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

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.

Let T be an orientation-preserving Lipschitz expanding map of the circle T. A pre-image selector is a map τ : T → T with finitely many discontinuities, each of which is a jump discontinuity, and such that τ (x) ∈ T(x) for all x ∈ T. The closure of the image of a pre-image selector is called a flower, and a flower with p connected components is called a p-flower. We say that a real-valued Lipsch...

‎In this paper‎, ‎we study the Hölder continuity of solution mapping to a parametric variational inequality‎. ‎At first‎, ‎recalling a real-valued gap function of the problem‎, ‎we discuss the Lipschitz continuity of the gap function‎. ‎Then under the strong monotonicity‎, ‎we establish the Hölder continuity of the single-valued solution mapping for the problem‎. ‎Finally‎, ‎we apply these resu...

In this article, we prove the existence of extremal positive solution for the distributed order fractional hybrid differential equation$$int_{0}^{1}b(q)D^{q}[frac{x(t)}{f(t,x(t))}]dq=g(t,x(t)),$$using a fixed point theorem in the Banach algebras. This proof is given in two cases of the continuous and discontinuous function $g$, under the generalized Lipschitz and Caratheodory conditions.

We use Baire categorical arguments to construct dramatically pathological locally Lipschitz functions. The origins of this approach can be traced back to Banach and Mazurkiewicz (1931) who independently used similar categorical arguments to show that \almost every continuous real-valued function deened on 0,1] is nowhere diierentiable". As with the results of Banach and Mazurkiewicz, it appears...

In this paper we consider the computability of the solution of the initialvalue problem for differential inclusions with semicontinuous right-hand side. We present algorithms for the computation of the solution using the “ten thousand monkeys” approach, in which we generate all possible solution tubes, and then check which are valid. In this way, we show that the solution of an upper-semicontin...

We propose two algorithms for solving global optimization problems on a hyperrectangle with an objective function satisfying the Vanderbei condition (this is also called $$\varepsilon $$ -Lipschitz continuous function). The belong to family of non-uniform covering methods. For we prove propositions about convergence -solution in terms function. illustrate performance using several test numerica...

We use the simple, but little-known, result that a uniformly continuous function on a convex set is -Lipschitz (as defined below) to extend Piyavskii’s algorithm for Lipschitz global optimization to the larger domain of continuous (not-necessarilyLipschitz) global optimization.