HAMMERSTEIN INTEGRAL EQUATIONS WITH INDEFINITE KERNEL

This paper deals with the problem of finding solutions of the --._rstein integral equation. It is shown that this problem can be reduced to the study of the critical points of certain functional defined on L2(). Existence of a solution of the Hammersteln integral equation is proved. Some other related results of interest are obtained.

We consider the problem of finding solutions ueL2(R I) L2(R) to the Hammerstein integral equation u(t) K(s,t)g(u(s),s)ds, where R is a bounded region in l q, K(s,t) K(t,s) e L2(RxR), and g: I / is a continuous function.We let KI: L2() / L2() be the operator defined by K! (u)(t) / K(s,t)u(s)ds.We denote by {i}ieZ and {i}ieZ the sequences of eigenvalues and corresponding eigenfunctions of u AK (u) (1.2) The equation (i.i) was discussed by Dolph in [2 ] and the following A. CASTRO results were obtained.If KI is positive definite the problem of finding the solutions of (i.i) can be reduced to the problem of finding the criti- cal points of certain functional J defined on L2().Moreover, if: there   exist two consecutive eigenvalues of (1.2), and AN+I, and real num- bers , y' and C such that (g(u,x) g(v,x))/(u-v) <_ 7' < AN+1 for all (u,x)eR n u 2 G(u,x) /O g(s,x)ds >_ (7/2)u + C for all (u,x) R x with > AN; and for every in X span {ilAi <_ A N } J has a unique minimum on the linear manifold + Xx; then (i.i) has a solution.
Here we prove that even in the case that K is indefinite the pro- blem of finding the solutions of (i.i) can be reduced to the study of the critical points of certain functional defined on L2().We show that (1.3)   alone implies the existence of a solution of (I.I).We also apply a re- sult on Liusternik-Schnierelmann theory due to Clark [ 2,p.71] to the study of (i.i) when g is odd in the first variable, mildly nonlinear and again not necessari{y positive definite.
Throughout this section H is a real separable Hilbert space, with inner product <,>, and f: H /I is a function of class C For each u E H we denote by Vf(u) the unique element of H such that lira f(u+tv) We say that a function is continuous with respect to weak convergence (CWC) if it takes weakly convergent sequences into convergent sequences.We say that a function w: H / l is weakly lower semicontinuous (WLSC) if x / x implies w(x) < lira W(Xn).
n Lemma 2.1.Suppose there exist X and Y, which are closed subspaces of H, such that H XY and for some m > 0 <Vf(x+y 1) Vf(x+Y2),yl-Y2 > >_ m[[y x-y2[[ 2 for ever7 x e X, Yl e Y' and Y2 Y" Assertion: There exists a continuous function : X / Y satisfying: i) iii) l__f, i__n addition, there exist an isomorphism A:H / H such that A(X)=X, A(Y)CY, for some ml> 0 <A(y),y > > yll 2 o= an y, an__d Vf A F i_s continuous with respect to weak convergen.ce (CWC),then is CWC when either dim Y <= or X and Y are orthogonal.
Proof: For each xX we define fx: Y/l by fx(y) f(x+y).From (2.1)   we have <Vfx(Yl ) Vfx(Y2)'Yl -Y2 "> > mll Yl-Y2 ][   (2.2) Let us see now that is continuous.Suppose, on the contrary, that there exist 6 > 0 and a sequence {Xn} n in X such that x / xeX and n II (x n) (x)II > 6.Since Vf is continuous, for n sufficiently large..
Thus is continuous, and this proves part i).
is CC.
Now we are ready to prove our variational, principle.
We want to consider the problem u(t) Kl(g(u(t),t)) ten.
(3.1) Let {Ai; i _+i, +2,... } be the sequence of eigenvalues of u AKI(U).Let {i}i be an orthonormal sequence of eigenfuctions corresponding to the se- quence of eigenvalues {A.}..We assume that A-2 < X-I < 0 < XI < 2"'" and that the {i}i form complete set in L2().Let X span{_l,_2,... } and Y span{l,2,...}.It follows that KI(X)CX and (Y) cY, and Y is the orthogonal complement of X. reover, K 1 restricted to Y is positive defi- nite and K I restricted to X is negative definite.
-i 1 Now we define the operators Q'QI: L2 () by Q(u) E../,A.<u,j>j and i Ql(U) r._7_ <u, @q >@ j.It is eily seen that Q and Q1 are compact linear j= operators.Suppose u y with xeX, yeY, That is, QI is a square root of K I on Y and Q a square root of -K I on X.From the definition of Q and QI it is easy to see that Q and QI are selfadjoint.
An elementary computation shows that u 0 is a solution of (3.1).And the theorem is proved.
From here on we consider the equation (3.1) assuming g(u,t) g(u) -g(-u).In our next theorem we make use of the following result which is specialization of a theorem due to Clark [ l,p.71].
Lemma 3.3.: Le___t H be a real Hilbert sce and f an even, real valued C 2 function defined o._.n H. Suppose that f has the property that whenever {x } = H is a bounded sequence such that f(x n) > 0 and f(Xn) / 0, n then {x } contains a convergent subsequ.e.nce.Suppose that f(0) 0, f is boun- ded above, there exists a subspace M o__f H o_f dimension > 0 such that <D2f(0)h,h> > 0 i__f hM with h + 0, an__d f(x) > 0 fo___/_r [ix II sufficiently Then there exist at least 2 + i solutions of Vf(x) O.
Theorem 3.4.Suppose g is a function of class C an__d g' i_s bounded, i_f there exist two integers N and r,N < r, such that: i') there exists a real number 7 with g' (u) _< 7 < for all ue]R,