In this article, we employ the lower regularized incomplete gamma functions to demonstrate the existence and uniqueness of solutions for fractional differential equations involving nonlocal fractional derivatives (GPF derivatives) generated by proportional derivatives of the form $$ D^{\rho }= (1-\rho )+ \rho D, \quad \rho \in [0,1], $$ D ρ = ( 1 − ρ ) + ρ D , ρ ∈ [ 0 , 1 ] , where D is the ordinary differential operator.

In this paper, using two different methods, we studied an open problem and obtained several results for Lyapunov-type and Hartman-Wintner-type inequalities for a Hadamard fractional differential equation on a general interval [a,b] , (1 a < b) with the boundary value conditions.

In this paper, we established some results concerning the existence and uniqueness of a nonlinear Volterra-Fredholm integro-differential equation of Caputo fractional order subject to the boundary value conditions. These new results are obtained by applying standard fixed point theorems. An example is presented to illustrate our main result.

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In this work, we investigate the condition of the given interval which ensures the existence and uniqueness of solutions for two‐point boundary value problems within conformable‐type local fractional derivative. The method of analysis is obtained by the principle of contraction mapping. Furthermore, benefiting from calculating the integral of the Green's function, we are able to improve a recent result by obtaining a sharper lower bound for an eigenvalue problem. Two examples are presented to clarify the obtained results. Finally, we present an open problem for the interested reader.