The Journal of the Indian Mathematical Society

R. P. Pant ¹, Vyomesh Pant ¹, A. B. Lohani ¹

Affiliations
1 Department of Mathematics, Kumaon University, D.S.B. Campus, Nainital-263002, IN

Abstract

While studying the fixed point theorems for four mappings A, B, S and T (say) a Meir-Keeler type (ε,δ) contractive condition alone is not sufficient unless some additional condition is imposed on δ or a ∅-contractive condition is also used together with it. In the following pages we prove two common fixed theorems assuming a Meir-Keeler type (ε,δ) contractive condition together with a plane contractive condition (Theorem 1) or Lipschitz type analogue of a plane contractive condition (Theorem 2); however, without imposing any additional restriction on δ or having a ∅-contractive condition used together with it. Simultaneously, we also show that none of the mappings involved in the following theorems is continuous at their common fixed point Thus we not only generalize the Meir-Keeler type and Boyd-Wong type fixed point theorems, but also provide one more answer to the problem (see Rhoades [22]) on the existence of a contractive definition, which is strong enough to generate a fixed point but does not force the map to be continuous at the fixed point. Our result extends the result of Pant [15].

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Vyomesh Pant ¹, K. Jha ²

Affiliations
1 Depaitment of Mathematics, Kumaon University, D.S.B. Campus, Nainital-263 002, IN
2 Department of Mathematics, Kathmandu University, P.O. Box No. 1250, Kathmandu, NP

Abstract

The present paper is aimed at proving a fixed point theorem under the (ε,δ) contractive condition, however without either imposing any additional restriction on δ or on the continuity of the map or assuming the Φ-contractive condition together with it. In the setting of the fixed point theorem we show that the map is discontinuous at the fixed point. Our theorem extends the results of Pant [Theorem 1, [13]] and Pant [17].

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