A {em Roman dominating function} on a graph $G$ is a function $f:V(G)rightarrow {0,1,2}$ satisfying the condition that every vertex $u$ for which $f(u)=0$ is adjacent to at least one vertex $v$ for which $f(v)=2$. A {em total Roman dominating function} is a Roman dominating function with the additional property that the subgraph of $G$ induced by the set of all vertices of positive weight has n...

‎We present some existence and convergence results for a general class of nonexpansive mappings in partially ordered hyperbolic metric spaces‎. ‎We also give some examples to show the generality of the mappings considered herein.

A nonnegative signed dominating function (NNSDF) of a graph G is a function f from the vertex set V (G) to the set {−1, 1} such that ∑ u∈N [v] f(u) ≥ 0 for every vertex v ∈ V (G). The nonnegative signed domination number of G, denoted by γ s (G), is the minimum weight of a nonnegative signed dominating function on G. In this paper, we establish some sharp lower bounds on the nonnegative signed ...

Let D be a finite and simple digraph with vertex set V (D) and arc set A(D). A signed Roman dominating function (SRDF) on the digraph D is a function f : V (D) → {−1, 1, 2} satisfying the conditions that (i) ∑ x∈N−[v] f(x) ≥ 1 for each v ∈ V (D), where N −[v] consists of v and all inner neighbors of v, and (ii) every vertex u for which f(u) = −1 has an inner neighbor v for which f(v) = 2. The w...

We discuss the superstability of generalized module left derivations and generalized module derivations on a Banach module. Let A be a Banach algebra and X a Banach A-module, f : X → X and g : A → A. The mappings Δ1 f,g , Δ2 f,g , Δ3 f,g , and Δ4 f,g are defined and it is proved that if ‖Δ1 f,g x, y, z,w ‖ resp., ‖Δ3 f,g x, y, z,w, α, β ‖ is dominated by φ x, y, z,w , then f is a generalized re...

BACKGROUND The composition of the intestinal microbiota seems to be an important factor in determining the clinical outcome in children with short bowel syndrome (SBS). Alterations in the microbiota may result in serious complications such as small bowel bacterial overgrowth (SBBO) and intestinal mucosal inflammation that lead to prolonged parenteral nutrition (PN) dependency with subsequently ...