THE CONTRACTION PROPERTY AND THE EQUIVALENC OF (LIPSCHITZ AND ARCWISE COMPLETENESS)

Researchers investigate the completeness and shrinkage properties in metric spaces in this paper as well as demonstrate that the shrinking characteristic entails Lipschitz- wholeness or arswise- wholeness in metric spaces. The contraction feature, on the other hand, does not entail completion in metric spaces. We demonstrate that a locally Lipschitz-linked metric area has the shrinkage characteristic if it is Lipschitz- whole and an arswise-linked metric area is arswise- entire if X has the intense shrinkage characteristic. The method applied by Borwein to generate a shrinkage map is crucial for showing Lipschitz-completeness of the space.