Multiplicity results for problems with uniform norms in boundary conditions

Nonlinear Analysis, Theoq. Methods &Applications, Vol. 30, No. 8, pp. 4781-4788, 1997 Proc. 2nd World Congress of Nonlinear Amlysfs Q 1997 Elsevier ScienceLtd

Pergamon

Printedin Great Britain. All rights reserved 0362-546X/97

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PII: SO362-546X(!W)OOOO7-2

MULTIPLICITY

RESULTS FOR PROBLEMS WITH UNIFORM NORMS IN BOUNDARY CONDITIONS t !%A. BRYKALOV

Institute

of Mathematics and Mechanics, Ural Branch of Russian Academy of Sciences, ul. Kovalevskoi 16, Ekaterinburg 620066, Russia

Key words and phrases: Method of monotone boundary conditions, nonlinear functional boundary conditions, Co-norms in boundary conditions, nonlinear boundary value problems for ordinary differential and functional differential equations; existence, nonuniqueness, and the number of solutions. 1. INTRODUCTION

AND NOTATION

Theory’of nonlinear boundary value problems with ordinary derivatives is one of highly developed branches of nonlinear analysis, see for example [l-4] and references therein. The earlier history of this theme can be traced with the help of [5,6]. The present paper is concerned with the method of monotone boundary conditions (see Section 2 below), which is applicable in the case of nonlinear functional boundary data. Our aim here is to demonstrate this method in application to some concrete problems. To this end, we use boundary conditions that contain uniform norms of the unknown functions and their derivatives. Such a condition may have the form ll~(*)ll@ = 9(d.)h where ]]z(.)[]c~ = maxLXt(z(t)] is the CO-norm of z(e), and g(z(.)) is some nonlinear mapping (possibly, a constant). Boundary data with uniform norms can provide samples of nonlinear functional boundary conditions and can be helpful in illustrating the corresponding technique. Data with uniform norms were dealt with in [7-111. The considerations in these articles are closely connected with monotonicity properties of boundary conditions. Let us also note that problems with nonlinear functional boundary conditions arise in the theory of control. Unless otherwise is stated explicitly, the functions considered in the present paper are defined on a fixed closed interval [a,b], w h ere --oo < a < b < +oo, and take values in R, the set of all real,numbers. We use the standard notation for the space of all continuous (CO), Lebesgue measurable with integrable absolute value (Lr), and absolutely continuous (AC) functions. The spaces Co, L1 are endowed in the standard way with the uniform (see above) and integral Ilx(‘)IIAC = Ils(-)llCo + ~~~(~)~~L,. F or an integer m > 1, we denote by CL? norms respectively, the space of all z(s) such that z(“)(.) E AC for k = 0,. . . , m - 1. The corresponding norm is m-1

Il4~>llCL~ = c ll~Y*)ll0 + llJ”‘(*)1lL,. k=O

tPartially

supported

by Russian Foundation

for Basic Research under grant 94-01-00310.

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2. METHOD

OF MONOTONE

BOUNDARY

C:ONDITIONS

In this section we formulate two theorems, which provide the technique used in Sections 3, 4. This technique of monotone boundary data can be applied to problems with nonlinear functional boundary conditions (not necessarily monotone ones) in order to establish the existence of solutions and estimate the number of them. The two theorems in this section are special cases of a more general result in [la], w Ilere the proofs are given. Some further developments of the method of monotone boundary conditions can be found in [13,9]. Topological properties (e.g., arc-wise connectedness) of the set of solutions were studied in [14]. Examples of application of the method to concrete problems can be found in [12,15,9,16] and in Sections 3, 4 below. Let us consider a second-order system of ordinary differential equations .l

=

fi(4

i2

=

A(4

x1,22),

(2.1)

5170)

(2.2)

on an interval t E [a, b]. The unknown functions z; : [a, b] --+ R, i = 1, 2, are absolutely continuous and should satisfy equations (2.1), (2.2) a 1most everywhere on [a, b]. The nonlinearities fi : [a, b] x R2 -+ R, i = 1, 2, are assumed to satisfy the Carathiodory conditions. That is, f;(t, x~,Q) is measurable in t for any fixed numbers ~1, x2 and is continuous in x1, x2 for almost every t. Besides that, we assume I_fiCt7

x1,

52)I

5

(2.3)

b(t)

almost everywhere on [a, b] for all x 1, ~2. Here
XI(.), x2(.)) = 0,

B2(xz(.),21(.),xz(.))

= 9.

(2.4) (2.5)

We assume that the mappings B; : Co x AC x AC -+ R, i = 1, 2, are continuous. Besides that, Bi are monotone in the first argument in the following sense. For any continuous functions u(e), w(.) and absolutely continuous functions .zr(.), .zs(.), if u(t) 2 w(t) for all t E [a,b], then B;(u(.),

ZI(.),

z2(.))

I

WJ(.),

a(.),

z2(.)).

Fix nondecreasing functions R; : [0, co) -+ [0, cm), i = 1, 2. We assume that for any absolutely continuous zr(.), z2(.) there exists a continuous function u;(v) that satisfies the equality Bi(ui(.), ZI(.), Q(.)) = 0 and the estimate Il”~(~)llCo

5

~i(llil(~)llL~

THEOREM 2.1. Under the above assumptions, (2.5) has at least one solution.

+

lli2(~)lIL~)~

boundary value problem (2.1), (2.2),

(2.6)

(2.4),

Let us also consider a scalar third-order ordinary differential equation xC3)= f(t, 2, k, 2)

(2.7)

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that should be satisfied almost everywhere on an interval t E [a, b]. A solution z : (a, b] -+ R continuous. The together with its first and second derivatives i, i ought to be absolutely right-hand side f satisfies the Carathdodory conditions. There exists a Lebesgue measurable and integrable function < : [a, b] + [0,00) such that the inequality If(G ~0,51,~2)1

5

t(t)

(2.8)

holds for almost all t E [e,b] and all 20, ~1, x2 E R. We will deal with solutions to (2.7) that satisfy boundary Bo(4*),

4.))

= 0,

(2.9)

&(i(.),

r(.))

= 0,

(2.10)

&(S(.),

z(.)) = 0.

(2.11)

The maps Bi : Co x CL; --t R, i = 0, 1, 2, are continuous. in the first argument. That is, for any admissibleu(.), u(.), then &(u(*), 4.)) 5 B;(2)(*), 4.)). Fix some nondecreasing Ri : [O,ca)+ exist some u;(e) E Co such that

We assume that Bi are monotone z(m), if u(t) < u(t) for all t E [a, b],

[O,co),i = 0, 1, 2. Let for any z(.) E CL: there

&(u;(.),z(.)) and the inequality

conditions

= 0

holds ll”i(~)llCo

(Here for i = 2 we assume THEOREM 2.2. Under at least one solution.

5 Qi(llz

(i+l)(.)llCL;-‘).

(2.12)

CL: = L1.)

the assumptions

made, boundary

value problem

(2.7), (2.9)-(2.11)

has

As was mentioned above, Theorems 2.1 and 2.2 are two particular cases of a more general result [12], which is valid for a system of functional differential equations under somewhat weaker restrictions. In the present paper we do not prove the two theorems and refer for the proof to (121. 3. A SECOND-ORDER

PROBLEM

Let us illustrate the proposed method of monotone boundary system (2.1), (2.2) with functional boundary data of the form ll~l(-)llc~

= 91(21(-hQ(-)),

11~2(-)llc~ = 92(~1(.),~2(‘)).

conditions

by a problem

for

(3.1) (3.2)

As above, we assume that the unknown functions are absolutely continuous, and the nonlinearities in the right-hand sides of the differential equations (2.1), (2.2) satisfy the Caratheodory conditions and the estimates (2.3). The mappings 9; : AC x AC + R, i = 1, 2, are continuous. Basing on Theorem 2.1, we can obtain the following sufficient conditions for the existence of at least four solutions.

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COROLLARY 3.1. Let the inequalities

i = 1, 2, hold for some fixed constants Ni and all absolutely continuous functions xi(+), zz(.). Then problem (2.1), (2.2), (3.1), (3.2) h as at least four solutions. Namely, for each of the th ere exists at least one solution to (2.1), following four systems of inequalities (3.4)-(3.7) (2.2), (3.1), (3.2) that satisfies this system of inequalities 7G2(3

>

0;

(3.4)

> 0,

77(x2(3)

<

0;

(3.5)

77CGC.J)

<

0,

77C~2C.l)

>

0;

(3.6)

77(~1(.))

<

0,

7+2(9)

<

03

(3.7)

V(“l(‘))

>

+I(.))

0,

where n(u(.)) = mpxu(t) t mtinu(t).

(3.8)

In the proof of Corollary 3.1 and in Section 4 we will employ the following simple properties. LEMMA 3.1. Let n be given by (3.8). For any continuous u : [a,b] -+ R we have 7(-u(.)) if n(~(.)) 2 0, For any absolutely continuous

then

= -rl(u(*)); IIu(.)ll~o = mpxu(t).

(3.10)

u : [a, b] -+ R we also have

v(u(.)) 2 2mpxu(t)

Proof of Lemma

(3.9)

-

* \ti(r)[dr. J a

(3.11)

3.1. Property (3.9) follows from the equalities mFx(-u(t))

= - mjnu(t),

mt;n(--u(t))

= - mfxu(t),

which are valid for any continuous u(.). To establish property (3.10), note that Ilu(.)llco = max{mpxu(t),

- mt;nu(t)}.

If v(u(*)) 2 0, then maxt u(t) 2 -mint u(t), and so 11u(a)ll~0 = maxt u(t), which gives (3.10). Due to (3.8), the desired inequality (3.11) is equivalent to mpx21(t) - mjnu(t)

<

Je

lti(7)jd~.

(3.12)

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Since

the

t min, tnl,,

minimal and we have that

So, inequality

maximal

values

of the function

(3.12) is valid, and Lemma

Proof of CoroHary conditions

3.1. Consider

u(s) are attained

at some

points

3.1 is proved.

an auxiliary

problem

for system

(2.1), (2.2) with boundary

mFxwlzl(t)

= 9r(xr(*),x2(.)),

(3.13)

yXW222(t)

=

(3.14)

g2(x:1(.),

x2(.)),

where the parameters WI = fl, ws = fl determine the corresponding conditions (3.13), (3.14) coincide with (2.4), (2.5) if we put &(G(*),

.Q(.),

Z2(‘))

=

mfXwzl(t)

-

sl(zl(.>,

z2(+)),

B2(Z2(.),

44,

z2(*))

=

yXw2z2(t)

-

s2(a(.),

22(e)).

signs.

Boundary

These expressions are monotone in xl(.), x2(.). (Th e y are monotone decreasing for w; = -1, which makes no difference here.) The mappings B; : Co x AC x AC --t R are continuous. For fixed zr(.), 22(a), we can make B;(~;(.),zl(.),zz(.)) equal to zero by taking x:(t) z w;gi(zl(.),z2(-)). Note that Is:(t)] 5 N;, so the estimates (2.6) hold. We see that all the assumptions of Theorem 2.1 hold, and so auxiliary problem (2.1), (2.2), (3.13), (3.14) has at least one solution. t o auxiliary problem (2.1), (2.2), (3.13), (3.14) Let us show that any solution x~(*),Q(.) satisfies the inequalities (3.15) w&h(*)) > 0, w277(52(.)) > 0 and is simultaneously

a solution

to (2.1), (2.2), (3.1), (3.2). Really, mpxwiG(t)

and due to estimates

=

as (3.16)

gi(~l(‘),zZ(‘))

(3.3), (2.3) on g; and fi, we have b IIlfLXUiZc;(t)

>

5

/[i(T)& a

2

5

j

l$i(T)ldT.

a

* t o account, we see that q(Wizi(.)) Taking inequality (3.11) m > 0. Property (3.9) implies n(wixi(*)) = wiv(zi(*)). Thus, th e d esired inequalities (3.15) hold. Now, putting u(t) = wiz;(t), from property (3.10) and boundary condition (3.16) we obtain jjzi(*)lloo = gi(xr(.),x2(.)). That is, the functions x1(.), zs(.) are also a solution to (2.1), (2.2), (3.1), (3.2). System of inequalities (3.15) for WI = fl, w2 = fl gives the four systems of inequalities (3.4)-(3.7). So we see that the four auxiliary problems (2.1), (2.2), (3.13), (3.14) with w1 = fl, wz = fl are solvable, and all their solutions satisfy (2.1), (2.2), (3.1), (3.2). Thus, boundary value problem (2.1), (2.2), (3.1), (3.2) h as at least four solutions, namely, at least one solution for each of the systems of inequalities (3.4)-(3.7). This completes the proof of Corollary 3.1.

Second World Congressof NonlinearAnalysts

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4. A THIRD-ORDER

In this section we will consider We require, as above, that x, i, conditions and inequality (2.8) on The boundary conditions are of

PROBLEM

a boundary value problem for differential equation (2.7). 2 be absolutely continuous and impose the Carathkodory the nonlinearity f. the form llx(.)ll0

= 90(x(%

Il~(~)llC~ =

(4.1)

91(x(*)),

(4.2)

IlWllC~= 92(x(*)).

(4.3)

The maps gi : CL: + R, i = 0, 1, 2, are assumed to be continuous. COROLLARY

4.1. Let N > 0 be a fixed constant. 1 * ij t(Tw a

J

Assume that inequalities

(4.4)

< 92(4.)),

;(h- 49zW) <91(4.)), ;(b- a)g1(4)) < I go(4.J)

(4.5) N

(4.6)

are valid for all functions z such that z, i, i are absolutely continuous. Then problem (2.7), (4.1)-(4.3) has at least eight solutions. Namely, for each of the following eight systems of inequalities ~oV(X(*)) > 0,

WM))

>

0,

w27?(5(*))

where & = fl, wr = fl, wz = fl, and n is defined by (3.8), problem (2.7), (4.1)-(4.3) that satisfies this system of inequalities.

> 0,

(4.7)

there is a solution to the

Proof of Corollary 4.1. We will use an auxiliary problem for equation (2.7) with boundary conditions mfxwox(t)

=

90(z(+)),

(4.8)

mpxwl$t)

= sr(x(.)),

(4.9)

mpxwzi(t)

=

(4.10)

92(x(-)),

where ws = fl, wr = fl, wz = fl are parameters. Problem (2.7), (4.8)-(4.10) be a particular case of (2.7), (2.9)-(2.11). To see that, it suffices to put &(u;(.),

z(.)> = mpxw;u;(t)

- 9i(z(.)),

i = 0, 1, 2.

appears to (4.11)

The mappings B; : Co x CL; + R given by (4.11) are continuous. They are monotone in ~;(a) (increasing for wi = +l, decreasing for wi = -1). For a fixed z(.), expressions (4.11) can be made equal to zero by putting ui(t) E UP = w;gi(Z(*)). Due to the estimates (4.4)-(4.6) on g;, we have 1~91 5 No with some No independent of z(.), So, estimates (2.12) hold with constant right-hand sides. All the assumptions imposed above on problem (2.7), (2.9)-(2.11)

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SecondWorldCongress of Nonlinear Analysts hold, Theorem

2.2 is applicable,

and the auxiliary

problem

(2.7), (4.8)-(4.10)

has at least one

solution. To complete the proof of Corollary 4.1, it suffices to show that any solution z(.) to the auxiliary problem (2.7), (4.8)-(4.10) satisfies the inequalities (4.7) and is also a solution to F rom the boundary condition (4.10) of the auxiliary problem and problem (2.7), (4.1)-(4.3). from estimates (2.8), (4.4) on f, 92, we see that

and employing

(3.11) in Lemma

inequality

3.1 with u = ~22, we obtain

rj(wg,i!(-)) > 0.

(4.12)

boundary So, it follows from property (3.10) that Il;(.)jlc o - maxt wz?(t). Consequently, condition (4.3) holds. Besides that, (4.12) and property (3.9) imply that w2r](?(e)) > 0. Thus the la& inequality in (4.7) is valid. Now, from the boundary condition (4.9) of the auxiliary problem, boundary condition (4.3), and inequality (4.5), which connects g1 with 92, we obtain

myJ&)

= T/1(4-)) > f@ -

a)g2(2(*))

= f(b-u)lliq.)llco .> ; j

IS(T)

a

With

the help of inequality

(3.11) for u=

WI?, we see that

rl(Wl”i(.)) > 0. From the last inequality, the following can be obtained. On the one hand, according to condition (4.2) holds. property (3.10), we see that Il~(.)llc~ = maxi wlh(+), and so, boundary On the other hand, property (3.9) implies that w~v(i(.)) > 0, which coincides with the second inequality in (4.7). The remaining case of boundary condition (4.1) and the first inequality in (4.7) is dealt with similarly. Namely, from the boundary condition (4.8) of th e auxiliary problem, the already obtained boundary condition (4.2), and estimate (4.6), which involves go, gl, it follows that

mpxw0z(t)

>

f(b- a)gl(s(-)) 2

k]

I+)ld~.

a

Using Lemma 3.1 as above, we obtain (4.7). Corollary 4.1 is proved. 5.

Remark

5.1.

the required

condition

(4.1) and the first inequality

in

CONCLUDINGREMARKS

The employed technique of monotone boundary conditions [12] allows to prove of the present paper also for the case of functional differential equations. Remark 5.2. As is clear from (2.6), the upper bounds in (3.3) can be made dependent on Il&(.)llt,, I~&(.)IIL,. Similarly, it follows from (2.12) that the upper bound in (4.6) can be taken dependent on Il~(.)llc~:. all the results

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Second World Congress of Nonlinear Analysts

Remark

5.3.

We see that for the considered

above second-order

problem

there

4 = 22 solutions and for the third-order one there are at least 8 = 23 solutions. can be shown that a k-order problem of this type has no less than 2” solutions. 5.4. We can merge Corollaries

Remark equation

with the highest

space of n-dimensional

derivative

vectors.

3.1 and 4.1. To do that, we can consider

of order m and with solutions

The solution

is subject

are at least Similarly a differential

that take values in

to boundary

conditions

it

R”, the

of the form

where xik) IS ’ the coordinate number i of the derivative of order k. This problem has the order mn and’ admits at least 2”” solutions provided inequalities similar to (3.3), (4.4)-(4.6) are imposed. REFERENCES 1. BERNFELD S.R. & LAKSHMIKANTHAM V., An Introduction to Nonlinear Boundary Value Problems. Academic Press, New York (1974). 2. GRANAS A., GUENTHER R.B. & LEE J.W., Nonlinear Boundary Value Problems for Ordinary Differential Equations, Dissertaliones Math., No. 244. Polish Acad. Sci., Warsaw (1985). 3. MAWHIN J., Topological Degree Methods in Nonlinear Boundary Value Problems, CBMS Regional Conference Series in Mathematics, No. 40. Am. Math. Sot., Providence, RI (1979). 4. O’REGAN D., Theory of Singular Boundary Value Problems. World Scientific Publishing Co., River Edge, NJ (1994). 5. CONTI R., Recent trends in the theory of boundary value problems for ordinary differential equations, Boll. Un. Mat. Ital. (3) 22, 135-178 (1967). 6. MAWHIN J., Boundary value problems for nonlinear ordinary differential equations: from successive approximations to topology, in Development of Mathematics 1900-1950 (Edited by J.-P. PIER), pp. 443-477. Birkhauser, Base1 (1994). 7. BRYKALOV S. A., Solvability of a nonlinear boundary value problem in a fixed set of functions, Dijferentsial’nye Uravneniya 27, 2027-2033 (1991). (In R ussian.) English translation in Diflerential Equations 27, 1415-1420 (1991). 8. FECKAN

M., The interaction of linear boundary value and nonlinear functional conditions, .Ann. 58, 299-310 (1993). 9. BRYKALOV S.A., Problems for ordinary differential equations with monotone boundary conditions, Diflerenisial’nye Uravneniya 32 (1996). (In Russian.) (to appear). Polon.

Math.

10. STANEK S., On a criterion for the existence of at least four solutions problems (preprint).

of functional

boundary

value

11. STANEK S., On the existence of two solutions of functional boundary value problems (preprint). 12. BRYKALOV S.A., Solvability of problems with monotone boundary conditions, Diflerenlsial’nye Urawneniya 29, 744-750 (1993) (In Russian.) English translation in Differential Equations 29, 633-639 (1993). 13. BRYKALOV S.A., Problems for functional differential equations with monotone boundary conditions, Diflerentsial’nye Uravneniya 32, 731-738 (1996). (In R ussian.) English translation in Diflerential Equations 32 (1996).

S.A., Properties of some plane sets and boundary value problems, Diflerentsial’nye Uravneniya 31, 739-746 (1995). (In R ussian.) English translation in Diflerential Equations 31 (1995). 15. BRYKALOV S.A., Solutions with a prescribed minimum and maximum, Diflerentsial’nye Urauneniya 29, 938-942 (1993). (In Russian.) English translation in Differential Equations 29, 802-805 (1993).

14. BRYKALOV

16. BRYKALOV Georgian

Math.

S.A., Nonconvex differential J. 4 (1997).

(to appear).

inclusions with nonlinear

monotone

boundary

conditions,