Nonlinear Analysis, Theory, Melhods & Applicafiom, Vol. 23, No. 6, pp. 797-823, 1994 Elsevier Science Lid Printed in Great Britain zyxwvutsrq EXISTENCE AND UNIQUENESS OF THE SOLUTIONS TO SECOND ORDER ABSTRACT EQUATIONS WITH NONLINEAR AND NONMONOTONE BOUNDARY CONDITIONS-f IRENA LASIECKA Applied Mathematics Department, University of Virginia, Charlottesville, VA 22903, U.S.A. (Received 1 May 1993; received for publication 14 July 1993) Key words and phrases: Abstract differential equations, nonlinear, multivalued boundary conditions, von Karman system of elasticity. 1. INTRODUCTION This paper is concerned with the wellposedness of abstract nonlinear differential equations of the form under the following assumptions: Mu,,(t) + Au(t) + AC aa, G*Au,(t) + AGf(u(t)) 3 S(u); t > 0; u(t = 0) = u,; u,(t = 0) = u1 U-1) zyxwvutsrq A is a realization of a closed, linear, positive selfadjoint operator a, acting of a Hilbert space H with a>(a) C H, as an operator ZD(A”‘) + a)(i1’2)‘. We shall denote by 1 1 and 11 II th e norm of Hand 9(A1’2), respectively, and we shall use the same symbol (* , *) to denote the scalar product on H and the duality pairing between a)(k?‘2) and ZD(A1’2)‘. (1.2) Let V be another Hilbert space such that ZD(x1’2) c V c H c V’ c LD(ii1’2)f, all injections being continuous and dense, M E .52(V, V’) and (MU, U) 2 c+,Iul2y, where a0 > 0 and ( , ) is understood as a duality pairing between V and I/‘. Hence, M-’ E e(V’, V). Setting A? = M(, with a>(k) = (U E V; Mu E H) we have D(ar”) = V. (1.3) Let U and U,, be other Hilbert spaces such that U, E U s UA. We shall denote by (-9 -) the scalar product on U and the duality pairing between U, and UA. The linear operator G: UA + H satisfies: A1’2G: UA -+ H is bounded, or equivalently, G*A: LD(a”2) + U,, is bounded, where (G*u, v) = (u, Gu). We shall also assume that G*A: a)(A”‘) -+ U. is surjective. CD: U + R’ is a proper, convex, lower semicontinuous function subgradient &D E U, x U;. with (I .4) (1.5) t Research supported by the National Science Foundation Grant NSF DMS 9204338.