*Correspondence: merturk3263@gmail.com 1Department of Mathematics, Yildiz Technical University, Davutpasa Campus, Esenler, Istanbul, Turkey Full list of author information is available at the end of the article Correction () Page , line : The statement ‘(ii) limr→t+ φ(r) ’ should be corrected as ‘(ii) limr→t+ φ(r) ’. () Page , line : The statement ‘. . . Condition is satisfied.’ should be rewritten ‘. . . Condition is satisfied and g is continuous.’ () Page , line : The statement ‘. . . and using (.)’ should be corrected as ‘. . . and using (.)’. () Page , line : The statement ‘≤ δj(k)+ + δl(k)+ + tk + n · φ( tk n )’ should be corrected as ‘≤ δj(k)+ + δl(k)+ + n · φ( tk n )’. () Page , line : The statement ‘From (.) and by . . .’ should be corrected as ‘From (.) and by . . . ’. () Page , line : ‘. . . now the assumption (b) holds.’ should be corrected as ‘. . . now the assumption (ii) holds.’ () Page , line (line in Corollary ) and Page , line (line in Corollary ): The statement ‘and there exist φ ∈ such that F ’ should be deleted. () Page , line (line in Corollary ) and Page , line (line in Corollary ): The statement ‘φ(F(x,x, . . . ,xn),F(y, y, . . . , yn))’ should be corrected as ‘d(F(x,x, . . . ,xn),F(y, y, . . . , yn))’. () Page , line (line in Corollary ) and Page , line (line in Corollary ): The statement ‘. . .F has the mixed g-monotone’ should be corrected as ‘. . .F has the mixed monotone’. That is, ‘g-’ should be deleted. () Page , line : ‘≤ φ( d(g(x),g(y))+d(g(x),g(y))+···+d(g(xn),g(yn)) n )’ should be corrected as ‘≤ φ( d(x,y)+d(x,y)+···+d(xn ,yn) n )’. () Page , line : ‘≤ n [d(g(x), g(y)) + d(g(x), g(y)) + · · · + d(g(xn), g(yn))]’ should be corrected as ‘≤ n [d(x, y) + d(x, y) + · · · + d(xn, yn)]’.