Applied Functional Analysis and Its Applications • Jen-Chih and Shahram Yao Rezapour Applied Functional Analysis and Its Applications Edited by Jen-Chih Yao and Shahram Rezapour Printed Edition of the Special Issue Published in Mathematics www.mdpi.com/journal/mathematics Applied Functional Analysis and Its Applications Applied Functional Analysis and Its Applications Editors Jen-Chih Yao Shahram Rezapour MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade • Manchester • Tokyo • Cluj • Tianjin Editors Jen-Chih Yao Shahram Rezapour China Medical University Hospital Azerbaijan Shahid Madani China University Iran China Medical University China Medical University Taiwan Taiwan Editorial Office MDPI St. Alban-Anlage 66 4052 Basel, Switzerland This is a reprint of articles from the Special Issue published online in the open access journal Mathematics (ISSN 2227-7390) (available at: https://www.mdpi.com/journal/mathematics/special issues/Applied Functional Analysis Its Applications). For citation purposes, cite each article independently as indicated on the article page online and as indicated below: LastName, A.A.; LastName, B.B.; LastName, C.C. Article Title. Journal Name Year, Article Number, Page Range. ISBN 978-3-03936-776-4 (Hbk) ISBN 978-3-03936-777-1 (PDF) c 2020 by the authors. Articles in this book are Open Access and distributed under the Creative Commons Attribution (CC BY) license, which allows users to download, copy and build upon published articles, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. The book as a whole is distributed by MDPI under the terms and conditions of the Creative Commons license CC BY-NC-ND. Contents About the Editors .............................................. vii Preface to ”Applied Functional Analysis and Its Applications” ................... ix Hsien-Chung Wu Informal Norm in Hyperspace and Its Topological Structure Reprinted from: Mathematics 2019, 7, 945, doi:10.3390/math7100945 ................. 1 Bing Tan, Shanshan Xu and Songxiao Li Modified Inertial Hybrid and Shrinking Projection Algorithms for Solving Fixed Point Problems Reprinted from: Mathematics 2020, 8, 236, doi:10.3390/math8020236 ................. 21 Jong Soo Jung Convergence Theorems for Modified Implicit Iterative Methods with Perturbation for Pseudocontractive Mappings Reprinted from: Mathematics 2020, 8, 72, doi:10.3390/math8010072 ................. 33 Jin Liang and Yunyi Mu Properties for ψ-Fractional Integrals Involving a General Function ψ and Applications Reprinted from: Mathematics 2019, 7, 517, doi:10.3390/math7060517 ................. 49 Badr Alqahtani, Hassen Aydi, Erdal Karapınar and Vladimir Rakoˇcevi´c A Solution for Volterra Fractional Integral Equations by Hybrid Contractions Reprinted from: Mathematics 2019, 7, 694, doi:10.3390/math7080694 ................. 63 Yinglin Luo, Bing Tan and Meijuan Shang A General Inertial Viscosity Type Method for Nonexpansive Mappings and Its Applications in Signal Processing Reprinted from: Mathematics 2020, 8, 288, doi:10.3390/math8020288 ................. 73 Lu-Chuan Ceng, Adrian Petru¸sel , Ching-Feng Wen and Jen-Chih Yao Inertial-Like Subgradient Extragradient Methods for Variational Inequalities and Fixed Points of Asymptotically Nonexpansive and Strictly Pseudocontractive Mappings Reprinted from: Mathematics 2019, 7, 860, doi:10.3390/math7090860 ................. 91 Lu-Chuan Ceng, Adrian Petru¸sel and Jen-Chih Yao On Mann Viscosity Subgradient Extragradient Algorithms for Fixed Point Problems of Finitely Many Strict Pseudocontractions and Variational Inequalities Reprinted from: Mathematics 2019, 7, 925, doi:10.3390/math7100925 .................111 Syed Shakaib Irfan, Mijanur Rahaman, Iqbal Ahmad, Rais Ahmadand Saddam Husain Generalized Nonsmooth Exponential-Type Vector Variational-Like Inequalities and Nonsmooth Vector Optimization Problems in Asplund Spaces Reprinted from: Mathematics 2019, 7, 345, doi:10.3390/math7040345 .................125 Liu He, Qi-Lin Wang, Ching-Feng Wen, Xiao-Yan Zhang and Xiao-Bing Li A Kind of New Higher-Order Mond-Weir Type Duality for Set-Valued Optimization Problems Reprinted from: Mathematics 2019, 7, 372, doi:10.3390/math7040372 .................137 v Elisabeth Kobis,¨ Markus A. Kobis¨ and Xiaolong Qin An Inequality Approach to Approximate Solutions of Set Optimization Problems in Real Linear Spaces Reprinted from: Mathematics 2020, 8, 143, doi:10.3390/math8010143 .................155 vi About the Editors Jen-Chih Yao (Professor) is Chair Professor of the Center for General Education at China University (Taiwan) and Director of the Research Center for Interneural Computing, China Medical University Hospital, China Medical University. He has published numerous papers in different fields (e.g., variational inequalities, complementarity problems, fixed point theorems, variational analysis, optimization, vector optimization problems, etc.). He is the editor-in-chief and a member of the editorial board of several journals. He was also the chairman of the organizing and scientific committees of several international conferences and workshops on nonlinear analysis and optimization. His name has been included on lists of highly cited researchers in 2014, 2015, 2016, 2017, 2018, and 2019. Shahram Rezapour (Professor) is a professor of mathematics at Azarbaijan Shahid Madani University (Iran). He simultaneously holds a visiting professor position at China Medical University (Taiwan). He has published numerous papers on different fields (e.g., approximation theory, fixed point theory, fractional integro-differential equations and inclusions, finite differences, numerical and approximate solutions of singular fractional differential equations, stability, and optimization). He is a member of the editorial boards of several journals. He was also the chairman of the organizing and scientific committees of several international conferences. His name has been included on lists of highly cited researchers in 2016 and 2017. vii Preface to ”Applied Functional Analysis and Its Applications” It is well known that applied functional analysis is very important in most applied research fields and its influence has grown in recent decades. Many novel works have used techniques, ideas, notions, and methods of applied functional analysis. Furthermore, applied functional analysis includes linear and nonlinear problems. The scope of this field is so wide that it cannot be expressed in a few books. This book covers a limited section of this field, namely, fixed point theory and applications, nonlinear methods and variational inequalities, and set-valued optimization problems. The most important application of fixed point theory is proving the existence of solutions for fractional integro-differential equations and, therefore, increasing our ability to model different kinds of phenomena. In most everyday matters, we seek to use optimization. The importance of optimization has attracted many researchers to this field over the past few decades and provided new ideas, concepts, and techniques. Jen-Chih Yao , Shahram Rezapour Editors ix mathematics Article Informal Norm in Hyperspace and Its Topological Structure Hsien-Chung Wu Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 802, Taiwan; [email protected] Received: 4 September 2019; Accepted: 8 October 2019; Published: 11 October 2019 Abstract: The hyperspace consists of all subsets of a vector space. Owing to a lack of additive inverse elements, the hyperspace cannot form a vector space. In this paper, we shall consider a so-called informal norm to the hyperspace in which the axioms regarding the informal norm are almost the same as the axioms of the conventional norm. Under this consideration, we shall propose two different concepts of open balls. Based on the open balls, we shall also propose the different types of open sets. In this case, the topologies generated by these different concepts of open sets are investigated. Keywords: hyperspace; informal open sets; informal norms; null set; open balls 1. Introduction The topic of set-valued analysis (or multivalued analysis) has been studied for an extensive period. A detailed discussion can refer to Aubin and Frankowska [1], and Hu and Papageorgiou [2,3]. Applications in nonlinear analysis can refer to Agarwal and O’Regan [4], Burachik and Iusem [5], and Tarafdar and Chowdhury [6]. More specific applications in differential inclusion can also refer to Aubin and Cellina [7]. On the other hand, the fixed point theory for set-valued mappings can refer to Górniewicz [8], and set-valued optimization can refer to Chen et al. [9], Khan et al. [10] and Hamel et al. [11]. Also, the set optimization that is different from the set-valued optimization can refer to Wu [12] and the references therein. Let P(X) be the collection of all subsets of a vector space X. The set-valued analysis usually studies the mathematical structure in P(X) in which each element in P(X) is treated as a subset of X. In this paper, we shall treat each element of P(X) as a “point”. In other words, each subset of X is compressed as a point, and the family P(X) is treated as a universal set. In this case, the original vector space X plays no role in the settings. Therefore, we want to endow a vector structure to P(X). Although we can define the vector addition and scalar multiplication in P(X) in the usual way, owing to lacking an additive inverse element, the family P(X) cannot form a vector space.