Traveling Wave Solutions in an Integrodifference Equation with Weak Compactness

This article studies the existence of traveling wave solutions in an integrodifference equation with weak compactness. Because of the special kernel function that may depend on the Dirac function, traveling wave maps have lower regularity such that it is difficult to directly look for a traveling wave solution in the uniformly continuous and bounded functional space. In this paper, by introducing a proper set of potential wave profiles, we can obtain the existence and precise asymptotic behavior of nontrivial traveling wave solutions, during which we do not require the monotonicity of this model.     

Monotone method for a system of nonlinear mixed type implicit impulsive integro-differential equations in Banach spaces

In this paper, by using a monotone iterative technique in the presence of lower and upper solutions, we discuss the existence of solutions for a new system of nonlinear mixed type implicit impulsive integro-differential equations in Banach spaces. Under wide monotonicity conditions and the noncompactness measure conditions, we also obtain the existence of extremal solutions and a unique solution between lower and upper solutions.     

Almost periodic linear differential equations with non-separated solutions

A celebrated result by Favard states that, for certain almost periodic linear differential systems, the existence of a bounded solution implies the existence of an almost periodic solution. A key assumption in this result is the separation among bounded solutions. Here we prove a theorem of anti-Favard type: if there are bounded solutions which are non-separated (in a strong sense) sometimes almost periodic solutions do not exist. Strongly non-separated solutions appear when the associated homogeneous system has homoclinic solutions. This point of view unifies two fascinating examples by Zhikov-Levitan and Johnson for the scalar case. Our construction uses the ideas of Zhikov-Levitan together with the theory of characters in topological groups.     

Traveling waves for chemotaxis–systems

In this paper we study the existence of traveling wave solutions to the Keller‐Segel model, a general model of chemotaxis, where the species do not reproduce. In the case of logarithmic sensitivity we show that various functionals modeling the reactive feedback on the chemo‐attractant do allow for traveling waves and a wide range of qualitatively different behavior is possible. We can find monotone fronts as well as pulse solutions in the densities of the population and the chemical. In particular, a new kind of solution exists, where both densities travel as pulses.     

A qualitative study on general Gause-type predator–prey models with non-monotonic functional response

In this paper, we study a diffusive predator–prey model with general growth rates and non-monotonic functional response under homogeneous Neumann boundary condition. A local existence of periodic solutions and the asymptotic behavior of spatially inhomogeneous solutions are investigated. Moreover, we show the existence and non-existence of non-constant positive steady-state solutions. Especially, to show the existence of non-constant positive steady-states, the fixed point index theory is used without estimating the lower bounds of positive solutions. More precisely, calculating the indexes at the trivial, semi-trivial and positive constant solutions, some sufficient conditions for the existence of non-constant positive steady-state solutions are studied. This is in contrast to the works in previous papers. Furthermore, on obtaining these results, we can observe that the monotonicity of a prey isocline at the positive constant solution plays an important role.     

Asymptotic bounds of solutions for a periodic doubly degenerate parabolic equation

This paper is concerned with a doubly degenerate parabolic equation with logistic periodic sources. We are interested in the discussion of the asymptotic behavior of solutions of the initial-boundary value problem. In this paper, we first establish the existence of non-trivial nonnegative periodic solutions by a monotonicity method. Then by using the Moser iterative method, we obtain an a priori upper bound of the nonnegative periodic solutions, by means of which we show the existence of the maximum periodic solution and asymptotic bounds of the nonnegative solutions of the initial-boundary value problem. We also prove that the support of the non-trivial nonnegative periodic solution is independent of time.     

Generalized quasilinearization method for reaction-diffusion equations under nonlinear and nonlocal flux conditions

In this paper we consider an initial boundary value problem for a reaction-diffusion equation under nonlinear and nonlocal Robin type boundary condition. Assuming the existence of an ordered pair of upper and lower solutions we establish a generalized quasilinearization method for the problem under consideration whose characteristic feature consists in the construction of monotone sequences converging to the unique solution within the interval of upper and lower solutions, and whose convergence rate is quadratic. Thus this method provides an efficient iteration technique that produces not only improved approximations due to the monotonicity of its iterates, but yields also a measure of the convergence rate.     

Traveling Wave Solutions In Coupled Chua'S Circuits,Part I: Periodic Solutions

We study a singularly perturbed system of partial di erential equations that models a one-dimensional array of coupled Chua's circuits. The PDE system is a natural generalization to the FitzHugh-Nagumo equation. In part I of the paper, we show that similar to the FitzHugh-Nagumo equation, the system has periodic traveling wave solutions formed alternatively by fast and slow flows. First, asymptotic method is used on the singular limit of the fast/slow systems to construct a formal periodic solution. Then, dynamical systems method is used to obtain an exact solution near the formal periodic soluion. In part II, we show that the system can have more complicated periodic and chaotic traveling wave solutions that do not exist in the FitzHugh-Nagumos equation.     

Traveling wave solutions in delayed lattice differential equations with partial monotonicity

In this paper, we investigate a system of delayed lattice differential equations with partial monotonicity. By using Schauder's fixed point theorem, a new cross-iteration scheme is given to establish the existence of traveling wave solutions. Our main results can deal with the existence of traveling wave solution for a class of delayed reaction diffusion system with partial monotonicity and generalize the results of Wu and Zou (J. Differential Equations 135 (1997) 315–357).     

Generalized functions solutions of monotone and semilinear parabolic equations

We investigate monotone semilinear equations in the setting of generalized functions. We derive existence and uniqueness results even in situations where no solution exists in the sense of distributions; next we show it is consistent with continuous solutions whenever they exist. Furthermore we prove that those transient generalized solutions stabilize toward steady state solutions the way weak solutions do. Lastly we point out an example how these techniques can be applied to different type of nonlinearities.     

Existence of equilibria via Ekeland's principle

In the literature, when dealing with equilibrium problems and the existence of their solutions, the most used assumptions are the convexity of the domain and the generalized convexity and monotonicity, together with some weak continuity assumptions, of the function. In this paper, we focus on conditions that do not involve any convexity concept, neither for the domain nor for the function involved. Starting from the well-known Ekeland's theorem for minimization problems, we find a suitable set of conditions on the function f that lead to an Ekeland's variational principle for equilibrium problems. Via the existence of ε-solutions, we are able to show existence of equilibria on general closed sets for equilibrium problems and systems of equilibrium problems.     

Existence results for degenerate p(x)-Laplace equations with Leray-Lions type operators

We show the existence and multiplicity of solutions to degenerate p(x)-Laplace equations with Leray-Lions type operators using direct methods and critical point theories in Calculus of Variations and prove the uniqueness and nonnegativeness of solutions when the principal operator is monotone and the nonlinearity is nonincreasing. Our operator is of the most general form containing all previous ones and we also weaken assumptions on the operator and the nonlinearity to get the above results. Moreover, we do not impose the restricted condition on p(x) and the uniform monotonicity of the operator to show the existence of three distinct solutions.     

Periodic problems and problems with discontinuities for nonlinear parabolic equations

In this paper we study nonlinear parabolic equations using the method of upper and lower solutions. Using truncation and penalization techniques and results from the theory of operators of monotone type, we prove the existence of a periodic solution between an upper and a lower solution. Then with some monotonicity conditions we prove the existence of extremal solutions in the order interval defined by an upper and a lower solution. Finally we consider problems with discontinuities and we show that their solution set is a compact R -set in (CT, L ²(Z)).     

The Spatially Homogeneous Relativistic Boltzmann Equation with a Hard Potential

In this paper, we study spatially homogeneous solutions of the Boltzmann equation in special relativity and in Robertson-Walker spacetimes. We obtain an analogue of the Povzner inequality in the relativistic case and use it to prove global existence theorems. We show that global solutions exist for a certain class of collision cross sections of the hard potential type in Minkowski space and in spatially flat Robertson-Walker spacetimes.     

Nonlinear elliptic differential equations with multivalued nonlinearities

In this paper we study nonlinear elliptic boundary value problems with monotone and nonmonotone multivalued nonlinearities. First we consider the case of monotone nonlinearities. In the first result we assume that the multivalued nonlinearity is defined on all ℝ. Assuming the existence of an upper and of a lower solution, we prove the existence of a solution between them. Also for a special version of the problem, we prove the existence of extremal solutions in the order interval formed by the upper and lower solutions. Then we drop the requirement that the monotone nonlinearity is defined on all of ℝ. This case is important because it covers variational inequalities. Using the theory of operators of monotone type we show that the problem has a solution. Finally in the last part we consider an eigenvalue problem with a nonmonotone multivalued nonlinearity. Using the critical point theory for nonsmooth locally Lipschitz functionals we prove the existence of at least two nontrivial solutions (multiplicity theorem).     

弱广义向量拟似变分不等式解的存在性

研究拓扑向量空间上弱广义向量拟似变分不等式解的存在性问题.与Khaliq和Rashid等在单调或伪单调假设条件下,利用KKM定理证明解的存在性不同,在集值映射是非空凸值映射并且有开下截口的条件下,利用Fan-Browder不动点原理证明解的存在性.     

A system of evolution hemivariational inequalities modeling thermoviscoelastic frictional contact

In this paper we prove the existence and uniqueness of the weak solution for a dynamic thermoviscoelastic problem which describes frictional contact between a body and a foundation. We employ the Kelvin–Voigt viscoelastic law, include the thermal effects and consider the general nonmonotone and multivalued subdifferential boundary conditions. The model consists of the system of the hemivariational inequality of hyperbolic type for the displacement and the parabolic hemivariational inequality for the temperature. The existence of solutions is proved by using a surjectivity result for operators of pseudomonotone type. The uniqueness is obtained for a large class of operators of subdifferential type satisfying a relaxed monotonicity condition.     

Large time behavior of solutions to semilinear systems of wave equations

This paper is concerned with the initial value problem for semilinear systems of wave equations. First we show a global existence result for small amplitude solutions to the systems. Then we study asymptotic behavior of the global solution. We underline that ``modified' free profiles are obtained for all global solutions to the systems even in the case where the free profile might not exist. Moreover, we prove non–existence of any free profiles for the global solution in some cases where the effect of the nonlinearity is strong enough. The first author was partially supported by Grant-in-Aid for Science Research (14740114), JSPS.     

Solutions with internal jump for an autonomous elliptic system of FitzHugh–Nagumo type

Systems of elliptic partial differential equations which are coupled in a noncooperative way, such as the FitzHugh–Nagumo type studied in this paper, in general do not satisfy order preserving properties. This not only results in technical complications but also yields a richer solution structure. We prove the existence of multiple nontrivial solutions. In particular we show that there exists a solution with boundary layer type behaviour, and we will give evidence that this autonomous system for a certain range of parameters has a solution with both a boundary and an internal layer. The analysis uses results from bifurcation theory, variational methods, as well as some pointwise a priori estimates. The final section contains some numerically obtained results.     

Periodic reiterated homogenization for elliptic functions

In this paper, we study reiterated homogenization for equations of the form . We assume that a is a Carathéodory function and satisfies some monotonicity and growth conditions and its reiterated unfolding converges almost everywhere to a Carathéodory type function. Under these assumptions, we show that the sequence of solutions converges to the solution of a limit variational problem. In particular this contains the case , where a is periodic in the second and third arguments, and continuous in each argument.