Consider the general sublinear ordinary differential equations of second order of theform X(t) + a(t)f(X(t)) =0, where a(t)∈C[t0,∞),f(x)∈C(R)and xf(x)>0,f'(x) ≥ 0 for x ≠ 0.Furthermore,f(x) also satisfies ...Consider the general sublinear ordinary differential equations of second order of theform X(t) + a(t)f(X(t)) =0, where a(t)∈C[t0,∞),f(x)∈C(R)and xf(x)>0,f'(x) ≥ 0 for x ≠ 0.Furthermore,f(x) also satisfies sublinear condition, which coversthe prototype nonlinear function f(x)=|x|γ sgnx with 0<γ<1 known as the EmdenFowler case. The coefficient α(t) is not assumed to be eventually nonnegative. A new oscillation criterion involving integral averages of α(t) due to Kamenev (Math. Zametki 23(1978), 249 - 251 ) for linear equation and Wong (Conference Proceedings, Canad. Math.Soc. 8(1987), 299-302) for Emder-Fowler equation with 0<γ<1 is shown to remainvalid for general sublinear equation.展开更多
In this paper the functional-differential equation of the distribution of primenumbers derived by de Visme [4] is strdied in details. It is proved that there existinfinitely many solutions of this equation , and all p...In this paper the functional-differential equation of the distribution of primenumbers derived by de Visme [4] is strdied in details. It is proved that there existinfinitely many solutions of this equation , and all positive strong solutions andpositive solutions of the Cauchy problem behave very much like the distrbutionfunction of prime numbers. The results are complementary to Driver [1] andwright [3].展开更多
文摘Consider the general sublinear ordinary differential equations of second order of theform X(t) + a(t)f(X(t)) =0, where a(t)∈C[t0,∞),f(x)∈C(R)and xf(x)>0,f'(x) ≥ 0 for x ≠ 0.Furthermore,f(x) also satisfies sublinear condition, which coversthe prototype nonlinear function f(x)=|x|γ sgnx with 0<γ<1 known as the EmdenFowler case. The coefficient α(t) is not assumed to be eventually nonnegative. A new oscillation criterion involving integral averages of α(t) due to Kamenev (Math. Zametki 23(1978), 249 - 251 ) for linear equation and Wong (Conference Proceedings, Canad. Math.Soc. 8(1987), 299-302) for Emder-Fowler equation with 0<γ<1 is shown to remainvalid for general sublinear equation.
文摘In this paper the functional-differential equation of the distribution of primenumbers derived by de Visme [4] is strdied in details. It is proved that there existinfinitely many solutions of this equation , and all positive strong solutions andpositive solutions of the Cauchy problem behave very much like the distrbutionfunction of prime numbers. The results are complementary to Driver [1] andwright [3].
