Solution of integral equations via new \(Z\)-contraction mapping in \(G_b\)-metric spaces. (English) Zbl 1480.54035
Summary: We introduce a new type of \((\alpha,\beta)\)-admissibility and \((\alpha,\beta)\)-\(Z\)-contraction mappings in the framework of \(G_b\)-metric spaces. Using these concepts, fixed point results for \((\alpha,\beta)\)-\(Z\)-contraction mappings in the framework of complete \(Gb\)-metric spaces are established. As an application, we discuss the existence of solution for integral equation of the form: \(x(t)=g(t)+\int^1_0K(t,s,u(s))ds\), \(t\in[0,1]\), where \(K:[0,1]\times[0,1]\times \mathbb R\to \mathbb R\) and \(g:[0,1]\to \mathbb R\) are continuous functions. The results obtained in this paper generalize, unify and improve results of Liu et al., Antonio-Francisco et al., F. Khojasteh et al. [Filomat 29, No. 6, 1189–1194 (2015; Zbl 1462.54072)], M. Kumar and R. Sharma [Bol. Soc. Parana. Mat. (3) 37, No. 2, 115–121 (2019; Zbl 1413.54134)] and others in this direction.
MSC:
| 54H25 | Fixed-point and coincidence theorems (topological aspects) |
| 47H10 | Fixed-point theorems |
| 47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
| 49J20 | Existence theories for optimal control problems involving partial differential equations |
| 49J40 | Variational inequalities |
Keywords:
\((\alpha, \beta)\)-\(Z_F\)-contraction; \((\alpha, \beta)\)-admissible type B mapping; fixed point; \(G_b\)-metric spaceReferences:
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