End point theorems of multivalued operators without continuity satisfying hybrid inequality under two different sets of conditions
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In the fixed point theory of contractive mappings, that is, mappings satisfying contractive inequalities, in partially ordered metric spaces the fixed point results are obtained under the assumption that the contractive inequality condition holds for pairs of points which are related by partial order rather than for arbitrary pairs of points. Alternatives to partial order for the purpose of restricting the contraction conditions in the fixed point results are the admissibility conditions. In this work we define a multivalued hybrid inequality by generalizing and combining two types of contractive inequalities and establish end point results for those operators which satisfy the hybrid inequality defined here under the two separate environments described above. For our purpose we define a new admissibility condition. The results are established in the most general structure of a metric space. The main theorems proved here are in the domain of setvalued analysis. The corresponding singlevalued cases are discussed. There is nowhere any assumption of continuity. The methodology is either a combination of analytic and order theoretic methods or purely analytic. Moreover the newly introduced method of proof in fixed point theory through Pata-type results are followed in the multivalued case. There are supporting examples of the main results.
KeywordsMetric space Partial order Multivalued cyclic \((\alpha , \beta )\)-admissible mapping \(\delta \)-distance End point
Mathematics Subject Classification54H10 54H25 47H10
The authors gratefully acknowledge the suggestions made by the learned referee.
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