Let (X, d) be a metric space and J ⊆ (0, ∞ ) be a nonempty set. We study the structure of the arbitrary intersection of VECTOR-VALUED LIPSCHITZ algebras, and define a special Banach subalgebra of ∩ {Lipγ (X, E): γ ∈ J}, where E is a Banach algebra, denoted by ILip J (X, E). Mainly, we investigate C− character amenability of ILip J (X, E).

In this paper, we study weighted composition operators on extended ANALYTIC LIPSCHITZ algebras ${\rm Lip}_{A}(X,K,\alpha)$ where $X$ is a compact plane set, $K$ is a closed subset of $X$ with nonempty interior and $0 < \alpha \leq 1$. We first give necessary conditions and sufficient conditions on a function $u \in \mathbb{C} ^{X}$ and self-map $\varphi$ of $X$ for which $T=uc_{\varphi}$ to be a weighted composition operator on ${\rm Lip}_{A}(X,K,\alpha)$. We next give the necessary conditions for these operators to be compact and provide some sufficient conditions for the compactness of such operators.

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We present a method to minimize locally LIPSCHITZ FUNCTIONS. At  rst, a local quadratic model is developed to approximate a locally LIPSCHITZ func-tion. This model is constructed by using the ϵ-subdifferential. We minimize this local model and compute a search direction. It is shown that this direc-tion is descent. We generalize the Wolfe conditions for  nding an adequate step length along this direction. Next, the method is equipped with a quasi-Newton approach to update the local model and its globally convergence is proposed. Finally, the proposed algorithm is implemented in MATLAB environment on some standard nonsmooth optimization test problems and compared with some algorithms in the literature.