Complete asymptotic analysis of positive solutions of odd-order nonlinear differential equation

We study the asymptotic behavior of solutions of the odd-order differential equation of Emden–Fowler type $$ {x^{{\left( {2n+1} \right)}}}(t)=q(t){{\left| {x(t)} \right|}^{\gamma }}\operatorname{sgn}x(t) $$ in the framework of regular variation under the assumptions that 0 < γ < 1 and q(t) : [a, ∞) → (0, ∞) is regularly varying function. We show that complete and accurate information can be acquired about the existence of all possible positive solutions and their asymptotic behavior at infinity.     

Asymptotic analysis of positive solutions of generalized Emden-Fowler differential equations in the framework of regular variation

Positive solutions of the nonlinear second-order differential equation $(p(t)|x'|^{\alpha - 1} x')' + q(t)|x|^{\beta - 1} x = 0,\alpha > \beta > 0,$ are studied under the assumption that p, q are generalized regularly varying functions. An application of the theory of regular variation gives the possibility of obtaining necessary and sufficient conditions for existence of three possible types of intermediate solutions, together with the precise information about asymptotic behavior at infinity of all solutions belonging to each type of solution classes.     

The Euler Method in the Field of Operators

Approximate solutions of a class of nonlinear differential equations are constructed in the field of Mikusiński operators by using a method similar to Euler's. Their character is analyzed and the error of approximation is estimated. The results are applied to a class of partial integro–differential equations.     

Asymptotic analysis of Emden-Fowler differential equations in the framework of regular variation

Sufficient conditions are established for the existence of slowly varying solution and regularly varying solution of index 1 of the second-order nonlinear differential equation $$x^{\prime\prime}(t)+q(t)|x(t)|^{\gamma}\,{\rm sgn}\, x(t)=0, \quad \quad (A)$$ where γ is a positive constant different from 1 and q : [a, ∞) → (0, ∞) is a continuous integrable function. We show how an application of the theory of regular variation gives the possibility of determining the precise asymptotic behavior of solutions of both superlinear and sublinear equation (A).     

Syntheses,Structures and Luminescent Properties of Metal Complexes with Imidazole‐Containing Polyamine Ligand

Four novel metal complexes [Cd(L)Cl](BF₄)·H₂O ( 1 ), [Cd(L)Br]₂[CdBr₄]·2H₂O ( 2 ), [Cu(L)Br]Br ( 3 ), and [Cu(L)](NO₃)₂ ( 4 ) were prepared by the reactions of ligand N¹‐(2‐aminoethyl)‐N¹‐(2‐imidazolethyl)‐ethane‐1,2‐diamine (L) with metal salts under different reaction conditions. All of these complexes exhibit 1D chains, but different structures. The results showed that the pH value of the reaction solution and counter ions have remarkable impact on the structure of the complexes. Furthermore, complexes 1 and 2 represent fluorescence properties in the solid state at room temperature.     

The numerical methods in the field of operators

The approximate solution of a class of nonlinear differential equations, in the field of Mikusiński operators is constructed by using the Euler method. The obtained results are applied to a class of partial integro–differential equations. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)