Preface

The aim of this book is to present a broad overview of the theory and applications related to functional calculus. The book is based on two main subject areas: matrix calculus and applications of Hilbert spaces.

Functional analysis is the most important branch of mathematics, whose foundation was laid by the great Persian polymath Muhammad ibn Mūsā al-Khwārizmī, also known as Algorithmi, during 973–1048. He named this branch the "Theory of Functions." Later, Newton and Leibnitz enriched this branch by introducing the concept of derivatives and integrals during 1665–1742 and thus gave birth to another name: calculus. This branch of mathematics has been recently divided into several subbranches, including differential calculus, integral calculus, stochastic calculus, etc. In mathematics, a functional calculus is a theory that permits someone to apply mathematical functions to mathematical operators. Now, functional calculus is a branch that connects operator theory, classical calculus, algebra, and functional analysis. In daily life, functionals are increasingly used to model real-world situations, for example if *f: R→R* is real valued functional from real to real number system. If we apply *f* on some function *x*∈*R*, then *f(x)* makes no sense but if we write it in equation form, then it makes sense, e.g. *f(x)= x*, which represents a physical process between two quantities such that there is direct proportionality. Similar problems occur daily in our surroundings. Therefore, it is necessary to understand what criteria should be satisfied by concerned functionals and operators used in modeling or in the description of daily life problems. It is functional calculus that guides and provides us with the path to how, when, and where particular functionals and operators may be used. Mostly, integral and differential equations are used when we wish to solve a technique or procedure that converts the mentioned equations into algebraic equations of known and unknown functions and functionals. Keeping these needs in mind, the editor of this book has been motivated to welcome international mathematicians and researchers to contribute various topics that address the areas of functional calculus and its applications in both pure and applied analysis. The editor has incorporated contributions from a diverse group of leading researchers in the field of functional calculus. This book aims to provide an overview of the present knowledge that addresses applications and results related to functional calculus. The main topics covered in this book are determinantal representations of the core inverse and its generalizations, which provides a foundation to solve matrix equations. Furthermore, new series formulae for matrix exponential series have been developed, which are used in solving algebraic equations. Also covered are results on fixed point theory, which is used for mapping the satisfying condition (DA) in Banach space. Results that address folding on chaotic graph operations and their fundamental groups are also introduced. Such algebraic structures are largely used in biology and chemistry. Elsewhere in the book, a brief review is considered of Hilbert space with its fundamental features and features of reproducing kernels in corresponding spaces. Spectral theory is an important area that is most applicable in quantum mechanics. Therefore, a number of fundamental concepts have been investigated regarding analytical applications and observations of PM10 fluctuations. Optimal control is a very important procedure, which is increasingly used in the study of mathematical models of real-world problems. It is helpful in developing future

**II**

**Chapter 10 149**

**Chapter 11 165**

Integral Inequalities and Differential Equations via Fractional Calculus

Approximate Solutions of Some Boundary Value Problems by Using

*by Kamal Shah,Thabet Abdeljawad, Hammad Khalil and Rahmat Ali Khan*

*by Zoubir Dahmani and Meriem Mansouria Belhamiti*

Operational Matrices of Bernstein Polynomials

predictions and control strategies of infectious diseases. Analytic and numerical results of the Euler–Bernoulli beam model with a two-parameter family of boundary conditions are also presented, where Chebyshev polynomial approximation has been used to approximate the solution. In recent times, fractional calculus has attracted great attention. Results on fractional integral inequalities are investigated. By using the principle of functional calculus, numerical analysis for boundary value problems of fractional differential equations are studied in the final chapter.

The theory of Hilbert spaces is the center around which functional analysis has developed. Hilbert spaces have a rich geometric nature as they are endowed with an inner product that permits the concept of orthogonality of vectors. Hilbert space methods are applied to several science and engineering areas such as optimization, variational and control problems, and to problems in approximation theory, nonlinear stability, and bifurcation as well as spectral theory and quantum mechanics. That is why a part of the book is devoted to a brief presentation and applications of Hilbert spaces. For the reader who has no previous experience in the theory of normed spaces with enough background for comprehending the theory of Hilbert spaces, there two chapters based on these topics in the book. An important application of the theory of Hilbert spaces to the reproducing kernels is also analyzed in this part. Spectral theory is an important area which is most applicable in quantum mechanics. In this content, a real-life application of Hilbert space where an investigation of the pollution and air quality in Caribbean region by the help of theoretical Hilbert frame aspect is also provided. Here some observations of PM10 fluctuations are analyzed by scaling and time-frequency properties of PM10 data in Hilbert frame and compared the functioning obtained in Hilbert space. Optimal control is also very important procedure which is increasingly used in study of mathematical models of real world problems. It is helpful in developing future predictions and control strategies of infectious disease. In this issue, analytic and numerical results of the Euler-Bernoulli beam model with a two-parameter family of boundary conditions have been presented where Chebyshev polynomial approximation has been used to approximate the solution.

We hope that this book will be of benefit to mathematicians, computational mathematicians, applied mathematicians, and researchers in the field of pure mathematics as well as in analysis. The book is written basically for those who have some knowledge of classical calculus and mathematical analysis. The authors of each section convey a strong emphasis on theoretical foundations.

#### **Kamal Shah**

**Chapter 1**

*Ivan I. Kyrchei*

**Abstract**

**1. Introduction**

. *<sup>m</sup>*�*<sup>n</sup>*

**1**

Determinantal Representations

Generalized inverse matrices are important objects in matrix theory. In particular, they are useful tools in solving matrix equations. The most famous generalized inverses are the Moore-Penrose inverse and the Drazin inverse. Recently, it was introduced new generalized inverse matrix, namely the core inverse, which was late extended to the core-EP inverse, the BT, DMP, and CMP inverses. In contrast to the inverse matrix that has a definitely determinantal representation in terms of cofactors, even for basic generalized inverses, there exist different determinantal representations as a result of the search of their more applicable explicit expressions. In this chapter, we give new and exclusive determinantal representations of the core inverse and its generalizations by using determinantal representations of the Moore-Penrose and Drazin inverses previously obtained by the author.

**Keywords:** Moore-Penrose inverse, Drazin inverse, core inverse, core-EP inverse,

In the whole chapter, the notations and are reserved for fields of the real and complex numbers, respectively. *<sup>m</sup>*�*<sup>n</sup>* stands for the set of all *<sup>m</sup>* � *<sup>n</sup>* matrices over

*<sup>r</sup>* determines its subset of matrices with a rank *r*. For **A** ∈ *<sup>m</sup>*�*<sup>n</sup>*, the symbols **<sup>A</sup>**<sup>∗</sup> and rkð Þ **<sup>A</sup>** specify the conjugate transpose and the rank of **<sup>A</sup>**, respectively, <sup>∣</sup>**A**<sup>∣</sup> or

For **<sup>A</sup>** <sup>∈</sup> *<sup>n</sup>*�*<sup>n</sup>* with index Ind**<sup>A</sup>** <sup>¼</sup> *<sup>k</sup>*, i.e., the smallest positive number such that

rk **<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> <sup>¼</sup> rk **<sup>A</sup>***<sup>k</sup>* , *the Drazin inverse* of **<sup>A</sup>**, denoted by **<sup>A</sup>***<sup>d</sup>*

matrix **X** that satisfies Eq. (2) and the following equations,

**AXA** ¼ **A** (1) **XAX** ¼ **X** (2) ð Þ **AX** <sup>∗</sup> <sup>¼</sup> **AX** (3) ð Þ **XA** <sup>∗</sup> <sup>¼</sup> **XA** (4)

, is called the unique

*det***<sup>A</sup>** stands for its determinant. A matrix **<sup>A</sup>** <sup>∈</sup> *<sup>n</sup>*�*<sup>n</sup>* is Hermitian if **<sup>A</sup>**<sup>∗</sup> <sup>¼</sup> **<sup>A</sup>**. **A**† means *the Moore-Penrose inverse* of **A** ∈ *<sup>n</sup>*�*<sup>m</sup>*, i.e., the exclusive matrix **X**

**2000 AMS subject classifications:** 15A15, 16W10

satisfying the following four equations:

of the Core Inverse and Its

Generalizations

Associate Professor, Department of Mathematics, University of Malakand, Khyber Pakhtankhawa, Pakistan

#### **Baver Okutmuştur**

 Assistant Professor, Department of Mathematics, Middle East Technical University (METU), Ankara, Turkey

### **Chapter 1**
