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Analytic Study on Boundary Value Problem of Implicit Differential Equations via Composite Fractional Derivative D. Vivek1 · E. M. Elsayed2,3

· K. Kanagarajan4

Received: 23 July 2019 / Revised: 2 February 2020 / Accepted: 21 March 2020 / © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2020

Abstract This paper is devoted to investigate implicit differential equations with boundary condition, which involves the composite fractional derivative in weighted space. The existence and uniqueness of the solution are obtained using the classic fixed point theorems. As an application, an example is presented to illustrate the main results. Keywords Composite fractional derivative · Boundary value problem · Existence · Fixed point Mathematics Subject Classification (2010) 26A33 · 33E12 · 34B10

1 Introduction In recent years, fractional order calculus has been one of the most rapidly developing areas of mathematical analysis. Indeed, a natural phenomenon may depend not only on the current time but also on its previous time history. Fractional calculus facilitates modeling of  E. M. Elsayed

[email protected] D. Vivek [email protected] K. Kanagarajan [email protected] 1

Department of Mathematics, PSG College of Arts, Science, Coimbatore, 641 014, India

2

Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

3

Faculty of Science, Mansoura University, Mansoura, 35516, Egypt

4

Department of Mathematics, Sri Ramakrishna Mission Vidyalaya College of Arts and Science, Coimbatore, 641 020, India

D. Vivek et al.

such phenomena via nonlocal fractional differential and integral operators. Fractional order differential equations naturally appear in the mathematical modeling of systems with memory. One can find numerous applications of fractional calculus in diverse fields such as mathematics, physics, chemistry, optimal control theory, finance, biology, and engineering [10, 12, 15]. Since it is clear that dealing with Riemann-Liouville (R-L) derivative in various applied problems is very difficult, therefore, certain modifications were introduced to avoid the difficulties. In this regard, some new type fractional order derivative operators were introduced in literature like Caputo and Hadamard. Recently, Hilfer [12] initiated extended R-L fractional derivative, named Hilfer fractional derivative, which interpolates Caputo fractional derivative and R-L fractional derivative. This said operator arose in the theoretical simulation of dielectric relaxation in glass-forming materials (see [24–27]). Followed by the work, Sousa and Oliveira [20] introduced the composite fractional derivative (ψ-Hilfer fractional derivative) with respect to another function, in order to unify the wide number of fractional derivatives in a single fractional operator and consequently, open a window for new applications, we refer to [19, 21]. At present, a great deal of efforts were spent in linear and nonlinear fractional differe