In this paper, we consider a class of time-dependent neutral stochastic evolution equations with the infinite delay and a fractional Brownian motion in a Hilbert space. We establish the existence and uniqueness of mild solutions for these equations under non-Lipschitz conditions with Lipschitz conditions being considered as a special case. An example is provided to illustrate the theory

We extend the definition of involutivity to non-Lipschitz tangent subbundles using generalized functions. prove Frobenius Theorem with sharp regularity estimate when subbundle is log-Lipschitz: if $${\mathcal {V}}$$ a log-Lipschitz involutive rank r, then for any $$\varepsilon >0$$ , locally there homeomorphism $$\Phi (u,v)$$ such that ,\frac{\partial \Phi }{\partial u^1},\dots u^r}\in C^{0,1-\...

Let $(X,d)$ be an infinite compact metric space, let $(B,parallel . parallel)$ be a unital Banach space, and take $alpha in (0,1).$ In this work, at first we define the big and little $alpha$-Lipschitz vector-valued (B-valued) operator algebras, and consider the little $alpha$-lipschitz $B$-valued operator algebra, $lip_{alpha}(X,B)$. Then we characterize its second dual space.

We consider concave and Lipschitz continuous preference functionals over monetary lotteries. We show that they possess an envelope representation, as the minimum of a bounded family of continuous vN-M preference functionals. This allows us to use an envelope theorem to show that results from local utility analysis still hold in our setting, without any further differentiability assumptions on t...

We prove a new local upper Lipschitz stability result and the associated local error bound for solutions of parametric Karush–Kuhn–Tucker systems corresponding to variational problems with Lipschitzian base mappings and constraints possessing Lipschitzian derivatives, and without any constraint qualifications. This property is equivalent to the appropriately extended to this nonsmooth setting n...

The existence of a mean-square continuous strong solution is established for vectorvalued Itô stochastic differential equations with a discontinuous drift coefficient, which is an increasing function, and with a Lipschitz continuous diffusion coefficient. A scalar stochastic differential equation with the Heaviside function as its drift coefficient is considered as an example. Upper and lower s...

By using the technique of factoring weakly compact operators through reflexive Banach spaces we prove that a class of ordinary differential equations with Lipschitz continuous perturbations has a strong solution when the problem is governed by a closed linear operator generating a strongly continuous semigroup of compact operators.

This paper presents a new result on absolute exponential stability (AEST) of a class of continuous-time recurrent neural networks with locally Lipschitz continuous and monotone nondecreasing activation functions. The additively diagonally stable connection weight matrices are proven to be able to guarantee AEST of the neural networks. The AEST result extends and improves the existing absolute s...

We prove logarithmic conditional stability up to the final time for backward-parabolic operators whose coefficients are Log-Lipschitz continuous in t and Lipschitz x. The result complements previous achievements of Del Santo Prizzi (2009) Santo, Jäh (2015), concerning (of a type intermediate between Hölder logarithmic), arbitrarily closed, but not time.