We introduce an iterative process which converges strongly to solutions of a certain variational inequity problem for $\eta$-inverse strongly accretive mappings in the set of common fixed points of finite family of strictly pseudocontractive mappings in Banach spaces. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.

Inspired by the note on split common fixed-point problem for quasi-nonexpansive operators presented by Moudafi (2011), based on the very recent work by Dang et al. (2012), in this paper, we propose an inertial iterative algorithm for solving the split common fixed-point problem for quasi-nonexpansive operators in the Hilbert space. We also prove the asymptotical convergence of the algorithm under some suitable conditions. The results improve and develop previously discussed feasibility problems and related algorithms.

We present a new extension of Serrin's lower semicontinuity theorem. We prove that the variational functional${\int }_{\mathrm{\Omega}}^{}f(x,u,{u}^{\prime })dx$ defined on ${W}_{loc}^{1,1}(\mathrm{\Omega})$ is lower semicontinuous with respect to the strong convergence in ${L}_{loc}^{1}$, under the assumptions that the integrand $f(x,s,\xi )$ has the locally absolute continuity about the variable $x$.

This paper focuses on the problem of multiplicative noise removal. Using a gray level indicator, we derive a new functional which consists of the adaptive total variation term and the global convex fidelity term. We prove the existence, uniqueness, and comparison principle of the minimizer for the variational problem. The existence, uniqueness, and long-time behavior of the associated evolution equation are established. Finally, experimental results illustrate the effectiveness of the model in multiplicative noise reduction. Different from the other methods, the parameters in the proposed algorithms are found dynamically.

We study the existence and monotone iteration of solutions for a third-order four-point boundary value problem with $p$-Laplacian. An existence result of positive, concave, and pseudosymmetric solutions and its monotone iterative scheme are established by using the monotone iterative technique. Meanwhile, as an application of our result, an example is given.

We introduce a new iterative scheme called Jungck-CR iterative scheme and study the stability and strong convergence of this iterative scheme for a pair of nonself-mappings using a certain contractive condition. Also, convergence speed comparison and applications of Jungck-type iterative schemes will be shown through examples.

The previously developed new perturbation-iteration algorithm has been applied to differential equation systems for the first time. The iteration algorithm for systems is developed first. The algorithm is tested for a single equation, coupled two equations, and coupled three equations. Solutions are compared with those of variational iteration method and numerical solutions, and a good agreement is found. The method can be applied to differential equation systems with success.

This paper deals with the extinction and nonextinction properties of the fast diffusion equation of homogeneous Dirichlet boundary condition in a bounded domain of $R^N$ with $N>2$. For , under appropriate hypotheses, we show that $m=p$ is the critical exponent of extinction for the weak solution. Furthermore, we prove that the solution either extinct or nonextinct in finite time depends strongly on the initial data and the first eigenvalue of $-\mathrm{\Delta }$ with homogeneous Dirichlet boundary.

This paper introduces the stability and convergence of two-step Runge-Kutta methods with compound quadrature formula for solving nonlinear Volterra delay integro-differential equations. First, the definitions of $(k,l)$-algebraically stable and asymptotically stable are introduced; then the asymptotical stability of a $(k,l)$-algebraically stable two-step Runge-Kutta method with is proved. For the convergence, the concepts of $D$-convergence, diagonally stable, and generalized stage order are firstly introduced; then it is proved by some theorems that if a two-step Runge-Kutta method is algebraically stable and diagonally stable and its generalized stage order is $p$, then the method with compound quadrature formula is $D$-convergent of order at least $\text{min}\{p,\nu \}$, where $\nu$ depends on the compound quadrature formula.

We will discuss some operator inequalities on chaotic order about several operators, which are generalization of Furuta inequality and show monotonicity of related Furuta type operator function.

We introduce an iterative process which converges strongly to a zero of a finite sum of monotone mappings under certain conditions. Applications to a convex minimization problem are included. Our theorems improve and unify most of the results that have been proved in this direction for this important class of nonlinear mappings.

Two iterative methods of order four and five, respectively, are presented for solving nonlinear systems of equations. Numerical comparisons are made with other existing second- and fourth-order schemes to solve the nonlinear system of equations of the Global Positioning System and some academic nonlinear systems.

Combining multivariate spectral gradient method with projection scheme, this paper presents an adaptive prediction-correction method for solving large-scale nonlinear systems of monotone equations. The proposed method possesses some favorable properties: (1) it is progressive step by step, that is, the distance between iterates and the solution set is decreasing monotonically; (2) global convergence result is independent of the merit function and its Lipschitz continuity; (3) it is a derivative-free method and could be applied for solving large-scale nonsmooth equations due to its lower storage requirement. Preliminary numerical results show that the proposed method is very effective. Some practical applications of the proposed method are demonstrated and tested on sparse signal reconstruction, compressed sensing, and image deconvolution problems.

We studied the approximate split equality problem (ASEP) in the framework of infinite-dimensional Hilbert spaces. Let $H_1$, $H_2$, and $H_3$ be infinite-dimensional real Hilbert spaces, let $C\subset H_1$ and $Q\subset H_2$ be two nonempty closed convex sets, and let $A:H_1\to H_3$ and $B:H_2\to H_3$ be two bounded linear operators. The ASEP in infinite-dimensional Hilbert spaces is to minimize the function $f\left(x,y\right)=\mathrm{(1}/\mathrm{2)}{∥Ax-By∥}_2^2$ over $x\in C$ and $y\in Q$. Recently, Moudafi and Byrne had proposed several algorithms for solving the split equality problem and proved their convergence. Note that their algorithms have only weak convergence in infinite-dimensional Hilbert spaces. In this paper, we used the regularization method to establish a single-step iterative for solving the ASEP in infinite-dimensional Hilbert spaces and showed that the sequence generated by such algorithm strongly converges to the minimum-norm solution of the ASEP. Note that, by taking $B=I$ in the ASEP, we recover the approximate split feasibility problem (ASFP).

We study the solvability of some nonlinear functional integral equations in the Banach algebra of real functions defined, continuous, and bounded on the real half axis. We apply the technique of measures of noncompactness in order to obtain existence results for equations in question. Additionally, that technique allows us to obtain some characterization of considered integral equations. An example illustrating the obtained results is also included.

An autonomy system with time-delayed feedback is studied by using the theory of functional differential equation and Hassard’s method; the conditions on which zero equilibrium exists and Hopf bifurcation occurs are given, the qualities of the Hopf bifurcation are also studied. Finally, several numerical simulations are given; which indicate that when the delay passes through certain critical values, chaotic oscillation is converted into a stable state or a stable periodic orbit.

This paper relies on the study of fixed points and best proximity points of a class of so-called generalized point-dependent $(K,\lambda )$-hybrid $p$-cyclic self-mappings relative to a Bregman distance $D_f$, associated with a Gâteaux differentiable proper strictly convex function $f$ in a smooth Banach space, where the real functions $\lambda$ and $K$ quantify the point-to-point hybrid and nonexpansive (or contractive) characteristics of the Bregman distance for points associated with the iterations through the cyclic self-mapping. Weak convergence results to weak cluster points are obtained for certain average sequences constructed with the iterates of the cyclic hybrid self-mappings.

We first introduce the concept of Bregman asymptotically quasinonexpansive mappings and prove that the fixed point set of this kind of mappings is closed and convex. Then we construct an iterative scheme to find a common element of the set of solutions of an equilibrium problem and the set of common fixed points of a countable family of Bregman asymptotically quasinonexpansive mappings in reflexive Banach spaces and prove strong convergence theorems. Our results extend the recent ones of some others.

This paper presents a hybrid spectral conjugate gradient method for large-scale unconstrained optimization, which possesses a self-adjusting property. Under the standard Wolfe conditions, its global convergence result is established. Preliminary numerical results are reported on a set of large-scale problems in CUTEr to show the convergence and efficiency of the proposed method.

The nonlinear conjugate gradient method is of particular importance for solving unconstrained optimization. Finitely many maximum functions is a kind of very useful nonsmooth equations, which is very useful in the study of complementarity problems, constrained nonlinear programming problems, and many problems in engineering and mechanics. Smoothing methods for solving nonsmooth equations, complementarity problems, and stochastic complementarity problems have been studied for decades. In this paper, we present a new smoothing nonlinear conjugate gradient method for nonsmooth equations with finitely many maximum functions. The new method also guarantees that any accumulation point of the iterative points sequence, which is generated by the new method, is a Clarke stationary point of the merit function for nonsmooth equations with finitely many maximum functions.

Modified Mann-Halpern algorithms for finding the fixed points of pseudocontractive mappings are presented. Strong convergence theorems are obtained.

Using the concept of common (E.A) property, we prove a common fixed point theorem for three pairs of weakly compatible self-maps satisfying a new contractive condition in the framework of a generalized metric space. Our results do not rely on any commuting or continuity condition of mappings. An example is provided to support our result. The results obtained in this paper differ from the recent relative results in the literature.

Let $E$ a reflexive Banach space having a uniformly Gâteaux differentiable norm. Let $C$ be a nonempty closed convex subset of $E$, $T:C\to C$ a continuous pseudocontractive mapping with $F(T)\ne \varnothing$, and $A:C\to C$ a continuous bounded strongly pseudocontractive mapping with a pseudocontractive constant $k\in (\mathrm{0,1})$. Let $\{{\alpha }_n\}$ and $\{{\beta }_n\}$ be sequences in $(\mathrm{0,1})$ satisfying suitable conditions and for arbitrary initial value $x_0\in C$, let the sequence $\{x_n\}$ be generated by $x_n={\alpha }_{n}A{x}_n+{\beta }_n{x}_{n-1}+(1-{\alpha }_n-{\beta }_n)T{x}_n,n\ge 1.$ If either every weakly compact convex subset of $E$ has the fixed point property for nonexpansive mappings or $E$ is strictly convex, then $\{x_n\}$ converges strongly to a fixed point of $T$, which solves a certain variational inequality related to $A$.

New △-convergence theorems of iterative sequences for asymptotically nonexpansive mappings in CAT(0) spaces are obtained. Consider an asymptotically nonexpansive self-mapping $T$ of a closed convex subset $C$ of a CAT(0) space $X$. Consider the iteration process $\{x_n\}$, where $x_0\in C$ is arbitrary and $x_{n+1}={\alpha }_n{x}_n\oplus (1-{\alpha }_n)T^n{y}_n$ or $x_{n+1}={\alpha }_n{T}^n{x}_n\oplus (1-{\alpha }_n)y_n,y_n={\beta }_n{x}_n\oplus (1-{\beta }_n)T^n{x}_n$ for $n\ge 1$, where $\{{\alpha }_n\},\{{\beta }_n\}\subset (\mathrm{0,1})$. It is shown that under certain appropriate conditions on ${\alpha }_n,{\beta }_n,\{x_n\}$ △-converges to a fixed point of $T$.

We establish some strong convergence theorems for a common fixed point of a finite family of relatively nonexpansive mappings by using a new hybrid iterative method in mathematical programming and the generalized projection method in a Banach space. Our results improve and extend the corresponding results by many others.

Nonlinear conjugate gradient method is one of the useful methods for unconstrained optimization problems. In this paper, we consider three kinds of nonlinear conjugate gradient methods with Wolfe type line search for unstrained optimization problems. Under some mild assumptions, the global convergence results of the given methods are proposed. The numerical results show that the nonlinear conjugate gradient methods with Wolfe type line search are efficient for some unconstrained optimization problems.

In 1986, Matthews generalized Banach contraction mapping theorem in dislocated metric space that is a wider space than metric space. In this paper, we established common fixed point theorems for a class of contractive mappings. Our results extend the corresponding ones of other authors in dislocated metric spaces.