Nonhecv Ana&~ir. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Tkeory. Merkodr & Apphzriom. Vol 10. No. 10. pp. 1083-1103. 19%. 0362-5%X,‘86 P3.M) - .M) Pnnred 1” Great Einraln. Perqamon Journals Ltd. zyxwvutsrqp GENERALIZATION OF FREDHOLM ALTERNATIVE FOR NONLINEAR DIFFERENTIAL OPERATORS Lucre BOCCARDO Department of Mathematics, Universita di Roma “La Sapienza”, P. A. Moro 2, 00185 Roma. Italy PAVEL DRABEK Department of Mathematics, Technical University of Plzeti, Nejedleho sady 14, 30614 Plzeti, Czechoslovakia DANIELA GIACHETTI Department of Mathematics, II Universita di Roma, Via Orazio Raimondo. 00173 Roma. Italy and MILAN KUCERA Mathematical Institute of Czechoslovak Academy of Sciences, iitna 25, 11567 Praha 1, Czechoslovakia zyxwvutsrqponm (Received for publication 21 October 1985) Key words and phrases: Nonlinear noncoercive equations, nonlinear eigenvalues, Fredholm alternative 1. INTRODUCTION WE SHALL consider the boundary valueproblem -(Iu’(t)lP-* u’(t))’ =f(t, u(t)) + g(r), t E (0, ‘2). u(0) = u(n) = 0. I It will be proved that if lie (roughly speaking) between eigenvalues of the eigenvalue problem - (Iu’IP-~u’)’ = AIulP-*u on (0,x) u(0) = U(ir) = 0 1 (EP) then (BP) is solvable for anyg E L, (theorem 2.1). Moreover, certain more general assumptions about f under which(BP) is solvable will be formulated using properties of the problem - (Iu’IP-‘u’)’ = ,uIuIP-~u+ - ~luIp-~u_ on(0, x) u(0) = u(n) = 0 1 (EP,, y) (theorem 2.2). Further, we shall prove underthe similar assumptions about f thesolvability of the equations asymptotically closeto (BP)(theorem 2.3). Certain weakerresults will be obtained for partialdifferential equations of the analogous type (theorems 2.4-2.6). Let us remark that for the special case of differential operators under 1083