Monotone iterative schemes for positive solutions of a fractional differential system with integral boundary conditions on an infinite interval

Strong convergence theorems for strongly monotone mappings in Banach spaces

Let $E$ be a uniformly smooth and uniformly convex real Banach space and $E^*$ be its dual space. Suppose $A : E\rightarrow E^*$ is bounded, strongly monotone and satisfies the range condition such that $A^{-1}(0)\neq \emptyset$. Inspired by Alber [2], we introduce Lyapunov functions and use the new geometric properties of Banach spaces to show the strong convergence of an iterative algorithm to the solution of $Ax=0$.

涉及移动球投影的伪单调变分不等式问题的收敛性分析

Convergence analysis for pseudo-monotone variational inequality problem involving projections onto a moving ball

针对伪单调变分不等式问题传统投影法的局限，提出一种向可行集内显式表达的球投影、无需预知代价算子Lipschitz常数的迭代格式，并通过数值实验验证了其有效性。

Tikhonov Regularized Iterative Methods for Nonlinear Problems

We consider the monotone inclusion problems in real Hilbert spaces. Proximal splitting algorithms are very popular technique to solve it and generally achieve weak convergence under mild assumptions. Researchers assume the strong conditions like strong convexity or strong monotonicity on the considered operators to prove strong convergence of the algorithms. Mann iteration method and normal S-iteration method are popular methods to solve fixed point problems. We propose a new common fixed point algorithm based on normal S-iteration method using Tikhonov regularization to find common fixed poin…

Existence and convergence theorems for monotone generalized alpa-nonexpansive mappings in uniformly convex partially ordered hyperbolic metric spaces and its application

In this paper, we generalize the existence result in [14] and prove convergence theorems of the iterative scheme in [12, 16] for monotone generalized alpa-nonexpansive mappings in uniformly convex partially ordered hyperbolic metric spaces. And we also give a numerical example to show that this scheme converges faster than the scheme in [14] and apply the result to the integral equation.

关于非线性单调和强凹算子在正常锥上固定点的研究

On Fixed Points of Nonlinear Monotone and Strongly Concave Operators Acting in Normal Cones

本文研究了在实巴纳赫空间正常锥上具有强凹性的非线性单调算子的新类别，建立了非零固定点存在的构造原理，并证明迭代过程以几何速度收敛，同时在宽泛的锥段内证明了固定点的唯一性。

反向惩罚方案用于单调包含问题

Backward Penalty Schemes for Monotone Inclusion Problems

本文提出两种反向惩罚方案，用于解决由单调算子和单值算子组成的单调包含问题，针对不同性质的单值算子采用不同步长策略。

Iterative Schemes for Bump Solutions in a Neural Field Model

We develop two iteration schemes for construction of localized stationary solutions (bumps) of a one-population Wilson-Cowan model with a smoothed Heaviside firing rate function. The first scheme is based on the fixed point formulation of the stationary Wilson-Cowan model. The second one is formulated in terms of the excitation width of a bump. Using the theory of monotone operators in ordered Banach spaces we justify convergence of both iteration schemes.

非线性系统的降维及其使用分数阶迭代方法的数值求解

Reduction of a nonlinear system and its numerical solution using a fractional iterative method

本文研究混合太阳能接收器微分方程系统导出的5变量非线性代数方程组，提出将其降维至2个变量的方法，并利用不依赖解析复杂度的分数阶迭代方法进行数值求解。

Solutions of Nonlinear Operator Equations by Viscosity Iterative Methods

Finding the solutions of nonlinear operator equations has been a subject of research for decades but has recently attracted much attention. This paper studies the convergence of a newly introduced viscosity implicit iterative algorithm to a fixed point of a nonexpansive mapping in Banach spaces. Our technique is indispensable in terms of explicitly clarifying the associated concepts and analysis. The scheme is effective for obtaining the solutions of various nonlinear operator equations as it involves the generalized contraction. The results are applied to obtain a fixed point of λ-strictly ps…

Some qualitative properties of solutions for nonlinear fractional differential equation involving two $Φ$--Caputo fractional derivatives

The momentous objective of this work is to discuss some qualitative properties of solutions such as the estimate on the solutions, the continuous dependence of the solutions on initial conditions as well as the existence and uniqueness of extremal solutions for a new class of fractional differential equations involving two fractional derivatives in the sense of Caputo fractional derivative with respect to a new function $Φ$. Firstly, by using the generalized Laplace transform method, we give an explicit formula of the solutions for the aforementioned linear problem which can be regarded as a n…

Strong convergence of a new composite iterative method for equilibrium problems and fixed point problems in Hilbert spaces

In this paper, first we introduce a new mapping for finding a common fixed point of an infinite family of nonexpansive mappings then we consider iterative method for finding a common element of the set of fixed points of an infinite family of nonexpansive mappings, the set of solutions of an equilibrium problem and the set of solutions of the variational inequality for $α$-inverse-strongly monotone mapping in a Hilbert space. We show that under suitable conditions, the sequence converges strongly to a common element of the above three sets. Our results presented in this paper improve and exten…

Monotonicity for entrywise functions of matrices

We characterize real functions $f$ on an interval $(-α,α)$ for which the entrywise matrix function $[a_{ij}] \mapsto [f(a_{ij})]$ is positive, monotone and convex, respectively, in the positive semidefiniteness order. Fractional power functions are exemplified and related weak majorizations are shown.

The Split Common Null Point Problem

We introduce and study the Split Common Null Point Problem (SCNPP) for set-valued maximal monotone mappings in Hilbert spaces. This problem generalizes our Split Variational Inequality Problem (SVIP) [Y. Censor, A. Gibali and S. Reich, Algorithms for the split variational inequality problem, Numerical Algorithms 59 (2012), 301--323]. The SCNPP with only two set-valued mappings entails finding a zero of a maximal monotone mapping in one space, the image of which under a given bounded linear transformation is a zero of another maximal monotone mapping. We present four iterative algorithms that s…

A New Iterative Projection Method for Approximating Fixed Point Problems and Variational Inequality Problems

In this paper, we introduce and study a new extragradient iterative process for finding a common element of the set of fixed points of an infinite family of nonexpansive mappings and the set of solutions of a variational inequality for an inverse strongly monotone mapping in a real Hilbert space. Also, we prove that under quite mild conditions the iterative sequence defined by our new extragradient method converges strongly to a solution of the fixed point problem for an infinite family of nonexpansive mappings and the classical variational inequality problem. In addition, utilizing this resul…

Non-instantaneous Impulsive Riemann-Liouville Fractional Differential Systems: Existence and Controllability Analysis

The article is dedicated towards the study of fractional order non-linear differential systems with non-instantaneous impulses involving Riemann-Liouville derivatives with fixed lower limit and appropriate integral type initial conditions in Banach spaces. First, mild solution of the system is constructed and subsequently its existence is proven using Banach's fixed point theorem. Then, results of approximate controllability are established using concept of fractional semigroup and an iterative technique. Suitable examples are given in the end supporting the methodology along with pointing out…

单调稳定性与不稳定性在希尔伯特空间中单调算子的算子方程中的条件稳定性与不稳定性

Conditional stability versus ill-posedness for operator equations with monotone operators in Hilbert space

本文研究了在希尔伯特空间中单调算子的算子方程的条件稳定性与不稳定性，通过拉文特修正式探讨了误差估计和收敛速率，并区分了线性问题中的well-posedness与ill-posedness，提出了非自伴线性单调正向算子的收敛速率扩展。