Eigenvalue problem for nonlinear elastic beam equation of fractional order

Recently, fractional differential calculus has attracted a lot of attention by many researchers of different fields, such as: physics, chemistry, biology, economics, control theory, biophysics, etc. [11,15,16]. Since the fractional integrals and derivatives have more abilities to describe phenomena, it means that they can decrease errors occurring in modeling of real-life events, thus, studying of fractional systems solutions becomes one of the most significant challenging part of applied mathematics. Also, fourth-order differential equations often used to describe the deformation of elastic beams and so are important in mechanics and engineering problems. Many authors have investigated fourth-order differential equations with different boundary conditions (see, for example, [1, 2, 5–7, 9, 17, 19, 25]). So, due to the importance of fractional differential equations and fourth-order differential equations, the existence of positive solutions of fractional-order beam equations has been studied by many authors. Xu et al. [24], considered the existence of positive solutions for fractional-order beam equation Du(t) = f ( t, u(t) ) , 0 < t < 1, 3 < α 6 4, (1)