An operator version of the Korovkin theorem has recently been obtained by D. Popa. With motivation this result, we have extended it using a more powerful convergence which also includes ordinary convergence. We presented an example to illustrate strength our theorem.

In this paper, we first study a hierarchical problem of Baillon’s type, and we study a strong convergence theorem of this problem. For the special case of this convergence theorem, we obtain a strong convergence theorem for the ergodic theorem of Baillon’s type. Our result of the ergodic theorem of Baillon’s type improves and generalizes many existence theorems of this type of problem. Two nume...

We propose a reformulation of the convergence theorem of monotone numerical schemes introduced by Zhang and Zhuo [32] for viscosity solutions of path-dependent PDEs (PPDE), which extends the seminal work of Barles and Souganidis [1] on the viscosity solution of PDE. We prove the convergence theorem under conditions similar to those of the classical theorem in [1]. These conditions are satisfied...

The semigroup-valued integral of M. Sion [S] is reformulated for a general notion of approximation by sums of values taken by a set function integrand. A convergence in measure theorem is established, which yields both his pointwise dominated convergence theorem as well as an integrability criterion which specializes to his existence theorem. In [S] M. Sion introduced and developed an "integral...

Integral and integro-differential equations are one of the most useful mathematical tools in both pure and applied mathematics. In this article, we present a variational iteration method for solving Fredholm integro-differential equations. This study provides an analytical approximation to determine the behavior of the solution. To show the efficiency of the present method for our proble...

Let I ⊂ P(N) stand for an ideal containing finite sets. We discuss various kinds of statistical convergence and I-convergence for sequences of functions with values in R or in a metric space. For real valued measurable functions defined on a measure space (X,M, μ), we obtain a statistical version of the Egorov theorem (when μ(X) < ∞). We show that, in its assertion, equi-statistical convergence...

In this paper we investigate the relationship between the statistical (or generally I-convergence) of a series and the usual convergence of its subseries. We also give a counterexample which shows that Theorem 1 of the paper by B.C. Tripathy “On statistically convergent series”, Punjab.Univ. J.Math. 32 (1999), 1–8, is not correct.

We establish two results in the pointwise convergence problem of a trigonometric series [An] £ cne inl with lim Hm £ I bTck | = 0 |n|< -x. * Jn-»oo \k\-n for some nonnegative integer m. These results not only generalize Hardy's theorem, the Jordan test theorem and Fatou's theorem, but also complement the results on pointwise convergence of those Fourier series associated with known 1}-convergen...

In this paper, we deal with the problem for finding a common element of finite sets in a Banach space. We first prove that an operator given by a convex combination of sunny generalized nonexpansive retractions in a Banach space is asymptotically regular. Using this result, we obtain a weak convergence theorem which is connected with the problem of image recovery. Further, using another weak co...