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1.
Existence of positive solutions for the nonlinear fractional differential equation
D
αu = f(x,u), 0 < α < 1 has been given (S. Zhang. J. Math. Anal. Appl. 252 (2000), 804–812) where D
α denotes Riemann–Liouville fractional derivative. In the present work we extend this analysis for n-term non autonomous fractional
differential equations. We investigate existence of positive solutions for the following initial value problem
with initial conditions
where
is the standard Riemann–Liouville fractional derivative. Further the conditions on a
j
’s and f, under which the solution is (i) unique and (ii) unique and positive as well, are given 相似文献
2.
We consider some classes of 2π-periodic functions defined by a class of operators having certain oscillation properties, which include the classical Sobolev class and a class of analytic functions which can not be represented as a convolution class as its special cases. Let be the largest integer not bigger than x. We prove that on these classes of functions the rectangular formula
is optimal among all quadrature formulae of the form
where the nodes 0 ≤ t
1 < ... < t
n
< 2π and the coefficients (weights) are arbitrary, i = 1,...,n, j = 0,1,..., ν
i
− 1, and (ν1,...,ν
n
) is a system of positive integers satisfying the condition . In particular, the rectangular formula is optimal for these classes of functions among all quadrature formulae of the form
with free nodes 0 ≤ t
1 < ... < t
N
< 2π and arbitrary weights . Moreover, we exactly determine the error estimates of the optimal quadrature formulae on these classes of functions.Project supported by the National Natural Science Foundation of China (Grant No. 10671019) and Research Fund for the Doctoral Program Higher Education (Grant No. 20050027007). 相似文献
3.
In this paper we prove that the inverse mean problem of geometric and golden means of positive definite matrices
is solvable (resp. uniquely solvable) if and only if
.
Received: 9 March 2006 相似文献
4.
For a trigonometric series
defined on [−π, π)
m
, where V is a certain polyhedron in R
m
, we prove that
if the coefficients a
k
satisfy the following Sidon-Telyakovskii-type conditions:
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 5, pp. 579–585, May, 2008. 相似文献
5.
We analyze polynomials P
n
that are biorthogonal to exponentials
, in the sense that
Here α>−1. We show that the zero distribution of P
n
as n→∞ is closely related to that of the associated exponent polynomial
More precisely, we show that the zero counting measures of {P
n
(−4nx)}
n=1∞ converge weakly if and only if the zero counting measures of {Q
n
}
n=1∞ converge weakly. A key step is relating the zero distribution of such a polynomial to that of the composite polynomial
under appropriate assumptions on {Δ
n,j
}.
相似文献
6.
We prove the following statement.
Let , and let . Suppose that, for all and , the sequence satisfies the relation
where e(u) : = e2πiu
.
Then
where
q is the set of q-multiplicative functions g such that . 相似文献
7.
For positive integers with a
r
= 2, the multiple zeta value or r-fold Euler sum is defined as [2]
. There is a celebrated sum formula [6, 10] among multiple zeta values as
,
where range over all positive integers with in the summation.
In this paper, we shall prove the so called restricted sum formula [4]. Namely, for all positive integers m and q with m ≥ q and a nonnegative integer p, that
. We prove the assertion by new expressions of multiple zeta values in terms of Drinfeld integrals.
This work was supported by the Department of Mathematics, National Chung Cheng University and by the National Science Council
of Taiwan, Republic of China. 相似文献
8.
We prove that the Cauchy problem for a hyperbolic, homogeneous equation with
coefficients depending on time, is well posed in every Gevrey class, although in general it is not well-posed in
provided the characteristic roots satisfy the condition
相似文献
9.
Giuseppe Maria Coclite Angelo Favini Gisèle Ruiz Goldstein Jerome A. Goldstein Silvia Romanelli 《Semigroup Forum》2008,77(1):101-108
The solution u of the well-posed problem
depends continuously on (a
ij
,β,γ,q).
Dedicated to Karl H. Hofmann on his 75th birthday. 相似文献
10.
On a periodic boundary value problem for cyclic feedback type linear functional differential systems
Sulkhan Mukhigulashvili 《Archiv der Mathematik》2006,87(3):255-260
Nonimprovable effective sufficient conditions are established for the unique solvability of the periodic problem
where ω > 0, ℓi : C([0, ω])→ L([0,ω]) are linear bounded operators, and qi∈L([0, ω]).
Received: 11 June 2005 相似文献