arXiv:1909.13681v1 [math.GM] 26 Sep 2019 On existence of solution to nonlinear ψ−Hilfer Cauchy-type problem Mohammed S Abdo 1 S K Panchal 2 and Sandeep P Bhairat 3∗ 1 Department of Mathematics, Hodeidah University, Al-Hodeidah-3114, Yemen. 2 Department of Mathematics, Dr Babasaheb Ambedkar Marathwada University, Aurangabad, (M.S.) India. 3 Institute of Chemical Technology Mumbai, Marathwada Campus, Jalna - 431 203 (M.S.) India. Abstract Considering the implicit fractional differential equation involving a general form of Hilfer fractional deriva- tive with respect to another function. We show that weighted Cauchy-type problem is equivalent to a Volterra integral equation. By employed a variety of tools of fractional calculus including Banach fixed point theorem, we prove the existence and uniqueness of solution of weighted Cauchy-type problem. Also through generalized Gronwall inequality, we prove the continuous dependence of data on the Cauchy-type problem. An example is provided to illustrate our main results. Keywords: Fractional integrals and derivatives; Picard iterative technique; singular fractional differential equation, Cauchy-type problem. Mathematics Subject Classification:26A33; 26D10; 34A08; 40A30. 1 Introduction Fractional differential equations [7, 13, 21] emerged as a rich and shapely field of research due to a variety of applications including numerous fields of science and engineering such as physics, mechanics chemistry, biology, engineering, and other, for instance, see [3, 11, 12, 17, 18, 20, 25]. and the references are given therein. The study of fractional differential equations ranges from the theoretical sides of some important qualitative properties of solutions such as existence, uniqueness, dependence continuous, and stability results to the analytic and numerical methods for finding solutions, see [14, 15, 16, 4, 5]. The properties of fractional integrals and fractional derivatives of a function with respect to another function have been introduced by Kilbas et al. in [13]. Howover, Almedia in [2] have recently introduced a fractional differentiation operator, which they called ψ- Caputo fractional operator. On the other hand, Sousa and Oliveira [23, 24] have recently proposed a ψ- Hilfer fractional operator and extended few previous works dealing with the Hilfer [8, 9, 10]. For example, Furati et al. in [9], considered nonlinear fractional differential equation involving Hilfer fractional derivative D α,β a + u(t)= f (t,u(t), ),t>a, 0 <α< 1, 0 ≤ β ≤ 1, (1) I 1−γ a + u(a + )= u a ,γ = α + β − αβ (2) where D α;ψ a + ,I 1−γ a + are Hilfer fractional derivative and Riemann-Liouville fractional integral, respectively, u a ∈ R. The authors used the Banach fixed point theorem to investigate the existence and uniqueness and stability of global solutions in the weighted space on the problem (1)-(2). Dheigude and Bhairat in [8], disccused the existence, uniqueness and continuous dependence of solution for problem (1)-(2) by using successive approximations and generalized Gronwall inequality. Lately, Sousa et al. in [22] proposed a generalized Gronwall inequality through the fractional integral with respect to another function I 1−γ;ψ a + (·). They considered Cauchy-type problem (1)-(2) involving the ψ-Hilfer fractional derivative D α,β;ψ a + (·) to obtain the existence uniqueness and continuous dependence of solutions. Oliveira and de Oliveira in [19], proposed a new fractional derivative the Hilfer–Katugampola fractional derivatives ρ D α,β a + (·) and * author for correspondence: sp.bhairat@marj.ictmumbai.edu.in