Bessem Samet, Ecole Supérieure des Sciences et Techniques de Tunis
3 September 2013
If we denote by beta((x,y),(u,v))=alpha((x,y),(u,v)) if (x,y) is comparable with (u,v) and 0 otherwise, then the condition (3.41) becomes a beta-psi contraction. From the mixed monotone property of F, clearly F is beta-admissible. So it will be more interesting and more general to consider condition (3.41) without order and to obtain the ordered case as an immediate consequence.
Remark
3 September 2013
If we denote by beta((x,y),(u,v))=alpha((x,y),(u,v)) if (x,y) is comparable with (u,v) and 0 otherwise, then the condition (3.41) becomes a beta-psi contraction. From the mixed monotone property of F, clearly F is beta-admissible. So it will be more interesting and more general to consider condition (3.41) without order and to obtain the ordered case as an immediate consequence.
Competing interests
No competing interest.