Mathematical analysis

Chemistry

Fixed-point theorem

Biochemistry

Uniqueness

Operator (biology)

Mathematics

Gene

Initial value problem

Boundary value problem

Integral equation

Fixed point

Picard–Lindelöf theorem

Transcription factor

Repressor

Mathematical Methods in the Applied Sciences

The problem of the existence and uniqueness of solutions of boundary value problems (BVPs) for a nonlinear fractional differential equation of order 2&lt; α ≤ 3 is studied. The BVP is transformed into an integral equation and discussed by means of a fixed point problem for an integral operator. Conditions for the existence and uniqueness of a fixed point for the integral operator are derived via b ‐comparison functions on complete b ‐metric spaces. In addition, estimates for the convergence of the Picard iteration sequence are given. An estimate for the Green's function related with the problem is provided and employed in the proof of the existence and uniqueness theorem for the solution of the given problem. Illustrative examples are presented to support the theoretical results.

On the solution of a boundary value problem associated with a fractional differential equation