Solvability of a coupled system of a fractional boundary value problem with fractional integral condition

Fractional differential equations arise in various fields of science and engineering such as rheology, fluid flows, electrical networks, viscoelasticity, chemical physics, biosciences, signal processing, systems control theory, electrochemistry, mechanics and diffusion processes. Fractional differential equations also serve as an excellent tool for the description of hereditary properties of various materials and processes. In consequence, fractional differential equations have become a very important and useful area of mathematics over the last few decades. For details, see [1-9, 22-40] and the references therein. Integral boundary conditions have various applications in applies fields such as blood flow problems, chemical engineering, thermoelasticity, underground water flow, population dynamics, and so on. we refer the reader to [10-15] for more details of nonlocal and integral boundary conditions. In recent years, some authors have studied fractional boundary value problem with integral boundary conditions. Bashir Ahmad and Juan J. Nieto [16] studied the existence results for a nonlinear fractional integrodifferential equation with integral boundary conditions