Prof. Mouffak Benchohra is a Full Professor at the department of mathematics, Djillali Liabes University of Sidi Bel Abbes since October 1994. Benchohra received the master's degree in Nonlinear Analysis from Tlemcen University, Algeria, 1994 and Ph.D. degree in Mathematics from Djillali Liabes University, Sidi Bel Abbes, Algeria. His research fields include fractional differential equations, evolution equations and inclusions, control theory and applications, etc. Benchohra has published more than 500 papers, and five monographs. He is a Highly Cited Researcher in Mathematics from Thompson Reuters (2014) and Clarivate Analytics (2017-2018). Benchohra has also occupied the position of head of department of mathematics at Djillali Liabes University, Sidi Bel Abbes. He is in the Editorial Board of 10 international journals.
Dr. Soufyane Bouriah is an Associate Professor at the faculty of Technology, Hassiba Benbouali university of Chlef. Bouriah received the master's degree in functional analysis and differential equations from Djillali Liabès University, Algeria, and the doctorate's degree in mathematical analysis and applications from Djillali Liabes University of Sidi Bel Abbes, Algeria. His research fields include fractional differential and applications.
Dr. Abdelkrim Salim is an Associate Professor at the faculty of Technology, Hassiba Benbouali university of Chlef since 2022. Salim received the master's degree in functional analysis and differential equations from Djillali Liabès University, Algeria, 2016, and the doctorate's degree in mathematical analysis and applications from Djillali Liabes University of Sidi Bel Abbes, Algeria, 2021. His research fields include fractional differential equations and inclusions, control theory and applications, etc.
Prof. Yong Zhou is a Full Professor at School of Mathematics and Computational Science, Xiangtan University since 2000. He is also a Distinguished Guest Professor at Macau University of Science and Technology since 2018. His research fields include fractional differential equations, functional differential equations, evolution equations and inclusions, control theory. Zhou has published seven monographs in Springer, Elsevier, De Gruyter, World Scientific and Science Press respectively, and more than three hundred research papers in international journals such as Mathematische Annalen, Journal of Functional Analysis, Inverse problems, Nonlinearity, Proceedings of the Royal Society of Edinburgh A, International Journal of Bifurcation and Chaos, Bulletin des Sciences Mathematiques, Comptes rendus Mathematique, Zeitschrift fur Angewandte Mathematik und Physik, Discrete and Continuous Dynamical System, etc. He has been uninterruptedly on the list of Highly Cited Researchers since 2014. Zhou has undertaken five projects from National Natural Science Foundation of China, and two projects of the Macau Science and Technology Development Fund of China. He won the second prize of Chinese University Natural Science Award in 2000, and the second prize of Natural Science Award of Hunan Province in 2017 and 2021. Zhou was the Editor-in-Chief of International Journal of Dynamical Systems and Differential Equations from 2007 to 2011. In addition, he had worked as an Associate Editor for IEEE Transactions on Fuzzy Systems, Journal of Applied Mathematics& Computing, Mathematical Inequalities& Applications, and an Editorial Board Member of Fractional Calculus and Applied Analysis.
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, wheren is a fraction. Apparently, L’Hopital discussed this idea in a letter with Leibniz. L’Hopital wrote to Leibniz in 1695 asking, “What ifn be12?” The study of fractional calculus was born from this question. Leibniz responded to the question, “d12x will be equal toxdx: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 Diazet al. [113] into account, where the authors presented thek-gamma andk-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 theψ-Hilfer fractional derivative with respect to a partic
