Existence and stability of solutions for maximal element theorem on Hadamard manifolds with applications

In this paper, maximal element theorem on Hadamard manifolds is established. First, we prove the existence of solutions for maximal element theorem on Hadamard manifolds. Further, we prove that most of problems in maximal element theorem on Hadamard manifolds (in the sense of Baire category) are essential and that, for any problem in maximal element theorem on Hadamard manifolds, there exists at least one essential component of its solution set. As applications, we study existence and stability of solutions for variational relation problems on Hadamard manifolds, and existence and stability of weakly Pareto-Nash equilibrium points for n-person multi-objective games on Hadamard manifolds.     

The existence of quasiperiodic solutions for coupled Duffing-type equations

In this paper, we prove the existence of infinitely many quasiperiodic solutions for a class of coupled Duffing-type equations via KAM theorem. Moreover, the set of quasiperiodic solutions is of infinitely Lebesgue measure in the phase space.     

The antipodal mapping theorem and difference equations in Banach spaces†

We employ the Borsuk–Krasnoselskii antipodal theorem to prove a new fixed point theorem in ordered Banach spaces. Then, the applicability of the result is shown by presenting sufficient conditions for the existence of solutions to initial value problems for first-order difference equations in Banach spaces. To prove that result we shall employ set valued analysis techniques.     

Nonlinear Dynamics Equation in p-Adic String Theory

We investigate nonlinear pseudodifferential equations with infinitely many derivatives. These are equations of a new class, and they originally appeared in p-adic string theory. Their investigation is of interest in mathematical physics and its applications, in particular, in string theory and cosmology. We undertake a systematic mathematical investigation of the properties of these equations and prove the main uniqueness theorem for the solution in an algebra of generalized functions. We discuss boundary problems for bounded solutions and prove the existence theorem for spatially homogeneous solutions for odd p. For even p, we prove the absence of a continuous nonnegative solution interpolating between two vacuums and indicate the possible existence of discontinuous solutions. We also consider the multidimensional equation and discuss soliton and q-brane solutions.     

Bistable wavefronts in a diffusive and competitive Lotka-Volterra type system with nonlocal delays

A diffusive Lotka-Volterra type model with nonlocal delays for two competitive species is considered. The existence of a traveling wavefront analogous to a bistable wavefront for a single species is proved by transforming the system with nonlocal delays to a four-dimensional system without delay. Furthermore, in order to prove the asymptotic stability (up to translation) of bistable wavefronts of the system, the existence, regularity and comparison theorem of solutions of the corresponding Cauchy problem are first established for the systems on R by appealing to the theory of abstract functional differential equations. The asymptotic stability (up to translation) of bistable wavefronts are then proved by spectral methods. In particular, we also prove that the spreading speed is unique by upper and lower solutions technique. From the point of view of ecology, our results indicate that the nonlocal delays appeared in the interaction terms are not sensitive to the invasion of species of spatial isolation.     

Almost automorphic solutions for some partial functional differential equations

In this work, we study the existence of almost automorphic solutions for some partial functional differential equations. We prove that the existence of a bounded solution on R⁺ implies the existence of an almost automorphic solution. Our results extend the classical known theorem by Bohr and Neugebauer on the existence of almost periodic solutions for inhomegeneous linear almost periodic differential equations. We give some applications to hyperbolic equations and Lotka-Volterra type equations used to describe the evolution of a single diffusive animal species.     

Asymptotic existence theorems for frames and group divisible designs

In this paper, we establish an asymptotic existence theorem for group divisible designs of type mn with block sizes in any given set K of integers greater than 1. As consequences, we will prove an asymptotic existence theorem for frames and derive a partial asymptotic existence theorem for resolvable group divisible designs.     

An intersection theorem for set-valued mappings

Given a nonempty convex set X in a locally convex Hausdorff topological vector space, a nonempty set Y and two set-valued mappings T: X ? X, S: Y ? X we prove that under suitable conditions one can find an x ∈ X which is simultaneously a fixed point for T and a common point for the family of values of S. Applying our intersection theorem, we establish a common fixed point theorem, a saddle point theorem, as well as existence results for the solutions of some equilibrium and complementarity problems.     

Existence of β-martingale solutions of stochastic evolution functional equations of parabolic type with measurable locally bounded coefficients

We prove a theorem on the existence of ??-martingale solutions of stochastic evolution functional equations of parabolic type with Borel measurable locally bounded coefficients. A ??-martingale solution of a stochastic evolution functional equation is understood as a martingale solution of a stochastic evolution functional inclusion constructed on the basis of the equation. We find sufficient conditions for the existence of ??-martingale solutions that do not blow up in finite time.     

Weighted pseudo-almost periodic solutions of a class of abstract differential equations

For abstract linear functional differential equations with a weighted pseudo-almost periodic forcing term, we prove that the existence of a bounded solution on R⁺ implies the existence of a weighted pseudo-almost periodic solution. Our results extend the classical theorem due to Massera on the existence of periodic solutions for linear periodic ordinary differential equations. To illustrate the results, we consider the Lotka-Volterra model with diffusion.     

Schrödinger-Poisson equations without Ambrosetti-Rabinowitz condition

We prove the existence of ground state solutions for a stationary Schrödinger-Poisson equation in R³. The proof is based on the mountain pass theorem and it does not require the Ambrosetti-Rabinowitz condition.     

Korn's inequality and Donati's theorem for the conformal Killing operator on pseudo-Euclidean space

We prove the Korn's inequality for the conformal Killing operator on pseudo-Euclidean space Rp,q, and an existence theorem for solutions to the non-homogeneous conformal Killing equation, which is a pseudo-Euclidean conformal generalization of Donati's theorem for Euclidean Killing operator.     

Density of the Extremal Solutions for a Class of Second Order Boundary Value Problem

In this paper, we prove a theorem on the existence of extremal solutions to a second-order differential inclusion with boundary conditions, governed by the subdifferential of a convex function. We also show that the extremal solutions set is dense in the solutions set of the original problem.     

Stability of solutions for variational relation problems with applications

In this paper, we prove that most of problems in variational relations (in the sense of Baire category) are essential and that, for any problem in variational relations, there exists at least one essential component of its solution set. As applications, we deduce the existence of essential components of the set of Ky Fan’s points based on Ky Fan’s minimax inequality theorem, the existence of essential components of the set of Nash equilibrium points for general n-person non-cooperative games, the existence of essential component of the set of solutions for vector Ky Fan’s minimax inequality, the existence of essential components of the set of KKM points and the existence of essential components of the set of solutions for Ky Fan’s section theorem.     

Existence of a time periodic solution for the compressible Euler equation with a time periodic outer force

We are concerned with a time periodic supersonic flow through a bounded interval. This motion is described by the compressible Euler equation with a time periodic outer force. Our goal in this paper is to prove the existence of a time periodic solution. Although this is a fundamental problem for other equations, it has not been received much attention for the system of conservation laws until now.When we prove the existence of the time periodic solution, we face with two problems. One is to prove that initial data and the corresponding solutions after one period are contained in the same bounded set. To overcome this, we employ the generalized invariant region, which depends on the space variables. This enable us to investigate the behavior of solutions in detail. Second is to construct a continuous map. We apply a fixed point theorem to the map from initial data to solutions after one period. Then, the map needs to be continuous. To construct this, we introduce the modified Lax–Friedrichs scheme, which has a recurrence formula consisting of discretized approximate solutions. The formula yields the desired map. Moreover, the invariant region grantees that it maps a compact convex set to itself. In virtue of the fixed point theorem, we can prove a existence of a fixed point, which represents a time periodic solution. Finally, we apply the compensated compactness framework to prove the convergence of our approximate solutions.     

Positive solutions for nonlinear periodic problems with concave terms

We consider a nonlinear periodic problem, driven by the scalar p-Laplacian, with a parametric concave term and a Carathéodory perturbation whose potential (primitive) exhibits a p-superlinear growth near +∞, without satisfying the usual in such cases Ambrosetti-Rabinowitz condition. Using critical point theory and truncation techniques, we prove a bifurcation-type theorem describing the nonexistence, existence and multiplicity of positive solutions as the parameter varies.     

An obstacle problem for nonlinear hemivariational inequalities at resonance

In this paper we examine an obstacle problem for a nonlinear hemivariational inequality at resonance driven by the p-Laplacian. Using a variational approach based on the nonsmooth critical point theory for locally Lipschitz functionals defined on a closed, convex set, we prove two existence theorems. In the second theorem we have a pointwise interpretation of the obstacle problem, assuming in addition that the obstacle is also a kind of lower solution for the nonlinear elliptic differential inclusion.     

Solitary waves for coupled nonlinear elliptic system with nonhomogeneous nonlinearities

In present paper we consider a class of coupled elliptic system with nonhomogeneous nonlinearities. This type of system is related to the Raman amplification in a plasma. We make rigorous study and find the threshold conditions to guarantee the existence, nonexistence and multiplicity of nontrivial solutions for both two and three coupled system by using Morse theory, direct analysis methods and Krasnosel’skii–Rabinowitz global bifurcation theorem. Moreover, we study the asymptotical behavior of positive solutions, and prove some interesting phenomena for these solutions. Comparing to our previous works Wang and Shi (standing waves for weakly coupled Schrödinger equations with quadratic nonlinearities. Preprint, 2015) on the homogeneous case, we encounter some new challenges in proving the existence and multiplicity of nontrivial solutions. We overcome these difficult by combining the Mountain–Pass theorem in convex set and the Nehari constraint methods.     

拟线性正对称方程组的边值问题及其对混合型方程的应用

<正> 1.引言 本文有两个目的.第一个目的是讨论拟线性正对称方程组的边值问题.如所知,线性的正对称方程组是一类相当广泛的偏微分方程组,许多常见的偏微分方程都可以化为正对称方程组去.对于这个方程组,可以利用能量不等式来证明许多边值问题的适定性.本文充分利用线性正对称方程组的结果,经过适当的估计,用压缩映照原理证明了拟线性正对称方程组的边值问题的解的存在性.对求解的区域而言,问题是大范围的,但     

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).