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We address existence and Ulam-Hyers and Ulam-Hyers-Mittag-Leffler stability of fractional nonlinear multiple time-delays systems with respect to two parameters’ weighted norm, which provides a foundation to study iterative learning control problem for this system. Secondly, we design PID-type learning laws to generate sequences of output trajectories to tracking the desired trajectory. Two numerical examples are used to illustrate the theoretical results.

In this paper, we consider the existence and uniqueness of solutions to the nonlocal boundary value problem for semi-linear differential equations involving Hilfer fractional derivative. With the help of properties of Hilfer fractional calculus, Mittag-Leffler functions, and fixed point methods, we derive existence and uniqueness results. Finally, examples are given to illustrate our theoretical results.

In this paper, we consider PID-type iterative learning control law for nonlinear impulsive ordinary differential equations. We obtain convergence results for open-loop and closed-loop iterative learning schemes with initial error in the sense of $$\lambda $$ -norm. Finally, a numerical example is given to demonstrate the validity of the design methods.

In this paper, we establish two fractional integral equalities involving once and twice differential functions. Then, we apply such equalities to give some fractional Hermite–Hadamard inequalities via s-convex and s-Godunova–Levin functions. Some applications to special means of positive real numbers are also given.

In this paper, we investigate Ulam-Hyers stability, Practical-Ulam-Hyers stability and Practical-Ulam-Hyers-Rassias stability of nonlocal fractional Langevin equations involving Riemann-Liouville fractional integral with positive constant coefficients. By utilizing an explicit bound of Mittag-Leffler function with two parameters and analytical methods, the practical stability results are presented via a generalized Gronwall inequality. Finally, two examples are given to illustrate our theoretical results.

In this paper, we apply asymptotic behavior on Mittag-Leffler functions $\mathbb{E}_{\alpha}(z)$ and $\mathbb{E}_{\alpha,\alpha}(z)$ for $z>0$ to discuss exp-type Ulam-Hyers stability of ${}^{\mathrm{c}} D_{t}^{\alpha}x(t)=\lambda x(t)+f(t,x(t))$ for the case $\lambda>0$ on a finite time interval $[0,1]$ and an unbounded interval $(1,\infty)$ .