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The main objective of this article is to introduce the notion of α,⊥F-contraction in the setting of orthogonal F-metric space and present some new fixed point results in such newly introduced space. We also furnish an example to manifest the originality of the obtained results. As application of our foremost result, we look into the solution of a nonlinear fractional differential equation.

The purpose of this paper is to define a rational orbitally (Θ,δb)-contraction and prove some new results in the context of b-metric spaces. Our results extend, generalize, and unify some known results in the literature. As application of our main result, we investigate the solution of Fredholm integral inclusion. We also provide an example to substantiate the advantage and usefulness of obtained results.

In this paper, we introduce new type of Θ-contraction in termof a self-mapping on a metric space to obtain common fixed point results. Some examples arealso provided to support the validity of our results and concepts presented herein.As an application of our results, periodic point results forthese Θ-contractions inmetric spaces are proved. Scopus

The aim of this paper is to introduce the notion of a Suzuki-Gerghaty type contractive mapping via simulation function along with $\mathcal{C}$-class functions and prove the existence of fixed point result. An example is given to show the validity of our results given herein. Moreover, we prove the existence of solution of nonlinear Hammerstein integral equation.

Within the S1 clan of human serine proteases is a group of proteases known as GASPIDs (Granule Associated Serine Peptidases of Immune Defense) that, although highly related, show differences in tissue distribution and substrates, implying different biological functions. The majority of GASPIDs have roles in immune defense, however there are some with unknown functions. Documented GASPID roles include cytotoxicity, cytokine signalling and extracellular matrix remodeling. GASPIDs are single domain proteases approximately 250-270 amino acids long encompassing a hydrophobic leader sequence, an N-terminal pro-dipeptide domain, a conserved [IV]-[IV]-G-G sequence at the beginning of the mature amino terminus, and the active site triad His-57, Asp-102 and Ser-195 (chymotrypsin numbering system). At the sequence level, GASPIDs can be distinguished from other S1 proteases by a family-specific conserved sequence P-H-S-[RMHYW]-P-[YF]-[MIV]-[AV] at amino acids 9 to 16 after the mature amino terminus. In addition, some GASPIDs encode a C-terminal prodomain of up to 20 amino acids that is removed during activation, although the exact role of this sequence is not known. With the exception of rodents, little is known about GASPIDs in species other than humans. Limited studies in fish have shown the presence of two GASPID genes, however, there has been no research into what other GASPID genes may exist in fish and how this family of serine proteases evolved during vertebrate evolution. The hypothesis of this study is that every vertebrate species has a distinct set of GASPID genes that changes during evolution. Conserved GASPIDs are involved in core immune function while species specific GASPIDs are involved in more specialized functions. The aims were to: 1) identify the primordial GASPID by tracing the emergence of GASPIDs during vertebrate evolution; 2) test the hypothesis that conserved GASPIDs are not involved in cytotoxicity, but rather have a role in core immune functions, while species-specific GASPIDs are involved in more specialized functions. It was found that some GASPIDs have been conserved during evolution while others have evolved in a species-specific manner. No GASPID genes were identified in primitive vertebrates (lamprey), indicating that GASPIDs appeared after the divergence of jawed and jawless vertebrates. In fish, there is a minimal complement of three GASPIDs, and evidence suggests that the primordial GASPID was a tryptase. During this analysis a previously unrecognised and conserved GASPID, PRSS57, was identified. The evolutionarily conserved GASPIDs are hypothesized to have core immune functions such as immune signalling or matrix remodeling, rather than more specialised functions such as cytotoxicity. PRSS57, a novel member of GASPIDs family was taken as a representative of evolutionary conserved GASPID in the tetrapod lineage. Studies described here show it to be a tryptase with matrix remodeling ability, suggesting that it is most likely involved in core immune functions. Granzyme H was taken as an example of a lineage specific GASPID. The extended substrate specificity of GZMH was determined by using a phage display system. The consensus sequence found for GZMH was Ile-[Ile/Val]-[Phe/Tyr]-[Tyr/Phe] ↓ Ser-[Gly/Ala]-X-[Glu/Asp]. Furthermore, based on contradictory literature reports regarding cytotoxicity, it was investigated whether GZMH has cytotoxic potential. It was found to be non-cytotoxic and to cleave extracellular matrix components, which is consistent with a more specialised role.

We define the notion of ${\alpha }^{\ast}\text{-}\psi $ -contractive mappings for cone metric space and obtain fixed points of multivalued mappings in connection with Hausdorff distance function for closed bounded subsets of cone metric spaces. We obtain some recent results of the literature as corollaries of our main theorem. Moreover, a nontrivial example of ${\alpha }^{\ast}\text{-}\psi $ -contractive mapping satisfying all conditions of our main result has been constructed.