In this paper, some new results are given on fixed and common fixed points of Geraghty type contractive mappings defined in b-complete b-metric spaces. Moreover, two examples are represented to show the compatibility of our results. Some applications for nonlinear integral equations are also given.

The clinical course of porous orbital implant infection may be prolonged, and the early symptom of recurrent discharge, a common problem for implant recipients, may delay diagnosis. Implant infection should be suspected when there is persistent conjunctival inflammation and discharge after implant placement despite antibiotic therapy, discomfort on implant palpation, and recurrent pyogenic granuloma (indicative of implant exposure). Implant removal is usually required in these cases. If orbital pain (not necessarily related to implant palpation) is the main complaint, without signs of conjunctival inflammation and with or without discharge, one should consider other reasons for the symptoms.

We describe a case of a 32-year-old man who presented with a visually apparent but otherwise asymptomatic mass in the right lateral fornix. Computerized tomography demonstrated the mass adjacent to the lacrimal gland. The mass was surgically removed, and histopathologic examination was consistent with a dacryolith of the lacrimal ductule. There were several cilia isolated from the dacryolith. This entity should be considered in the differential diagnosis of patients presenting with a localized mass in the region of the lacrimal gland.

In this paper, we consider and extend some fixed point results in F-complete F-metric spaces by relaxing the symmetry of complete metric spaces. We generalize α,β-admissible mappings in the setting of F-metric spaces. The derived results are supplemented with suitable examples, and the obtained results are applied to find the existence of the solution to the integral equation. The analytical results are compared through numerical simulation. We pose certain open problems for extending and applying our results in the future.

In this paper, we discuss the existence of fixed points of mappings defined on uniform spaces generated by a family of b-pseudometrics. We also give some sufficient conditions under which the fixed point is unique.