On Osgood theorem in Banach spaces

Let X be a real Banach space, ω : [0, +∞) → ? be an increasing continuous function such that ω(0) = 0 and ω(t + s) ≤ ω(t) + ω(s) for all t, s ∈ [0, +∞). According to the infinite dimensional analog of the Osgood theorem if ∫¹₀ (ω(t))^?1 dt = ∞, then for any (t₀, x₀) ∈ ?×X and any continuous map f : ?×X → X such that ∥f(t, x) – f(t, y)∥ ≤ ω(∥x – y∥) for all t ∈ ?, x, y ∈ X, the Cauchy problem (t) = f(t, x(t)), x(t₀) = x₀ has a unique solution in a neighborhood of t₀. We prove that if X has a complemented subspace with an unconditional Schauder basis and ∫¹₀ (ω(t))^?1 dt < ∞ then there exists a continuous map f : ? × X → X such that ∥f(t, x) – f(t, y)∥ ≤ ω(∥x – y∥) for all (t, x, y) ∈ ? × X × X and the Cauchy problem (t) = f(t, x(t)), x(t₀) = x₀ has no solutions in any interval of the real line.     

On an extension of Rolle's theorem to locally convex spaces

This work concerns the extension of a weak form of the Rolle's theorem to locally convex spaces that satisfy an axiom of separation. The result provides a condition for asserting the uniqueness of a solution to nonlinear functional equations, including nonlinear integro-differential equations. We use the extended Rolle's theorem to prove the uniqueness of a solution to a nonlinear, fractional differential equation.     

Alternative solution of the inhomogeneous linear differential equation of order two

In this paper, we present an alternative method for solving the general inhomogeneous linear ordinary differential equation (ODE) of order two. The solution appears in the standard form as the sum of the solution of the equivalent homogeneous problem and the particular solution of the inhomogeneous problem at hand. The main advantage of the method exposed herein is that the particular solution is computable from two different integrals. This allows the problem solver to choose the simplest integral with which to work with in order to get the final solution. For illustrative purposes we employ the method presented to aid in the solution of some example problems including the inhomogeneous Klein-Gordon equation.     

A comparison theorem and uniqueness theorem of backward doubly stochastic differential equations

In this paper, we deal with one dimensional backward doubly stochastic differential equations (BDSDEs). We obtain a comparison theorem and a uniqueness theorem for BDSDEs with continuous coefficients.     

L_p-DICHOTOMY OF LINEAR DIFFERENTIAL EQUATIONS IN AN ARBITRARY BANACH SPACE

The notion of Lp-dichotomy for linear differential equations with possibly unbounded operator is introduced. By help of Banach fixed point theorem sufficient conditions for the existence of bounded solutions of nonlinear differential equations with an Lp-dichotomous linear part are obtained.     

A Schauder-type fixed point theorem

In this brief note we study Schauder's second fixed point theorem in the space $(\mathit{BC},6\cdot 6)$ of bounded continuous functions $\varphi :[0,\infty )\to {\Re }^n$ with a view to reducing the requirement that there is a compact map to the requirement that the map is locally equicontinuous. Several examples are given, both motivating and applying the theory.     

A note on Malmquist-Yosida type theorem of higher order algebraic differential equations

In this article, we give a simple proof of Malmquist-Yosida type theorem of higher order algebraic differential equations, which is different from the methods as that of Gackstatter and Laine [2], and Steinmetz [12].     

Multiple Solutions to Stochastic Differential Delay Equations and a Related Comparison Theorem

In this article, we discuss the existence of multiple solutions to a one-dimensional stochastic differential delay equation with continuous drift coefficients and derive a related comparison theorem.     

A Liouville theorem for the axially-symmetric Navier-Stokes equations

Let v(x,t)=vrer+vθeθ+vzez be a solution to the three-dimensional incompressible axially-symmetric Navier-Stokes equations. Denote by b=vrer+vzez the radial-axial vector field. Under a general scaling invariant condition on b, we prove that the quantity Γ=rvθ is Hölder continuous at r=0, t=0. As an application, we prove that the ancient weak solutions of axi-symmetric Navier-Stokes equations must be zero (which was raised by Koch, Nadirashvili, Seregin and Sverak (2009) in [15] and Seregin and Sverak (2009) in [26] as a conjecture) under the condition that b∈L^∞([0,T],BMO−1). As another application, we prove that if b∈L^∞([0,T],BMO−1), then v is regular.     

Decreasing and fast solutions for a second-order difference equation related to Fisher-Kolmogorov's equation

We prove the existence of so-called fast solutions of a second-order difference equation related to traveling wave solutions of Fisher-Kolmogorov's equation. Variational approach is used and the fast solutions are obtained as minimizers of an energy functional on a weighted Hilbert space. Numerical experiments are presented.     

A generalized existence theorem of backward doubly stochastic differential equations

In this paper, we deal with a class of one-dimensional backward doubly stochastic differential equations （BDSDEs）. We obtain a generalized comparison theorem and a generalized existence theorem of BDSDEs.     

共振条件下的常微分方程组2π-周期解的存在性

本文获得了常微分方程组在G（t,u）次线性并满足一定的强制性条件下的2π-周期解的存在性．     

Homoclinic Solutions and Chaos in Ordinary Differential Equations with Singular Perturbations

Ordinary differential equations are considered which contain a singular perturbation. It is assumed that when the perturbation parameter is zero, the equation has a hyperbolic equilibrium and homoclinic solution. No restriction is placed on the dimension of the phase space or on the dimension of intersection of the stable and unstable manifolds. A bifurcation function is established which determines nonzero values of the perturbation parameter for which the homoclinic solution persists. It is further shown that when the vector field is periodic and a transversality condition is satisfied, the homoclinic solution to the perturbed equation produces a transverse homoclinic orbit in the period map. The techniques used are those of exponential dichotomies, Lyapunov-Schmidt reduction and scales of Banach spaces. A much simplified version of this latter theory is developed suitable for the present case. This work generalizes some recent results of Battelli and Palmer.

     

Application of the Kudryashov method for finding exact solutions of the high order nonlinear evolution equations

The application of the Kudryashov method for finding exact solutions of the high order nonlinear evolution equations is considered. Some classes of solitary wave solutions for the families of nonlinear evolution equations of fifth, sixth and seventh order are obtained. The efficiency of the Kudryashov method for finding exact solutions of the high order nonlinear evolution equations is demonstrated.     

Multi-dimensional reflected backward stochastic differential equations and the comparison theorem

In this article, we study the multi-dimensional reflected backward stochastic differential equations. The existence and uniqueness result of the solution for this kind of equation is proved by the fixed point argument where every element of the solution is forced to stay above the given stochastic process, i.e., multi-dimensional obstacle, respectively. We also give a kind of multi-dimensional comparison theorem for the reflected BSDE and then use it as the tool to prove an existence result for the multi-dimensional reflected BSDE where the coefficient is continuous and has linear growth.     

A new conservation theorem

A general theorem on conservation laws for arbitrary differential equations is proved. The theorem is valid also for any system of differential equations where the number of equations is equal to the number of dependent variables. The new theorem does not require existence of a Lagrangian and is based on a concept of an adjoint equation for non-linear equations suggested recently by the author. It is proved that the adjoint equation inherits all symmetries of the original equation. Accordingly, one can associate a conservation law with any group of Lie, Lie-Bäcklund or non-local symmetries and find conservation laws for differential equations without classical Lagrangians.     

A mean value theorem in Asplund spaces

In this paper a nonsmooth mean value theorem in Asplund spaces, under invexity, is provided.