Nonlinear Analysis and Optimization with Applications

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: closed (31 December 2020) | Viewed by 25802

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Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 82444, Taiwan
Interests: nonlinear analysis and its applications; fixed point theory; variational principles and inequalities; optimization theory; equilibrium problems; fractional calculus theory
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Guest Editor
Department of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan
Interests: nonlinear analysis and its applications; fixed point theory; variational principles and inequalities; optimization theory; equilibrium problems

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Guest Editor
School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China

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Guest Editor
Department of Mathematics, Babeş-Bolyai University, 400084 Cluj, Romania
Interests: nonlinear boundary value problems for ODEs and PDEs; theory of nonlinear operators; topological fixed point theory; critical point theory; mathematical modeling in biology and medicine
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Nonlinear analysis has wide and significant applications in many areas of mathematics, including functional analysis, variational analysis, nonlinear optimization, convex analysis, nonlinear ordinary and partial differential equations, dynamical system theory, mathematical economics, game theory, signal processing, control theory, data mining, and so forth. Optimization problems have been intensively investigated, and various feasible methods in analyzing convergence of algorithms have been developed over the last half century. In this Special Issue, we will focus on the connection between nonlinear analysis and optimization as well as their applications to integrate basic science into the real world.

We cordially and earnestly invite researchers to contribute their original and high-quality research papers which will inspire advances in nonlinear analysis, optimization, and their applications. Potential topics include but are not limited to:

  • Functional analysis;
  • Critical point theory;
  • Bifurcation theory;
  • Set-valued analysis;
  • Calculus of variations and PDEs;
  • Variational and topological methods for ODEs and PDEs;
  • Fxed point, coincidence point, and best proximity point theory;
  • Nonsmooth analysis and optimization;
  • Graph theory and optimization;
  • Game theory;
  • Convex analysis;
  • Matrix theory;
  • Control theory;
  • Inverse and ill-posed problems;
  • Finite element method;
  • Dynamical systems;
  • Image and signal processing;
  • Data mining

Prof. Dr. Wei-Shih Du
Prof. Dr. Liang-Ju Chu
Prof. Dr. Fei He
Prof. Dr. Radu Precup
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Axioms is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Functional analysis
  • Critical point theory
  • Bifurcation theory
  • Set-valued analysis
  • ODEs and PDEs
  • Fxed point, coincidence point, and best proximity point theory
  • Nonsmooth analysis
  • Convex analysis
  • Matrix theory
  • Control theory
  • Dynamical systems
  • Image and signal processing
  • Data mining
  • Graph theory and optimization

Related Special Issue

Published Papers (11 papers)

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Research

10 pages, 282 KiB  
Article
New Generalized Ekeland’s Variational Principle, Critical Point Theorems and Common Fuzzy Fixed Point Theorems Induced by Lin-Du’s Abstract Maximal Element Principle
by Junjian Zhao and Wei-Shih Du
Axioms 2021, 10(1), 11; https://doi.org/10.3390/axioms10010011 - 20 Jan 2021
Cited by 2 | Viewed by 1617
Abstract
In this paper, by applying the abstract maximal element principle of Lin and Du, we present some new existence theorems related with critical point theorem, maximal element theorem, generalized Ekeland’s variational principle and common (fuzzy) fixed point theorem for essential distances. Full article
(This article belongs to the Special Issue Nonlinear Analysis and Optimization with Applications)
10 pages, 285 KiB  
Article
A Remark for the Hyers–Ulam Stabilities on n-Banach Spaces
by Jaeyoo Choy, Hahng-Yun Chu and Ahyoung Kim
Axioms 2021, 10(1), 2; https://doi.org/10.3390/axioms10010002 - 29 Dec 2020
Cited by 4 | Viewed by 1610
Abstract
In this article, we deal with stabilities of several functional equations in n-Banach spaces. For a surjective mapping f into a n-Banach space, we prove the generalized Hyers–Ulam stabilities of the cubic functional equation and the quartic functional equation for f [...] Read more.
In this article, we deal with stabilities of several functional equations in n-Banach spaces. For a surjective mapping f into a n-Banach space, we prove the generalized Hyers–Ulam stabilities of the cubic functional equation and the quartic functional equation for f in n-Banach spaces. Full article
(This article belongs to the Special Issue Nonlinear Analysis and Optimization with Applications)
18 pages, 351 KiB  
Article
An Inertial Generalized Viscosity Approximation Method for Solving Multiple-Sets Split Feasibility Problems and Common Fixed Point of Strictly Pseudo-Nonspreading Mappings
by Hammed Anuoluwapo Abass and Lateef Olakunle Jolaoso
Axioms 2021, 10(1), 1; https://doi.org/10.3390/axioms10010001 - 24 Dec 2020
Cited by 8 | Viewed by 1844
Abstract
In this paper, we propose a generalized viscosity iterative algorithm which includes a sequence of contractions and a self adaptive step size for approximating a common solution of a multiple-set split feasibility problem and fixed point problem for countable families of k-strictly [...] Read more.
In this paper, we propose a generalized viscosity iterative algorithm which includes a sequence of contractions and a self adaptive step size for approximating a common solution of a multiple-set split feasibility problem and fixed point problem for countable families of k-strictly pseudononspeading mappings in the framework of real Hilbert spaces. The advantage of the step size introduced in our algorithm is that it does not require the computation of the Lipschitz constant of the gradient operator which is very difficult in practice. We also introduce an inertial process version of the generalize viscosity approximation method with self adaptive step size. We prove strong convergence results for the sequences generated by the algorithms for solving the aforementioned problems and present some numerical examples to show the efficiency and accuracy of our algorithm. The results presented in this paper extends and complements many recent results in the literature. Full article
(This article belongs to the Special Issue Nonlinear Analysis and Optimization with Applications)
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20 pages, 297 KiB  
Article
A Self-Adaptive Shrinking Projection Method with an Inertial Technique for Split Common Null Point Problems in Banach Spaces
by Chibueze Christian Okeke, Lateef Olakunle Jolaoso and Regina Nwokoye
Axioms 2020, 9(4), 140; https://doi.org/10.3390/axioms9040140 - 2 Dec 2020
Cited by 1 | Viewed by 1726
Abstract
In this paper, we present a new self-adaptive inertial projection method for solving split common null point problems in p-uniformly convex and uniformly smooth Banach spaces. The algorithm is designed such that its convergence does not require prior estimate of the norm [...] Read more.
In this paper, we present a new self-adaptive inertial projection method for solving split common null point problems in p-uniformly convex and uniformly smooth Banach spaces. The algorithm is designed such that its convergence does not require prior estimate of the norm of the bounded operator and a strong convergence result is proved for the sequence generated by our algorithm under mild conditions. Moreover, we give some applications of our result to split convex minimization and split equilibrium problems in real Banach spaces. This result improves and extends several other results in this direction in the literature. Full article
(This article belongs to the Special Issue Nonlinear Analysis and Optimization with Applications)
38 pages, 670 KiB  
Article
Nonlinear Approximations to Critical and Relaxation Processes
by Simon Gluzman
Axioms 2020, 9(4), 126; https://doi.org/10.3390/axioms9040126 - 28 Oct 2020
Cited by 10 | Viewed by 3697
Abstract
We develop nonlinear approximations to critical and relaxation phenomena, complemented by the optimization procedures. In the first part, we discuss general methods for calculation of critical indices and amplitudes from the perturbative expansions. Several important examples of the Stokes flow through 2D channels [...] Read more.
We develop nonlinear approximations to critical and relaxation phenomena, complemented by the optimization procedures. In the first part, we discuss general methods for calculation of critical indices and amplitudes from the perturbative expansions. Several important examples of the Stokes flow through 2D channels are brought up. Power series for the permeability derived for small values of amplitude are employed for calculation of various critical exponents in the regime of large amplitudes. Special nonlinear approximations valid for arbitrary values of the wave amplitude are derived from the expansions. In the second part, the technique developed for critical phenomena is applied to relaxation phenomena. The concept of time-translation invariance is discussed, and its spontaneous violation and restoration considered. Emerging probabilistic patterns correspond to a local breakdown of time-translation invariance. Their evolution leads to the time-translation invariance complete (or partial) restoration. We estimate the typical time extent, amplitude and direction for such a restorative process. The new technique is based on explicit introduction of origin in time as an optimization parameter. After some transformations, we arrive at the exponential and generalized exponential-type solutions (Gompertz approximants), with explicit finite time scale, which is only implicit in the initial parameterization with polynomial approximation. The concept of crash as a fast relaxation phenomenon, consisting of time-translation invariance breaking and restoration, is advanced. Several COVID-related crashes in the time series for Shanghai Composite and Dow Jones Industrial are discussed as an illustration. Full article
(This article belongs to the Special Issue Nonlinear Analysis and Optimization with Applications)
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19 pages, 1062 KiB  
Article
Modified Viscosity Subgradient Extragradient-Like Algorithms for Solving Monotone Variational Inequalities Problems
by Nopparat Wairojjana, Mudasir Younis, Habib ur Rehman, Nuttapol Pakkaranang and Nattawut Pholasa
Axioms 2020, 9(4), 118; https://doi.org/10.3390/axioms9040118 - 15 Oct 2020
Cited by 8 | Viewed by 2326
Abstract
Variational inequality theory is an effective tool for engineering, economics, transport and mathematical optimization. Some of the approaches used to resolve variational inequalities usually involve iterative techniques. In this article, we introduce a new modified viscosity-type extragradient method to solve monotone variational inequalities [...] Read more.
Variational inequality theory is an effective tool for engineering, economics, transport and mathematical optimization. Some of the approaches used to resolve variational inequalities usually involve iterative techniques. In this article, we introduce a new modified viscosity-type extragradient method to solve monotone variational inequalities problems in real Hilbert space. The result of the strong convergence of the method is well established without the information of the operator’s Lipschitz constant. There are proper mathematical studies relating our newly designed method to the currently state of the art on several practical test problems. Full article
(This article belongs to the Special Issue Nonlinear Analysis and Optimization with Applications)
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20 pages, 1573 KiB  
Article
Reconstruction of Piecewise Smooth Multivariate Functions from Fourier Data
by David Levin
Axioms 2020, 9(3), 88; https://doi.org/10.3390/axioms9030088 - 24 Jul 2020
Cited by 3 | Viewed by 2337
Abstract
In some applications, one is interested in reconstructing a function f from its Fourier series coefficients. The problem is that the Fourier series is slowly convergent if the function is non-periodic, or is non-smooth. In this paper, we suggest a method for deriving [...] Read more.
In some applications, one is interested in reconstructing a function f from its Fourier series coefficients. The problem is that the Fourier series is slowly convergent if the function is non-periodic, or is non-smooth. In this paper, we suggest a method for deriving high order approximation to f using a Padé-like method. Namely, we do this by fitting some Fourier coefficients of the approximant to the given Fourier coefficients of f. Given the Fourier series coefficients of a function on a rectangular domain in Rd, assuming the function is piecewise smooth, we approximate the function by piecewise high order spline functions. First, the singularity structure of the function is identified. For example in the 2D case, we find high accuracy approximation to the curves separating between smooth segments of f. Secondly, simultaneously we find the approximations of all the different segments of f. We start by developing and demonstrating a high accuracy algorithm for the 1D case, and we use this algorithm to step up to the multidimensional case. Full article
(This article belongs to the Special Issue Nonlinear Analysis and Optimization with Applications)
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15 pages, 291 KiB  
Article
A Discussion on the Existence of Best Proximity Points That Belong to the Zero Set
by Erdal Karapınar, Mujahid Abbas and Sadia Farooq
Axioms 2020, 9(1), 19; https://doi.org/10.3390/axioms9010019 - 11 Feb 2020
Cited by 9 | Viewed by 2265
Abstract
In this paper, we investigate the existence of best proximity points that belong to the zero set for the α p -admissible weak ( F , φ ) -proximal contraction in the setting of M-metric spaces. For this purpose, we establish [...] Read more.
In this paper, we investigate the existence of best proximity points that belong to the zero set for the α p -admissible weak ( F , φ ) -proximal contraction in the setting of M-metric spaces. For this purpose, we establish φ -best proximity point results for such mappings in the setting of a complete M-metric space. Some examples are also presented to support the concepts and results proved herein. Our results extend, improve and generalize several comparable results on the topic in the related literature. Full article
(This article belongs to the Special Issue Nonlinear Analysis and Optimization with Applications)
17 pages, 321 KiB  
Article
Admissible Hybrid Z-Contractions in b-Metric Spaces
by Ioan Cristian Chifu and Erdal Karapınar
Axioms 2020, 9(1), 2; https://doi.org/10.3390/axioms9010002 - 21 Dec 2019
Cited by 20 | Viewed by 2476
Abstract
In this manuscript, we introduce a new notion, admissible hybrid Z -contraction that unifies several nonlinear and linear contractions in the set-up of a b-metric space. In our main theorem, we discuss the existence and uniqueness result of such mappings in the [...] Read more.
In this manuscript, we introduce a new notion, admissible hybrid Z -contraction that unifies several nonlinear and linear contractions in the set-up of a b-metric space. In our main theorem, we discuss the existence and uniqueness result of such mappings in the context of complete b-metric space. The given result not only unifies the several existing results in the literature, but also extends and improves them. We express some consequences of our main theorem by using variant examples of simulation functions. As applications, the well-posedness and the Ulam–Hyers stability of the fixed point problem are also studied. Full article
(This article belongs to the Special Issue Nonlinear Analysis and Optimization with Applications)
15 pages, 316 KiB  
Article
Exact Solutions for a Class of Wick-Type Stochastic (3+1)-Dimensional Modified Benjamin–Bona–Mahony Equations
by Praveen Agarwal, Abd-Allah Hyder, M. Zakarya, Ghada AlNemer, Clemente Cesarano and Dario Assante
Axioms 2019, 8(4), 134; https://doi.org/10.3390/axioms8040134 - 3 Dec 2019
Cited by 22 | Viewed by 2518
Abstract
In this paper, we investigate the Wick-type stochastic (3+1)-dimensional modified Benjamin–Bona–Mahony (BBM) equations. We present a generalised version of the modified tanh–coth method. Using the generalised, modified tanh–coth method, white noise theory, and Hermite transform, we produce a new set of exact travelling [...] Read more.
In this paper, we investigate the Wick-type stochastic (3+1)-dimensional modified Benjamin–Bona–Mahony (BBM) equations. We present a generalised version of the modified tanh–coth method. Using the generalised, modified tanh–coth method, white noise theory, and Hermite transform, we produce a new set of exact travelling wave solutions for the (3+1)-dimensional modified BBM equations. This set includes solutions of exponential, hyperbolic, and trigonometric types. With the help of inverse Hermite transform, we obtained stochastic travelling wave solutions for the Wick-type stochastic (3+1)-dimensional modified BBM equations. Eventually, by application example, we show how the stochastic solutions can be given as white noise functional solutions. Full article
(This article belongs to the Special Issue Nonlinear Analysis and Optimization with Applications)
11 pages, 264 KiB  
Article
Best Proximity Points for Monotone Relatively Nonexpansive Mappings in Ordered Banach Spaces
by Karim Chaira, Mustapha Kabil, Abdessamad Kamouss and Samih Lazaiz
Axioms 2019, 8(4), 121; https://doi.org/10.3390/axioms8040121 - 1 Nov 2019
Cited by 2 | Viewed by 2107
Abstract
In this paper, we give sufficient conditions to ensure the existence of the best proximity point of monotone relatively nonexpansive mappings defined on partially ordered Banach spaces. An example is given to illustrate our results. Full article
(This article belongs to the Special Issue Nonlinear Analysis and Optimization with Applications)
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