In this paper, we prove a general common fixed point theorem for two pairs of weakly compatible self-mappings of a (possibly non complete) metric space such that one of them satisfies the property (E.A) under a contractive condition of Lipschitz type. Our result provides a generalization and some improvements to a result obtained by K. Jha, R.P. Pant and S.L. Singh in 2003 and a recent result o...

We investigate the asymptotic properties as t → ∞ of the differential equation ẍ(t) + a(t)ẋ(t) +∇G(x(t)) = 0, t ≥ 0 where x(·) is R-valued, the map a : R+ → R+ is non increasing, and G : R → R is a potential with locally Lipschitz continuous derivative. We identify conditions on the function a(·) that guarantee or exclude the convergence of solutions of this problem to points in argminG, in the...

We study the global inversion of a continuous nonsmooth mapping f : R → R, which may be non-locally Lipschitz. To this end, we use the notion of pseudo-Jacobian map associated to f , introduced by Jeyakumar and Luc, and we consider a related index of regularity for f . We obtain a characterization of global inversion in terms of its index of regularity. Furthermore, we prove that the Hadamard i...

Under standard Lipschitz conditions, trajectories of systems described by ordinary differential equations are well defined in both forward and reverse time. (The flow map is invertible.) However for hybrid systems, uniqueness of trajectories in forward time does not guarantee flow-map invertibility, allowing non-uniqueness in reverse time. The paper establishes a necessary and sufficient condit...

The phrase convex optimization refers to the minimization of a convex function over a convex set. However the feasible convex set need not be always described by convex inequalities. In this article we consider a convex feasible set which are described by inequality constraints which are locally Lipschitz and not necessarily convex and need not be smooth. We show that if the Slater’s constraint...

The ergodic properties of SDEs, and various time discretizations for SDEs, are studied. The ergodicity of SDEs is established by using techniques from the theory of Markov chains on general state spaces. Application of these Markov chain results leads to straightforward proofs of ergodicity for a variety of SDEs, in particular for problems with degenerate noise and for problems with locally Lip...

In this paper, a distributed average tracking problem is studied for Lipschitz-type nonlinear dynamical systems. The objective is to design distributed average tracking algorithms for locally interactive agents to track the average of multiple reference signals. Here, in both the agents’ and the reference signals’ dynamics, there is a nonlinear term satisfying the Lipschitz-type condition. Thre...

In this paper, we are interested in the numerical solutions of stochastic functional differential equations (SFDEs) with jumps. Under the global Lipschitz condition, we show that the pth moment convergence of the Euler-Maruyama (EM) numerical solutions to SFDEs with jumps has order 1/p for any p ≥ 2. This is significantly different from the case of SFDEs without jumps where the order is 1/2 for...

It is well-known that a stochastic differential equation (SDE) on a Euclidean space driven by a Brownian motion with Lipschitz coefficients generates a stochastic flow of homeomorphisms. When the coefficients are only locally Lipschitz, then a maximal continuous flow still exists but explosion in finite time may occur. If – in addition – the coefficients grow at most linearly, then this flow ha...

In this paper, we prove the existence and uniqueness result of the reflected BSDE with two continuous barriers under monotonicity and general increasing condition on y, with Lipschitz condition on z.