On a new structure of the pantograph inclusion problem in the Caputo conformable setting

• Published in: Boundary value problems Vol. 2020; no. 1; pp. 1 - 21
• Main Authors: Thabet, Sabri T. M., Etemad, Sina, Rezapour, Shahram
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 11.11.2020; Hindawi Limited; SpringerOpen
• Subjects: Analysis; Approximations and Expansions; Difference and Functional Equations; Differential equations; Inclusion problem; Mathematics; Mathematics and Statistics; Operators (mathematics); Ordinary Differential Equations; Pantograph differential equation; Pantographs; Partial Differential Equations; The Caputo conformable derivative; The lower and upper semi-continuity; The lower and upper semi-continuity; Inclusion problem; 34A12; Pantograph differential equation; The Caputo conformable derivative; 34A08
• ISSN: 1687-2770; 1687-2762; 1687-2770
• DOI: 10.1186/s13661-020-01468-4

In this work, we reformulate and investigate the well-known pantograph differential equation by applying newly-defined conformable operators in both Caputo and Riemann–Liouville settings simultaneously for the first time. In fact, we derive the required existence criteria of solutions corresponding to the inclusion version of the three-point Caputo conformable pantograph BVP subject to Riemann–Liouville conformable integral conditions. To achieve this aim, we establish our main results in some cases including the lower semi-continuous, the upper semi-continuous and the Lipschitz set-valued maps. Eventually, the last part of the present research is devoted to proposing two numerical simulative examples to confirm the consistency of our findings.