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In this manuscript we discuss, consider, generalize, improve and unify several recent results for so-called F-contraction-type mappings in the framework of complete metric spaces. We also introduce ( φ , F ) -weak contraction and establish the corresponding fixed point result. Using our new approach for the proof that a Picard sequence is a Cauchy in metric space, our obtained results complement and enrich several methods in the existing literature. At the end we give one open question for F-contraction of Ćirić-type mapping.

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Introduction Gastric cancer (GC) represents one of the most common cancers worldwide, frequently diagnosed at advanced stages with poor prognosis, indicating on need for new diagnostic and prognostic markers. The aim of the study was to determine the expression of IL-32, proinflammatory and angiogenic mediators, in patients with diffuse and intestinal gastric cancer and the relationship with clinicopathological aspects. Material and Methods The tissue samples of diffuse and intestinal types of tumor of 70 patients with gastric cancer were analyzed. Expression of IL-32, VEGF, IL-17, and CD31 was measured by immunohistochemistry. Results IL-32 expression was significantly lower in tissue samples from patients with diffuse type of gastric cancer that is also a severe and more progressive form (TNM stages III and IV, poor histological differentiation, and higher nuclear grade III). Expression of IL-17 was also decreased in patients with diffuse type of gastric cancer. Microvascular density was diminished in diffuse type of gastric cancer. Conclusions Downregulated expression of IL-32 in tumor tissue of patients with diffuse type of gastric cancer may implicate on its role in limiting ongoing proinflammatory and proangiogenic processes. This emphasizes on unrecognized role of IL-32 in biology of diffuse type of gastric cancer.

In this paper we consider ( s − q ) -graphic contraction mapping in b-metric like spaces. By using our new approach for the proof that a Picard sequence is Cauchy in the context of b-metric-like space, our results generalize, improve and complement several approaches in the existing literature. Moreover, some examples are presented here to illustrate the usability of the obtained theoretical results.