On fixed points of set-valued directional contractions (Q1071346)

Let (V,d) be a metric space, B(V) (C(V)) denotes the class of all non- empty bounded (compact) subsets of V and H denotes the Hausdorff pseudometric. Let S be a non-empty subset of V. A map \(T: S\to C(V)\) is called a weak directional contraction if there exists a \(k\in [0,1)\) such that for each \(x\in S\) either \(x\in Tx\) or there exists a \(y\in Tx\) with \(d(x,y)=d(x,Tx)\) and there exists a \(z\in S\) with \(d(x,y)=d(x,z)+d(z,y)\) such that \(H(Tx,Tz)<kd(x,z)\). A map \(T: S\to B(V)\) is called a directional contraction if there exists a \(k\in [0,1)\) such that for each \(x\in S\), \(y\in Tx\) H(Tx,Tz)\(\leq kd(x,z)\) for all \(z\in S\) such that \(d(x,y)=d(x,z)+d(z,y).\) Fixed point theorems for directional contractions and for weak directional contractions are proved.