Existence theorems for m-accretive operators in Banach spaces

Abstract

In 1985, the second author proved a surjective result for m-accretive and φ-expansive mappings for uniformly smooth Banach spaces. However, in this case, we have been able to remove the uniform smoothness of the Banach space, without any additional assumption.

The asymptotic behavior of the solutions of the Cauchy problem generated by ϕ-accretive operators

“…The uniqueness of a zero for an operator φ-accretive at zero is an immediate consequence of (4). On the other hand, it is an easy consequence of [13,Theorem 8] that every m-ψ-strongly accretive operator is φ-accretive at zero with φ = ψ • . .…”

Existence of fixed points for the sum of two operators

“…Therefore, by Theorem 8 in [19] or Remark 3.8, we know that R(I − T ) = X, which means that, in particular, there exists x ∈ X such that (I − T )x = 0. Consequently, we obtain the following result (see Corollary 2.4 in [33], and [27]).…”

“…We assume that ψ is a nondecreasing function, otherwise this is Lemma 3 of [19]. Suppose that (T x n ) is not a Cauchy sequence.…”

“…8 The main result of [19] establishes that if X is a Banach space and A : D(A) ⊆ X → 2 X is an m-accretive and ψ-expansive (with ψ a continuous function) operator, then A is surjective. By using Lemma 3.2 along with the arguments given in [19] it is not difficult to see that the same conclusion holds when A is maccretive and ψ-expansive with ψ a nondecreasing function.…”

“…Since I − T is an m-accretive and ψ-expansive mapping in X, hence, by using either Theorem 8 of [19] when ψ is a continuous function or Remark 3.8 when ψ is a nondecreasing function, we obtain that I − T : X → X is bijective. Which means, among other things, that there exists…”