1 Introduction

Fractional calculus is a branch of mathematics that encompasses real- and complex-order derivatives and integrals. It is an extension of integer differential calculus [10], [11], [13], [14], [15], [32], [36], [58], [75], [77], [79], [217], [219], [220], [221], [222]. The notion of fractional differential calculus has a lengthy history. One may question what meaning the derivative of a fractional order might have, that is, dnydxn, where n is a fraction. Apparently, L’Hopital discussed this idea in a letter with Leibniz. L’Hopital wrote to Leibniz in 1695 asking, “What if n be 12?” The study of fractional calculus was born from this question. Leibniz responded to the question, “ d12x will be equal to xdx:x. This is an apparent paradox from which, one day, useful consequences will be drawn.”

“Fractional calculus” was born on September 30, 1695. Subsequently, many well-known mathematicians have contributed to the development of this theory throughout the years. As a result, fractional calculus has its origins in the work of Leibniz, L’Hopital (1695), Bernoulli (1697), Euler (1730), and Lagrange (1772). Some years later, Laplace (1812), Fourier (1822), Abel (1823), Liouville (1832), Riemann (1847), Grünwald (1867), Letnikov (1868), Nekrasov (1888), Hadamard (1892), Heaviside (1892), Hardy (1915), Weyl (1917), Riesz (1922), P. Levy (1923), Davis (1924), Kober (1940), Zygmund (1945), Kuttner (1953), J. L. Lions (1959), and Liverman (1964) and several more contributed to the fundamental principles of fractional calculus.

Ross organized the first fractional calculus conference at the University of New Haven in June 1974 and edited the proceedings [197]. Thereafter, Spanier published the first monograph devoted to “Fractional Calculus” in 1974 [181]. The integrals and derivatives of noninteger order, as well as fractional integro-differential equations, have seen various applications in studies in theoretical physics, mechanics, and applied mathematics. The exceptionally extensive encyclopedic-type monograph by Samko, Kilbas, and Marichev was published in Russian in 1987 and in English in 1993 [227] (for more details see [165]). The works devoted to fractional differential equations include the books of Miller and Ross (1993) [172], Podlubny (1999) [188], Kilbas et al. (2006) [143], Diethelm (2010) [114], Ortigueira (2011) [184], Abbas et al. (2012) [14], and Baleanu et al. (2012) [57].

As is generally known, the origins of fixed point theory may be traced back to the system of successive approximations (or the Picard iterative approach) used to resolve some differential equations. Banach derived the fixed point theorem through a series of repeated approximations. Over the last few decades, fixed point theory has grown enormously and independently of differential equations. However, the results of fixed points have recently been discovered to be the instruments for the solutions of differential equations. Differential fractional-order equations have recently been demonstrated to be an excellent tool for exploring several phenomena in various domains of science and engineering, including electrochemistry, electromagnetics, viscoelasticity, and economics. It is common in the literature to propose a solution to fractional differential equations by combining several types of fractional derivatives; see, e. g., [7], [8], [9], [14], [15], [18], [22], [26], [27], [32], [36], [48], [58], [62], [140], [141], [253]. On the other hand, there are more findings about the issue of boundary values for fractional differential equations [32], [53], [67], [69], [253].

Mawhin’s coincidence degree theory [126], [169] has been widely used in the study of many classes of nonlinear differential equations. Coincidence degree theory is useful in the study of nonlinear fractional differential equations (NFDEs). We can use this technique when other techniques do not work, such as the fixed point principle. For example, in the works [68], [73], the authors obtained the results by using coincidence degree theory.

The pantograph equations have been widely used in the fields of quantum mechanics and dynamical systems [190], [198]. Actually, several researchers have investigated some new existence and uniqueness results for NFDE pantograph problems by applying fixed point theorems or the coincidence degree theorem [19], [19], [65], [68], [70], [73], [180], [190].

Many articles and monographs have been written recently in which the authors investigate numerous results for systems with different types of fractional differential equations and inclusions and various conditions. One may see [16], [39], [42], [130], [160], [162], [226] and the references therein.

In this monograph, a new generalization of the well-known Hilfer fractional derivative is given. We took the publications of Diaz et al. [113] into account, where the authors presented the k-gamma and k-beta functions and demonstrated a number of their properties, many of which can also be found in [101], [175], [177], [178]. In addition, we were inspired by Sousa’s numerous publications [102], [103], [104], [105], [106], [107], [108], in which the authors established another sort of fractional operator known as the ψ-Hilfer fractional derivative with respect to a particular function and provided several essential properties about this type of fractional operator. Our work in this monograph may be viewed as a continuation and generalization of the preceding studies, i. e., several results in the fractional calculus literature.

In the following we give an outline of the organization of this monograph, which consists of 11 chapters.

Chapter 2 provides the notation and preliminary results, descriptions, theorems, and other auxiliary results that will be needed for this study. In the first section we give some notations and definitions of the functional spaces used in this book. In the second section, we give the definitions of the elements from fractional calculus theory, and then we present some necessary lemmas, theorems, and properties. In the third section, we give some properties of the measure of noncompactness. We finish the chapter by giving all the fixed point theorems that are used throughout the book and some definitions and necessary results of coincidence degree theory in the last section.

Chapter 3 deals with some existence results for a class of problems for nonlinear implicit fractional differential equations (IFDEs) with Caputo-type fractional derivatives and periodic conditions by using coincidence degree theory. The chapter is divided into five sections. Section 3.1 provides an introduction and some motivations. Section 3.5 contains some remarks and suggestions. The main results of the chapter begin in Section 3.2, where we provide the existence of periodic solutions for the following nonlinear IFDE:

where cDα is the fractional derivative of Caputo and f:T‾×R×R⟶R is a continuous function. The results are based on coincidence degree theory. Next, some examples are included to show the applicability of our results. In Section 3.3, we establish some existence and uniqueness results for the following functional fractional differential problem:

where b>0, cD0α is the Caputo fractional derivative of order α, 0<β≤1<α≤2, 1+β<α, f:T‾×Rn×Rn×Rn→Rn is a given continuous function, and A is an Rn×n invertible matrix. In addition, an example is given to justify our results. Section 3.4 deals with the existence of solutions for the following nonlinear fractional differential problem:

where ceDa+α and ceDa+μ are the Caputo exponential fractional derivatives of order α>0 and μ>0, respectively, and g:T‾×R×R⟶R is a continuous function.

The aim of Chapter 4 is to prove the existence and uniqueness of periodic solutions to a large class of problems for NFDEs with ψ-Caputo fractional derivative. As always, we...