Existence theorems for nonlinear problems by continuation methods

Nonlinear

Analysis,

Theory, Methods

PII: SO362-546X(96)00333-1

EXISTENCE

THEOREMS FOR NONLINEAR CONTINUATION METHODS RADU

Department Key words differential

and phrases: equations,

& Applications, Vol. 30. No. 6, pp. 3313-3322, 1997 Proc. 2nd World Congress of Nonlinear Analysts 0 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362-546X.97 $17.00 + 0.00

of Mathematics,

BY

PRECUP

University

Continuation methods, fixed singular differential equations,

PROBLEMS

“BabekBolyai”,

3400

point., essential map, periodic solutions.

Cluj,

coincidence

Romania theorems,

impulsive

1. INTRODUCTION

Continuation methods of Leray-Schauder type represent an important tool in the theory of differential and integral equations. Roughly speaking, by means of a continuation theorem we can obtain a solution of a given equation starting from one of the solutions of a more simpler equation. There are two main approaches to the theory of continua,tion methods. One uses the subtle notion of Leray-Schauder degree for compact perturbations of t.he identity and its extensions to various classes of mappings. We only mention a few names of contributors as follows: J. Leray, J. Schauder, E.H. Rothe, H. Amann, J. Mawhin, F.E. Browder, R.D. Nussbaum and W.V. Petryshyn. The other approach is based upon the fixed point theorem of Schauder and its generalizations, and on the notion of an essential mapping. In this direction, we mention the names of H. Schafer, A. Granas, M. Furi, M. Martelli, A. Vignoli, I. Massabb and W. Krawcewicz. In this report, we adopt the second approach and we describe recent developments both in theory and applications. We first present an abstract continuation principle which makes possible to understand unitary particular continuation theorems for a great variety of single and set-valued mappings in metric, locally convex or Banach spaces. We then present several existence principles of coincidence type which complement the results obtained by K. Geba, A. Granas, T. Kaczynski and W. Krawcewicz [8] and by A. Granas, R.B. Guenther and J.W. Lee [lo]. Using these principles we can give no degree versions to some continuation theorems in the absence of a priori bounds, recently obtained by A. Capietto, J. Mawhin and F. Zanolin [2]. Finally, following the ideas from [2] and [3], we describe two applications to the existence of periodic solutions of impulsive differential equations and of singular superlinear differential equations. 2. AN

Let X mapping Assume least on

ABSTRACT

CONTINUATION

PRINCIPLE

and Y be two sets and A and B two proper subsets of X and Y. respectively. Consider a H : X x [0, l] -+ Y and a set A of functions from X into [0, 11 which are constant on A. the constant functions 0 and 1 belong to A. Also consider a function v which is defined at the following family of subsets of X

{H(.>a(.))-l(B); The nature

of the values of v does not import.

S = {ZEE X;

a E d} u (8). Denote

H(x,X) E B for some X E [O,l])

and HA = H(., X) for each X E [0, 11.

3313

3314

Second World Congress of Nonlinear Analysts

THEOREM2.1 [19]. Suppose that the following

conditions

are satisfied

(i) for each a E A, there exists a* E A such that u”(z) (ii)

the ma,pping

u(z) for z E s 0 for z E A;

=

F = Ho satisfies

v(H(., a(.))-‘(B)) for any a E A with

= dF-‘(B))

# 40)

(2-l)

H(., u(.)) IA= F ;A.

Then there exists at least one z E X \ A solution (2.1) and v(H,-l(B))

to HI(z)

E B. Moreover,

F = HI also satisfies

= v(H,-l(B)).

A mapping F of the form H(.,a(.)), a E A, satisfying (2.1), is said to be essential with respect to (H,d, v). If F is essential with respect to (H,d, v), then F-‘(B) # 0, i.e. the inclusion F(z) E B has at least one solution. Theorem 2.1 says that, under assumption (i), the property of Ho of being essential with respect to (H, d, V) spreads to HI. To make theorem 2.1 useful for applications we need to identify the essentiality property. This can be done by using, for example, methods from fixed point theory or degree theory. The function v is in the first case the simple indicator function having only two values: ~(‘)=

{

lif0fCCX OifC=0

while for the second case, its values are integers obtained The continuation A special (particularly,

principle

in normal

version of theorem a metric space).

topological

2.1 is obtained

COROLLARY2.2. Assume X is a normal the following (i) (ii)

conditions

spaces if we assume that X is a normal

topological

topological

space

space, V is a closed set, 3; c V c X \ A, and

[O,l]) with 8(z) = 0 for z E x and e(z) = 1 for z E V;

with respect to (H,d,r/).

Then HI is also essential

with respect to (A,d, v(H;l(B))

For example,

by means of the degree.

hold

Bu E A for every a E A and B E C(X; Ho is essential

(2.2)

condition

v), and = v(H,-l(B)).

(i) is satisfied for A = {a E C(X;

This is the case of most particular Granas [9] (see also [5]).

continuation

[O,l]);

a 1~ is constant}.

theorems

but not of the following

result due to A.

Second World Congress of Nonlinear Analysts The continuation THEOREM a mapping

principle

for contractions

2.3 [9]. Let U be an open subset of a complete such that

(a)

T(z, X) # z for all 2 E dU and X E [O,l];

(b)

for each E > 0 there is 6 > 0 with d(T(z,

(c) there is (Y E [O, l[ such that d(T(z, Then, if To = T(.,O)

3315

has a (unique)

X),T(y,

and T : u x [0, l] --, 2

2 E for 1 X - p 15 6 and 2 E u;

X)) 2 ad(z, y) for all 2, y E ?? and X E [0, 11.

fixed point in U, Tl = T(., 1) also has.

Notice that, under the assumptions with X = v’, Y = 2 x 2, A = %I, and

A

X),T(s,p))

metric space (Z,d)

of theorem B = ((2,~);

= {u E C(g;

2.3, all the hypothesis of corollary 2.2 are satisfied z E Z}, H(z, X) = (T(z, X),z), Y given by (2.2)

[O, 11); a ]a~ and a 1~ are constant}

where V = w and W is the union of the open balls B(z; T), z E S, with T > 0 sufficiently small that S C W C m C U. In this case, the essentiality of Ho with respect to (H, A, V) is equivalent with the equality {A 6 [O,l]; T(z,X) = 2 for some z E U} = [O,l]. The continuation

principle

in completely

regular

spaces

Suppose X is a completely regular space (particularly, the type of corollary 2.2 is true, if in addition

a locally

convex space). Then,

a result of

V or ;ii is compact. For details

and other consequences

of the abstract

continuation

principle

we send to [17], (181 and

P91* 3. CONTINUATION

THEOREMS

FOR

COINCIDENCES

Throughout the Sections 3-5, X and Y are Banach spaces and L : D(L) c X -+ Y is a linear Fredholm mapping of index zero. We let X = X1 $ X2 and Y = Ye $ Yz, where X1 = kerL and Yz =Im L. We also consider continuous linear projectors P : X -+ X1 and Q : Y -+ Ye and we fix a linear isomorphism J : X1 + Ye. If 2 is a metric space, then a mapping N : 2 + Y is said to be L-compact on 2 provided that (L + JP)-‘N is compact (continuous and with relatively compact range). Also, we say that N is L-completely continuous on 2 if it is L-compact on each bounded subset of 2. These definitions do not depend on the choice of P, Q and J. Finally, we consider a subset K,-, of X, a nonempty open bounded subset U of KO and a nonempty convex set K c Y. We denote by ?? and 1)U the closure and boundary of U with respect to Ko. Let M = {F : i? + K; F is L-compact

and Lx # F(z)

on au}.

We say that the mapping F E M is essential in M provided that for each G E M with G Jau= F (au, there is at least one z E U with Lz = G(z). The following result is known as the topological transversality theorem for coincidences. It is also a simple consequence of theorem 2.1. THEOREM conditions

3.1 [12]. Assume hold

H : v x [0, l] -+ K is L-compact

on u x [O, l] and that the following

3316

Second World1 Congress of Nonlinear Analysts

(a)

Lr # H(z,

X) for ail z E dU and X E [O! l];

(b)

He is essential in M.

Then there exists z E U with As an example PROPOSITION be L-compact

of essential

HI is essential in M too.

LZ = H(z, 1). Moreover, mapping,

we have the following

proposition.

3.2 [21]. Suppose lie is convex and (L + JP)-l(K on n and x0 E U. Assume -

+ JP(g))

C I<,,. Let Fc, : u---f Yr

L,to + Fe(U) c K,

Fe(x) # 0 and (Fe(x), J(x - 20)) 5 0 for x E (20 + Xl) n au. Then

the mapping

More about

Lzo t Fn is essential in hf.

this subject

4. A CONTINUATION

can be found in [7], [8], [lo], [12] and [al]. PRINCIPLE

FOR

FAMILIES

OF MAPPINGS

WITH

DIFFERENT

DOMAINS

Let U c Kn x [O,l] be a nonempty open bounded subset of Ku x [0, l] and H : a + K a mapping. If V is any subset of X x [O,l], we write VA = {Z E X; (x,X) E V}. We also denote HA : ,?i~ --t K, HA(X) = H(x, X). In this section we deal with the family {HA; X E [0, l]} of mappings with the different domains a~. The main idea is to reduce the study of this family to that of a certain family of mappings from the same domain. u into K x [O,l].Thus, we pass from mappings acting between X and Y, to mappings acting between X x R and Y x R. Such an idea has already been used by M. Furi and M.P. Pera [6] and P.M. Fitzpatrick, I. Massabb and J. Pejsachowicz [4]. In addition to Section 3, let C : D(L) x It + Y x R be defined by C(Z~ X) = (Lz, X), and let

M=(F:ii THEOREM

+ K x [0, 11; F i;s L-compact

4.1 [21]. Assume H : u -+ K is L-compact

(a)

Ls # H(z, A) for any (z, X) E 8U;

(b)

Ho :a 3 li x [O,l],

‘&(x,X)

Then there exists x E Ur with essential in M too.

and C(Z, X) # F(x, A) on 824 1 . on u and

essential in M.

= (H(a,X),O),is

Lx = H(r, 1). Moreover,

the mapping

31r(o, X) = (H(x,X),

The next result is concerning with a sufticient condition for that (b) holds, namely homotopic on l/e with a mapping of the form Lx0 + F(x) like that in proposition 3.2. THEOREM

1) is

that HO be

4.2 [21]. Suppose that K. is convex and

(L t JP)-‘(K Let Fo : KO --f Yr be L-completely

continuous

+ JP(Ko)) c Ko. on Ke and 10 E Ue and the following

(4.1) conditions

hold

Second World

3317

Congress of Nonlinear Analysts

Lxo + Fo(Ko) c K, (4.2)

~~(~1 # 0 ad (F~(~), J(X - z~)) I 0 for z E (x0 + xl) n au,. If H : u -+ K is L-compact,

satisfies (a) and

Lx# (1- P)(LXO t Fe(x)) +pH(z,O) for(x,0)EdU,P~10, I[, then there exists z E Us with Lz = H(z, 1). Notice that for X = Y, ri, = K and L = I (the identity), tively, proposition 2 and corollary 1 in [20]. 5. NO DEGREE

VERSIONS

OF SOME

CONTINUATION

THEOREMS

theorems

4.1 and 4.2 become,

respec-

OF CAPIETTO-MAWHIN-ZANOLIN

Let H : lie x [0, I] + K be an L-completely continuous mapping and try to find z E KO such that Le = H(z: 1). For this we could apply theorem 4.2 if we should be able to find a set U c K. x [0, l] with the required properties. Such a method of finding U was given in [2] in the frame of the coincidence degree theory. In this section we present a no degree approach to that method. Suppose Ko is convex, 20 E Ko, FO : Ko + Ye is L-completely continuous and (4.1)-(4.2) hold. In addition, assume that Fo(~) # 0 for x E 20 + X1 with 2 # zo,

(Fo(x), J(x - 20)) I 0 for 2 E 20 t Xl. Denote S = {(x,X)e

K. x [O,l];

Lz = H(x,X)}

S(xo) = {(x,0); x E Ko, Lz = (1 - P)(LZO t Fob(z)) + pH(z,O) for some P E [0, l]}. Also consider

a continuous

functional

CJ : ri,

x

[0, l] -+ R.

5.1 [21]. Assume there are constants c- and c+, c- < c+, such that if we denote c+[), the following conditions are satisfied:

THEOREM

a-‘(]~-, (il)

S n U is bounded;

(i2)

a(S)

(i3)

S(Q)

V =

n {c-, c+} = 0; is bounded

and included

Then there exists z E & with

by V.

Lz = H(z,l).

The proof is based upon theorem 4.2 where U is constructed as a bounded open subset of the level set V. A simple consequence of theorem 5.1 is the following result. Recall that the functional @ is said to be proper on S provided that @-‘([a, b]) is compact for each bounded real interval [a, b]. COROLLARY (il’)

5.2 [21]. Suppose

9 is proper

on S;

Second World

3318

(i2’)

+ is lower bounded cj sf a(S) for ah j;

(i3’)

S(Q)

Congress

of Nonlinear

on S and there is a sequence

Analysts

(cj) of real numbers

with

cI +

co and

is bounded.

Then there exists 2 E Kn such that Lz = H(z,

1).

For X = Y, Kn = K and L = I, the results of this section reduce to theorem from [20]. 6. PERIODIC

SOLUTIONS

FOR

IMPULSIVE

In the next sections we describe two applications the existence of solutions to the following problem

DIFFERENTIAL

1 and corollary

2

EQUATIONS

of corollary 5.2. The first one is concerning with impulses:

u” = f(t,u,u’) a.e. t E [O,l] u(0) = U(l),, u’(0) = a’(1) u($) = (Yk(u(tk)) u’(tt) = Pk(u(tk), u’(tk)), k = 1, . .. . m

with

(6.1)

where the points (T): 0 = tn < tr < ... < t,, < tm+r = 1 are fixed and (hl)

Cyk : R --t R and @k : R2 + R are continuous functions, while f : [O, l] x R2 + R is an L’-Caratheodory function (f( ., U.,ZJ) is Lebesgue measurable for each (u, V) E R2, f(t, ., .) is continuous for a.e. t E [O,l], and for each r > 0, there exists yr E L1[O, l] such that If(t, u, v)l 5 -yr(t) for a.e. t E [0, l] and u2 + v2 5 r”).

We look for solutions

in the following

space

C* = {u : [O, l] -+ R; u and U’ are everywhere continuous except possibly discontinuity of first kind, at which u and u’ are left continuous} endowed

with norm

]] u ]I= sup{(U2(t)

+ ~“(t))l/~;

W,‘J = {u E q; u’ is absolutely continuous u(0) = u(l), U’(0) = U’(l)}, and for a given number

c > 0, consider

t E [0, l]}. Denote on each ]tk, &+I[,

and its inverse L-l

NA : C; --+ L’[O, 11 x R” NA(u)

=

x(f(-,

u,u’)

t

c2u,

{Qk(?‘(tk)))~z~,

A) = 4 Is,l [u’2 -

The next assumptions (h2)

a&t)

xuf(t,

x Rm,

{u’(tjk)}&). is a linear

bounded

mapping.

Also consider

x R”, {Pk(‘lL(tk),‘IL’(tk))}~=CT).

Then the mapping H : C* x [0, l] -+ C$., H(., X) = L-‘Nx, equivalent to the equation u = H(u, 1). Now we consider the functional Q : C+ x [0, l] -+ R, qu,

li = 0, .. .. m, and

L : WZJ -+ L1[O, l] x R”

the linear mapping

Lu = (u” t c2u, {U(t;r.)}pl, It is easy to see that L is invertible

at points (T) of

is completely

u,u’) + (1 - X)c2u2] min{l,

l/( cw

continuous

+

ut2)}dtl

while

(6.1) is

.

are as follow:

= 0 if and only if t = 0 and there is q1> 0 such that tPk(O,t) > 0 for ) t I> ql (1 < k 5 m);

Second World

(h3)

there exist

Congress

of Nonlinear

and q2 > 0 such that

0 5 6 < c7r/(2m)

Analysts

3319

] @k(t,s)/ok(t)

- s/t

]i

d whenever

t # 0, t2 + s2 2 qz, 1 5 k 5 m. Then,

if (u, A) E S (i.e. u = H(u, A)) and c2u2 + u’~ 2 1 on [O,l], by (h2), we have

where n is the number

of zeroes of u in 10, 11. By (h3), we deduce that

Q(u, A) E [n - ms/(m),

n + ms/(cn)]

C ]n - l/2, n + 1/2[

whenever u* + uf2 2 q2on [0, 11, where q = max{ 1,1/c, ql, qz}. Thus (i2’) holds with cj = j +-l/2, jo and ju sufficiently large, provided that the following condition is satisfied: (ha)

for each rr > 0 there is rz 2 ~1 such that if (u, A) E S and inf(u2(t) then ]] u ]I< ~2.

FinaUy, for (il’) (h5)

we need the following

6.1 [20]. If (hl)-(h5)

t E [0, l]} 5 T:,

assumption:

for each n E N there is R, 2 0 such that if (u,X) then inf{u2(t) + u’2(t); t E [O,l]} < R;.

THEOREM

$ uf2(t);

j 2

E S and @(u,X)

E [n-m6/(c?r),n+m6/(c~)]

hold for some c > 0, then there exists at least one solution

u 6 C+

to (6.1). For example, conditions (h4) and (h5) are fulfilled when f(t, U, u) = -c2u + g(t, u, w), where c > 0 and g(t, XL,v)/(u2+ w2)1/2+ 0 as u2 + v* + 00 uniformly a.e. in t E [0, 11, provided that the following nonresonance-like condition is satisfied: C/K $ [n - mS/(m),

n + mh/(cr)]

for ah n E N.

For details, see [PO].See also [ll] for f(t, U,D) superlinear the results in [ll] and [20] reduce to those in [2]. 7. PERIODIC

SOLUTIONS

OF SINGULAR

SUPERLINEAR

We now show how to use corollary 5.2 to establish singular periodic boundary value problem b(pu’)’

= -g(u)

in u and w. In the absence of impulses,

f f(t, u,pu’)

DIFFERENTIAL

the existence

conditions

of solutions

for the following

for a.e. t E [0, l]

40) = U(l), (P')(O) = (Pal). We shall assume that the following

EQUATIONS

(7.1)

hold

(al) p E C[O, l]nC’(]O,

l[), p > 0 on ]O,l[, g E C(R), g(u)/u -+ cc as ] u I---* 00, f : [0, l] xR2 + R is a Carathkodory function and ] f(t,u,v) ]I c(] u ] + ] w I) + k(t) for aII U,V E R and a.e. t E [O,l], where c 2 0;

(a2)

l/p E L’[O, 1] and k E L’[O, 11.

Without

loss of generality,

we suppose that

p 5 1 on [0, l] and ug(u) > 0 for u # 0.

3320

Second World

Congress

of Nonlinear

Analysts

We look for solutions to (7.1) in C[O, l] n Cl(]O, l[), with pu’ E C[O, l] and pu’ differentiable a.e. on [0, 11. Because we do not require p(O)p(l) # 0, the differential equation in (7.1) may be singular at t=Oandt=l. In order to apply corollary 5.2 we let X = CA, where

c:, = {U E C[o,i]n Cl(]o,i[); pi' E C[o,i],~(o) = I, with norm 11u /II= max[o,l] (u*(t)

$ (p~‘)~(t))‘/~,

= (pd)(i))

Y = Co,

co = {u E C[O, 11; U(0) = O} with norm 1 u jo3. We consider

I, : CA --f CO, W)(t)

= (Pu’)(t)

- (PaJ)

and H : C$, x [O,l] ---f CO,

H(u,X)(t)

= s;{-(1

- x)[;g(u) t &]

- &7(u)

t $‘f(s,%P~‘))~s.

Next we define FO : CA --f tR,

Fe(u)(t) = -t J~Mu)Ip+ P’u’/(~ + Ipu’l)Vs, and @ : CA x [0, 11 --+ R, @(u,X)

= $y IJo'{P~'2t(1

&It

- +[$s(4t

-XPzlf'(s,u,PU')}X(2L)(S)dS where

x(u)(t) = min{l,

We have the following THEOREM

existence (al)

I

+ (pu’)2(t))}.

l/(G(t)

7.1 [22]. Assume

~P'1LS(~)

principle

for (7.1).

and (a2). Also suppose

(as)

there is R > 1 such that II u Ill< R for each (u,X)

(a4)

for each n E N there is R, > 1 such that II u Ill< R, for any (u, A) E S satisfying and min[o,l] (u* + (p~‘)~) > 1.

Then

(7.1) has at least one solution.

E S with

min[,,,]

(u’ + (pu’)“) < 1; Q(u, X) = n

Remarks. 1) It is not hard to prove that for each n. E N there is rn 2 1 such that min[o,l] (PU’)~)~/~< r, for every (u, X) E S with @(u, X) = n. 2) Let us state the following (a5)

(~2 +

condition:

for every P > 1 there is R(r) > 1 sucJn that 111~ II*< R(r) f or each (u, X) E S with min[o,l]

(u” +

(pu’)2)1/2 5 I-. By remark 1, (a5) implies both (a3) and (a4). For example, seams that this does not occur for singular equations.

(6)

holds when p E 1 (see [2]), but it

Second World Congress of Nonlinear Analysts A suficient

condition

for (a3)

For x = (XI, ~2) E R2, we denote G(zi)

= Jo”’ g(s)&, fl(h

fi(C

27 A) =

Also, let p(r) = sup{V(z); PROPOSITION (as)

-[Cl

-

X)/P

+

x, A) =

XPl!J(Z*)

-

V(z)

= G(zl)

+

xf/2,

x2/p

(1 -

X)pxzl(l

t

1x21)

+ XPf(&Xl,

x2).

]z] 5 r} for r > 1.

7.2 [22]. Suppose

there exist ti E L’(0,

1; R+) and $ : [0, co[-IO, LJ(x1)h(k

for all 2 E R2,

x7 A) +

x2f2(t!

W[ continuous, x> A) I

such that

q+fqv/(x))

X E [0, l] and a.e. t E [O,l], and m

J 41) Then

3321

dYlti,(Y)

>

(7.2)

I 29 I1 '

(a3) is satisfied.

Remark. In case that Jo” dy/$(y) = co, inequality (7.2) with p( 1) replaced by p(r) is true for each T 2 I. Hence, in this case, even (a5) is satisfied. This happens for p E 1, when +(y) = y + 1 (see

PI). A suficient PROPOSITION (a2*) Then

condition

for

(aA)

7.3 [22]. Suppose

l/p E Lq[O, 11 for some q > 1 and k E Lo3[0, 11. (a4) is satisfied.

This result follows from the following

lemmas.

Let (u, X) E S. Then:

1) Suppose 0 < t2 - tl 5 1 and u2(to) + (pu’)2(to) 5 r2 for some to E [tl, t2]. Also assume that pu’ > 0 (or 2 0) on [tl, t2]. Then there exists r > 1 only depending on r, such that I (p4(t)

II T on [tl,t21.

2) If& > 0 (or I) on some interval [tl, tz], 0 < t2 - tl 5 1, (pu’)(tl) = (pu’)(t2) (or < 0) on ]tl, t2[, then there is r, 2 1 independent of tl, t2, u, and X, such that I (mu’)

II 7, on [h,t21.

3) If ] (pu’)(t) ]I r on [tr,tz], 0 < t2 - tl 5 1, and minltl,tzl r* 2 r only depending on r and r such that maq,,,,,]

= 0 and u > 0

(u” t (P.u’)~)‘/~

For existence results concerning singular boundary solutions, see [l], [15], [16] and their references.

(u2 $ (~u’)~)r/~

5 T, then there is

I r*.

value problems

which admit

a priori

bounds

of

3322

Second

World

Congress

of Nonlinear

Analysts

REFERENCES 1. BOBISUD

L. E. & O’REGAN

resonance,

J. math.

2. CAPIE’ITO

A.,

problems,

Am.

4. FITZPATRICK Bull.

5. FRIGON

spaces,

7. GAINES

Accad.

Vol.

K., GRANAS

9. GRANAS

A., Continuation

10. GRANAS

A., GUENTHER

nonlinear 11.

KIRR

differential

13.

W.,

14.

V.,

MAWHIN Methods Kluwer,

global

multivoques,

Top&

Mat.

and

Paris

300,

nonlinear

67,

31-38

differential

pares

W.,

303-306

for nonlinear

equations

in

(1979). equations,

in Lecture

Notes

in

Some

appl.

Topol. general

Methods

et Cquations Nonlinear

existence

‘70, 153-196

of superlinear

of Cluj

Homotopie

non

lin&ires

dans

les

(1985).

maps,

J. W.,

solutions

Anal.

principles

3, 375-379

in the

(1994).

CarathCodory

theory

of

(1991).

impulsive

differential

systems,

preprint,

Department

of

(1995).

B la thkorie

des

263.

Warsaw

BAINOV

set of solutions

Natur.

PWN,

Qquations

non

kin&&es

dans

les espaces

de Banach,

in

(1988).

D. D. & SIMEONOV

P. S., Theory

of Impulsive

Diflerential

Equations.

theorems

and

Equations

and

of Singular

Boundary

periodic

IncJusio

solutions ns (Edited

of ordinary

differential

by A. GRANAS

and

equations, M. FRIGON),

in

Topological

pp.

291-376.

(1995).

15.

O’REGAN

D.,

16.

O’REGAN

D., Differential

17.

PRECUP

Theory

R., On

the some

18. PRECUP

R., On

19. PRECUP

R., Foundations

on

and

20.

degree

J. Math.

J., Continuation

Geometry

and

(1989).

in Differential Dordrecht

value

of autonomous

continuation

des contractions

connected

SC. Fis.

for contractive

ae, Vol.

Singapore

boundary

perturbations

maps,

pour

T. & KRAWCEWICZ

Sci.

Contribution

LAKSHMIKANTHAM

at

(1977).

University

Mathematic

periodic

for periodic

.I., Complementing

of an unbounded

Berlin

R., Periodic

Dissertationes

to superlinear

theorems

de Leray-Schauder

Rend.

R. B. & LEE

Babes-Bolyai

Scientific,

existence

KACZYNSKI

systems,

KRAWCEWICZ

World

du type

Lincei

method

E. & PRECUP

Mathematics, 12.

the

C. R. Acod.

problems

(1994).

Springer, A.,

de Banach,

value

(1983).

J., Coincidence

568.

boundary

(1992).

9, 79-81

Naz.

singular

approach

F., Continuation 41-72

4, 197-208

R. E. & MAWHIN

Mathematics, espaces

Sot.

M. P., On

Atti

329,

A., Rksultats

Anal.

M. & PERA

F., A continuation

I. & PEJSACHOWICZ

moth.

of nonlinear

(1994).

(1990).

MASSABb

Am.

Nonlinear

Banach

8. GEBA

Sac.

for a class

263-284

J. & ZANOLIN

M. & GRANAS

Methods

347-395

math.

P. M.,

bifurcation,

6. FUR1

88,

MAWHIN

Trans.

solutions

184,

J. & ZANOLIN

Eqm.

A.,

systems,

Applic.

MAWHIN

J. dig.

3. CAPIE’ITO

D., Positive

Anolyaia

equations topological fixed

PRECUP

R., A Granas Topol.

21.

PRECUP

R., Continuation

22.

PRECUP

R., Periodic

Methods

principle,

theorems

of Deimling,

point

September

type

approach

Nonlinear

principles 1993,

Cluj,

to some Anal.

principles solutions

Problems.

Scientific,

math.

Nonlinear

Analysis

Nonlinear

Analysis

Romania,

continuation

5, 385-396

singular

pp.

Carol. 20, 23,

type,

136-140.

theorems

Singapore

(Ink.

of Leray-Schauder

for coincidences,

of superlinear

World

Commentat.

transversality

of the continuation

Topology,

impulses,

Value

at resonance,

and

Univ.

(1994).

36, l-9

673-694

(1995).

(1993).

1315-1320 Proceedings

(1994). of the 23rd

BabesBolyai,

periodic

boundary

Cluj value

(1995). Mathematic ordinary

a (Clt?i) differential

(to appear). equations

(to appear).

Conference (1994).

problems

with