Existence of solutions to ibvp for nonlinear second order equations with functional dependence

Nodimw

Analysis.

Theory,

Merho&

Pergamon

& Applicmiom. Printed

Vol. 28, No. IO, pp. 17Wl717, 1997 @ 1997 Elsetier Scicncc Ltd m Great Britain. All rights reserved 0362-546X/97 $17.00 + 0.00

PII:SO362-546X(96)00009-0

EXISTENCE OF SOLUTIONS TO IBVP FOR NONLINEAR SECOND ORDER EQUATIONS WITH FUNCTIONAL DEPENDENCE HENRYK

LESZCZYrjSKI

Institute of Mathematics, University of Gdalisk, ul. Wita Stwosza 57, SO-952 Gdafisk, Poland (Received

Key words andphrases:

12 June 1995;

receivedfor

publication

I8 January

1996)

Green function, Banach contraction principle, impulsive conditions,

functional dependence.

INTRODUCTION

This paper can be placed in the existence theory of impulsive ordinary differential equations. The foundation and the most vital impact on this theory is closely related to two mathematicians: V. Lakshmikantham and D. D. Bainov, whose scientific output is represented in Ref. [l]. It is observed that this particular branch of differential equations has been constantly developed since the early 1980s. Being directly inspired by Ref. [2], we have stated the question of natural sufficient condition for the existence of impulsive solution to a BVP with functional dependence added not only to the right-hand side of the differential equation, but also to all impulsive conditions. The Hale-type functionals used here cause the need to give initial conditions on a ‘thick’ set JO, which leads to a natural jump at 0 and 1 to be included into the definition of impulsive solutions. This is the first difference between functional and nonfunctional problems. In order to show the second we will give examples of the functional dependence from two models: deviated variables and integrals. The first model especially shows that jumps of the unknown function and its first derivative in the right-hand side of the differential equation induce a multitude of jumps of its second derivative. Consequently, contrary to what was stated in Ref. [2], we cannot demand that solutions have their second derivatives everywhere exept for jump points. That is the reason why we apply a simpler proof technique, namely, we use the Banach contraction principle. Put JO = [-T, 0) u [ 1, 1 + T], J = [0, 11, J’ = J\Ji,,, where .&,,r = {tr, . . . , tm] with 0 = to < fl < * * . < tm < f,,,+r = 1, and j = JO u J. Suppose that E is a Banach space. If x : J - E, we denote x(s) - x+/p 0) X+/-O) = ,J$ x(s) t x:,-w = pm s-t The notation x’(t) is obvious except the jump points, where, if there is no indication, we understand x’(t) to be x>(t). Define

c&JJ: E) = x : J - E

XI4 1 x;a 1 XII, ,,,,+,I * x;[ ,(,,,+,) are continuous~ there exist X-(ti), x+(I~) and xL(ti), x:(t;) fori= 1,...I m 1709

1.

1710

HENRYK

LESZCZYI;JSKI

In a similar way we define Ci,r(j, R). For x E C,,,,(.!, E) and ti E Jimp we write AxI,=,~ = X+ (ti) - X- (ti) and Ax;,=,, = xl (ti) - XL (ti). If x E Cimr(J, E) and t E J, then we define a Haletype operator x, : [-T, T] - E by putting x,(s) = x(t + s) for s E [--7, T]. For t E J we write xt = {w : i--T, Tl - E 1 3xE~,cj,~j x;

=

{w

: [-T,

T]

XI = w 1.

E 1 3xCC,,,Cj,Ej x; = w }

-

Let R = &J {t} X X, X Xl. Take 4 : Jo - E, f : R - E and hi, ii : X,, - E for i = 1,. . . , m. We will assume throughout the paper that the function 4 has an extension of class C’ onto 70. Of the function f, we assume that for every x E CImp(~, E) the function J’ 3 t - f(t, xf, xi) E E is bounded piecewise continuous. Consider the impulsive boundary-value problem (IBVP) with the functional dependence expressed by the above given Hale-type operator

(1)

AXI,=,,= hi(xt!, xi,),

(t 6 J’), (i = 1,. , ml.

Axi,=,, =

(i=

(3)

-x”(t)

= fk

ii(+,

xr, XI), Xi,),

l,...,m),

cx-(1) +dx’_(l)

ax, (0) - bx: (0) = zo,

x(t) = 4(t),

(2)

= ZI.

(4) (5)

(t E Jo).

By an impulsive solution or briefly by a solution to problem (l)-(5) we will understand a function x E Cimp(J, E) which satisfies (2)-(5), has its second order derivative almost everywhere (a.e.) on J’ and which satisfies (1) on J’, possibly except a set of Lebesgue’s measure 0. The paper is organized in the following way. Section 1 provides the construction of an integral analogue to (l)-(5). In Section 2 we give some a priori bounds for solutions to the integral equation, which will be used in Section 3 to obtain a unique impulsive solution to (l)-(5) by means of the Banach contraction principle. The main result will be followed by a couple of examples that show the functional character of the differential problem. 1. PRELIMINARIES

Define p = ac + ad + bc. We shall assume throughout known to make the problem -x”(r)

= 0,

ax(O) - bx’ (0) = zo,

the paper that p # 0, which is well cx(1) + dx’(1) = 21,

(6)

have a unique regular solution. This solution is defined by x(t)

= ;[zo(c+d-ct)+z,(b+at)l,

(t E J).

(7)

The above statement is expressed in the following Lemma. LEMMA 1. If p # 0, the function x : J - E defined by (7) is the only C2 solution to problem (6). For j, 1 = 1,. . . , m we define functions Hi.1 and Hj,I by -(at Hi,,(t)

=

+ b)(c + d) + p

(at + b)(cP+ d) P ’ 0,

tj, t

, whenlrl~jandtE[tj,tj+,),

when j < I I m and c E [ when

& [tj, tj+i).

tj+i ),

(8)

1711

Solutions to IBVP

(at + b)[Q(C + d) - c] + p(1 - f,)

when 1 I I< jandt

E [tj,tj+i),

’ Cur + b)[r/(c + B 1 - cl when j < 1 5 m and t E [tj, P when r & [tj, tj+i). 0, Given any hi, & E E, consider the following auxiliary impulsive differential BVP -x”(f) = 0, AXI ,=,, = hi, AX; ,=,, = JIi (i = 1, . , m), ax(O) -bx’(O) = 0, cx(1) +dx’(l) = 0. It is very easy to observe that the following statement holds true. fij,,(t)

=

tj+l),

(9)

(10) (11) (12)

LEMMA 2. If p # 0, the function x : J - E defined by X(t) = i {Hj,l(t)hl + fi,.,(t)h] /=I for t E [tj, t,+i ), where Hi,/ and Rj,l are defined by (8) and (9) is the only impulsive problem (lOE(12). Next, we consider the BVP -x”(f) = F(t), ax(O) - bx’(0) = 0, cx(1) + dx’(1) = 0.

(13) solution to

(14) (15)

LEMMA 3. If p # 0, the function x : J - E defined by x(t)

=

I

1

0

GO, s)F(s)

where the Green function G E C(J x J, R) is given by (b + at)(c + d - cs) p-l, G(t,s) = (b+as)(c+d-alp-‘, I

(16)

t-h

when 0 5 r 5 s I I, whenOss
(17)

is the only C2 solution to problem (14) and (15). Having done the above preparations, we can represent all impulsive solutions to a nonhomogeneous IBVP in the following form 1

x(t) = ;[zo(c + d - ct) + zl(b + at)] +

for

I0

GO, s)F(s) d.s

/=I E [tj, tj+i ), where G, Hj,/ and FZj,, are given by (17), (8) and (9), respectively. Based on this formula, we can write a fix-point equation for (l)-(5) which looks as follows I x(t) = (7’x)(t) := $[zo(c + d - ct) + zl(b + at)] + GO, s) j-h x,, xi, ds I0 m (19) + C{ffj,l(t) h(&,, Xi,) + fijJ(t) %(&,P Xi,)} t

I=1

for t E [tj, tj+i),

j = 1,. . . , m; we obviously put (TX)(?)

:= 4(t) on Jo.

1712

HENRY K LESZCZY r(lSKI 2. BASIC PROPERTIES OF THE FIX-POINT

LEMMA 4. If x E Ci,r(J,

E), then 7’~ is differentiable

OPERATOR

and we have

cm)

for f E [tj, tj+r ), and almost everywhere on J’ we have -$m where

-u(c + d) H;,,(t)

=

0) := -fk

XI, x:1>

(21)

, when 1 5 1 I j and t E [fl, t,+r 1,

4c ,Pd) when j < 1 5 m and t E [ti, lj+t), P ’ when t & [rj, tj+r). 1 0, U[l/(C + d) - cl , whenIrlIjandtE[tj,tj+r),

&y,(t)

(22)

t [t,,

u[t,(c 45, - c] (23) when j < I I m and E tj+r), P ’ when t 4 [tI, tj+l). I 0, We hope that there will be no confusion if we denote by II e 1) the natural norm in E and any supremum norm in a functional space. Suppose that we are given two real functions XO, XI defined on j such that (Xj)l, = Ild(j)/dt(j) +I]. Define =

(24) Assume that x E G,,,r(.?, E; &Xl). We will estimate II (TX)(t)//, JJd/dr (Tx)(t)/J lld2/dt211 (TX)(Z) Il. First, we define some auxiliary functions and constants. &(r) = (Ial f + I4 1 Ic+ dl + Ipl, Qr) = (Ial f + Ibl 1 [ICI + IdI1 + lPl(I - 0) IPI IPJ & = lal Ic+ dl , p = Ial (ICI + IdI) lpl ’ IPI ’ (JaJ t + 161) (Ic+ dl + ICI s) Ip-*I, when 0 I t I s I 1. X”(z’S)= i (lals+16l)(Ic+d[+~c~t)lp-‘I, whenO
when 0 I t 5 s I 1, whenOIsrt< 1,

llfk

O,O,II 5 i%(t),

llh/(O, OIlI 5 MI,

llJI,(Q OIlI 5 a,

(25) (26) (27) (28)

ZoOI = llzoll (ICI (1 - t) + IdI) + lb1 II (lb1 f I4 t), Zl = 11~011 ICI + IIZIII Ia{ Assumption 1. Suppose that there are integrable functions MO, I+ : J ML, iI?/, I+/, Li, for i = 0, 1 and 1 = 1,. . . , m, such that

and

(29)

R, and constants (30)

Solutions

1713

to IBVP

and for all functions x, z E Cimp(J, E) and all t E J’, I = 1,

, m we have (31’)

Ilf(t. XI, xl) - j-k Zf,2:) II 5 Lo(t) llxt - z, II + LI (2) lb; - 6 II Ilh(x,,, XI,) - ml,, $,I II 5 Lor Ilxr, - zr,II + h 11x1, - 4, IL llJ&,~ XI,) - h(z,,, &II 5 Lo, lb,, - ZI,IILl/ 11x:,- 4, II. 5. Suppose that Assumption

LEMMA

(31”) (31”‘)

1 is satisfied and x E G,,,r(j, E; X0, Xl), then we get the

estimates II (~x)W

II 5 Loft; Xo, XI),

Ildldt(fx)(t)

(32)

II I LI 0; Xo, X 1

for t E J’, where Zo(r) fo(r;Xo, Xl) := lp,l + )dW I

[MO(S)+~ob)IIMo)slI +~1(~)IIW~)sllld~

+ CIYo(t) [M + ~o/ll(xoh,ll +hIlwl),,II /=I

(33)

1

+El’olO)[a +~o,llwo)r,lI +~dIWdt,II I) I ZI := j--$+ o I Kl (t, s) [MO(S)+ Lo(s) IIwo)sll + Ll b) II WI ),I11ds

fl(~;~o,Xl)

+~I~,~M~+~o~II~Xo~r,II

(34)

+hIIwlh,Ill

I=1

+E [a, +Eo,ll(~oh,II +~lrllwdt,Il

I}

and a.e. on J’ we have

s

L2(CXO,Xl)

:= [MO(t) +Lo(r)Il(Xo)ltll

+LI(~)Ilwl)1IIl.

(35)

Take two auxiliary functions B, (v = 0, 1) such that (&)I~ = Ild(Y)/dt(Y) 411. To bear their right sense in our mind, we can think of them as if they were defined by B,(t) = max,Et-T,T] XV for v = 0, 1. We adopt the following assumption. Assumption 2. Suppose that the functions BO and B, satisfy the inequalities

(Bo) la 2 II 4 I(,

(&)I,, 2 Ildldt4ll, and 1 II(Zo)r II BoOI 2 1,~’ + I o K20,

s) [MO(S) + Lo(s) BobI + Ll(s) Bl(s) 1ds

+ xi II(Yo)r II [ M + LO/BOO/)+ LuBl (t/I 1 /=I

+Il@drll [ti,+Lo,Bo(t,)

(36)

+~,,Bl(t,H}

for t E J’, where &(t, s) = s~p~~(-~,r~ &(t + 5, s), and I ZI Bl(t) I jg--+ o K3(t, s) [MO(S) + Lo(s) Bob) + LI (s) Bl(s) 1ds I

+~{~,~~/+~o/Bo~t/~+L~/B~~t/~lll /=I

+iil [ fi, + ~o,Bo(t,) + t,,Bl (t,) I}

(37)

1714

HENRYK

LESZCZYhSKI

for t E J’, where Ks(t, s) = ~trp~~t-~,r~ Kr (t + 5, s). LEMMA

6. If Assumption

2 is satisfied, then there exists a pair (X0, Xi ) which fulfills the inequal-

ities X0(t) 2 fo(t; x0, XI 1 I

Xl 0)

2 L1(1;

x0,

Xl)

7

(38)

Proof It is enough to define decreasing sequences of functions X0.0 = Bo, Xl,0 = BI and Xr,j+l = fr(.;Xo,,,Xi,j) for j = I,..., m. Next, we take Xv = x0.,+1 = Lo(.;Xo,j,Xi,)), limi-, Xv,j, v = 0, 1. Observe that we apply a sort of monotone iterative technique in this proof, which basically comes from [3]. Assumption

3. Suppose that there exist 8 > 0 and

1 > 8 L sup max IEJ

1 IJ 0

&(t,

s) [Lo(s)

+ K-l

LI

> 0 such that

K

(~11

ds

J; K~(~.S)[~LO(~)+LI(J)~~~+~{Y,[KLOI+LI/I+~,[K~~~+~~,~}}~ I=1 (39) Note that Assumption 3, though technical, becomes an effective sufficient condition for the existence of a pair (Bo, Bi) that satisfies the inequalities in Assumption 2. We take in the space %J’, R2), of all bounded measurable functions, the norm II(Bo,BI)II~,)

:= max {

llBoII,

K ilBlIl 1.

(40)

Due to Assumption 3 the right-hand sides of these inequalities are then contractive with the Lipschitz constant 8, and we apply the Banach contraction principle. In practice, we can think of some strengthening and simplifying of Assumption 3 by putting K = 1. 3. THE

MAIN

RESULT-EXISTENCE

THEOREM

As to our main tool, the Banach contraction principle is applied, thus we shall show that a Banach space is mapped into itself and that our fix-point operator 7’ restricted to that space becomes a contraction. 7. Suppose that Assumptions 1 and 2 are satisfied, then there is a pair of functions Xl E Cimp(wf,R+) such that I’ : Cmp(J: E;Xi, Xl) - Cimp(J,E;Xo, Xl).

LEMMA XO,

Proof In view of Lemma 7 there are functions X0, Xi E Ci,,(j, R,) which satisfy inequalities in formula (38). Given a function x E Cimr(J, E; X0, Xi), we apply Lemma 5 and we get (32), which leads to the estimates II(T

This completes the proof.

5 Lo(t;xI,X1)

5 X0(t),

Ildldt (TX)(~) II 5 LI (t; Xo, XI 1 5 Xl (t) H

(41) (42)

Solutions

1715

to IBVP

LEMMA 8. Suppose that Assumptions 1 and 3 are satisfied, then there is a pair of functions x0, XI E Gmp (J, R+) and a constant K > 0 such that the operator f mapping Grnp (j, E; X0, XI ) into C&p (j, E; X0, XI ) is a contraction with respect to the norm

IlXll(~, := SUP { IlX(f)II,

KIIX’tt)ll

td

Proof Remember that Assumption 3 implies Assumption we obtain two functions X0. Xi E Cimr(J, R,) such that

(43)

1.

2. Consequently,

from Lemma 7,

f : CimptX E;Xo, XI) - Cimp(J E;~o, XI). Take any x, z E Cmp(j, E; X0, Xl ), and a constant proof. In view of Assumption 1 we derive II((fz)(t)ll

I

I

> 0 to be specified in a due course of the

K

~~(?,s)[~o(s)l~x,-z~~l+K-‘~~~~)KllX~-z~llldu

+ f IYOW [Lo/ II&, - Zt,11+ K-‘h

(44)

K lb;, - Z:,111

I=1 +6,(t)

[Lo,

II&,

-&,/I

+ K-‘h,

K 11X;, - Z;,IIl}

9

+htdKIIx:--Z:IIlds KII$tfX)(f) - -$Z)tt)ll 5 (’0 ~~tf,~)[K~o(~)IIx,-zZ,ll + i{

YI [KLo/

I=1

We can choose we then obtain

K

II

- (~z)(t)ll

K @X)(t) This completes the proof.

- -$tfd(t)ll

Zr, 11 + LI/ -Zt,II

K 11X;,

-

Z;, 111

(45)

+hh;,--;,IIl},

in Assumption

3. From estimates (44), (45)

5 f3 lb - zll(K)

(46)

s 0 IIX - ZII(K)

(47)

n

We formulate our fundamental C&,(~

-

+%[K~o,IIXt,

> 0 so that it realizes the inequality II

THEOREM

II-%,

existence theorem.

1. Suppose that Assumptions 1 and 2 are satisfied, then there exists a function x E E; X0, Xl ) which is a solution to (l)-(5) unique in Cimp(J, E; X0, Xl 1.

Proof It follows from Lemmas 7 and 8 that there are functions X0, Xi E C&p (.6 R+ ) which satisfy inequalities given in (38) and the integral operator T is a contractive mapping acting on GP (J, E; X0, Xl ) and taking values in ChP (J’, E; X0, Xr ) . Applying the Banach contraction principle, we get a unique fix-point x E Cimp ($ E; X0, Xi ). Due to Lemma 4 this function is twice differentiable almost everywhere on J’, and wherever there exists x”(t), we have also -x”(t) = ftt. xt, xi ). The remaining boundary and impulsive conditions follow immediately from (19) and (20). This finishes the proof. w

1716

HENRYK

LESZCZYI%Kl

4. EXAMPLES

Take as, ui E R and /J E (0, 1). Consider -x”(t)

= aex(t/2)

+ a],

(48)

x(0) = x(l) = 0. (49) A-q,=,, = E, AQ, = 6 t First, we solve to the problem -x”(t) = g(t) with boundary impulsive condition (49). The solution is obviously given by t 1 6(/J - 1)t - Et + g(s)s( 1 - I) d.s + when t E [O, P), gb)(l -s)tds, I0 If x(t) = I (t - l)@ + (1 - t)E + g(s)s(l - t) d.r + g(s)(l - s)td.s, when t E [P, 11. It (50) Example 1. We consider the problem of (48) and (49) with E = 6 = 0, i.e. losing its impulsive character and preserving a functional dependence, namely: a deviation on the right-hand side. Here, we start iterations x0(t), xi(t), . with any continuous function x0(t) such that x0(0) = x0( 1) = 0. The iterative formula can be rewritten as follows 1-s

112

=

aI

-

2

+ 4u,(l - t)

I/Z

xi(s) (1 -2s)ds. (51) r 112 It is easily seen how to weaken the general convergence conditions. For instance, if we consider (51) with the maximum norm, then an adequate sufficient condition looks as follows xi+l(t)

I0

x, (s) s ds + 2tu0

I/Z

I/Z

4luol(l

-t)

o

sd.s+2t

I with some 8 E (0, 1). Compare (52) with (39).

laoI

I 112

(1 -2s)ds

I 8 < 1.

(52)

Example 2. We consider the problem of (48) and (49) with ~12 l/2 and d2 + &2 > 0. This is the simplest impulsive case, because all iterations are divided only into two formulae: the one on the interval [O, P) and the other one on the interval [p, 11. Obviously, the first one at a particular stage of the iteration influences both of them at the next stage. Here, and in our next example, we start iterations with the function 6(/J- 1)t ---Et, when t E [0, P), (53) xo(f) = i (t - 1)~6 + (1 - t)~, when t E [/A, 11. In this particular case, the iterative formula takes the concise form t/2 112 1 - t2 + 4uo(l -f) Xi(S)SdS + 2tUO Xi(S) (1 - 2s) dS, Xi+1 (f) = X0(t) + 01 (54) 2 I t/7. I0 and, once again, condition (52) guarantees convergence with regard to the supremum norm. The limit function satisfies (48) on (0, 1) \ {p}. Example 3. We consider problem (48), (49) with /J < l/2 and S2 + ~~ > 0. If we start our iterations from a function given by (53), then iterative formula (54) defines approximate solutions to the problem of (48) and (49) everywhere except t E {p 2j I j = 0,. , jo ], where Jo2j” < 1 I p2j”+‘. This case differs from the former one in that the solution cannot be classical for there

1717

Solutions to IBVP

are some additional points where x”(t) does not exist, e.g. in the present example we have at least one such point, namely t = p 2’. This illustrates the main difference between our definition of solutions and that in [2]. Example 4. Take F E C((0, 1) x R x R, R), H, l? E C(R2, R), O(O,al, &, fil, &, 61 E C’((0, l), (0, 1)) and p E (0, 1). Consider

-x”(r)

= F(t, x(ao(t)), x’(a,(t))),

(55)

(f # /J),

x(O) = x(1) = 0. (56) AXI,=,, = H(x(Bo(p)), x’U3, (p))), AxI,=~ = J%x(/~~(P)), x’(Bt (p))), However the jumps of the function and its derivaive are well defined, equation (55) cannot be satisfied on flO’(p) u “i’(p). We can imagine the posibility that this set is infinite. Note that IBVP (55) and (56) can be specified from (1) to (5) by writing &,r = Iv}, and f(t, w, w’) = F(t, w(q(t)

for w E X,, w’ E X,’

(57)

for wEX~, w’EXj,. - p)) for w E X,, w’ E XL.

(58)

- t), w’(ocl(t) - I)),

h (w, w’) = H(w(Bobd - p), w’(B, (~1 - ~11,

&(w, w’) = B(w&(p) In particular, j = 0, 1.

- p), w’@,(p)

this mode1 contains the case of no deviations, i.e. oCj(I) = flj(t)

Example 5. Take F E C( (0, 1) x R x R, R), and T > 0, jf E (0,l). differential IBVP

[[-;-$j;;:;;\;

(59)

= Bj(t) = f for

Consider the integro-

(61)

AXI,=,, = k X(t)

=

0,

(t

E

[-T,O]

U

[l

+

T]).

(62)

To see that problem (60)-(62) comes within (l)--(5), we specify the functions f, h, it as follows: f(r,w.w’)=F(f.SI,w(~)dF.II,w’(~)dF) h(w,w’)=H(~~~W(~)ds.~~~h.l(~)dS).

(63) (64)

where (t, w, w’) and (w, w’) belong to the natural domains of adequate functions; the function i is defined in a similar way. REFERENCES I. LAKSHMIKANTHAM V., BAINOV D. D. & SIMEONOV P S., Theory of Impulsive DifferentiulEquntions. World, Singapore (I 989). 2. DAJUN GUO, Existence of solutions of Boundary value problems for nonlinear second order impulsive differential equations in Banach spaces, 1 Math. Anal. Appl. 181, 407-421 (1994). 3. LADDE G. S., LAKSHMIKANT’HAM V. & VATSALA A. S., Monotone Iterative Techniques for Non-linear Dr%ferential Equations. Pitman Advanced Publishing Program, Boston (1985).