Newly existence of solutions for pantograph a semipositone in $ \Psi $-Caputo sense

Abstract

In the present manuscript, the BVP problem of a semipostone multipoint $ \Psi $-Caputo fractional pantograph problem is addressed. $ \mathcal{D}_{r}^{\nu;\psi}\varkappa(\varsigma)+\mathcal{F}(\varsigma , \varkappa(\varsigma), \varkappa(r+\lambda\varsigma)) = 0, \ \varsigma \mbox{ in }(r, \mathcal{\Im}), $ $ \varkappa(r) = \vartheta_{1}, \ \varkappa(\mathcal{\Im}) = \sum\limits_{i = 1}^{m-2} \zeta_{i}\varkappa(\mathfrak{\eta}_{i})+\vartheta_{2}, \ \vartheta_{i} \in\mathbb{R}, \ i\in\{1, 2\}, $ and $ \lambda $ in $ \left(0, \frac{\mathcal{\Im}\mathfrak{-}r}{\mathcal{\Im} }\right) $. The seriousness of this research is to prove the existence of the solution of this problem by using Schauder's fixed point theorem (SFPT). We have developed our results in our research compared to some recent research in this field. We end our work by listing an example to demonstrate the result reached.

Keywords

• $ \psi $-caputo derivative
• bvp
• pantograph problem
• changing sign nonlinearity
• schauder fixed point theorem