In this paper, we introduce a new concept of generalised /-projection operator which extends the generalised projection operator -KK • B* -» K, where B is a reflexive Banach space with dual space B* and K is a nonempty, closed and convex subset of B. Some properties of the generalised /-projection operator are given. As an application, we study the existence of solution for a class of variational inequalities in Banach spaces.

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In this paper, we introduce two new concepts of relaxed η-α pseudomonotonicity and relaxed η-α demipseudomonotonicity as well as two classes of vector variational-like inequalities with relaxed η-α peseudomonotone mappings and relaxed η-α demipseudomonotone mappings in Banach spaces. By using KKM technique, we obtain the existence of solutions for vector variational-like inequalities with relaxed η-α pseudomonotone mappings in reflexive Banach spaces. We also show the solvability of vector variational-like inequalities with relaxed η-α demipseudomonotone mappings in reflexive Banach spaces by means of Kakutani-Fan-Glicksberg fixed-point theorem. The results presented in this paper extend and improve some corresponding results of several authors.

In this paper, we study the existence of nonzero solutions for a class of generalized variational inequalities involving setcontractive mappings by using the fixed point index approach in reflexive Banach spaces. Under some suitable assumptions, we show some new existence theorems of nonzero solutions for this class of generalized variational inequalities in reflexive Banach spaces.

Bladder cancer (Bca) is a common urinary tract malignancy with frequent recurrences after initial resection. Submucosal injection of gemcitabine prior to transurethral resection of bladder tumor (TURBT) may prevent recurrence of urothelial cancer. However, the underlying mechanism remains unknown. In the present study, ultra-performance liquid chromatography Q-Exactive mass spectrometry was used to profile tissue metabolites from 12 BCa patients. The 48 samples included pre-and post-gemcitabine treatment Bca tissues, as well as adjacent normal tissues. Principal component analysis (PcA) revealed that the metabolic profiles of pre-gemcitabine BCa tissues differed significantly from those of pre-gemcitabine normal tissues. A total of 34 significantly altered metabolites were further analyzed. Pathway analysis using MetaboAnalyst identified three metabolic pathways closely associated with Bca, including glutathione, purine and thiamine metabolism, while glutathione metabolism was also identified by the enrichment analysis using MetaboAnalyst. In search of the possible targets of gemcitabine, metabolite profiles were compared between the pre-gemcitabine normal and post-gemcitabine Bca tissues. Among the 34 metabolites associated with Bca, the levels of bilirubin and retinal recovered in Bca tissues treated with gemcitabine. When comparing normal bladder tissues with and without gemcitabine treatment, among the 34 metabolites associated with Bca, it was observed that histamine change may be associated with the prevention of relapse, whereas thiamine change may be involved in possible side effects. Therefore, by employing a hypothesis-free tissue-based metabolomics study, the present study investigated the metabolic signatures of Bca and found that bilirubin and retinal may be involved in the mechanism underlying the biomolecular action of submucosal injection of gemcitabine in urothelial Bca.