Existence of solutions for a semipositone fractional boundary value pantograph problem

Abstract

The boundary value problem (BVP) for a nonlinear non positone or semi-positone multi-point Caputo-Hadamard fractional differential pantograph problem is addressed in this study. $ \begin{equation*} \mathfrak{D}_{1}^{\upsilon}x(\mathfrak{t})+\mathrm{f}(\mathfrak{t}, x( \mathfrak{t}), x(1+\lambda\mathfrak{t})) = 0, \ \mathfrak{t}\in(1, \mathfrak{b}) \end{equation*} $ $ \begin{equation*} x(1) = \delta_{1}, \ x(\mathfrak{b}) = \sum\limits_{i = 1}^{m-2}\zeta_{i}x(\mathfrak{\eta } _{i})+\delta_{2}, \ \delta_{i}\in\mathbb{R}, \ i = 1, 2, \end{equation*} $ where $ \lambda\in\left(0, \frac{\mathfrak{b-}1}{\mathfrak{b}}\right) $. The novelty in our approach is to show that there is only one solution to this problem using the Schauder fixed point theorem. Our results expand some recent research in the field. Finally, we include an example to demonstrate our findings.

Keywords

• caputo-hadamard fractional derivative
• bvp
• changing sign nonlinearity
• pantograph problem
• schauder fixed point theorem