we introduce a new concept of general $g$-$eta$-monotone operator generalizing the general $(h,eta)$-monotone operator cite{arvar2, arvar1}, general $h-$ monotone operator cite{xiahuang} in banach spaces, and also generalizing $g$-$eta$-monotone operator cite{zhang}, $(a, eta)$-monotone operator cite{verma2}, $a$-monotone operator cite{verma0}, $(h, eta)$-monotone operator cite{fanghuang}...

in this article, we develop the distributed order fractional hybrid differential equations (dofhdes) with linear perturbations involving the fractional riemann-liouville derivative of order $0 < q < 1$ with respect to a nonnegative density function. furthermore, an existence theorem for the fractional hybrid differential equations of distributed order is proved under the mixed $varphi$-lipschit...

and Applied Analysis 3 Definition 2.2 see 16 . Let ψ : R → R be a locally Lipschitz function, then ψ◦ u;v denotes Clarke’s generalized directional derivative of ψ at u ∈ R in the direction v and is defined as ψ◦ u;v lim sup y→u t→ 0 ψ ( y tv ) − ψ(y) t . 2.4 Clarke’s generalized gradient of ψ at u is denoted by ∂ψ u and is defined as ∂ψ u { ξ ∈ R | ψ◦ u;v ≥ 〈ξ, v〉, ∀v ∈ Rn}. 2.5 Let f : R → R b...

Lipschitz extensions were recently proposed as a tool for designing node differentially private algorithms. However, efficiently computable Lipschitz extensions were known only for 1-dimensional functions (that is, functions that output a single real value). In this paper, we study efficiently computable Lipschitz extensions for multi-dimensional (that is, vector-valued) functions on graphs. We...

Remark 1. The Lipschitz condition (4) is called global because it holds for all x, y ∈ R with the same constant L. Klebaner Theorem 5.4 assumes only a local Lipschitz condition, where the Lipschitz constant may depend on the size of x, y. We use a global condition in order to make the presentation simpler. Later on we will also assume a Lipschitz condition with respect to t, see (21). (There is...

We characterize the local single-valuedness and continuity of multifunctions (set-valued mappings) in terms of their submonotonicity and lower semicontinuity. This result completes the well-known condition that lower semicontinuous, monotone multifunctions are single-valued and continuous. We also show that a multifunction is actually a Lipschitz single-valued mapping if and only if it is submo...

There exists a calculus for general nondifferentiable functions that englobes a large part of the familiar subdifferential calculus for convex nondifferentiable functions [1]. This development started with F.H. Clarke, who introduced a generalized gradient for functions that are locally Lipschitz, but (possibly) nondifferentiable. Generalized gradients turn out to be the subdifferentials, in th...

We study rank functions (also known as graph homomorphisms onto Z), ways of imposing graded poset structures on graphs. We first look at a variation on rank functions called discrete Lipschitz functions. We relate the number of Lipschitz functions of a graph G to the number of rank functions of both G and G× E . We then find generating functions that enable us to compute the number of rank or L...

We introduce a new concept of general $G$-$eta$-monotone operator generalizing the general $(H,eta)$-monotone operator cite{arvar2, arvar1}, general $H-$ monotone operator cite{xiahuang} in Banach spaces, and also generalizing $G$-$eta$-monotone operator cite{zhang}, $(A, eta)$-monotone operator cite{verma2}, $A$-monotone operator cite{verma0}, $(H, eta)$-monotone operator cite{fanghuang}...