Ulam–Hyers Stability for Fractional Differential Equations in Quaternionic Analysis

Abstract

Using the fixed point method and the weakly Picard operator technique, we obtain some abstract Ulam-Hyers stability results of the initial value problem of fractional differential equations in quaternionic analysis. Sufficient conditions for the existence of solutions of the initial value problem are given by the application of the method of associated spaces. An example is provided to illustrate these results.

Hyers-Ulam stability of a nonautonomous semilinear equation with fractional diffusion

“…inequality (9) is just the previous inequality(7). Let us see that inequality (9) is also true for + n 1.…”

“…On the other hand, there are some preliminary studies on the Hyers-Ulam stability of differential equations involving Riemann-Liouville-type fractional operators, see [6,7]. The study of partial differential equations with fractional Laplacian diffusion is relatively recent.…”

Hyers-Ulam Stability of a Nonlinear Multi-Term Fractional Differential Equation on Bounded Time Interval

“…Besides, in many papers, the Ulam stability of classical differential equations has been extended to the several types of fractional differential equations. For more details, one can see [17][18][19][20][21][22].…”

Hyers-Ulam-Rassias stability of high-dimensional quaternion impulsive fuzzy dynamic equations on time scales

“…Beyond that, it is powerful to use quaternions to model rotation and orientation in engineering, physics and molecular biology, etc. (see [8, 10, 11, 20-22, 48, 49] and the book [6] of Topics in Clifford Analysis), many superiorities over real-valued vectors in the applications of these related fields were demonstrated adequately in the fuzzy environment (see [50,51]).…”

“…Since then a series of mathematical questions related to this stability theory was collected in the book and studied by Ulam (see [38]) and improved by Rassias (see [31]). From then on, many new results were reported on this topic and a new comprehensive stability theory of various types of differential equations was developed under the framework of Hyers-Ulam-Rassias's stable structure including ordinary and impulsive ordinary differential equations (see [32,47]), functional equations (see [14,16]), fuzzy differential equations (see [23,36]), differential operators (see [27]) and quaternion differential equations (see [50]).…”