A fixed point problem with constraint inequalities via an implicit contraction
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Author list: Ansari A.H., Kumam P., Samet B.
Publisher: Springer
Publication year: 2017
Journal: Journal of Fixed Point Theory and Applications (1661-7738)
Volume number: 19
Issue number: 2
Start page: 1145
End page: 1163
Number of pages: 19
ISSN: 1661-7738
eISSN: 1661-7746
Languages: English-Great Britain (EN-GB)
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Abstract
Recently, Jleli and Samet [Fixed Point Theory Appl. 2016 (2016), doi:10.1186/s13663-016-0504-9] established an existence result for the following problem: Find x∈ X such that x= Tx, Ax⪯ 1Bx, Cx⪯ 2Dx, where (X, d) is a metric space equipped with the two partial orders ⪯ 1 and ⪯ 2, and T, A, B, C, D: X→ X are given mappings. This existence result was obtained under a continuity assumption imposed on the mappings A, B, C and D. In this paper, we prove that the result of Jleli and Samet holds true by supposing that only A and B are continuous (or only C and D are continuous). Moreover, we prove that the considered problem has one and only one solution. We provide an example to show that our result is a significant generalization of that of Jleli and Samet. Moreover, we consider a more large class of mappings T: X→ X satisfying a certain implicit contraction. © 2016, Springer International Publishing.
Keywords
constraint inequalities, partial order
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