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1

Iterates of q-Bernstein operators on triangular domain with all curved sides

Iliyas Mohammad, Khan Asif, Arif Mohd, Mursaleen Mohammad, Lone Mudassir Rashid

Demonstratio Mathematica

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2022

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Vol. 55, nr 1

891-899

EN

In this article, Phillips-type Bernstein operators (Btm,qF)(t,s) and (Bsn,qF)(t,s) , their products, and Boolean sum based on q-integer have been studied on a triangle with all curved sides. Furthermore, convergence of iterates of these operators have been analyzed using the weakly Picard operators technique and the contraction principle.

2

Boundary value problems for nonlinear fractional differential equations and inclusions with nonlocal and fractional integral boundary conditions

Ntouyas S. K.

Opuscula Mathematica

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2013

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Vol. 33, no. 1

117-138

EN

This paper studies the boundary value problem of nonlinear fractional differential equations and inclusions of order q ∈ (1, 2] with nonlocal and integral boundary conditions. Some new existence and uniqueness results are obtained by using fixed point theorems.

3

Asymptotic behavior of some random splitting schemes

Białkowski M., Wesołowski J.

Probability and Mathematical Statistics

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2002

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Vol. 22, Fasc. 1

181--191

EN

We consider three new schemes of random splitting of a unit interval.These schemes are related to settings considered earlier in literature. Essentially we are concerned with asymptotic behavior of sequences of subdivisions. In all three cases almost sure or weak limits are obtained for a sequence of points of divisions. The two of the schemes considered are dual to each other in the sense of the contraction principle of Chamayou and Letac [2].

4

A general contraction principle for vector-valued martingales

Geiss S.

Probability and Mathematical Statistics

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2000

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Vol. 20, Fasc. 1

93--120

EN

We prove a contraction principle for vector-valued martingales of type [formula] where X is a Banach space with elements x1, ‧‧, xn, (Δi)ni=1 ⊂ L1(Q,P) a martingale difference sequence belonging to a certain class, [formula] a sequence of independent and symmetric random variables exponential in a certain sense, and Ai operators mapping each Δi into a non-negative random variable. Moreover, special operators Ai are discussed and an application to Banach spaces of Rademacher type α (1<α ≤ 2) is given.

5

Operators on martingales, ф-summing operators, and the contraction principle

Geiss S.

Probability and Mathematical Statistics

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1998

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Vol. 18, Fasc. 1

149--171

EN

For the absolutely Ф-summmg operators T: X→Y between Banach spaces X and Y we consider martingale inequalities of the type…[formula] where ..[formula]…is a martingale difference sequence and i is a sequence of normalized functionals on X, and we show that these inequalities are useful in different directions. For example, for a Banach space X, x1…xn ∈X, independent standard Gaussian variables gn, and 1 < r < ∞ we deduce that..[formula]… where is a scale-valued martingale difference sequence such that [formula]…is predictable ..[formula].. is a sequence of stopping times and [formula].