Internat. J. Math. & Math. Sci. Vol. 8 No. 2 (1985) 241-246 241 PERTURBATIONS OF NORMALLY SOLVABLE NONLINEAR OPERATORS, WILLIAM O. RAY and ANITA M. WALKER Department of Mathematics The University of Oklahoma Norman, OK 73019 (Received January 16, 1984) ABSTRACT. X Let Y and F + F and be Banach spaces and let X differentiable mappings from Y to be Gateaux In this note we study when the operator of a surjective is surjective for sufficiently small perturbations F operator The methods extend previous results in the area of normal solva- bility for nonlinear operators. Solvab2llty, normal solvabit2ty, inverse function theorems, Theorem, contractor detions. KEY WORDS AND PHRASES. Ekgand’ s AMS/MOS Subject classification. Let We call F be a nonlinear mapping from a Banach space X to a Banach space Y F Gateaux differentiable if for each x X there is an operator F’ (x), not necessarily linear, mapping lim X h X to =(x) F(x+th) This operator set of adjoint operators F F’(x)* Y for which F’(x)(h) t t-O for all 47H15. is nory solvable if the injectivity of the implies surjectivity of F In this light, the theory of normally solvable operators has been investigated by, among others, S.I. Phohzhayev [i], F.E. Browder [2], [3], [4] and [5], W.A. Kirk and J. Caristi [6], D. Downing and W.A. Kirk and W.O. Ray [9] and J. Kolomy [i0]. each of the operators F’ (x)* that each =(x)* [7], M. Altman [8], W.J. Cramer Throughout these works the injectivity of has been implicit assumption. If it is assumed also has a bounded inverse, then similar surjectivity results are obtained by weakening an assumption on the closure (or weak enclosure) of When F’ (x)* equation has a bounded inverse, F(x) 0 Newton’s method F(X) can be employed to solve the if the bounded inverse assumption is removed, J. Moser [ii] Newton’s method (See also W.O. Ray [12]). showed that can be modified to still yield a solution of F(x) In this note we avoid the explicit use of adjoint operators; instead we con- F which we perturb Under the assumptions we make sider a uniformly surjective Gateaux differentiable operator by a "small" Gateaux differentiable operator on 6’ (x) we demonstrate, via a transfinite the operator F + Hence Newton’s Newton’s method, the surjectivity of method is sufficiently regular that small O. 242 W. O. RAY AND A. M. WALKER perturbations of a differentiable operator do not hinder the convergence of the iterates. As the primary tool in deriving our main result, we make use of an extension of Caristi’s Theorem formulated in [13]: (M,d) Let THEOREM i. M _ Let g (0,) and let be be a continuous nonx M 0 be fixed. If g into itself which satisfies c(d(x,Xo))d(x,g(x)) then [0, (R)) c fc(u)du increasing function for which is a mapping of M[O,(R)) be a complete metric space and let a lower semicontinuous function. (g(x)) (x) () (x E M) M has a fixed point in Our main result is the following theorem, the proof of which is similar to those found in the earlier paper [13]. X Let and Y be Banach spaces and let differentiable mappings from X to THEOREM 2. ’(x) Y Let c fc(u)du non-increasing function for which F and [0,(R)) be Gateaux (0,) be a continuous Suppose for each X is a bounded linear operator from to x X that Y (i) and B(O;c([[x[[)) F’(x)(B(O;I)) Suppose, in addition, for some (II[I)-II P If the mapping (x)l[ F + m .. E (0,i) (2) and each has closed graph, then X x P that X is an open mapping from Y onto IIF’(x)-lll (Note that if F’(x) is linear, invertible and if c(llxll) is then (2) automatically fulfilled.) As a simple consequence it is enough to assume that F and each have closed graph. F Let THEOREM 3. differentiable operator P The proof that P graph, then so does be given as in Theorem 2. and P XY by P in particular, P F + If F Define the Gateaux have closed and is an open mapping from X onto Y is open in Theorems 2 and 3 follows readily from a direct application of Theorem 2.1 of [9]. Instrumental in verifying both the surjectivity and openness conclusions above are a pair of "contractor inequalities" (cf. [8]). These inequalities are given by: LEMMA. P by a t F + (0,i] F and Let be given as in Theorem 2. Then there is a E X and an [[P Px (,I) q Let such that for each XY y be defined Y there is such that t(y-Px)[[ qt[ly Pxl[ (4) and [IX x[[) c(l[xll)-itlly pxll (5) Similar inequalities can be deduced from injectivity hypotheses on (see the survey [14] for a discussion). It is noteworthy that the uniform surJectivity hypothesis (2) is somewhat weaker than the corresponding assumptions on the adjoint P’(x)* PERTURBATIONS OF NORMALLY SOLVABLE NONLINEAR OPERATORS, I When specialized to the case that the function and Y be Banach spaces and let differentiable mappings from X to Let THEOREM 4. e’(x) is a constant, the I. Rosenholtz and W.O. Ray [15]. above extends a result of X c(u) 243 Y e F and x E X Suppose for each X is a bounded and linear operator from be Gateaux that (6) Y to and F’(x)(B(O;l)) B(O;5) E (0,i) Suppose, in addition, for some (7) 5 > 0 for some that 6-111G ’(x) If the operator particular, if (8) P F + e has closed graph, then P is an open surjection. In F and e each have closed graph, then P is an open surjection. We now present the proofs of the above theorems; we first verify the Lemma and then use it to prove our main results. PROOF OF LEMMA. then choose Y y Fix (,i) q and and the conclusions follow for any x y So without loss of generality we may assume c(llxll)llY- PxlI-I(y-Px) F’(x)(w) c(llxll)llY F’(x) geneity of 6 (0,i] B(O;c(llxll)) F Set e and Px y for some x t P(X) c(llxll)-llly h F’(x)(h) implies that P(x) For each By (2) there exists a Pxll-l(y-Px) By hypothesis both t 6 y If with Pxllw llhll X x w E B(O;l) so that Then the homo- c(llxll)-lllY Pxll are Gateaux differentiable, so we can choose so small that llF(x+th) tF’ (x) (h)II F(x) (q-)tIly Pxll (9) and ]Je(x+th) e(x) <i(q-)tllY (10) and combining (9) and (i0) yields, via the triangle and (3), x+th Setting te’ (x) P(x)ll that +tllF’ (x) (h) (y-Px)ll + tile’ (x) (h)ll 1/2(q-)tlly i Pxll + (q-)tllY Pxll + 0 + tile’ (x) ll-llhll -< (q-)tllY Pxll + t’c(llxll)c(llxll)-lllY qtll xl1 -< xll giving (4). II xll To derive (5) observe that tllhll c(llxll)-itllY Pxll We now prove Theorem 2. PROOF. We begin by demonstrating the surjectivity of Define a metric Since p X on P has closed graph, (x,p) Fix y Y and is continuous from q (X,p) (,i) to p(x,y) by max{(l+q)llx P yll f + 6 c(o)-lll Px is a complete metric space. Set [0, ) @(x) since (l+q)(l-q)-lily- Pxll P has closed graph. Then PylI}. W. O. RAY AND A. M. WALKER 244 We proceed by contradiction, so suppose y g X-X by g(x) it follows that - fining llP(g(x)) Px t(y-Px)ll and qtlly P(x) Then by the Lemma, de- Pxll (ii) c(llxll)Ilg(x) xll -< tllY PxII Note that since y # Px it follows that g(x) # x for (12) each X x Applying the triangle inequality to (ii) gives llP(g(x)) Pxll -< (l+q)tlly Pxll (13) A second application of the triangle inequality gives IIP(g(x)) (l-t)IPx y yll -< qtlly xll or Pxll -< (l-q)-l(IIx tllY Yll llP(g(x)) Yl]) (14) Now, (13) and (14) together imply llP(g(x)) Pxll -< (l+q) (l-q) -I (llPx (x) yll llP(g(x)) (g(x)) (15) Also (12) and (14) imply (l+q)c (llxll) llg (x) (l+q) (l-q) -I x (llx(x) (g(x)) Now let Then x llxll -< (l+q)llxll llxll c(0)-lllpx either case, c(p(x,0)) _< Now, if P(0)I c(llxll (16) Consider first if c((l+q)llxll) implies that c(0)-lllpx _< ]l(g(x)) (l+q)llxll c(llxll) since c is nonincreasing. (and thus (l+q)llxll _< c(0)-lllpx P(0)I P(0)II) c(c(0)-lllpx P(0)II) -< c(llxll) Hence in X be fixed in p(x,0) Likewise, if then 0 0 Yll- so p(x,g(x))= (l+q)llx- g(x)II p(x,O) _< then (16) implies c(p(x,0))p(x,g(x)) _< c(llxll)(l+q)llx _< (x) (g(x)) so (*) holds, while if p(x,g(x)) c(p(x,0))p(x,g(x)) _< c(0)-iIIpx- P(g(x))ll c(llxll)c(0)-lllpx -< Ii’x _< (x) then (15) implies (g (x))II (g(x)) so again (*) holds. Thus Theorem I implies y P(X) and P g has a fixed point, a contradiction. Hence is surjective. Now, to show P is open as well, fix w X and let 6 > 0 It suffices B(Pw;) c_ (B(w;6)) for a sufficiently small choice of So let y then fly- Pwll <B(Pw;) Define a mapping Y B(w;6) to show that by (x) Theorem 2. We will show 0 (B(w;6)) thereby completing the proof of We accomplish this by applying Theorem 2.1 of [9]. y- Px By hypothesis, for q for which (4) and (5) hold. (,i) fixed, there is a Hence (4) implies t (0,I] and an X PERTURBATIONS OF NORMALLY SOLVABLE NONLINEAR OPERATORS, I [[P- (l-t)xll 245 P- fly- -< qtllY qllII while (5) implies xll -< c(llxll) -ltlly I1 -tllxll =(llxll) Now observe that llxll-< yields that c(llxl[) B(w;6) x + ]lwll 6 (17) II wll implies -< Applying the function so 6. c IIII -< llx wll c(6+llwl[) -< llxll gives that So (17) becomes I1 c(/llwll)-tllxll xll -< (IS) In order to apply Theorem 2.1 of [9] we must verify that (l-q) -I Fa s-iB(s)ds M c(6+]lw]]) 0 < e < -I B(s) [0, )-[0,) a,B for appropriate choices of and s 6(l-q)c(6+llwll)e q-I (19) _< and If we choose a c(6+llwll)-lllwlle l-q a 6 a and if then (19) follows: a (l-q)-if0 s-iB(s)ds (l-q)-ifoa ds (l-q) -1 a (l_q)-lc(6+llwll)-lllxlle l-q (l-q)-lc(6+llwll) -1 e l-q (l-q)-ic (6+l]w]])-16 (l-q) c (6+]]wl]) eq-le l-q _< _< =6 x B(w;6) 0 as required. Hence Theorem 2.1 applies to give the existence of an Px 0 0 Px 0 y Thus and y 6 P(b(x;6)) for which Theorem 3 is an easy consequence of the Mean Value Theorem. PROOF. {xn} Let lim Fx + n nN > 0 so that y xn {Xn} [Ixn implies Value Theorem of Mcleod [16"1 to (Xm) for which Xn-X, > 0 is Cauchy, for every m,n >_ N (Xn) X be a sequence in Since Xmll E -ic(0) -I and PXn -y there exists an Applying the Mean yields that E -{’(txn + (l-t)Xm)(Xn m) x 0 < t < i} from which it follows, via (3), that llxn xmll -< llxn -< llxn- xmllsup{ll’ (tXn xmllsup{c(lltXn _< IIxn xmll.c(O) + + (l-t)xmll (1-t)xmll) 0 < t < I} 0 < t < i) < E if m,n >_ N Hence converges. {xn is Cauchy in By assumption Y so the completeness of has closed graph and so lim n- Y xn implies x {Xn} i.e. W. O. RAY AND A. M. WALKER 246 Fx, graph, so xn y Fx n + Since y x, it follows that Therefore y Fx n y Fx, + x x F has closed giving P closed But Px graph, as desired. By Theorem 2, P is an open surjection. We conclude by remarking that most of our conclusions are fairly direct consequences of earlier results which have used inequalities (4) and (5) as their main assumptions. Theorem 3.2 in Thus, for example, surjectivity in Theorem 2 is a special case of [13], while openness was inferred directly from Theorem 2.1 of [9]. Our main goal here has been to expose a further class of operators to which these more general results apply. The inequalities (4) and (5) have come to play a central role in the theory of normal solvability, and the above results show that these inequalities and the mapping properties they imply are stable under small perturbations. REFERENCES i. POHOZHAYEV, S.I. 2. BROWDER, F.E. Nonlinear operators and nonlinear equations of evolution in Banach spaces, Proc. Symp. Pure Math. 18, part 3, Amer. Math Soc., Providence, R.I., 1976. BROWDER, F.E. 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