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One question of long standing interest is whether the space L,(#,X) of Bochner integrable functions has the Dunford-Pettis property if the Banach space X has the Dunford-Pettis property. The lack of a characterization of weakly compact subsets of LI(#,X ) has impeded a successful solution to this problem. In an attempt to circumvent this difficulty we shall study those sets in LI(#,X) that are mapped into norm compact sets by weakly compact operators on LI(#,X ). The paper opens with a study of those sets in an arbitrary Banach space X that mapped into relatively norm compact sets by all weakly compact operators on X. These sets are called Dunford-Pettis sets. An internal characterization of Dunford-Pettis sets is given. In the second section we give a rather general sufficient condition for a subset of LI(#,X) to be a Dunford-Pettis set. It is then deduced that if X is a Banach space with the Dunford-Pettis property and contains no copy of E 1, then LI(#,X) has the Dunford-Pettis property for all finite measures #. The third section deals with specific sets in L~(#,X) that are known to be weakly compact and shows that if X has the Dunford-Pettis property, then these specific sets are Dunford-Pettis sets.Throughout this paper (D, X, #) is a finite measure space and X and Y are Banach spaces with duals X* and Y* respectively. The space of all bounded linear operators from X to Y under the usual operator norm will be denoted by L(X, Y). The space of #-Bochner integrable functions on D with values in X will be denoted by L~(#,X). A subset M of L,(#,X) is called uniformly integrable if lim ~(~)-o ! IIflld/z=0 uniformly in fs M. * This work will constitute a portion of the author's Ph.D. thesis now in preparation at the University of Illinois under the direction of Professor J. J. Uhl, Jr. 0025-5831/79/0241/0035/$01.40for all feM. Choose a set EeS with #(E)<6 such that for each co$E there is a Dunford-Pettis set D(c0) with f(co)eD(co) for all feM. Then since limg.,(o))=0 J weakly Theorem 1 guarantees that lim g.j(a))fj(o)=0 for o)¢E. Since the sequence J (g,,(c0)fj (o)) is a uniformly integrable sequence, the Vitali convergence theorem ensures that lira I [g,,fjld#=0. Hence for j sufficiently large I Ig.Jjld#

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Existence and uniqueness results are established for weak formulations of initial-boundary value problems which model the dynamic behavior of an Euler-Bemoulli beam that may come into frictional contact with a stationary obstacle. The beam is assumed to be situated horizontally and may move both horizontally and vertically, as a result of applied loads. One end of the beam is clamped, while the other end is free. However, the horizontal motion of the free end is restricted by the presence of a stationary obstacle and when this end contacts the obstacle, the vertical motion of the end is assumed to be affected by friction. The contact and friction at this end is modelled in two different ways. The first involves the classic Signorini unilateral or nonpenetration conditions and Coulomb's law of dry friction; the second uses a normal compliance contact condition and a corresponding generalization of Coulomb's law. In both cases existence and uniqueness are established when the beam is subject to Kelvin-Voigt damping. In the absence of damping, existence of a solution is established for a problem in which the normal contact stress is regularized.

A model for the dynamics of a Gao elastic nonlinear beam, which is subject to a horizontal traction at one end, is studied. In particular, the buckling behavior of the beam is investigated. Existence and uniqueness of the local weak solution is established using truncation, approximations, a priori estimates, and results for evolution problems. An explicit finite differences numerical algorithm for the problem is presented. Results of representative simulations are depicted in the cases when the oscillations are about a buckled state, and when the horizontal traction oscillates between compression and tension. The numerical results exhibit a buckling behavior with a complicated dependence on the amplitude and frequency of oscillating horizontal tractions. (Résumé d'auteur