Fixed Point Theorems for Monotone Mappings in Ordered Banach Spaces Under Weak Topology Features

Abstract

We present several fixed point theorems for monotone nonlinear operators in ordered Banach spaces. The main assumptions of our results are formulated in terms of the weak topology. As an application, we study the existence of solutions to a class of first-order vectorvalued ordinary differential equations. Our conclusions generalize many well-known results.

Random Fixed Point Theorems and Applications to Random First-Order Vector-Valued Differential Equations

“…We combine the advantages of the strong topology (continuity of random operators with respect to the strong topology) with the advantages of the weak topology (the random operators will satisfy some compactness conditions relative to the weak topology) to draw new conclusions about random fixed points for a given monotone random operator. Our results are random versions of the results in [39]. We start this section by recalling some definitions and auxiliary results which will be used further on.…”

“…Lemma 17 (see [39], Lemma 1.8). Let E be an ordered real Banach space with a normal order cone P. Suppose that fx n g is a totally ordered sequence which is contained in a relatively weakly compact set, then it converges strongly in E.…”

“…Clearly, h 1 ð:,ωÞ, h 2 ð:,ωÞ ∈ CðI, EÞ and for each t ∈ I, we have Step 4. The reasoning in [39], Theorem 3.2 yields that each Φðω, :Þ satisfies the condition ðP ðn 0 ÞÞ. Therefore, the random operator Φ satisfies the condition ðP ðn 0 ÞÞ.…”