... and only if P has a fixed point.Proof. Suppose x·x· T0, then x0xT0Px0. This implies that x0 is afixed point of P. On the other hand, if Px0 x0, x0∈ X, then for the ... [H2.1], and [H2.2] hold. Then, operator P is a continuous operator of x on X.Proof. Letx, y ∈ Ξ ⊂ X, where Ξ is a bounded subset of X.Supposex·, x and x·, y are the PC-mild solutions of Cauchy ... Z0 and ck∈ X, there exists δ ∈ N such that ckδ ck.Now, we state the following result which asserts the existence of PC-mild solutionfor Cauchy problem 3.9 and gives the estimate of...