Existence and properties of nontrivial solutions for novel fractional differential equations with ϕ-Hilfer operators and boundary conditions

This study investigates a class of fractional differential equations of the form $$ D^{\mu ,\phi}_{0^{+}} q(n) + f(n, q(n)) = 0, \quad 0 < n < 1, \quad 1 < \mu \leq 2, $$ where $D^{\mu ,\phi}_{0^{+}} $ represents the Caputo fractional derivative. The nonlinear function $f: [0,1] \times [0,\infty ) \rightarrow [0,\infty ) $ is assumed to be continuous, and $\phi \in C^{2}[0,1] $ satisfies $\phi '(n) > 0 $ . The problem is supplemented with nonlocal boundary conditions: $$ q(0) = 0, \quad q(1) = \rho \int _{0}^{1} p(r) q(r) \phi '(r) \, dr, \quad 0 < \rho < 1. $$ By constructing an equivalent integral representation using Green’s function, the existence of nontrivial positive solutions is established through the application of fixed point theorems. The analysis provides new insights into the solvability and properties of solutions for this class of fractional boundary value problems.