In this research article, by the utilize of the well-known monotone iterative technique along with the technique of upper and lower solutions, we investigate the existence of extremal solutions for a new category of nonlinear fractional differential equations involving a novel Caputo version so-called }${\footnotesize \psi }$% {\footnotesize –Caputo fractional derivative with nonlinear boundary conditions. As an application of this work, two pertinent examples are offered to highlight our theoretical results.

The work reported here concerns with study a generalized nonlinear fractional boundary value problems involving $ \vartheta $- fractional derivative in the Riemann-Liouville sense. The existence and uniqueness of positive solutions to the problem at hand are proved. Our discussion relies on the properties of the Green’s function, the upper and lower solutions method, and the classical fixed point theorems in a cone. Moreover, building upper and lower control functions have an effective role in the analysis. Some examples are offered to justify the validity of theoretical findings.