On a certain type of nonlinear differential equations admitting transcendental meromorphic solutions

We study the differential equations w ²+R(z)(w ^(k))² = Q(z), where R(z),Q(z) are nonzero rational functions. We prove

1. if the differential equation w ²+R(z)(w′)² = Q(z), where R(z), Q(z) are nonzero rational functions, admits a transcendental meromorphic solution f, then Q ≡ C (constant), the multiplicities of the zeros of R(z) are no greater than 2 and f(z) = √C cos α(z), where α(z) is a primitive of $\tfrac{1} {{\sqrt {R(z)} }}$ such that √C cos α(z) is a transcendental meromorphic function.
2. if the differential equation w ² + R(z)(w ^(k))² = Q(z), where k ? 2 is an integer and R,Q are nonzero rational functions, admits a transcendental meromorphic solution f, then k is an odd integer, Q ≡ C (constant), R(z) ≡ A (constant) and f(z) = √C cos (az + b), where $a^{2k} = \tfrac{1} {A}$ .

     

The transcendental meromorphic solutions of a certain type of nonlinear differential equations

In this paper, we study the differential equations of the following form w²+R(z)²(w(k))=Q(z), where R(z), Q(z) are nonzero rational functions. We proved the following three conclusions: (1) If either P(z) or Q(z) is a nonconstant polynomial or k is an even integer, then the differential equation w²+P²(z)²(w(k))=Q(z) has no transcendental meromorphic solution; if P(z), Q(z) are constants and k is an odd integer, then the differential equation has only transcendental meromorphic solutions of the form f(z)=acos(bz+c). (2) If either P(z) or Q(z) is a nonconstant polynomial or k>1, then the differential equation w²+(z−z₀)P²(z)²(w(k))=Q(z) has no transcendental meromorphic solution, furthermore the differential equation w²+A(z−z₀)²(w^′)=B, where A, B are nonzero constants, has only transcendental meromorphic solutions of the form , where a, b are constants such that Ab²=1, a²=B. (3) If the differential equation , where P is a nonconstant polynomial and Q is a nonzero rational function, has a transcendental meromorphic solution, then k is an odd integer and Q is a polynomial. Furthermore, if k=1, then Q(z)≡C (constant) and the solution is of the form f(z)=Bcosq(z), where B is a constant such that B²=C and q^′(z)=±P(z).     

Julia Sets of Polynomials are Uniformly Perfect

Let f be a polynomial of degree at least two. We shall showthat the Julia set J(f) of f is uniformly perfect. This meansthat there is a constant c(0,1) depending on f only such thatwhenever zJ(f) and 0 < r < diam J(f) then J(f) intersectsthe annulus {w:cr |w — z| r}.     

Dynamics of meromorphic functions with direct or logarithmic singularities

Let f be a transcendental meromorphic function and denote byJ(f) the Julia set and by I(f) the escaping set. We show thatif f has a direct singularity over infinity, then I(f) has anunbounded component and I(f)J(f) contains continua. Moreover,under this hypothesis I(f)J(f) has an unbounded component ifand only if f has no Baker wandering domain. If f has a logarithmicsingularity over infinity, then the upper box dimension of I(f)J(f)is 2 and the Hausdorff dimension of J(f) is strictly greaterthan 1. The above theorems are deduced from more general resultsconcerning functions which have ‘direct or logarithmictracts’, but which need not be meromorphic in the plane.These results are obtained by using a generalization of Wiman–Valirontheory. This method is also applied to complex differentialequations.     

Uniform Lattice Point Estimates for Co-Finite Fuchsian Groups

In this paper we obtain uniform estimates for the lattice pointproblem in the hyperbolic plane H under the assumption thatthe action is by a Fuchsian group which is co-finite. We fixa point w from H and set N_t(z, w) equal to the number of translatesof w by the group which lie in a geodesic ball of radius tcentred at a point z of H. The behaviour of N_t(z, w) is thenexamined when t is large and z is allowed to vary over H. Weshow that the finite quantity depends crucially on the point w, and indeed can become arbitrarilylarge with w. On the other hand, for the average of this quotientwe derive the estimate as t , where the implied constant is an explicit function ofw. In this formula, vol(F) is the hyperbolic volume of a Dirichletfundamentaldomain F for , and |_w| denotes the number of elements from fixing w. This estimate is then combined with a recent samplingtheorem of K. Seip to obtain an inequality which decides whetheror not the orbit . w forms a set of interpolation for a givenweighted Bergman space in H. 1991 Mathematics Subject Classification:11F72, 11P21, 30D35, 30E05, 30F35.     

Normal Families and Shared Values

For f a meromorphic function on the plane domain D and a C,let _f(a) = {z D: f(z) = a}. Let F be a family of meromorphicfunctions on D, all of whose zeros are of multiplicity at leastk. If there exist b 0 and h > 0 such that for every f F,_f(0) = _f^(k)(b) and 0 < |f^(k+1)(z)| h whenever z _f(0), thenF is a normal family on D. The case _f(0) = Ø is a celebratedresult of Gu [5]. 1991 Mathematics Subject Classification 30D45,30D35.     

Common Borel Radii of an Algebroid Function and Its Derivative

In this article, by comparing the characteristic functions, we prove that for any ν-valued algebroid function w(z) defined in the open unit disk with ${\limsup_{r\rightarrow1-}T(r,w)/\log\frac{1}{1-r}=\infty}$ and the hyper order ρ ₂(w)?=?0, the distribution of the Borel radii of w(z) and w′(z) is the same. This is the extension of G. Valiron’s conjecture for the meromorphic functions defined in ${\widehat{\mathbb{C}}}$ .     

A reverse Denjoy theorem

Suppose that C₁ and C₂ are two simple curves joining 0 to ,non-intersecting in the finite plane except at 0 and enclosinga domain D which is such that, for all large r, has measure at most 2, where 0 < < .Suppose also that u is a non-constant subharmonic function inthe plane such that u(z) = B(|z|, u) for all large z C₁ C₂.Let A_D(r, u) = inf { u(z):z D and | z | = r }. It is shownthat if A_D(r, u) = O(1) (or A_D(r, u) = o(B(r, u))), then lim_r B(r, u)/r^/2 > 0 (or lim_r log B(r, u)/log r /2).     

Superconvergence of a Collocation-type Method for Hummerstein Equations

This paper considers the numerical solution of Hammerstein equationsof the form by a collocation method applied not to this equation, but ratherto an equivalent equation for z(t) :=g(t, y(t)). The desiredapproximation to y is then obtained by use of the (exact) equation In an earlier paper, questions of existence and optimal convergenceof the respective approximations to z and y were examined. Inthis sequel, collocation approximations to z are sought in certainpiecewise polynomial function spaces, and analogous of knownsuperconvergence results for the iterated collocation solutionof (linear) second-kind Fredhoim integral equations are statedand proved for the approximation to y.     

The Existence of Periodic and Subharmonic Solutions of Subquadratic Second Order Difference Equations

By critical point theory, a new approach is provided to studythe existence of periodic and subharmonic solutions of the secondorder difference equation where f C(R x R^m, R^m), f(t+M,z)+f(t,z) for any (t, z)R x R^mand M is a positive integer. This is probably the first timecritical point theory has been applied to deal with the existenceof periodic solutions of difference systems.     

Conformal Conjugacies in Baker Domains

Let F be a meromorphic function in the complex plane. We investigatethe behaviour of the iterates of F in a Baker domain B. In particular,we describe the dynamics of the orbits with the help of conformalconjugacies; that is, we determine a function which is univalentin a large simply connected subdomain of B such that (F(z))=T((z))holds throughout B. Here T is either a parabolic or hyperbolicMöbius transformation mapping either a half plane or Conto itself. This functional equation is always solvable ina Baker domain if F has only finitely many poles. Moreover,there is an example of a function with infinitely many poleswhere one cannot find an appropriate conformal conjugacy inan invariant Baker domain.     

Stability of Gorenstein categories

We show that an iteration of the procedure used to define theGorenstein projective modules over a commutative ring R yieldsexactly the Gorenstein projective modules. Specifically, givenan exact sequence of Gorenstein projective R-modules

such that the complexes Hom_R(G, H) and Hom_R(H,G) are exact for each Gorenstein projective R-module H, themodule Coker() is Gorensteinprojective. The proof of this result hinges upon our analysisof Gorenstein subcategories of abelian categories.     

Isomorphisms between Semisimple Weighted Measure Algebras

It is shown that, if is an isomorphism between semisimple weightedmeasure algebras M(w₁) and M(w₂), then maps L¹(R⁺, w₁) ontoL¹(R⁺, w₂). This is used to describe all the automorphisms ofM(R⁺, w). A necessary and sufficient condition is given forM(w₁) and M(w₂) to be isomorphic.     

The Hausdorff Dimension of Julia Sets of Entire Functions IV

It is known that, for a transcendental entire function f, theHausdorff dimension of J(f) satisfies 1 dimJ(f) 2. For eachd (1, 2), an example of a transcendental entire function fwith dimJ(f) = d is given. It is then indicated how this functioncan be modified to produce a transcendental meromorphic functionF with one pole with dimJ(F) = d. These appear to be the firstexamples of Julia sets with non-integer dimensions whose dimensionshave been calculated exactly.     

Finiteness of integrals of functions of Levy processes

We prove necessary and sufficient conditions for the almostsure convergence of the integrals

and thus of ,where M_t = sup{|X_s|: s t} is the two-sided maximum processcorresponding to a Lévy process (X_t)_{t 0}, a(·)is a non-decreasing function on [0, ) with a(0) = 0, g(·)is a positive non-increasing function on (0, ), possibly withg(0 + ) = , and f(·) is a positive non-decreasing functionon [0, ) with f(0) = 0. The conditions are expressed in termsof the canonical measure, (·), of the process X_t. Thespecial case when a(x) = 0, f(x) = x and g(·) is equivalentto the tail of (at zero or infinity) leads to an interestingcomparison of M_t with the largest jump of X_t in (0, t]. Some results concerning the convergence at zero and infinityof integrals like t g(a(t) + |X_t|) dt, t g(S_t) dt,and t g(R_t) dt, where S_t is the supremum process and R_t= S_t – X_t is the process reflected in its supremum, arealso given. We also consider the convergence of integrals suchas , etc.     

The Cauchy Transform * (Mathematical Surveys and Monographs 125) * By Joseph A. Cima, Alec L. Matheson and William T. Ross: * 272 pp., {pound}43.75 (US$75), ISBN 0-8218-3871-7 * (American Mathematical Society, Providence, RI, 2006).

In the border country between complex analysis, harmonic analysisand differential equations, there can be found many populartransforms, of which those of Fourier and Laplace are probablythe most commonly used. This book is concerned with anotherone, which is more rarely sighted. The Cauchy transform of a finite complex measure µ definedon the unit circle in the complex plane is the analytic functionKµ defined by

forz in the unit disc . Its name derives from the observation thatCauchy's integral formula, when applied to an analytic functionf, is equivalent to the statement that the Cauchy transformof the measure is just     

Simultaneous determination of the source terms in a linear hyperbolic problem from the final overdetermination: weak solution approach

The problem of determining the pair w:={F(x, t);f(t)} of sourceterms in the hyperbolic equation u_tt = (k(x)u_x)_x + F(x, t) andin the Neumann boundary condition k(0)u_x(0, t) = f(t) from themeasured data µ(x):=u(x, T) and/or (x):=u_t(x, t) at thefinal time t = T is formulated. It is proved that both componentsof the Fréchet gradient of the cost functionals J₁(w)= ||u(x, t;w) – µ(x)||₀² and J₂(w) = ||u_t(x, T;w)– (x)||₀² can be found via the solutions of correspondingadjoint hyperbolic problems. Lipschitz continuity of the gradientis derived. Unicity of the solution and ill-conditionednessof the inverse problem are analysed. The obtained results permitone to construct a monotone iteration process, as well as toprove the existence of a quasi-solution.     

Malmquist theorem for solutions of differential equations in a neighborhood of a logarithmic singular point

The Malmquist theorem (1913) on the growth of meromorphic solutions of the differential equation f = P(z,f) / Q(z,f), where P(z,f) and Q(z,f) are polynomials in all variables, is proved for the case of meromorphic solutions with logarithmic singularity at infinity.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 4, pp. 476–483, April, 2004.