A note on fuzzy differential equations

We study the effect of the forcing term to the solution of a fuzzy differential equation.     

A note on the oscillatory property for nonlinear difference equations and differential equations

Criteria are given to determine the oscillatory property of solutions of the nonlinear difference equation: Δ^du_n + ∑_{i = 1}^mp_inf_i(u_n, Δu_n,…,Δ^{d ? 1}u_n) = 0, n = 0, 1, 2,…, where d is an arbitrary integer, generalizing results that have been obtained by B. Szmanda (J. Math. Anal. Appl.79 (1981), 90–95) for d = 2. Analogous results are given for the differential equation: u^(d) + ∑_{i = 1}^mp_i(t)f_i(u, u′,…, u^{(d ? 1)}) = 0, t ? t₀, which coincide with the criteria given by 2., 3., 599–602) and 4., 5., 6., 715–719) for the case m = 1.     

A note on the positive solutions of second order differential equations

We obtain an existence theorem for monotone positive solutions of nonlinear second-order nonlinear differential equations via principal and nonprincipal solutions of the corresponding homogeneous equation.     

A note on non-separable solutions of linear partial differential equations

A method has been presented for constructing non-separable solutions of homogeneous linear partial differential equations of the type F(D, D′)W = 0, where $\text{D =}\text{?}\text{?x}\text{, D′ =}\text{?}\text{?y}$,

$\text{F(D,D′)=}\text{∑}\text{n}\text{r+s=0}\text{C}{}_{\mathrm{rs}}\text{D}{}^{\mathrm{r}}\text{D′}{}^{\mathrm{s}}\text{,}$

where c_rs are constants and n stands for the order of the equation. The method has also been extended for equations of the form Φ(D, D′, D″)W = 0, where $\text{D =}\text{?}\text{?x}\text{, D′ =}\text{?}\text{?y}\text{, D″ =}\text{?}\text{?z}$ and

$\text{Φ(D,D′,D″)W=}\text{∑}\text{n}\text{r+s+t=0}\text{C}{}_{\mathrm{rst}}\text{D}{}^{\mathrm{r}}\text{D′}{}^{\mathrm{s}}\text{D}{}^{\mathrm{t}}\text{D″}{}^{\mathrm{s}}\text{.}$

As illustration, the method has been applied to obtain nonseparable solutions of the two and three dimensional Helmholtz equations.     

A note on the Gautschi-type method for oscillatory second-order differential equations

The Gautschi-type method has been proposed by Hochbruck and Lubich for oscillatory second-order differential equations. They conjecture that this method allows for a uniform error bound independent of the size of the system. The conjecture is proved in this note.     

A note on exponential stability of quasi-linear ordinary differential equations

It is shown that the exponential stability of the zero solution of a quasi-linear ordinary differential equation can be characterized completely by its linear part.     

A note on exponential stability of quasi-linear ordinary differential equations

It is shown that the exponential stability of the zero solution of a quasi-linear ordinary differential equation can be characterized completely by its linear part.     

A note on sharply bounded arithmetic

Summary We prove some independence results for the bounded arithmetic theoryR ₂ ⁰ , and we define a class of functions that is shown to be an upper bound for the class of functions definable by a certain restricted class of ₁ ^b in extensions ofR ₂ ⁰ .Mathematics subject classification: 03F30This article was processed by the author using the LATEX style filepljour1 from Spinger-Verlag.     

A note on arithmetic Diophantine series

Czechoslovak Mathematical Journal - We consider an asymptotic analysis for series related to the work of Hardy and Littlewood (1923) on Diophantine approximation, as well as Davenport. In...