A fixed point problem with constraint inequalities via an implicit contraction

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 = T x , A x ⪯ 1 B x , C x ⪯ 2 D x , 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.