Abstract
Fuzzy differential equation driven by Liu process is an important tool to deal with dynamic system in fuzzy environment. Stability for a fuzzy differential equation plays a key role in differential equation, which means influence of the state of a system to small changes in the initial state. In order to discuss the influence of different initial value on the solution, this paper proposes a concept of stability in mean for fuzzy differential equation driven by Liu process. Some stability theorems for fuzzy differential equation being stable in mean are given. In addition, the concept of stability in mean for fuzzy differential equation driven by Liu process is extended to the case of multi-dimensional. A sufficient condition for multi-dimensional fuzzy differential equation being stable in mean is also provided in this paper.
Highlights
There are various types of fuzzy phenomena in the world
Tian and Guo [14] considered the initial value of fuzzy differential equation and a sufficient condition for stability of the trivial solution of equation was obtained by using Lyapunov function
Some concepts of stability for fuzzy differential equations driven by Liu process have been studied, for example, by finding the equilibrium point xe of the fuzzy differential equation that satisfies f (t, xe) = 0 and g(t, xe) = 0 for all t, Zhu [21] defined some relationships between the equilibrium point xe and the solution xt of the fuzzy differential equation when any crisp initial vector x0 is close enough to the equilibrium point xe, those relationships were called Lyapunov stable almost surely, Lyapunov stable in mean, Lyapunov stable in credibility
Summary
In order to address fuzziness, credibility theory was founded by Liu [6] and was refined by Liu [7] in 2007, it is a branch of mathematics for studying the behavior of fuzzy phenomena. Huo and Wang [15] extended Liu formula and Liu integral to the case of multi-dimensional and the existence and properties of Liu integral were studied by You and Wang [16]. Ma and Huo [17] presented a definition of generalized Liu integral, and the properties of this kind of generalized fuzzy integral were proved. Some properties of complex fuzzy integral were studied by You and Wang [19].
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