In this paper we introduce a new general class of multi-valued Lipschitz hemicontractive mappings. We then prove strong convergence theorems for finding a fixed point of the mapping using a new two steps averaged algorithm. The method used in the proof is new and enables us to systematically avoid so many strong assumptions in the contemporary literature. The theorems obtained generalize and improve many results in the literature.

The changes in facial appearance resulting from the effect of plastic surgery may manifest in the form of textural variations, and/or geometric variations with respect to the facial structural features such as eyes, nose, mouth, jaw, nose, brow, etc. This study argues that despite these variations, there exist facial features that are insensitive to the effect of plastic surgery. This paper therefore attempts to investigate this assertion through mathematical connotations/modeling based on some quasiconformal geometrical properties. Also, the study presents some experimental grounds using sample images of pre-surgery and post-surgery faces of different individuals selected from different aesthetic plastic surgery procedures from plastic surgery database and testing using an algorithm coined: conformity to similarity test algorithm for validating our claim. Based on the experimental result, it is evident that the intrinsic structural information of the interior curvature of facial features expressed as edge elements of the global appearance of a face image, is insensitive to the effect of aesthetic surgery induced transformations.

We introduce a new iterative scheme for finding common fixed points of a finite family of nonextensive mapping and zeros of strongly monotone mappings in Lp spaces, which yields a solution to a convex optimization problem. This provides a partial extension of a theorem of Yamada and some other authors from Hilbert spaces to the more general Banach spaces.

A relatively new method called q-Homotopy Analysis Method (q-HAM) is adopted in this paper to obtain an analytical solution of the time fractional Rosenau–Hyman equation in series form. Our analysis shows the simplicity nature of the application of q-HAM to nonlinear fractional differential equations. The convergence rate of the method used is faster in the sense that just very few terms of the series solution are needed for a good approximation due to the presence of the auxiliary parameter h comparable to exact solutions. Numerical solution obtained by this method is compared with the exact solution and solutions obtained by other analytical methods of the equation under various conditions. The numerical results are obtained using Mathematica 9 and MATLAB R2012b.

A very important general class of split feasiblity problem was introduced by Moudafi and Al-Shamas 13], in the case when the mappings are firmly nonexpansive defined on real Hilbert spaces. We propose in this paper a new Krasnoselskii’s-type algorithm to solve the problem in the more general case when the mappings are Lipschitz hemicontractive. We show that the proposed algorithm converges weakly to a solution of the problem. We also show that the iterative sequence obtained converges strongly to a solution of the problem under suitable compactness assumptions.

This paper introduces a new averaged algorithm for finding a common fixed point of a countably infinite family of generalized k-strictly pseudocontractive multi-valued mappings. The new iterative sequence introduced is proved to be an approximating fixed point sequence for common fixed points of a countably infinite family of this class of mappings. Furthermore, under some mild assumptions, strong convergence theorems are also proved for this class of mappings. The method of proof used here is new and enables to overcome many strong restrictions appearing in contemporary literature. The stated theorems improve and generalize many recent works in iterative scheme for multi-valued mappings.

Let H be a real Hilbert space, K a nonempty subset of H, and \(T:K\rightarrow \mathit{CB}(K)\) a multi-valued mapping. Then T is called a generalized k-strictly pseudo-contractive multi-valued mapping if there exists \(k\in[0,1)\) such that, for all \(x,y\in D(T)\), we have \(D^{2}(Tx,Ty)\leq\|x-y\|^{2}+kD^{2}(Ax,Ay)\), where \(A:=I-T\), and I is the identity operator on K. A Krasnoselskii-type algorithm is constructed and proved to be an approximate fixed point sequence for a common fixed point of a finite family of this class of maps. Furthermore, assuming existence, strong convergence to a common fixed point of the family is proved under appropriate additional assumptions.

Motivated by the recent work of Moudafi and inspired by Xu , Censor et al. , and Yang,we investigate a Krasnoselskii-type iterative algorithm for solving the split equality fixed pointproblem recently introduced by Moudafi et al.. Weak and strong convergence theorems areproved for the class of demi-contractive mappings in Hilbert spaces. Our theorems extend andcomplement some recent results of Moudafi and a host of other recent important results.

Given a Lipschitz pseudocontractive mapping T from a closed convex and bounded subset K of a real Hilbert space H onto itself, and an arbitrary x1 ∈ K, a Krasnolselskii-type sequence defined by xn+1 = (1− λ)xn + λTyn, yn = (1− λ)xn + λTxn is proved to be an approximate fixed point sequence of T , for a suitable λ ∈ (0, 1). Under some suitable compactness assumptions on K or on T , the sequence converges strongly to a fixed point of T . The algorithm is simple and natural, and the theorems presented here improve the theorem of Ishikawa [1] and other similar results in the literature.

This paper presents a fourth-order nonlinear conjugate gradient method in equality constrained optimization.The idea is to transform the constrained problem into unconstrained type through the Lagrange multipliers scheme.Using four terms of Taylor series development, we approximate the transformed function (augmented Lagrange function).Lastly, we employ the new fourth-order nonlinear conjugate gradient method in equality constrained optimization to solve the optimization problem.We present the algorithm in steps and some properties of the gradients are proved, using classical results.Also, the convergence analysis has been proved under classical and known assumptions.Furthermore, we present the obtained numerical results and compare them to some existing results.The analysis of results confirms that the new method is accurate.

In this paper, a general class of multi-valued strictly pseudocon- tractive mappings, which properly includes the class of multi-valued k???strictly pseudocontractive mappings, is introduced. Furthermore, it is proved that if T belongs to this class of mappings and the set of fixed points of T is nonempty, a Krasnoselskii-type sequence is constructed and proved to be an approximate fixed point sequence of T . Finally, convergence of the sequence to a fixed point of T is proved under appropriate additional conditions.