sequencesHölder inequalityMinkowski inequality.In this paper, a general theorem on absolute Riesz summability factors of infinite series is proved under weaker conditions. Also we have obtained some new and known results.doi:10.1515/tmj-2017-0025Hüseyin Bor...
monotone convergence theorem monotone decreasing function monotone decreasing sequence monotone function monotone increasing function monotone increasing sequence monotone nondecreasing function monotone nondecreasing sequence monotone nonincreasing function monotone nonincreasing sequence ...
We establish a one-to-one correspondence between (i) exchangeable sequences of random variables whose finite-dimensional distributions are minimum (or maxi
Theorem 3.8.4 Let q > 0, r ⩾ 0, α > 0, β > 0 be real numbers with 1α+1β=1 and qα ⩾ 1, and {bk}, k∈ ℕ0,n, be a sequence of nonnegative real numbers. Let {xk} and {yk}, k∈ ℕ0,n, be sequences of real numbers with x0 = y0 = xn = yn = ...
It is well known that the distribution of extreme values of strictly stationary sequences differ from those of independent and identically distributed sequ
If u does not vanish in (0, ∞), then u is decreasing on (0, ∞). Integrating the equation from 0 to s∈ (0, r) we find (2.19)−u′(s)=ϕ−1(1 sN−1∫0sξN−1(λψ(u(ξ))+g(λ,u(ξ)))dξ). Integrating (2.19) from r /2 to r yields (2.20)u(r/...
0þ h With this definition, we have the following Lyapunov– Razumikhin theorem: Theorem 2.1: Suppose f : ! Rn maps bounded sub- sets of into bounded subsets of Rn and consider (1). Suppose v, w : Rþ ! Rþ are continuous, non-decreasing functions, v(s) positive for ...
Thus, bringing a quasi-deterministic system to a more stochastic regime by decreasing β to the critical value βc (with ΔJ smaller than ΔJc(∞)) increases the diversity of irreversible patterns and hence entropy production. However, further reduction of β makes the system more random (i....
wherep,q>0,u0(x),v0(x)∈C(Ω¯)are nonnegative and nontrivial functions,Ω∈RNa bounded connected and smooth domain. We determine the existence and uniqueness of the solution. The blow-up phenomenon is considered and the blow-up rates are obtained. ...
In this case, by property (P3) of the previous lemma, we obtain that the non-decreasing sequence {αn} is contained in [12,1). Also, since T is non-expansive and with at least one fixed point, it follows by standard arguments that {xn} is Fejér-monotone with respect to Fix(...