Approximation and solution of a general symmetric functional equation

In this paper, we investigate the Hyers–Ulam–Rassias stability of a general equation f(_φ1(x,y))=_φ2(f(x),f(y))$f\left({\phi }_1(x,y)\right)={\phi }_2(f\left(x\right),f\left(y\right))$ in metric spaces. As a consequence, we obtain some stability results in the sense of Hyers–Ulam–Rassias.     

A functional equation with contracting argument

Novosibirsk. Translated from Sibirskii Mathematicheskii Zhurnal, Vol. 31, No. 2, pp. 216–218, March–April, 1990.     

A functional equation

The translational functional equation (1) with right-hand side analytic in a finite convex region D is investigated. It is proved that a solution exists which is analytic in a finite convex region G₁ determined by G. A corollary of this result is the existence of analytic solutions of differential-difference equations of finite order and of difference equations of infinite order with constant coefficients.Translated from Matematicheskie Zametki, Vol. 5, No. 6, pp. 733–742, June, 1969.     

A functional equation

Existence theorems for and the determination of continuous solutions, defined on the real axis R, of the functional equationf (t)=A[t,f (at–b),f (at–c)], wherea, b, and c are real parameters, A:R×E×E E is a continuous operator, and E is a Banach space.Translated from Matematicheskie Zametki, Vol. 9, No. 2, pp. 161–170, February, 1971.The author wishes to thank B. N. Sadovskii for his interest and help in this work.     

A non-radially symmetric solution to a class of elliptic equation with Kirchhoff term

We consider the following equation with Kirchhoff term $-(a+b\int_{\mathbb{R}^3} {|\nabla u|^2} dx)$ $\Delta u + u =|u|^{p-2}u$, $u \in H^1 (\mathbb{R}^3)$, where $a, b$ are positive constants and $2 < p < 6$. By deducing a variant variational identity and a constraint set, we are able to prove the existence of a non-radially symmetric solution $u(x_1, x_2, x_3)$ for the full range of $p\in (2,6)$. Moreover this solution $u(x_1, x_2, x_3)$ is radially symmetric with respect to $(x_1,x_2)$ and odd with respect to $x_3$.     

A linear functional equation with a singularity at a fixed point

Summary. Local solutions of the functional equation¶¶z^k f( z) = ?_k=1ⁿG_k( z) f( s_kz ) +g( z) z{^\kappa} \phi \left( z\right) =\sum_{k=1}^nG_k\left( z\right) \phi \left( s_kz \right) +g\left( z\right) ¶with k > 0 \kappa > 0 and | s_k| \gt 1 \left| s_k\right| \gt 1 are considered. We prove that the equation is solvable if and only if a certain system of k \kappa conditions on G_k (k = 1, 2, ... , n) and g is fulfilled.     

A remark on a logarithmic functional equation

We revisit the logarithmic functional equation of Heuvers and Kannappan [K.J. Heuvers, Pl. Kannappan, A third logarithmic functional equation and Pexider generalizations, Aequationes Math. 70 (2005) 117-121] and give a simple proof of the result and discuss the locally integrable solutions of the equation.     

Spherically symmetric solutions of a reaction-diffusion equation

Two theorems are proved for the spherically symmetric solutions of the “bistable” reaction-diffusion equation $\text{u}{}_{\mathrm{t}}\text{= Δ}{}_{\mathrm{x}}\text{u + ?(u)}$, where ? is cubic-like and x ∈ Rⁿ. The first theorem says that, for a suitable class of initial data, there are only two types of asymptotic behavior, u(x, t) tends to an equilibrium solution as t → + ∞ or u(x, t) → 1 uniformly on compact sets. The second theorem says that in the latter case, if the solution is followed out along any ray, it approaches, in shape, the one-dimensional travelling wave.     

A curious functional equation

For Israel Gohberg, outstanding analyst, with affection and admiration.     

A sine functional equation

Aequationes mathematicae -