Random fixed points for ψ-contractions with application to random differential equations

In this paper, some random common fixed point and coincidence point results are proved with PPF dependence for random operators in separable Banach spaces. Our results present stochastic versions and extensions of recent results of Dhage [J. Nonlinear Sci. Appl. 5 (2012) and Differ. Equ. Appl. 2 (2012)], Kaewcharoen [J. Inequal. Appl. 2013:287] and many others. We also establish results concerning iterative approximation of PPF dependent random common fixed points. Moreover, an application to random differential equations is given here to illustrate usability of the obtained results.

Lagrangian Descriptors for Stochastic Differential Equations: A Tool for Revealing the Phase Portrait of Stochastic Dynamical Systems

In this paper, we introduce a new technique for depicting the phase portrait of stochastic differential equations. Following previous work for deterministic systems, we represent the phase space by means of a generalization of the method of Lagrangian descriptors to stochastic differential equations. Analogously to the deterministic differential equations setting, the Lagrangian descriptors graphically provide the distinguished trajectories and hyperbolic structures arising within the stochastic dynamics, such as random fixed points and their stable and unstable manifolds. We analyze the sense in which structures form barriers to transport in stochastic systems. We apply the method to several benchmark examples where the deterministic phase space structures are well-understood. In particular, we apply our method to the noisy saddle, the stochastically forced Duffing equation, and the stochastic double gyre model that is a benchmark for analyzing fluid transport.