Some variants of fibre contraction principle and applications: from existence to the convergence of successive approximations

Let (X1,→) and (X2, ↪→) be two L-spaces, U be a nonempty subset of X1×X2 such that Ux1 := {x2 ∈ X2 | (x1, x2) ∈ U} is nonempty, for each x1 ∈ X1. Let T1 : X1 → X1, T2 : U → X2 be two operators and T : U → X1 ×X2 defined by T (x1, x2) := (T1(x1), T2(x1, x2)). If we suppose that T (U) ⊂ U , FT1 6= ∅ and FT2(x1,·) 6= ∅ for each x1 ∈ X1, the problem is in which additional conditions T is a weakly Picard operator ? In this paper we study this problem in the case when the convergence structures on X1 and X2 are defined by metrics. Some applications to the fixed point equations on spaces of continuous functions are also given.