In this paper, we investigate the conditions on the control mappings psi ,varphi :(0,infty )rightarrow mathbb{R} that guarantee the existence of the fixed points of the mapping T:Xrightarrow P(X) satisfying the following inequalities: ψ(H(Tx,Ty))≤φ(d(x,y))∀x,y∈X,provided that H(Tx,Ty)>0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\psi \\bigl(H(Tx,Ty)\\bigr)\\leq \\varphi \\bigl(d(x,y)\\bigr) \\quad \\forall x,y\\in X, \\text{provided that } H(Tx,Ty)>0, $$\\end{document} and ψ(H(Tx,Ty))≤φ(A(x,y))∀x,y∈X,provided that H(Tx,Ty)>0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\psi \\bigl(H(Tx,Ty)\\bigr)\\leq \\varphi \\bigl(A(x,y)\\bigr) \\quad \\forall x,y\\in X, \\text{provided that } H(Tx,Ty)>0, $$\\end{document} where A(x,y)=max { d(x,y), d(x,Tx), d(y,Ty), (d(x,Ty) +d(Tx,y))/2 } , and (X, d) is a metric space. The obtained fixed point results improve many earlier results on the set-valued contractions. As an application, we consider the existence of the solutions of an FDE.