On the solvability of some operator equations with angle-conditions

Nonlinear dnalysis, Theory, Printed in Great Britain.

Methods

& Application,

ON THE SOLVABILITY

Vol. 6, No. 9, pp. 935-941,

0362-546X%2/090935 Pergamon

1982.

OF SOME OPERATOR ANGLE-CONDITIONS

EQUATIONS

$03,00/O Press Ltd.

WITH

SERGIO INVERNIZZI Istituto di Matematica dell’Universit8, I-34100 Trieste, Italy (Received for publication 16 March 1982) Key words and phrases: Fredholm map, coincidence degree, periodic solutions.

1. INTRODUCTION

LET 2 BE a real normed space, with norm ( . (; suppose that 1. ( is induced by an inner product

( ., a); the following theorem can be easily proved by a degree argument: THEOREM

1.1. If G: Z+

2 is a completely continuous map such that

(G(z),z)lzl-*

-=I 1;

then Z - G is onto (I = identity), i.e. for any fixed w E Z, the equation z = G(z) + w has at least one solution 2 E Z. Our aim is to extend the above result, proving a similar existence theorem for the more general equation Lx = (F + G)(x),

(1.1)

where L is a linear Fredholm map of index zero, defined on a subspace domL of a real normed space X with values in Z, and where F and G are two maps from X to Z such that their sum F + G is L-completely continuous (for the terminology see [l]). The proof will be based on the extended version of the Leray-Schauder continuation principle established by J. Mawhin (see [l]), that is, roughly speaking, on the comparison of equation (1.1) with an uniquely solvable linear equation of the form Lx = Ax, via a linear homotopy. As application of our abstract theorem, we shall study the problem of existence of p-periodic solutions of a vector ordinary differential equation of Rayleigh type: (x E RN;’ = dl&;V = grad).

-x” = f(t, x, x’) + V@(Y) + g(x)

(1.2)

This problem was already considered by the author in [2], assuming that g satisfies an ‘angle condition’ and is infinite at infinity, and thatfis uniformly bounded. Under these assumptions, in that paper it is proved that (1.2) has at least one p-periodic solution provided that the following ‘nonresonance’ condition holds: K: =

my sup x,yEnN;lIJJll=l

2Y>-=-c AZ,

where 12,:- (27r/p)* is the first nonzero eigenvalue of the linear problem - x” = h, x p-periodic. 935

(1.3)

S.INVERNIZZI

936

Here we shall consider equation (1.2) allowing f to be ‘quasibounded’ in x’, uniformly with respect to the other variables (for example, when N = 3, the map f(t, x, x’) = B(t, x)Ax’, Ml < const., can be considered). We shall prove that the same conclusion of [2] holds, provided that both the constant K and the ‘quasinorm’ off

&If:= ,,p$PmIlf(bx>Y)II.IIYII-’ satisfy an inequality related to the values A, ,Ufor which the linear 2-dimensional problem - XI’= pJx’ + Ax,

xp-periodic,

J=

has nontrivial solutions. This inequality becomes (1.3) when the constant M is 0, and is, in some sense, ‘the best possible’.

2,ANABSTRACTEXISTENCETHEOREM

Let X be a real normed space, with norm (1.I], and let 2 be a real inner product space, with inner product ( * , . ) and corresponding norm ( . I. Let L:domLCX+Z be a linear Fredholm map of index zero, and let F, G:X+Z

be two (possibly nonlinear) maps such that their sum F + G is L-completely [l]). We shall use the following notations. For any constant d > 0, let

continuous

(see

Ad:={x~domL:]Lxj~d}. For any map T: X+

Z, let

a, + w }). If x E R, the inequality 0 T Cl < x means that there exist two (UTUERU{real constants d, a, with d > 0, a < x, such that (T(x), Lx) s a ILxl*

for all x E dom L\Ad. In the special case dom L = X = Z, L = identity, we have Cl TO = h~:u_p (T(x),x)

/xl-*.

2.1. Suppose that: (i) for any d > 0, the set F(Ad) is bounded; (ii) q F + G 0 < 1; (iii) there exists an L-completely continuous linear map A :X+ that: (relationship with L) (iiir) ker (L - A) = (0)

THEOREM

Z, with •i A 0 < 1, and such

Solvability of some operator equations with angle-conditions

937

(ii&) there exists C > 0 such that, for all x E domL, ]]x]]s C(]Axl + ]LxI) (relationship with G) (ii&) there exist (Y=Z0 with (Y+ 1 > 0, and p c 0, such that, for all x E X, (G(x),Ax)

2 (~(G(x)[.jAx

+ p

(ii&) for every d > 0 and every r > 0, there exists m = m(d, I) > 0 such that, if x E Ad, then either [AxI c m or (G(x)] > T. Then there exists x E dom L such that Lx = (F + G)(x). Proof. By a well known result of J. Mawhin, it is enough to show the existence of a constant R > 0 such that, if (x, A) E domL X [0, l] satisfies Lx = (1 - il)Ax + A(F + G) (x),

(2.1)

then (Ix]]< R. We refer to [l] for details. Suppose that (2.1) holds. The assumptions q A 0 < 1, q F + G 0 < 1 imply the existence of constants d1 > 0, d2 > 0, al < 1, u2 < 1, such that (Ax, Lx) s allLxl* for x E if x E dom L\Ad, , and ((F + G)(x), Lx) s a2(Lx]*- for x E dom L\Adz. Therefore, domL\(Ad, UA&, we have ILxj* = (1 - ii)(Ax,Lx)

which is a contradiction.

+ A((F+

G)(x),

Lx) G max{ui,a2}lLx]‘<

ILx(*,

Hence (2.1) implies x EAdl U Adz, i.e. ILx/ c d : = max{di, d2).

(2.2)

Using the assumption (i) we obtain, from (2.1), (2.2), I(1 - /i)Ax + /lG(x)I’s

p,

(2.3)

with p: = (d + sup{)F(x)]:x E A,})*. N ow we can repeat the argument developed in [2]. If we define 5: = (1 - A)(Axl, q: = nlG(x)l, we obtain from (2.3), using (ii&), the inequality p 2 p + 7J2+ 24;rl+ &j*+f*+2&+2/3~(cu+

2@( 1 - n) 1)(52+$)+*

(remark that the eigenvalues of the quadratic form g* + q* + 2& pi:= 1 + a< ~~~12: = 1 - or). Therefore we have (o-2@)(a+

1)-‘*C*+

are ~1, ,u2 with 0 <

?J*

2 min{A*+ (1 - A)*:A E [O,l]}. min{lAx]*, /G(x)]? = (l/2) min{]Axl*, [G(x)]‘}, i.e. min{)Axj, ]G(x)J} c r: = (2(p - 2/3)(a + l)-‘)“*. If lAxI c r, then IAx\ + /Lx/ c r + d; if IG(x)) < r, we can use the constant m = m(d, r) of assumption (ii&), and then (AxI c m. Recalling (iii*), we find R : = C(max{r, m} + d).

S. INVERNIZZI

938

Remark 2.2. Theorem 1.1 follows easily from the above result; dom L. = X = Z, 15 = identity, A = 0, F = w (A, F constant maps). 3. APPLICATION

TO PERIODIC SOLUTIONS OF ORDINARY EQUATIONS IN [WN

we need

only

to set

DIFFERENTIAL

Let IWNbe the real N-dimensional Euclidean space, with inner product (x, y): = ~:xiyi and solutions of the vector ordinary norm llxll: = (x, x) *. Let p > 0. We look for p-periodic differential equation -x”

= f(t, x,x’)

+ V$(x’)

(x E [WN.’ = dldt; V = grad)

+ g(x)

We assume that f is a continuous function iw x RN + RN, p-periodic in the first variable, @:IWN+ [w is C’, and that g:LWN+ [WNis C’, too. For short, w: = (21rfp). THEOREM 3.1. Suppose (a) the upper limit

(3.1) that

that:

M:=

llf(L?Y>II

lim su r-m rEFApl.Ilx R+lIyll>r

IIYII

is finite;

(b) Ik(4II - 00asIlxll ---,03; (c)there exists an N X N unitary

real matrix

U such that the lower limit

(g(x),ux>

D:=!if ,,$A llg(x)ll is # -1; (d) K:= x yE;~$y,,_l (D&)Y,Y)

is finite; let

Ko:=

mado,

(e) K. + ‘MO L a? < 0. Then the equation (3.1) has at least one p-periodic

. (Iux(l

KI;

solution.

Remark 3.2. A simple and physically significant example of a map f which satisfies (a) is f(t, x, y): = B(t, x)y, where B(t, x) is an N x N skew-symmetric matrix with norm uniformly bounded by M. For instance, if N = 3, the Lorentz force induced on a charge particle with velocity y by a bounded magnetic field is of this type. Remark 3.3. For the definition of the constant D in (c) we use the convention ull~ll-~ = 0 if u = 0 E [WN,which agrees with the usual definition of the sign of a real number: sign 0 = 0. Therefore the constant D is well defined, even if g = 0 in some point. When N = 1, for a continuous g: [w -+ [w the assumption (c) is equivalent to the sign condition g(x)x 2 0 (or G 0) for 1x1 > r. It is obvious that, if g is monotone, the sign condition holds if and only if g has a zero. This is true in higher dimension: suppose that a continuous g: IWN+ IWNis monotone, i.e. for all x1, x2 E (WN @(Xl) -

&2),

Xl

-

x2) ?= 0;

(3.2)

then (c) holds if and only if there exists x0 E [WNsuch that g(xo) = 0. In fact, if g(xo) = 0, let US fix 4 : = (1 + ~)~~xO~~ with E > 0, SO that 0 3 - (1 + &)-I > - 1. If /XII > 9 we have, using (3.2), (g(x), x> s (g(x), x0) 2 - Ild~>ll~ll~oll = - Il&)lldl + Cl 3 - (1 + C’llg(x)ll~ IIxII. On

Solvability

of some operator

equations

939

with angle-conditions

the other hand, if (c) holds, we can see that the equation g(x) = 0 is solvable by a simple application of the Brouwer degree in RN. Some examples of functions g which verify (b), (c) and (d) can be founded in [2]. Remark 3.4. The inequality (e) is the ‘best possible’ solutions of (3.1), assuming that conditions (a)-(d) in (e), we find a counterexample just in,dimens;lon f(t, x, y) : = M./y - e(t), J =

g(x) : = KZX, z =

The equation

with respect to the existence of p-periodic holds onf and g. In fact, if equality holds N = 2. Let us consider the functions

, M > 0, e continuousp

-periodic;

,K>O.

(3.1) is now the 2-dimensional

linear

x” + MJx

equation

+ KZx = e(t).

(3.3)

The constants M, K in the above definitions are the same of assumptions (a), (d); the hypothesis (b) holds trivially, and (c) holds with U = I. If we suppose K + Mw - a? = 0, we have o = (M/2) + (M2/4 + K)i. An elementary computation shows that the complex number I; = io is a root of the 4th-degree polynomial in f: det( f”Z + CM.I + KZ). This implies (Fredholm alternative: see [3], for example) that we can choose e in such a way that (3.3) does not have p-periodic solutions. Proof

of theorem 3.1. We shall apply theorem

2.1. Let us define

X: = {x E C’(R, R”‘),p-periodic},

]lx]l : = sup/x(t)ll

2 : = {z E C”(R, RN),p-periodic},

(z, rv)

dom L : = X n C2(R, RN), F(x) :=f(.,x,x’)

: = I0

p (44,

+ supllx’(r)]]; w(t))

d;

Lx 1= -xII;

+ V$(x’);

G(x) : = g(x). Then L is a linear Fredholm map of index zero, and F + G is L-completely continuous. us prove (i), (ii) and (iii). (i). If d > 0 is given and if x E A d, by Almansi-Wirtinger and Sobolev inequalities obtain sup]]x’(t)ll s d : = (1 + (1/2n))pid The assumption

(e) implies

the existence

we

(3.4)

of E > 0 so small that

Kou-* + Ma-’ Using

Let

+

Em-l

<

1.

(a) we can find r > 0 such that, for all (t, x, y) E R x RN x RN, Ilfk -“c,Y)]] c (M + s) 11~ it+ 6,

(3.5)

S. INVERNIZZI

940

with 6 : = max{]]f] : t E [0, p], (x, y) E RN x RN, /lx]] + ]/y/l c r}. Now we have, if x E Ad and recalling (3.4), that + ‘5+ 6 + suPw~wll

]@)I Q(M

: IIYIIc 4).

(ii). Let us compute ((F + G) (x), Lx)

P$w, -x7 dl + 1 I)rc t, s op I Ilf(t,x, x’>ll-IIx”ll~+ Iff@%(x)x’,x’) df x, x’), -x”)

=

s

dl + [

(g(x), -x”) dl

[ ((M+ ~)II~‘II + 4 . II41dt+ [ Klb’112 d

c ((A4+ E)IX’I+ dppf) *lx”1+ zqxq2 c (&wP + MO.-’+ &d)lX”l*+ d&q The inequality (3.5) enable us to choose t > 0 such that y : = K0U2 +Mw-‘+Ew-‘ p’d/ t ( i.e. if x EdomL\A,,&),

we have

((F + G) (x), Lx) 7 ylLx]*. (iii). We define Ax := VUX, with 0 < v < o 2. The estimate of q A 0 is easy: we have (Ax, vo-*)x”)2 for every x E dom L, so that OAR < 1. Suppose now that Lx = Ax, i.e. -x” = VUX. We obtain lx”12< VW-*~X”~~, and this implies x(t) = 0 for all t E R. In order to prove (iii*) we can apply straightforward arguments, based again on the Almansi-Wirtinger and Sobolev inequalities. Let us prove (ii&). The assumption (c) implies that we can fix a constant ~2< 0, with Cu > - 1, and we can find a constant q > 0 such that Lx) s

(g(x), wll&)ll-

IWxII-’ 2 Lf

whenever llxlj > q. Let p := min{O, min{(g(x), Ux): llx]]c q}}. Then we have p s 0 and Q?(X)?Ux) 3

@il&)II. IWI + s

for all x E RN. Now we can compute (G(x),Ax)

= $.&P)dfP

v~(~ig(x)lI.Ilr/xli+B)dt

3 GIG(x)) . lAx[ + v& and (ii&) holds with cx = 5, /I = v-p. The last check is (iii4); we shall prove it by a contradiction.

Suppose that there exist d*

Solvability >

of some operator

equations

with angle-conditions

0, I-* > 0, and, for every 12E N, a map x,, E C*(R, RN), p-periodic,

([

941

such that

G d*,

(3.6)

I.&I = v(I,” llxnII*dt)’ > a,

(3.7)

IZ,x,l =

tG(xn)t=

11x:/1* dr)*

(~/lg-hl12df)i~r*. 0

(3.8)

Now tj& := [(x,1/*IS ’ a sequence of real-valued functions. The inequality (3.7) implies that the sequence of the mean values of qn on [0, p] satisfies (l/p) ff I& dt > (nl~)*/p. Therefore

there exists a sequence t,, E [0, p] such that w&I) =

II&(hJ II2’ (4*/p.

(3.9)

Using (b) we can find r > 0 such that

Il&)ll > r*4+

(3.10)

whenever llxll > r. Let us fix now II* E N such that (n*lv)/p* >p(l

+ (1/2zr))p*d* + l-.

(3.11)

Recall that (3.6) implies supl]x’(t)ll G (1 + (1/2Jc))p$d*. Therefore,

for all t E[t,p, tn* +p]

(3.12)

we have, using (3.9), (3.12), (3.11),

llx,*(t)ll 1 IIXn*(~n*)ll - I/I”‘M4 Q11 > (n*lv)/p* - p( 1 + (1/2n))p*d * > I-.

By the periodicity we obtain

of x,,‘, the above inequality holds for all t E [0, p], so that, using (3.10),

(~ll~(x.*)il’df)‘>r*. which is a contradiction with (3.8). The proof is complete. REFERENCES 1. GAINES R. E. & MAWHIN J., Coincidence Degree, and Nonlinear Differential Equations, Springer, Heidelberg (1977). 2. INVERNIZZI S., Periodic Solutions of Nonlinear Rayleigh Systems, Anahi Funz. Applic. Suppl. B. U. M.I. 1, 45 52 (1980). 3. ROUCHE N. & MAWHIN J., Equations Diffkrentielles Ordinaires, Masson, Paris (1973).