Existence Results for a Class of Nonlinear Hadamard Fractional with p-Laplacian Operator Differential Equations

Recently, fractional differential equations have been acquired much attention due to its applications in a number of fields such as physics, mechanics, chemistry, biology, signal and image processing, see for example the books (Baleanu et al., 2012; Kilbas et al., 2006; Lakshmikantham et al., 2009; Yang et al., 2015). Some recent works on fractional differential equations involving Riemann Liouville and Caputo-type fractional derivatives are studied using nonlinear analysis methods such as Krasnoselskii fixed-point Theorems (Agarwal and O'Regan, 1998; Ghanmi and Horrigue, 2018; Guo et al., 2007; Guo et al., 2008), Leray-Schauder alternative (Ghanmi and Horrigue, 2019; Qi et al., 2017), sub-solution and super-solution methods (Wang et al., 2019; Mâagli et al., 2015) and iterative techniques (Liu et al., 2013). Hadamard (1892) introduced an important fractional derivative, which differs from the above-mentioned ones because its definition involves logarithmic function of arbitrary exponent and named as Hadamard derivative. In the last few decades many authors are paying more and more attention to fractional differential equation involving Hadamard derivative, the study of the topic is still in its primary stage. For details and recent developments on Hadamard fractional differential equations, see (Huang and Liu, 2018; Wang et al., 2018; Zhai et al., 2018) and references therein. Recently, some researches have extensively interested in the study of the fractional differential equations with p-Laplacian operators see for examples (Chamekh et al., 2018; Ding et al., 2015). From the above review of the literature concerning fractional differential equations, most of the authors investigated only the existence of solutions or positive solutions for Hadamard fractional differential equations without considering the pi-Laplacian operator. A very few authors established results along with p-Laplacian operator, us example in (Wang and Wang, 2016), the authors considered the following nonlinear Hadamard fractional differential problem: