A study of fractional differential equations with proportional boundary conditions

This article deals with a new concept of nonlocal proportional segmental boundary conditions with respect to initial and terminal sections of the given domain. Equipped with these conditions, we investigate the existence and uniqueness of solutions for a Caputo-type fractional differential equation with the nonlinearity depending upon the unknown function together with its lower order fractional derivative. We apply the standard tools of the fixed point theory to accomplish the desired results. Our study is useful in the given configuration as it helps to comprehend the fractional boundary value problems in the sense of proportional boundary conditions.