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For finite-dimensional linear differential systems with bounded coefficients, we prove that their exponential dichotomy on $\mathbb{R}$ is equivalent to their Ulam–Hyers stability on $\mathbb{R}$ with uniqueness. We also consider abstract non-autonomous evolution equations which are exponentially bounded and exponentially dichotomic and prove that Ulam–Hyers stability with uniqueness is maintained when perturbing them with a nonlinear term having a sufficiently small Lipschitz constant.

For a large class of nonautonomous linear delay equations with distributed delay, we obtain the equivalence of hyperbolicity, with the existence of an exponential dichotomy, and Ulam–Hyers stability. In particular, for linear equations with constant or periodic coefficients and with a simple spectrum these two properties are equivalent. We also show that any linear delay equation with an exponential dichotomy and its sufficiently small Lipschitz perturbations are Ulam–Hyers stable.

For finite-dimensional linear differential systems with bounded coefficients we prove that their exponential dichotomy on R is equivalent to their Ulam-Hyers stability on R with uniqueness. We also consider abstract non-autonomous evolution equations which are exponentially bounded and exponentially dichotomic and prove that Ulam-Hyers stability with uniqueness is maintained when perturbing them with a nonlinear term having a sufficiently small Lipschitz constant. [ABSTRACT FROM AUTHOR]

We add relevant references about which we learned after the completion of the initial work. We mainly show how the concept of exponential trichotomy can successfully replace the one of exponential dichotomy in some results from the paper in the title.

For a large class of nonautonomous linear delay equations with distributed delay, we obtain the equivalence of hyperbolicity, with the existence of an exponential dichotomy, and Ulam-Hyers stability. In particular, for linear equations with constant or periodic coefficients and with a simple spectrum these two properties are equivalent. We also show that any linear delay equation with an exponential dichotomy and its sufficiently small Lipschitz perturbations are Ulam-Hyers stable. [ABSTRACT FROM AUTHOR]

We add relevant references about which we learned after the completion of the initial work. We mainly show how the concept of exponential trichotomy can successfully replace the one of exponential dichotomy in some results from the paper in the title. [ABSTRACT FROM AUTHOR]

In the present paper, we investigate, using the Picard operator technique, some existence and Ulam type stability results for the Darboux problem associated to some partial fractional order differential inclusions.

Using the weakly Picard operator technique, we will present Ulam-Hyers stability results for integral inclusions of Fredholm and Volterra type and for the Darboux problem associated to a partial differential inclusion.

In this paper, we study existence of solutions to a Cauchy problem for non-linear ordinary differential equations involving two Caputo fractional derivatives. The existence and uniqueness of solutions are obtained by using monotonicity, continuity and explicit estimation of Mittag-Leffler functions via fixed point theorems. Further, we present Ulam-Hyers stability results by using direct analysis methods. Finally, examples are given to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

In the present paper, we investigate, using the Picard operator technique, some existence and Ulam type stability results for the Darboux problem associated to some partial fractional order differential inclusions. [ABSTRACT FROM AUTHOR]

Using the weakly Picard operator technique, we will present Ulam-Hyers stability results for integral inclusions of Fredholm and Volterra type and for the Darboux problem associated to a partial differential inclusion. [ABSTRACT FROM AUTHOR]