Generalized Kantorovich modifications of positive linear operators

<p style='text-indent:20px;'>Starting with a positive linear operator we apply the Kantorovich modification and a related modification. The resulting operators are investigated. We are interested in the eigenstructure, Voronovskaya formula, the induced generalized convexity, invariant measures and iterates. Some known results from the literature are extended.</p>

SOLUTION OF A NONLINEAR OPERATOR EQUATION OF THE FIRST KIND WITH A CONTINUOUS POSITIVE LINEAR OPERATOR

In the space H, a nonlinear operator equation of the first kind is studied, when the linear, nonlinear operator and the right-hand side of the equation are given approximately. Based on the method of Lavrent'ev M.M. an approximate solution of the equation in Hilbert space is constructed. The dependence of the regularization parameter on errors was selected. The rate of convergence of the approximate solution to the exact solution of the original equation is obtained.

Nonlocal q-fractional boundary value problem with Stieltjes integral conditions

In this paper, we are dedicated to investigating a new class of one-dimensional lower-order fractional q-differential equations involving integral boundary conditions supplemented with Stieltjes integral. This condition is more general as it contains an arbitrary order derivative. It should be pointed out that the problem discussed in the current setting provides further insight into the research on nonlocal and integral boundary value problems. We first give the Green's functions of the boundary value problem and then develop some properties of the Green's functions that are conductive to our main results. Our main aim is to present two results: one considering the uniqueness of nontrivial solutions is given by virtue of contraction mapping principle associated with properties of u0-positive linear operator in which Lipschitz constant is associated with the first eigenvalue corresponding to related linear operator, while the other one aims to obtain the existence of multiple positive solutions under some appropriate conditions via standard fixed point theorems due to Krasnoselskii and Leggett–Williams. Finally, we give an example to illustrate the main results.

The iterates of positive linear operators with the set of constant functions as the fixed point set

Let Ω ⊂ Rp, p ∈ N∗ be a nonempty subset and B(Ω) be the Banach lattice of all bounded real functions on Ω, equipped with sup norm. Let X ⊂ B(Ω) be a linear sublattice of B(Ω) and A: X → X be a positive linear operator with constant functions as the fixed point set. In this paper, using the weakly Picard operators techniques, we study the iterates of the operator A. Some relevant examples are also given.

A Spline Group – Korovkin Approximation Theorem

In this paper, using homogeneous groups, we prove a Korovkin type approximation theorem for a spline groupby using the notion of a generalization of positive linear operator.

Quantitative Estimates for Positive Linear Operators in terms of the Usual Second Modulus

We give accurate estimates of the constantsCn(A(I),x)appearing in direct inequalities of the form|Lnf(x)-f(x)|≤Cn(A(I),x)ω2 f;σ(x)/n,f∈A(I),x∈I, and n=1,2,…,whereLnis a positive linear operator reproducing linear functions and acting on real functionsfdefined on the intervalI,A(I)is a certain subset of such functions,ω2(f;·)is the usual second modulus off, andσ(x)is an appropriate weight function. We show that the size of the constantsCn(A(I),x)mainly depends on the degree of smoothness of the functions in the setA(I)and on the distance from the pointxto the boundary ofI. We give a closed form expression for the best constant whenA(I)is a certain set of continuous piecewise linear functions. As illustrative examples, the Szàsz-Mirakyan operators and the Bernstein polynomials are discussed.

Approximate Iteration Algorithm with Error Estimate for Fixed Point of Nonexpansive Mappings

The purpose of this article is to present a general viscosity iteration process{xn}which defined byxn+1=(I-αnA)Txn+βnγf(xn)+(αn-βn)xnand to study the convergence of{xn}, whereTis a nonexpansive mapping andAis a strongly positive linear operator, if{αn},{βn}satisfy appropriate conditions, then iteration sequence{xn}converges strongly to the unique solutionx*∈f(T)of variational inequality〈(A−γf)x*,x−x*〉≥0,for allx∈f(T). Meanwhile, a approximate iteration algorithm is presented which is used to calculate the fixed point of nonexpansive mapping and solution of variational inequality, the error estimate is also given. The results presented in this paper extend, generalize, and improve the results of Xu, G. Marino and Xu and some others.

Invariant probability measures and dynamics of exponential linear type maps

We consider the problem of the asymptotic size of the random maximum-weight matching of a sparse random graph, which we translate into dynamics of the operator in the space of distribution functions. A tight condition for the uniqueness of the globally attracting fixed point is provided, which extends the result of Karp and Sipser [Maximum matchings in sparse random graphs. 22nd Ann. Symp. on Foundations of Computer Science (Nashville, TN, 28–30 October, 1981). IEEE, New York, 1981, pp. 364–375] from deterministic weight distributions (Dirac measures μ) to general ones. Given a probability measure μ which corresponds to the weight distribution of a link of a random graph, we form a positive linear operator Φμ (convolution) on distribution functions and then analyze a family of its exponents, with parameter λ, which corresponds to the connectivity of a sparse random graph. The operator 𝕋 relates the distribution F on the subtrees to the distribution 𝕋F on the node of the tree by 𝕋F=exp (−λΦμF). We prove that for every probability measure μ and every λ<e, there exists a unique globally attracting fixed point of the operator; the probability measure corresponding to this fixed point can then be used to compute the expected maximum-weight matching on a sparse random graph. This result is called the e-cutoff phenomenon. For deterministic distributions and λ>e, there is no fixed point attractor. We further establish that the uniqueness of the invariant measure of the underlying operator is not a monotone property of the average connectivity; this parallels similar non-monotonicity results in the statistical physics context.