We introduce a new three-step iteration scheme and prove that this new iteration scheme is convergent to fixed points of contractive-like operators. Also, by providing an example, we show that our new iteration method is faster than another iteration method due to Suantai (2005). Furthermore, it is shown that this new iteration method is equivalent to some other iteration methods in the sense of convergence. Finally, it is proved that this new iteration method is $T$-stable.

We introduce and study a new system of generalized $H\left(·,·\right)-\eta \text{-}$cocoercive operator inclusions in Banach spaces. Using the resolvent operator technique associated with $H\left(·,·\right)-\eta \text{-}$cocoercive operators, we suggest and analyze a new generalized algorithm of nonlinear set-valued variational inclusions and establish strong convergence of iterative sequences produced by the method. We highlight the applicability of our results by examples in function spaces.

Let $X$ be a real Banach space and $K$ a nonempty closed convex subset of $X$. Let $T_i:K\to K\mathrm{}\mathrm{}\mathrm{}\mathrm{}\hspace{0.17em}(i=\mathrm{1},\mathrm{}\mathrm{}\mathrm{2},\mathrm{}\dots ,\mathrm{}\mathrm{}m)$ be $m$ asymptotically nonexpansive mappings with sequence $\{k_n\}\subset [\mathrm{1},\hspace{0.17em}\mathrm{\infty })$, , and $ℱ={\bigcap }_{i=\mathrm{1}}^m\mathrm{‍}F(T_i)\ne \mathrm{\varnothing }$, where $F$ is the set of fixed points of $T_i$. Suppose that $\{a_{in}{\}}_{n=\mathrm{1}}^{\mathrm{\infty }}$, $\{b_{in}{\}}_{n=\mathrm{1}}^{\mathrm{\infty }}$, $i=\mathrm{1,2},\dots ,m$ are appropriate sequences in $[\mathrm{0,1}]$ and $\{u_{in}{\}}_{n=\mathrm{1}}^{\mathrm{\infty }}$, $i=\mathrm{1,2},\dots ,m$ are bounded sequences in $K$ such that for $i=\mathrm{1,2},\dots ,m$. We give $\{x_n\}$ defined by $x_{\mathrm{1}}\in K,$ $x_{n+\mathrm{1}}=(\mathrm{1}-a_{\mathrm{1}n}-b_{\mathrm{1}n})y_{n+m-\mathrm{2}}+a_{\mathrm{1}n}{T}_{\mathrm{1}}^n{y}_{n+m-\mathrm{2}}+b_{\mathrm{1}n}{u}_{\mathrm{1}n},\mathrm{}{y}_{n+m-\mathrm{2}}=(\mathrm{1}-a_{\mathrm{2}n}-b_{\mathrm{2}n})y_{n+m-\mathrm{3}}+a_{\mathrm{2}n}{T}_{\mathrm{2}}^n{y}_{n+m-\mathrm{3}}\hspace{0.17em}+$ $b_{\mathrm{2}n}{u}_{\mathrm{2}n},\dots ,$ $y_{n+\mathrm{2}}=(\mathrm{1}-a_{(m-\mathrm{2})n}-b_{(m-\mathrm{2})n})y_{n+\mathrm{1}}+a_{(m-\mathrm{2})n}{T}_{m-\mathrm{2}}^n{y}_{n+\mathrm{1}}+b_{(m-\mathrm{2})n}{u}_{(m-\mathrm{2})n},$ $y_{n+\mathrm{1}}=(\mathrm{1}-a_{(m-\mathrm{1})n}\hspace{0.17em}-$ $b_{(m-\mathrm{1})n})y_n+a_{(m-\mathrm{1})n}{T}_{m-\mathrm{1}}^n{y}_n+b_{(m-\mathrm{1})n}{u}_{(m-\mathrm{1})n},$ $y_n=(\mathrm{1}-a_{mn}-b_{mn})x_n+a_{mn}{T}_m^n{x}_n+b_{mn}{u}_{mn},\mathrm{}\mathrm{}\mathrm{}\mathrm{}m\ge \mathrm{2},\mathrm{}\mathrm{}\mathrm{}n\ge \mathrm{1}.$ The purpose of this paper is to study the above iteration scheme for approximating common fixed points of a finite family of asymptotically nonexpansive mappings and to prove weak and some strong convergence theorems for such mappings in real Banach spaces. The results obtained in this paper extend and improve some results in the existing literature.

The concept of $\lambda$-statistical convergence for sequences of real numbers was introduced in Mursaleen (2000). In this paper, we prove decomposition theorem for $\lambda$-statistical convergence. We also define and study $\lambda$-statistical convergence, $\lambda$-statistically Cauchy, and strongly ${\lambda }_p$-summability in Paranormed Space.

We investigate the modified Mellin transform on certain function space of generalized functions. We first obtain the convolution theorem for the classical and distributional modified Mellin transform. Then we describe the domain and range spaces where the extended modified transform is well defined. Consistency, convolution, analyticity, continuity, and sufficient theorems for the proposed transform have been established. An inversion formula is also obtained and many properties are given.

We introduce the concept of an ${\alpha }_c$-admissible non-self-mappings with respect to ${\eta }_c$ and establish the existence of PPF dependent fixed and coincidence point theorems for ${\alpha }_c{\eta }_c$-$\psi$-contractive non-self-mappings in the Razumikhin class. As applications of our PPF dependent fixed point and coincidence point theorems, we derive some new fixed and coincidence point results for $\psi$-contractions whenever the range space is endowed with a graph or with a partial order. The obtained results generalize, extend, and modify some PPF dependent fixed point results in the literature. Several interesting consequences of our theorems are also provided.

We give refinement of Jensen’s type inequalities given by Bakula and Pečarić (2006) for the co-ordinate convex function. Also we establish improvement of Jensen’s inequality for the convex function of two variables.

We investigate the Krätzel transform on certain class of generalized functions. We propose operations that lead to the construction of desired spaces of generalized functions. The Krätzel transform is extended and some of its properties are obtained.

Recently, Alghamdi and Mursaleen (2013) used the Hankel matrix to determine the necessary and suffcient condition to find the sum of the Walsh-Fourier series. In this paper, we propose to use the Hankel matrix as well as any general nonnegative regular matrix to obtain the necessary and sufficient conditions to sum the derived Fourier series and conjugate Fourier series.

The purpose of this paper is to define some new types of summability methods for double sequences involving the ideas of de la Vallée-Poussin mean in the framework of probabilistic normed spaces and establish some interesting results.

Using Lyapunov-Krasovskii functional approach, we establish a new result to guarantee the existence of periodic solutions of a certain multidelay nonlinear functional differential equation of second order. By this work, we extend and improve some earlier result in the literature.

In the present paper we introduce some sequence spaces over n-normed spaces defined by a Musielak-Orlicz function $ℳ=(M_k)$. We also study some topological properties and prove some inclusion relations between these spaces.

We study some new strongly almost lacunary $I$-convergent generalized difference sequence spaces defined by an Orlicz function. We give also some inclusion relations related to these sequence spaces.

We prove a result on the existence and uniform attractivity of solutions of an Urysohn integral equation. Our considerations are conducted in the Banach space consisting of real functions which are bounded and continuous on the nonnegative real half axis. The main tool used in investigations is the technique associated with the measures of noncompactness and a fixed point theorem of Darbo type. An example showing the utility of the obtained results is also included.

This paper investigates a class of self-adjoint compact operators in Hilbert spaces related to their truncated versions with finite-dimensional ranges. The comparisons are established in terms of worst-case norm errors of the composite operators generated from iterated computations. Some boundedness properties of the worst-case norms of the errors in their respective fixed points in which they exist are also given. The iterated sequences are expanded in separable Hilbert spaces through the use of numerable orthonormal bases.