2<1. The associated Green''s function for the higher order m-point boundary value problem is first given, and growth conditions are imposed on f which yield the existence of multiple positive solutions by using the Leggett–Williams fixed point theorem. [Copyright &y& Elsevier]

This paper studies the global behaviors of the periodic logistic system with periodic impulsive perturbations. The results of D.D. Bainov and P.S. Simeonov (1993) are extended and dynamics different from the corresponding continuous system are found. It is shown that the system may have a unique positive periodic solution which is globally asymptotically stable, or go extinct when the two periods are rational dependent. When they are rational independent, the system has no periodic solutions, however, still has a global attractor or go extinct under some conditions. [Copyright &y& Elsevier]

In this paper we study the boundedness and the asymptotic behavior of the positive solutions of the system of difference equations xn+1=A+∑i=1kaixn−pi/∑j=1mbjyn−qj,yn+1=B+∑i=1kciyn−pi/∑j=1mdjxn−qj, yn+1=B+∑i=1kciyn−pi/∑j=1mdjxn−qj, where k,m∈{1,2,…}, A,B,ai,ci,bj,dj, i∈{1,…,k}, j∈{1,…,m}, are positive constants, pi, qj, i∈{1,…,k}, j∈{1,…,m}, are positive integers such that p1, q1 and the initial values xi,yi, i∈{−π,−π+1,…,0}, π=max{pk,qm} are positive numbers. [Copyright &y& Elsevier]

By means of generalized averaging pair technique and Riccati transformation method, oscillation criteria for self-adjoint differential matrix system of the form r(t)P(t)X′(t)′+p(t)P(t)X′(t)+Q(t)FX′(t)GX(t)=0 are obtained. This paper generalizes and improves the existing results including those obtained by Yang (2002), Li and Agarwal (2000) and Rogovchenko (1999). [Copyright &y& Elsevier]

This paper investigates the effects of a degenerate diffusion term in reaction–diffusion models ut=[D(u)ux]x+g(u) with Fisher-KPP type g. Both in the case when D(0)=0 and when D(0)=D(1)=0, with D(u)>0 elsewhere, we obtain a continuum of travelling wave solutions having wave speed c greater than a threshold value c&ast; and we show the appearance of a sharp-type profile when c=c&ast;. These results solve recent conjectures formulated by Sa´nchez-Gardun˜o and Maini (J. Differential Equations 117 (1995) 281) and Satnoianu et al. (Discrete Continuous Dyn. Systems (Series B) 1 (2000) 339). [Copyright &y& Elsevier]

In this paper, we propose algorithms for the following problems in the implicitization of a set of differential rational parametric equations P. (1) To find a characteristic set for the implicit prime ideal of P. (2) To find a canonical representation for the image of P. (3) To decide whether the parameters of P are independent, and if not, to re-parameterize P so that the new parametric equations have independent parameters. (4) To compute the inversion maps of P, and as a consequence, to decide whether P is proper. If the implicit variety is of dimension one and P is not proper, to find a proper re-parameterization for P. [Copyright &y& Elsevier]

In this paper we first establish a new general criterion for the permanence of Kolmogorov-type systems of nonautonomous functional differential equations. Then, as applications of this criterion we study the permanence of a class of n-species general nonautonomous food chain systems with delay and new sufficient condition are established. [Copyright &y& Elsevier]

In this paper we define a non-commutative residue for the algebra of classical anisotropic pseudo-differential operators on Rn modulo regularizing operators. Furthermore, by means of the Weyl formula, we show that this trace coincides with the Dixmier trace on operators of order −|M|. [Copyright &y& Elsevier]

New existence and uniqueness proofs are given for the solutions of the equations governing the self-similar compressible boundary layer (Falkner–Skan equations). The properties of the solutions are studied and some bounds on important quantities are concluded. The paper is restricted to favourable pressure gradients and to wall cooling. [Copyright &y& Elsevier]

In this paper, we develop Kaplan–Yorke''s method and consider the existence of periodic solutions for delay differential equations with two delays. Especially, we study Hopf and saddle-node bifurcations of periodic solutions for the equation with parameters, and give conditions under which the bifurcations occur. [Copyright &y& Elsevier]