An Elementary Proof of Surjectivity for a Class of Accretive Operators

Abstract

Abstract. An operator A defined on a real Banach space X is said to be locally accretive if, for each A > 0, x e X and each v near x, \\x -v|| < ||jc -y + \(Ax -Ay)\\. It is shown that if A : X -» X is locally accretive and locally Lipschitzian then (/ + A)(X) = X.Let X be a real Banach space. A nonlinear operator A : X -» X is said to be accretive if, for each X > 0 and for each w, y E X,while A is said to be locally accretive if (1) is solvable when A is locally Lipschitzian and accretive on X, a result which was subsequently generalized by R. H. Martin [12] to the continuous accretive operators. The existence theory for equation (2) [12] showed that if A is locally Lipschitzian (in Martin's theorem only continuity is required) and accretive, then A is m-accretive, i.e., (/ + A)(X) = X. Recently M. Crandall and A. Pazy [7] have given a proof of Martin's result which does not rely on solvability of (2), but which remains rather complex, being based on an intricate iteration scheme. It is our purpose in this note to present an elementary and straightforward proof of Browder's theorem for locally accretive operators.

Iterative solution of nonlinear equations of the monotone type in Banach spaces

“…Then {*n}^Lo converges strongly to a solution of the equation (16) x -XAx = f in K. PROOF: The existence of a solution to equation (16) follows from Theorem B. Let q denote a solution.…”

“…Recently, Ray [16] gave an elementary proof of Theorem B by employing a fixed point theorem of Caristi, [5].…”

Ishikawa and Mann iteration methods for nonlinear strongly accretive mappings

“…Let X be a real Banach space. A mapping T with domain D(T) and range R(T) in X is said to be accretive (see for example [2,11,16,23]) if the inequality, (1) \\x-y\\^\\x-y + t[Tx-Ty)\\, holds for each x, y in D(T) and for all t > 0. T is said to be m-accretive if T is accretive and (I + rT)(X) = X for all r > 0, where / denotes the identity operator on X.…”

“…x + Tx = f has a solution in X. Recently, Ray [23] gave an elementary proof of this result by employing a fixed point theorem of Caristi [7]. In [19] Martin proved that (2) is solvable if T is continuous and accretive, and utilising the result he proved that if T is continuous and accretive, then T is m-accretive.…”

“…In terms of the duality mapping, a mapping T with domain D(T), and range R{T) in X is accretive (see for example, [2,16,23]…”