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 $${x \in X}$$x∈X such that $${x = Tx, Ax \preceq_1 Bx}$$x=Tx,Ax⪯1Bx, $${Cx \preceq_2 Dx}$$Cx⪯2Dx, where (X, d) is a metric space equipped with the two partial orders $${\preceq_1}$$⪯1 and $${\preceq_2}$$⪯2, and $${T,A,B, C,D : X \rightarrow X}$$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 $${T : X \rightarrow X}$$T:X→X satisfying a certain implicit contraction.