Boundedness, monotonicity and asymptotic behavior of solutions of Caputo fractional differential equations (Q6607681)

Bearing in mind the importance of nonlinear fractional differential equations in applied mathematics, the author studies the asymptotic behavior of solutions to the following problem: \N\[\N\left\{ \begin{array}{l} D^{\beta}_{C}x(t)=l(t)(\phi(x(t)))+k(t),\quad \beta\in(0,1),\,t\in[0,+\infty),\\\Nx(0)=x_0, \end{array} \right.\N\]\Nwhere \(D^{\beta}_{C}\) represents the standard Caputo fractional derivative of order \(\alpha_i\in (0,1)\). The existence of at least one nonnegative, bounded and nondecreasing continuous solution on \([0,+\infty)\) is established using the Schauder fixed point theorem. Furthermore, using the boundedness property of the solution, the asymptotic behavior of the solution is also analyzed. Finally, the results are illustrated with examples.