We present a new quadrature rule based on the spline interpolation along with the error analysis. Moreover, some error estimates for the reminder when the integrand is either a Lipschitzian function, a function of bounded variation or a function whose derivative belongs to L are given. We also give some examples to show that, practically, the spline rule is better than the trapezoidal rule. Key...

Let Ω be an open subset of R where 2 ≤ n ≤ 7; we assume n ≤ 2 because the case n = 1 has been treated elsewhere (see [Alli]) and is quite different from the case n > 1; we assume n ≤ 7 is that our work will make use of the regularity theory for area minimizing hypersurfaces. Let F(Ω) = L1(Ω) ∩ L∞(Ω)). Suppose s ∈ F(Ω) and Suppose γ : R→ [0,∞) is locally Lipschitzian, positive on R ∼ {0} and zer...

Let {Ti}i=1 be a finite family of nonexpansive self-maps of H . Denote the common fixed points set of {Ti}i=1 by ⋂N i=1 Fix(Ti). Let F : H → H be a mapping such that for some constants k,η > 0, F is k-Lipschitzian and η-strongly monotone. Let {αn}n=1 ⊂ (0,1), {λn}n=1 ⊂ [0,1) and take a fixed number μ ∈ (0,2η/k2). The iterative schemes concerning nonlinear operators have been studied extensively...

In this paper, characterizations of the degree to which a mapping $mathcal{T} : L^{X}longrightarrow M$ is an $(L, M)$-fuzzy topology are studied in detail.What is more, the degree to which an $L$-subset is an $L$-open set with respect to $mathcal{T}$ is introduced.Based on that, the degrees to which a mapping $f: Xlongrightarrow Y$ is continuous,open, closed or a quotient mapping with respect t...

Let E be a real q-uniformly smooth Banach space with constant dq, q ≥ 2. Let T : E → E and G : E → E be a nonexpansive map and an η-strongly accretive map which is also κ-Lipschitzian, respectively. Let {λn} be a real sequence in 0, 1 that satisfies the following condition: C1: limλn 0 and ∑ λn ∞. For δ ∈ 0, qη/dqk 1/ q−1 and σ ∈ 0, 1 , define a sequence {xn} iteratively in E by x0 ∈ E, xn 1 Tn...

in an earlier work we showed that for ordered fields f not isomorphic to the reals r, there are continuous 1-1 unctions on [0, 1]f which map some interior point to a boundary point of the image (and so are not open). here we show that over closed bounded intervals in the rationals q as well as in all non-archimedean ordered fields of countable cofinality, there are uniformly continuous 1-1 func...

Let G be a semitopological semigroup, C a nonempty subset of a real Hilbert space H , and = {Tt : t ∈ G} a representation of G as asymptotically nonexpansive type mappings of C into itself. Let L(x)= {z ∈H : infs∈G supt∈G ‖Ttsx−z‖ = inf t∈G ‖Ttx−z‖} for each x ∈ C and L( )= ⋂x∈C L(x). In this paper, we prove that ⋂s∈G conv{Ttsx : t ∈ G}⋂L( ) is nonempty for each x ∈ C if and only if there exist...

This paper consists of two main results. The first one shows that if S is a left reversible semigroup of selfmaps on a complete metric space (M,d) such that there is a gauge function φ for which d(f(x),f (y)) ≤ φ(δ(Of (x,y))) for f ∈ S and x,y in M , where δ(Of (x,y)) denotes the diameter of the orbit of x,y under f , then S has a unique common fixed point ξ in M and, moreover, for any f in S a...

Let E be a real q−uniformly smooth Banach space with constant dq, q ≥ 2. Let T : E → E and G : E → E be a nonexpansive map and an η−strongly accretive map which is also κ− Lipschitzian, respectively. Let {λn} be a real sequence in [0, 1] satisfying some appropriate conditions. For δ ∈ (0, ( qη dqκ )q−1), define a sequence {xn} iteratively in E by x0 ∈ E, xn+1 = T n+1xn = Txn − δλn+1G(Txn), n ≥ ...

1. Introduction We will be considering the existence of solutions of ordinary differential equations in Banach spaces, taking these to have the nominal form (1.1) ˙ x = Ax + f (x), x(0) = ˆ ξ 0 on some interval [0, T ]. Here x(·) takes values in the Banach space X and A is the infinitesimal generator of a C 0 semigroup S(·) of linear operators on X .f : X → X. We are indebted to [8] for an exce...