On meromorphic solutions of generalized algebraic differential equations

Summary In this paper we investigate the rate of growth of meromorphic functions f which are solutions of certain algebraic differential equation whose coefficients a(z) are arbitrary meromorphic functions. By method based on Nevanlinna's theory of meromorphic functions, it has been shown that if the zeros and poles of f satisfy the condition N(r, f′/f)=S(r, f′/f) then the ratio T(r, f′/f)/(T(r, a(z)), as r → ∞ outside a set of r values of finite measure, is bounded for at least one of the coefficients a(z). The content of an invited address delivered by the author on March 27, 1971 to the 683th meeting of the American Mathematical Society of the University of Illinois at Chicago Circle, Chicago, Illinois, U.S.A. Entrato in Redazione il 16 novembre 1970.     

On the algebraic properties of solutions of inhomogeneous hypergeometric equations

Generalized hypergeometric differential equations of arbitrary order are considered. Necessary and sufficient conditions for the algebraic independence of solutions of collections of such equations, as well as of their values at algebraic points, are obtained.     

On the growth of entire solutions of algebraic differential equations

This is the opening memorial plenary lecture on memorial meeting in honor of Professor Shlomo Strelitz on Conference of Differential Equations and Complex Analysis, University of Haifa, Israel, December 2000. Shlomo Strelitz: 7.1.1923–27.9.1999, teached at Vilnius University 1946–1973, professor 1967–1973.__________Published in Lietuvos Matematikos Rinkinys, Vol. 45, No. 1, pp. 57–63, January–March, 2005.     

On the order of growth of solutions of algebraic differential equations

Assume thatf is an integer transcendental solution of the differential equationP _n (z, f, f′)=P _n−1(z, f, f′, ... f ^(p)), whereP _n andP _n−1 are polynomials in all variables, the degree ofP _n with respect tof andf′ is equal ton, and the degree ofP _n−1 with respect tof, f′, ... f ^(p) is at mostn−1. We prove that the order ρ of growth off satisfies the relation 1/2≤ρ<∞. We also prove that if ρ=1/2, then, for a certain real ν, in the domain {z: ν<argz<ν+2π}/E _*, whereE _* is a certain set of disks with finite sum of radii, the estimate lnf(z)=z ^1/2 (β+o(1)), β∈C, holds forz=re ^iϕ,r≥r(ϕ)≥0. Furthermore, on the ray {z: argz=ν}, the following relation is true: ln‖f(re ^iν)‖=o(r ^1/2),r→+∞,r>0, , where Δ is a certain set on the semiaxisr>0 with mes Δ<∞. “L'vivs'ka Politekhnika” University, Lvov. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 1, pp. 69–77, January, 1999.     

On the unbounded extendability of solutions to nonlinear differential algebraic equations

We consider a system of nonlinear ordinary differential equations that are unsolved for the derivative of the unknown vector function and identically degenerate on the domain. We prove a theorem on the unbounded extendability of solutions to this system, allowing arbitrarily high unsolvability index with respect to the derivative.     

Birkhoff-recurrent solutions to algebraic equations

The paper considers algebraic equations whose coefficients are continuous functions given on a connected topological group. It is proven that if the vector function made up of the coefficients is recurrent in the sense of Birkhoff and if the discriminant of the equation never vanishes, then each continuous solution is recurrent. The proof is based on the theory of extensions of dynamic systems.Translated from Matematicheskie Zametki, Vol. 12, No. 4, pp. 617–623, April, 1973.     

On meromorphic solutions of algebraic differential equations in angular domains

We obtain asymptotic estimates of meromorphic solutions to the differential equationP _n (z, , )=P _n–1 (z, , ,..., ^(m) ) in the angular domain P={z: arg z · }. Here P_n(z, w, w) is a polynomial in all variables, and of degree n with respect to w and w; P_n–1(z, w, w, ..., w^(m)) is a polynomial in all variables, and of degree n –1 with respect to w, w, ..., w^(m) In the particular case, when the solutions are entire functions, these estimates are more precise than the known estimates that are obtained by using the method of Wiman-Valiron, which cannot be applied to meromorphic solutions in the domain P.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 4, pp. 514–523, April, 1992.     

Existence and uniqueness of positive solutions of a system of nonlinear algebraic equations

In this paper we consider the nonlinear system \(\gamma _i(x_i)=\sum _{j=1}^{m}g_{ij} (x_j)\), \( 1\le i \le m\). We give sufficient conditions which imply the existence and uniqueness of positive solutions of the system. Our theorem extends earlier results known in the literature. Several examples illustrate the main result.     

代数微分方程的代数解

Using value distribution theory and techniques,the problem of the algebroid solutions of second order algebraic differential equation is investigated. Examples show that the results are sharp.     

Existence and multiplicity of positive solutions for multi-parameter three-point differential equations system

In this paper, we study the existence and multiplicity of positive solutions for the differential equations system

     

Spectral multiplicity of the solutions of polynomial operator equations

In the note one considers operators T, acting in a Hilbert space and satisfying an equation of the form (T)=A, where is a polynomial, while A is a given normal operator, assumed to be either reductive or unitary. Under these conditions one computes some spectral characteristics of the operator T (spectral multiplicity, disc, lattice of invariant subspaces, etc.). Fundamental examples are the weighted substitution operators (TL²(X,)L²(X,), Tf=·(f·), where is a periodic automorphism of (X,), L (X, ).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 157, pp. 157–164, 1987.The author expresses his sincere gratitude to N. K. Nikol'skii for the formulation of the problem and for the useful discussion of the results.     

Lamé equations with algebraic solutions

In this paper we study Lamé equations L_n,By=0 in so-called algebraic form, having only algebraic functions as solution. In particular we provide a complete list of all finite groups that occur as the monodromy groups, together with a list of examples of such equations. We show that the set of such Lamé equations with is countable, up to scaling of the equation. This result follows from the general statement that the set of equivalent second-order equations, having algebraic solutions and all of whose integer local exponent differences are 1, is countable.     

On the solutions of a system of two Diophantine equations

We obtain all positive integer solutions(m1,m2,a,b)with ab,gcd(a,b)=1 to the system of Diophantine equations km21-lat1bt2a2r=C1,km22-lat1bt2b2r=C2,with C1,C2∈{-1,1,-2,2,-4,4},and k,l,t1,t2,r∈Z such that k0,l0,r0,t10,t2 0,gcd(k,l)=1,and k is square-free.     

Recurrent relations for the solutions of an infinite system of linear algebraic equations

We obtain recurrent relations for bounded solutions of the system of equations $$X_k - \sum\limits_{n = 0}^\infty {\frac{{(k + n)!}}{{k!n!}}} \alpha ^{k + n + 1} x_n = f_{k,} k = 0,1,..., \alpha \in (0,1/2),$$ with right-hand sides {f _k } _k=0 ^∞ ={δ _kj } _k=0 ^∞ ,j=0,1,..., where δ _kj is the Kronecker symbol.