On the asymptotic behavior and radial symmetry of positive solutions of semilinear elliptic equations in R n II. Radial symmetry

Communicated by J. Serrin     

Necessary and Sufficient Conditions for the Invertibility of the Nonlinear Difference Operator \left( {\mathcal{D}x} \right)\left( t \right) = x\left( {t + 1}\right) - f\left( {x\left( t \right)} \right) in the Space of Functions Bounded and Continuous on the Axis

We find necessary and sufficient conditions for the nonlinear difference operator $\left( {\mathcal{D}x} \right)\left( t \right) = x\left( {t + 1} \right) - f\left( {x\left( t \right)} \right)$ $t \in \mathbb{R}$ , where $f:\mathbb{R} \to \mathbb{R}$ is a continuous function, to have the inverse in the space of functions bounded and continuous on $\mathbb{R}$ .     

Exact solution of the non-linear differential equation RR̈ + 32R2 − AR−4 + B = 0

The non-linear equation $\text{R}\text{R}\text{?}\text{+}\text{3}\text{2}\text{R}{}^2\text{- AR}{}^{\mathrm{?4}}\text{+ B = 0}$ is shown to represent simply periodic motion with a minimum at R₁ and a maximum at R₁R₀ or a maximum at R₁ and a minimum at R₁R₀^?1. R₀ is a function of the ratio $\text{A}\text{B}$ and is greater than 1 for $\text{A}\text{B}$ > 1 and less than 1 for $\text{A}\text{B}$ > 1. The period of the motion satisfies the simple relation T(R₀^?1) = R₀^?1T(R₀). The exact solution to the above equation is represented in terms of elliptic integrals of the first and second kinds and a simple algebraic function.