Adv. Math. Econ. 5, 23-54 (2003)

Functional evolution equations governed by m-accretive operators

Charles Castaing^{1} and Ahmed Gamal Ibrahim^{2}

1 Département de Mathématiques, Case 051, Université Montpellier II, F-34095 Montpellier cedex 5, France

2 Department of Mathematics, Faculty of Sciences, Cairo University, Cairo, Egypt

Received: September 1, 2002 Revised: December 1, 2002

JEL classification: C62, C72

Mathematical Subject Classification (2000): 28A33, 28A50

Abstract. Let r > 0 be a finite delay and C_{0} = C_{E}([−r, 0]) be the Banach space of continuous vector-valued functions defined on [−r, 0] taking values in a separable reflexive Banach space E such that its strong dual E′ is uniformly convex. We discuss the existence of strong solutions for an evolution equation of the form u̇(t) ∈ −A(t)u(t) + F (t, τ(t)u) a.e in [0, 1], where u : [−r, 1] → E is a continuous mapping from [−r, 1] into E such that its restriction to [−r, 0] is equal to φ ∈ C_{0}, and its restriction to [0, 1] is absolutely continuous, (i.e u(t) = u(0)+∫_{0}^{t} u̇(s) ds, ∀t ∈ [0, 1] with u̇ ∈ L_{E}^{1}([0, 1]), and satisfies the preceding inclusion, A(t) is an m-accretive multivalued operator on E, F : [0, 1]×C_{0} → E is a convex weakly compact valued, separately scalarly measurable and separately scalarly upper semicontinuous multifunction and (τ(t)u)(s) = u(t + s), ∀s ∈ [−r, 0]. Some applications to the sweeping process (or Moreau’s process) and Optimal Control involving Young measures are also presented.

Key words: accretive operator, ball-compact, functional differential inclusion, maximal monotone operator, multifunction, original control, relaxed control, relaxation, Young measure