?41q?1?Dvv?Vol.41, No. 1 1998?1?ACTA MATHEMATICA SINICAJan., 1998?X?C?dH?KZUP?M?(?wEw?U?250014)?g?q?y?o?t?h?p?wvrm?si?lk?xz?nmj?n?u?q?y?o MR(1991)fN?D47H15, 34B15eR?DO175Bifurcation of Nonlinear Problems Modeling Flows through Porous MediaLiu Xiyu (Department of Mathematics, Shangdong Normal University, Jinan 250014, China)Abstract This paper deals with a class of nonlinear boundary value problems whichappears in the study of models of flows through porous media. Existence results of asymptotic bifurcation and continua are reported both for operator equations and for boundary value problems.Keywords Bifurcation, Continua, Operator theory, FLows through porous media.1991 MR Subject Classification 47H15, 34B15Chinese Library Classification O1751?Bt?k?ym?l?a?jM?XN?yf?kym?XN?d dtG(x(t) p(t)H(x(t)? = p?(t)H(x(t) + f(,t,x(t),t (0,1),x(0) = a,x(1) = b;a,b 0(1.1)?W15.?XN?M?V?cn?D. ORegan?W3?fjC?t?n?XN(1.1)?Bernstein-Nagumo?XN?XN(1.1)C?tQx?n?dP?c?K?G?XNw?W6, 11.?W?L?cnH?n?Kw?AXN?Z?cG?TD?Rabinowitz9, Amann10,QE?s?Il?1214?cnM?V?Z?XN(1.1)dP?R?F?k?a?1996-06-20,a?1996-12-19?vwNh?NhRFX?vwNh?g?108Cuu?41q?WL?G,H C1(R1), H?Gx?G() = , G(0) = H(0) = 0, p C0,1 C1(0,1), t (0,1)?p(t) 0, f C0,) (0,1) (R10),0,).2?W?L?FQ?b?b?Z?H?T?icG?M?Z?ObP?R?n?S?k?Q?E2.1?XVip?xZ?an, a X, an a, En XV?R?xZ?an En.?E = limnEn= x X:?EnkQxnk Enk?xnk x.?EV?R?xZ?XVBanachxZ?PVX?XVcnxZ?u?H?A : D(A) X,?D(A)VR P?P?R= 0,).?Z?D(A)VtPM?P?u?H?A(,x) = 0,(,x) D(A),(2.1)?0 D(A)V(2.1)?dP?V0?n?i?P? = 0?W0= x D(A) : (0,x) .?0?= .?x0 0,WE(x0)V?H(0,x0)?R?E = cl(E(x0) : x0 0),Z?u?kH?An,?An: R P P?s?W?H?x = An(,x)(2.2)?dPVn,?0n= x P : (0,x) n.?x0 0nWEn(x0)Vn?H(0,x0)?R?En= cl?E n(x0) : x0 0n?.Y?Q?(N1) An?R P?s?n N.(N2)lim ?x?An(0,x)? ?x?= 0, n N.(N3)?(n,xn) nV?er?(nk,xnk)?(nk,xnk) (,x) .?bZ?D?D?xZR P?e?E2.2?(N1), (N2)?En?n Nr?e?E2.3?n?i?xZRP?H00?tPG, G?x?E?e?Y?(N1)(N3)? = limnn?Q?(N3)? 0.?E2.4?Q?(N1)(N3)?n?i?E2.5?Q?(N1)(N3)?n=10n?e?E?e?aIZ?pz?2.2?Q?Sw?Q?(N3)?0?= .?0?e?E?e?0?e?GV?etP?H0 0.?n?0n G.?(0,xn) 0n?G,?xn?e?B?Q?(N3)w?(0,xn) (0,x) .?1?do?zg?lYO?109?(0,x) (R P)G, (0,x) 0.?0 G?N?0n G, n N.?En?e?B?x0n 0n?En?x0n? G ?= .?n G ?= .?Q?(N3)? G ?= .B?2.2?c?E2.6?n=10n?e?(N1)(N3)?x0 0?H(0,x0)?R?E(x0)?e?aI?Y?J?E(x0)?e?x0 0.?2.5?xn 0?E(xn)?e?2.4w?xn x0 0.WE(x0)VH(0,x0)?R?E(x0)?e?R 0?E(x0) QR, E(x0) QR= ,?QR= 0,R BR.?X = QRV?i?xZ?E(x0)VX?i?xZ?Y = QR,?E(x0),Yi?X?i?PK1,K2?X = K1 K2, E(x0) K1, Y K2,K1 QR= .B?etPU R P?K1 U QR, U (K2 QR) = .?etPG?K1 G clG U QR,?G = .?xn x0Q(0,x0) E(x0) K1,B?n?(0,xn) GB?E(xn)?en?E(xn) G ?= .?(n,yn) E(xn) G.?n?iB?(?n,y?n) (,x) G?u?m?H?A:A(,x) = G(x) A(,x).(2.3)?A : D(A) P, D(A) R P, G : P P.WB(X,X)V?P?X?X?cn?eH?J?BanachxZ?P= v X: (v,x) 0, x P.?B B(X,X),?B?H?Y?Q?(A1) A(,x) J(x) + ()B(Gx), (,x) D(A),?J : P X, B B(X,X),?B : P P. (A2) () 0, 0?lim () = .(A3)? 0, v P?Bv= v.(A4)?BM,m 0?Jx? M(1 + ?Gx?),x P,(2.4)(v,Gx) m?Gx?,(,x) 0.(2.5)?E2.7?Q?(A1)(A4)? 0,?0() 0?(,x) 0,?Gx? ?0 0.?Q?2.6w?E2.8?n=10n?e?Q?(N1)(N3)Q(A1)(A4)?x0 0?R?E(x0)?c?(1) E(x0)?R 0e?(2) E(x0) 0, P?H?k? 0V?B?110Cuu?41q3?B?cG?JYTO?bZ?ic?L?XN(1.1).?W8J?n?Y?(H0) p?(t) 0, t (0,1)Mp?(t) 0, t (0,1).(H1) f(,t,x) (t)(,x), t (0,1), (0,), x R10, C(0,1),0,), 0,) (R10), 0,)?10s(1 s)(s)ds 0, = d C(0,1),0,)?10s(1 s)(s)ds 0, t (0,1).?(,x) D(A),?A(,x)(t) =?d dtG(x(t) p(t)H(x(t)? + p?(t)H(x(t) + f(,t,x(t),?0 D(A)V(1.1)?dP?JS?b?E3.1?Q?(H0)(H3)? = 0?XN(1.1)?dx0,?H(0,x0)?R?0?e?L?XN?V?2.8,Z?u?o?(2.3)?m?H?A.?E3.2?Q?(H0)(H3)?(,x) 0V?(1.1)?d?10s(1 s)f(,s,x(s)ds 0, t (0,1)?,A(,x)(t) =(1 t)G(a) + tG(b) + (1 t)?t0p(s) + sp?(s)H(x(s)ds1?do?zg?lYO?111+ t?1tp(s) + (1 s)p?(s)H(x(s)ds +?10k(t,s)f(,s,x(s)ds.?H?J(x)(t) =tG(b) + (1 t)?t0p(s) + sp?(s)Hx(s)ds+ (1 t)G(a) + t?1tp(s) + (1 s)p?(s)H(x(s)ds,(3.3)?Y?(H4) fC?f(,t,x) ()G(x); (,t,x) R(0,1)(0,).?Ulim +() =+.?H?B B(X,X)?(Bx)(t) =?10k(t,s)x(s)ds,(3.4)?Q?(A1), (A2)?VH?B?M?M?h?Vv P7.?(v,x) =?10vxds, x X.?x X?k(t,s)?n?(Bv,x) = (