where p > , φp(x) = |x|p–x for x =  and φp() = , c ∈ Cn(R,R) and c(t + T) ≡ c(t), f is a continuous function defined in R and periodic in t with f (t, ·) = f (t + T , ·) and f (t, ) = , g(t,x) = g(x) + g(t,x), where g : R × (, +∞) → R is an L-Carathéodory function, g(t, ·) = g(t + T , ·), g ∈ C((,∞);R) has a singularity at x = , e : R→ R is a continuous periodic function with ...

We study the existence of almost periodic mild solutions of a class of partial functional differential equations via semilinear almost periodic abstract functional differential equations of the form (∗) x′ = f(t, x, xt) . To this end, we first associate with every almost periodic semilinear equation (∗∗) x′ = F (t, x) a nonlinear semigroup in the space of almost periodic functions. We then give...

‎In this paper‎, ‎we obtain exact solutions involving parameters of some nonlinear PDEs in mathmatical physics; namely the one-‎dimensional modified complex Ginzburg-Landau equation by using the $ (G'/G) $ expansion method‎, homogeneous balance method, extended F-expansion method‎. ‎By ‎using homogeneous balance principle and the extended F-expansion, more periodic wave solutions expressed by j...

Let p be a prime number and f ∈ Zp[x] a polynomial over the p-adic integers lifting the Frobenius in the sense that f(x) ≡ xp (mod p) and deg(f) = p. Let Πf := {ζ ∈ Q p : f◦n(ζ) = ζ for some n ∈ Z+} be the set of f -periodic points. We show that an irreducible algebraic varieties X ⊆ Am which meet Γ×m f in a Zariski dense set only in the case that X is defined by equations of the form xi = a fo...

The main feature of this paper concerns extensions of the Liouville theorem to the following class of elliptic equations in non-divergence form: aij(x)∂iju + bi(x)∂iu + c(x)u = 0 in R N , with c ≤ 0. We show that the Liouville property holds (that is, the space of bounded solutions has at most dimension one) if the coefficients aij , bi and c are periodic, with the same period, and it does not ...

Let M be a compact oriented Riemannian surface with non-positive curvature. Suppose that F : R× [0, ε] → M is an isometric immersion. We show that either M is isometric to flat torus or the image of F is periodic, i.e., it is a periodic flat strip in M. Introduction The results of this paper are related to a well-known problem to the effect that the geodesic flow on the unit tangent bundle of a...

In this paper will consider diffeomorphisms f : M → M of a compact surface M , isotopic to the identity map, and study the growth of their periodic points. An important approach to studying such properties of diffeomorphisms is via the rotation set ρ(f) and the associated rotation vectors. For the particular case of homeomorphisms of tori Franks has shown that if ρ(f) has non-empty interior the...

It is well known that (see [9]) Cesaro means of 2π-periodic functions f ∈ Lp(T) (1 ≤ p ≤ ∞) converges by norms. Hereby T is denoted the interval (−π,π). The problem of the mean summability in weighted Lebesgue spaces has been investigated in [6]. A 2π-periodic nonnegative integrable function w : T→R1 is called a weight function. In the sequel by L p w(T), we denote the Banach function space of ...