Among primary liver carcinoma cases, the proportion of liver hepatocellular carcinoma (LIHC) cases is 75%–85%. Current treatments for LIHC include chemotherapy, surgical excision, and liver transplantation, which are effective for early LIHC treatment. Nevertheless, the early symptoms of liver carcinoma are atypical, so a large proportion of LIHC patients are diagnosed at an advanced stage. Histocompatibility minor 13 (HM13), located in the endoplasmic reticulum, is responsible for catalysing the hydrolysis of some signal peptides after cleavage from the precursor protein. Here, we studied the role of HM13 in LIHC development through bioinformatics analysis. Database analysis showed that HM13 was of great significance for LIHC tumorigenesis. Compared to normal liver tissues, HM13 expression was increased to a greater extent in LIHC tissues. After analysis of Kaplan‒Meier plotter and Gene Expression Profiling Interactive Analysis (GEPIA) datasets, we discovered that highly expressed HM13 exhibited an association with shorter overall survival (OS), disease-free survival (DFS), and disease-specific survival (DSS). We conducted Gene Ontology (GO) and Kyoto Encyclopedia of Genes and Genomes (KEGG) analyses to analyse HM13-related genes, and the data indicated that these genes obviously participated in rRNA processing, ribosome biogenesis, spliceosome, Huntington's disease, and ATP-dependent helicase activity. The Cell Counting Kit-8 (CCK-8) assay and Transwell assay showed that reducing HM13 expression hindered LIHC cell proliferation, migration, and invasion. In conclusion, these findings indicate that HM13 is a biomarker and is related to the poor prognosis of LIHC. Our results are conducive to discovering new targets for LIHC treatment.

The South Altyn Orogenic Belt (SAOB) is one of the most important orogenic belts in NW China, consisting of the South Altyn Continental Block and the Apa–Mangya Ophiolitic Mélange Belt. However, its Palaeozoic tectonic evolution is still controversial. Here, we present petrological, geochemical, zircon U–Pb and Lu–Hf isotopic data for the Mangya plutons with the aim of establishing the Palaeozoic tectonic evolution. We divide the Early Palaeozoic magmatism in the Apa–Mangya Ophiolitic Mélange Belt into four episodes and propose a plate tectonic model for the formation of these rocks. During 511–494 Ma, the South Altyn Ocean (SAO) was in a spreading stage, and some shoshonite series, I-type granitic rocks were generated. From 484 to 458 Ma, the oceanic crust of the SAO subducted northward, accompanied by large-scale magmatic events resulting in the generation of vast high-K calc-alkaline series, I-type granitic rocks. During 450–433 Ma, the SAO closed, and break-off of the subducted oceanic slab occurred, with the generation of some high-K calc-alkaline series, I–S transitional type granites. The SAOB was in post-orogenic extensional environment from 419 to 404 Ma, and many A-type granites were generated.

The organic light-emitting (OLE) materials have attracted great interest due to their potential applications in sensors, biodetectors and OLE devices. However, highly efficient emission from organic solids is still a great challenge because of the aggregation-caused quenching effect. In this article, a three-dimensional (3D) organic-inorganic hybrid nanoparticle, based on polyhedral oligomeric silsesquioxane (POSS), was precisely fabricated via click chemistry with high yield, and its structure was characterized by Fourier transform infrared spectroscopy, 1H, and 29Si nuclear magnetic resonance spectroscopies, and Matrix-Assisted Laser Desorption Ionization Time-of-Flight Mass Spectrometry, respectively. The resultant 3D organic-inorganic nanohybrid showed significantly enhanced emission in solid film (Φfilm 80%) with a slight red-shift as compared with its organic counterpart, (Φfilm 14%), which exhibits a large red-shift in solid film, due to the deaggregation effect of POSS. Simultaneously, the resultant nanohybrid also exhibited good film formability, excellent spectrum and thermal stability (>250 °C) due to the introduction of POSS.

We construct a sequence {An} of maximal monotone operators with a common domain and converging, uniformly on bounded subsets, to another maximal monotone operator A; however, the sequence {t−1nAn} fails to graph-converge for some null sequence {tn}.

Moudafi and Maingé [Towards viscosity approximations of hierarchical fixed-point problems, Fixed Point Theory Appl. (2006), Art. ID 95453, 10pp] and Xu [Viscosity method for hierarchical fixed point approach to variational inequalities, Taiwanese J. Math.13(6) (2009)] studied an implicit viscosity method for approximating solutions of variational inequalities by solving hierarchical fixed point problems. The approximate solutions are a net (xs,t) of two parameters s,t∈(0,1), and under certain conditions, the iterated lim t→0lim s→0xs,t exists in the norm topology. Moudafi, Maingé and Xu stated the problem of convergence of (xs,t) as (s,t)→(0,0) jointly in the norm topology. In this paper we further study the behaviour of the net (xs,t); in particular, we give a negative answer to this problem.

Consider a nonexpansive self-mapping T of a bounded closed convex subset of a Banach space. Banach's contraction principle guarantees the existence of approximating fixed point sequences for T. However such sequences may not be strongly convergent, in general, even in a Hilbert space. It is shown in this paper that in a real smooth and uniformly convex Banach space, appropriately constructed approximating fixed point sequences can be strongly convergent.

We establish a Banach space version of a theorem of Suzuki [8]. More precisely we prove that if X is a uniformly convex Banach space with a weakly continuous duality map (for example, lp for 1 < p < ∞), if C is a closed convex subset of X, and if F = {T (t): t ≥ 0} is a contraction semigroup on C such that Fix(F) ≠ ∅, then under certain appropriate assumptions made on the sequences {αn} and {tn} of the parameters, we show that the sequence {xn} implicitly defined by

for all n ≥ 1 converges strongly to a member of Fix(F).

A contractive mapping on a complete metric space may fail to have a fixed point. Diametrically contractive mappings are introduced and it is shown that a diametrically contractive self-mapping of a weakly compact subset of a Banach space always has a fixed point.

Turbnikov/s property (U, λ, α, β) is investigated. In particular, it is shown that property (U, λ, α, α, − 1) with α > 1 is equivalent to α-uniform smoothness. It s also shown that property (U, 1, α, 1) with α > 1 is equivalent to the space being a Hilbert space. The dual property (U*, γ α α − 1) is also introduced and it is shown that a Banach space X has (U*, γ α, α − 1) if and only if X is α-uniformly convex.

Iterative algorithms for nonexpansive mappings and maximal monotone operators are investigated. Strong convergence theorems are proved for nonexpansive mappings, including an improvement of a result of Lions. A modification of Rockafellar’s proximal point algorithm is obtained and proved to be always strongly convergent. The ideas of these algorithms are applied to solve a quadratic minimization problem.

Let  be a commutative family of nonexpansive self-mappings of a closed convex subset C of a uniformly smooth Banach space X such that the set of common fixed points is nonempty. It is shown that if a certain regularity condition is satisfied, then the sunny nonexpansive retraction from C to F can be constructed in an iterative way.

We prove the convergence of an iterative method to a fixed point of a nonexpansive mapping in a uniformly smooth Banach space. We are able to relax one of the control conditions of P.L. Lions (1977).

A Banach space X is said to have uniform property (K) if there exists a constant (k ∈ [0,1) such that whenever xn ⇀ 0, ∥xn∥ → 1, and we have lim sup ∥ym∥ ≤ k. This property is the uniform version of property (K) recently introduced by B. Sims (Bull. Austral. Math. Soc. 50(1994), 523–528). Sufficient conditions for uniform property (K) are given. Some examples are presented to separate various Banach space properties. Applications to nonlinear operators are also included.

Let X be a real uniformly convex Banach space satisfying the Opial's condition, C a bounded closed convex subset of X, and T: C → C an asymptotically non-expansive mapping. Then we show that for each x in C, the sequence {Tnx} almost converges weakly to a fixed point y of T, that is,

This implies that {Tnx} converges weakly to y if and only if T is weakly asymptotically regular at x, that is, weak- . We also present a weak convergence theorem for asymptotically nonexpansive semigroups.

As a generalization of Kiang and Tan's proximately nonexpansive semigroups, the notion of a proximately uniformly Lipschitzian semigroup is introduced and an existence theorem of common fixed points for such a semigroup is proved in a Banach space whose characteristic of convexity is less than one.

An Lp inequality for l < p < 2 is established and applications to fixed points of uniformly Lipschitz mappings and strongly unique best approximations are given.