Abstract
The present paper is aimed at weakening the continuity conditions while proving fixed point theorems for a pair of maps and then using the results to provide applications in dynamic programming. Generalizing the concept of reciprocal continuity introduced by Pant (Indian J Pure Appl Math 30(2):147–152, 1999), we introduce the notion of g-reciprocal continuity and prove common fixed point theorems under minimal type contractive conditions. The results proved by us can be extended to the nonexpansive or Lipschitz type maps. Simultaneously, we provide more answers to the problem posed by Rhoades (Contemp Math 72:233–245, 1988) regarding existence a contractive definition which is strong enough to generate a fixed point, but which does not force the map to be continuous at the fixed point. We also provide an application of our results in dynamic programming.
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