Nonlinear Andysir, Theory. Methods & Applicdmu. Vol. 5, No. 9. pp. 1037-1042, 1981. 0362446x/81/09103746 W2.00/0

Printed in Great Britain. @ 1981 Pcr~amon Rcss Ltd.

USING A NONLINEAR SPECTRAL THEORY TO SOLVE BOUNDARY VALUE PROBLEMS*

GIUSEPPE CONTI Istituto di Matematica Applicata della Facolta di Ingegneria dell’I_Jniversit~ di Firenze, Italy

and

RITA IANNACCI

Dipartimento di Matematica deIl’Universit8 della Calabria, C.P. 9, Roges di Rende, Cosenza, Italy

(Receiued 24 June 1980)

Key words and phrases: Boundary value problems, Schauder fixed point theorem, Caratheodory’s conditions.

1. INTRODUCTION

IN THIS PAPER we are concerned with solutions of

x’ - A(+ = f(t, x)

which satisfy the following linear condition

(1)

LJ = r, (2)

where A(t) is a real matrix 12 X n defined on [a, b] C 54, the real n vectorf(t, x) is defined on [a, b] x R”, L is a linear continuous map from C([a, b], OX”) into R” and r E R”.

Generally the existence of a solution to problem (l), (2) is reduced to proving the existence of a fixed point for a suitably defined map. To this purpose Theorems of Banach, Schauder, Leray and Schauder, Tychonov and others are frequently used. It is possible to find a nearly exaustive reference in Conti [2], Opial [5], Bemfeld and Lakshmikantham [l].

In this paper in order to show that problem (l), (2) has at least one solution, we use the spectral theory for nonlinear maps introduced in [3] by Furi and Vignoli. In this way we obtain an existence theorem for (l), (2) which improves a result of [2].

Finally we furnish an application of our result in the context of the existence of a periodic solution for a second order differential equation.

2. NOTATIONS AND DEFINITIONS

Let [a, b] be a fixed compact interval of the real line IR. Let IR” be the n-dimensional Euclidean space with the norm ]) .]I. We denote by d the algebra of n X n matrices with the

mum IlAll = ,,yl IIAxll-

* Paper written with the sponsorship of the C.N.R., Italy.

1037

1038 G. Corn AND R. IANNACCI

Let C([a, b], Rn) be the space of all continuous maps X: f + x(t) from [a, b] into R” with the usual norm ]lxllc = t~,o bl max Ilx(t)]l and L’([a, b], R”) the space of all measurable and integrable

maps y: [a, b] + R” with the norm

II Ylll = lb IlY(~)ll dr ll

Let E be a real Banach space with the norm ]I. 11’. A continuous map F: E + E is said to be compact if it sends bounded sets into relatively

compact sets. A continuous map F: E + E is called quasibounded if it sends bounded sets into bounded

sets and

IFI = limsupw< a lMl’-+ - z

(see [4]). The real number IFI is called the quusinorm of F. Notice that any continuous linear map L is quasibounded and 1 LI = II LII.

Let F: E ---, E be quasibounded. We put

d(F) = lim inf w lM’-+ m z

(see [3]). The spectrum of F is the set defined by C(F) = {A E R: d(A - F) = 0}, (see [3]).

3. THE MAIN RESULT

Let us consider the following ordinary differential equation

x’ - A(t)x = f(r, x) (1)

with the condition

under the following hypotheses:

Lx = r (2)

(a) A: [a, b] + SB is measurable and integrable on [a, b]; (b) fi [a, b] x Iw” + Fi” verifies the Caratheodory’s conditions; (c) there are two nonnegative maps (Y, p E L’([a, b], R) such that Ilf(~ XIII s 44 + B(Wll,

x E R”, a.e. on [a, b]; (d) L is a linear continuous map from C([a, b], W) into R” and r E Im L.

It follows from condition (a) that there exists only one continuous map U: (t, s) + U(t, s) from [a, b] x [a, b] into Sp such that

U(t, s) = I + I

‘A(+& s) dt, s, t E [a, bl, S

where Z is the identity of ~4. In particular U(. , a) is a linear continuous map from R” in to C([a, b], W) and hence the

composition product Lu = LoU(. , a) is a linear map from R” into [Wm. By Lc we denote the generalised inverse of Lu, i.e., LuLtLu =Lu . We put M =

Using a nonlinear spectral theory to solve boundary value problems 1039

THEOREM. Assume that conditions (a)-(d) hold. Suppose that the problem

[ X’ - A(t)x = f(t, w), Lx = r,

has solutions for any w E C[a, b], 54”) with Lw = r. If

then problem (l), (2) has solutions.

Proof. Since r E Im L, there exists 00 E C([a, b], R’) such that LUO = r. Let V, denote the linear variety defined by V, = {u E C([a, b], IFP): Lu = r}.

For any u E V, we set

*) T(u)(t) = -U(f, a)LCL I’U(r, t)flt, u(t)) dt+ /‘U(& r)j’(r, u(r)) dt+ Hr, cl 0

where Hr = U(t, s) (c + Lfir) and c E Ker L & Clearly T(u) E V, for any u E V,. The map T: V, + V, defined by .) is compact (see, e.g., [2]).

Let us write V, = u. + W, where W denotes the kernel of L. Since L is continuous, W and V, are closed. Thus W is a real Banach space with the norm induced by C([a, b], UP).

For any w E W we set

(a) F(w) = T(w + ua) - U@

Since T(u) E V, for any u E V,, we have F(w) E W for any w E W. We observe that the map F: W-+ W defined by (.) is compact. Moreover Fis quasibounded. Indeed, by using hypothesis (c), we can write

II~(w)(Oll s Mll4l + llLtilll4 + M211L6il IILKII 41

+ lIPIll IMIC) + wl4ll + lIPIll IIWIIC). SO that IFI s IlPh (1 + M* IlLtll IMI).

We claim that under our hypotheses we get Z(F) C (-1,l). Let w # 0 and y = Aw - F(w). Equivalently we can write Aw = y + F(w). For any A E R with IAl 5 1 we have

IMOII c PI IIw(t)ll c IIYIIc + IIuoIIc + IIHrllc

+ ~IlLtll IIL j-’ ( If( I/ t, r T, w t + udt)) dt(( + M ( 1 ‘I&, w(t) + uo(t)) II h. L1

By using (c) we obtain

Ilw(t>ll c (IIYIIc + llHrllc + lbollc

+ MlbZril jlL j-’ W 9.f-t t, w(r) + ud$> dt + Ml141 a II

+ MM% Ibollc) + M I’ B(r) IlwWll dr. (I

1040 G.Cotm AND R. IANNACCI

We get by Gronwall’s Lemma

IM)ll c (II Y c + Ilfwc + Il~ollc + Ml41 + MlPlll II~OIIC II

+ MIIWI /IL I’ UC Wh ~(4 + 44) dr/l) evWW%). (I Therefore we have

exp(--~IlPlId llwll~ s IIAlc + lip+ + IIVOIIC + Ml141

+ MlPlll Il~ollc + MlkIl /I L j’ wt. T) a

x f(t, w(t) + uo(4) dt . II By dividing by ]]w]lc and taking lim inf as ]lw]lc + CO we obtain

exp(--MIIPIId 6 d@ - F)

x f(r, w(r) + uo(r)) dr

It follows from hypotheses that d(A - F) > 0 for any A E R with IAl 3 1, hence 2 (F) c

(-1, 1). Assuming a result shown in [3] it follows that I - F is onto. In particular F has a fixed point

in W, i.e., there exists wg E W such that wg = F(wo) = T( wo + UO) - ug. Therefore T has a fixed point belonging to V,. This implies that problem (.I), (2) has at least one solution.

COROLLARY. Under the hypotheses of previous theorem, (e) can be replaced by the stronger condition

(0 ew(-MIIPIId > M211Gll IMI lIPIll.

Then the same conclusion holds good.

Proof. The thesis follows from the inequality

IIL j' WC r)f(r, W(T)) drii c IILll~(ll~Il~ + IlPll~ llwllc)~ II

Remark 1. Theorem 16.2 in [2] insures the existence of solutions for problem (l), (2) under the assumption

@“I 1 - WPlll > ~211L2111 IILII lIPIll

Using a nonlinear spectral theory to solve boundary value problems 1041

instead of (e’). Observe that (e) and (e’) are both more general than (e”) since exp(-~ll~l~)

2 1 - ~138111~

Remark 2. Define the linear operator

f I

I-: x-+x(f) = -U(t,s)LtL U(t, t)x(t) dr+ U(t, $x(r) dt.

It is easily seen that (e”) is a sufficient ~onditioo in order to have IfI)\ &3//r < 1, which allows use of Schauder’s fked point theorem. Oniy rarely is it possible to evaluate ][I]/. However it is easy to see that in some simple cases condition (e) is verified, but f/I’ll lla)/i & 1.

Now an application of our theorem is given. Consider the following boundary value problem

(

U*+U=g(t,U,u’),

(‘) 40) X5 r&r), U’(0) = 24’(3t),

where g(t, u, w) is periodic in t of period JZ and the Caratheodory’s conditions are verified. Moreover assume that

where a; /3 are nonnegative maps E Lr([O, x], R) and 11jI{\i = &.

Setting

x=~~)=(~,), A=(_; +;)

we can write the problem (*) in the following form

I x’ -AX =f(t,x), Lx =o.

It is easy to see that

Il(t, s) = cos(r - s) sin(t - s)

-sin@ - f) > cos(I. - s) ’

hence IIU(t, s>ll = 1 = M, Lu = U(JZ, 0) - f = -21 and L.5 =Ls;’ = -41. We have

!I I L ’ u(t, r)f(v(~>) dt 0

1042 G. CONII AND R. IANNACCI

It follows that

exp(-M IIPII~) = exp (- $1 > $2 WG?II

Hence the problem (*) has solutions under our assumptions. It is easy to see that in this example we cannot apply either

results of [5]. theorem 16.2 of [2] or the

REFERENCES

1. BERNFELD, S. R. & LAKSHMIKANTHAM V., An Introduction to Nonlinear Boundary Value Problems, Academic Press, New York, (1974).

2. CONTI R., Recent trends in the theory of boundary value problems for ordinary differential equations, Boll. (In. Mat. Ital. XXII 3, 135-178 (1967).

3. FURI, M. & VIGNOLI A., A nonlinear spectral approach to surjectivity in Banach spaces, J. Funct. Anal. 20, 304-318 (1975).

4. GRANAS A., On a class of nonlinear mappings in Banach spaces, Bull. Acad. Pol. Sci., Cl. III, vol. 5, no. 9, 867-870 (1957).

5. OPIAL Z., Linear problems for system of nonlinear differential equations, J. Difl Equat. 3, 580-594 (1967).