Some novel analysis on two different Caputo-type fractional-order boundary value problems

Nowadays, a number of classical order results are being analyzed in the sense of fractional derivatives. In this research work, we discuss two different boundary value problems. In the first half of the paper, we generalize an integer-order boundary value problem into fractional-order and then we demonstrate the existence and uniqueness of the solution subject to the Caputo fractional derivative. First, we recall some results and then justify our main results with the proofs of the given theorems. We conclude our results by presenting an illustrative example. In the other half of the paper, we extend the Banach's contraction theorem to prove the existence and uniqueness of the solution to a sequential Caputo fractional-order boundary value problem.