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1 proceedings of the mericn mthemticl society Volume 104, Number 4, December 1988 SYSTEMS OF FUNCTIONAL DIFFERENTIAL EQUATIONS WITH ASYMPTOTICALLY CONSTANT SOLUTIONS WILLIAM F. TRENCH AND TAKASI KUSANO (Communicted by Kenneth R. Meyer) Abstrct. Sufficient conditions re given for nonliner system of differentil equtions with deviting rguments to hve solutions which pproch finite limits s t > oo. No specific ssumptions other thn continuity re imposed on the deviting rguments. The nonlinerities my be superliner, subliner, or singulr in form, or mixture of these. Some of the results re globl. We consider the system of functionl differentil equtions with deviting rguments (!) x'i(t) = fi(t,xi(gu (t)),...,xn(gin(t))), 1 (í)mm0)píj'sgnfoíffyít))], l<i<n, 3 = 1 where j : [, oo) + R re continuous nd 7^ re nonzero rel constnts, 1 < i,j < n. For nottionl convenience, we bbrevite (1) s x'i(t) = fl(t;x), or, in terms of vector nottion, s X'(t) = F(t;X). l<i<n, There hs been considerble recent interest in the study of the oscilltory behvior of solutions of differentil systems of the form (1); see, e.g. [1-7]. To the best of the uthors' knowledge, however, systemtic existence theory for solutions of (1) hs not yet been fully developed. Our purpose here is to provide conditions gurnteeing the existence of solutions of (1) which re symptoticlly constnt t infinity. We mke no ssumptions (other thn continuity) on the deviting rguments {gij(t)}. For convenience in the cse where some or ll of these my be retrded for some vlues of t (perhps even liminft_(00 gij(t) = 00 for some i,j), we mke the following definition. Received by the editors Jnury 4, Mthemtics Subject Clssifiction (1985 Revision). Primry 34K15, 34K25. Key words nd phrses. Functionl differentil equtions, symptoticlly constnt, deviting rguments, subliner, superliner, singulr, globl Americn Mthemticl Society /88 $ $.25 per pge

2 1092 W. F. TRENCH AND TAKASI KUSANO DEFINITION 1. If -oo < to < oo, then ^n(to) is the spce of continuous n-vector functions X = (xi,...,xn) on (-00,00) which re constnt on ( 00,in], with the topology induced by the following definition of convergence: if lim J->oo Xj X s j * 00 sup -oo<t<t \\Xj(t) - X(t)\\ for every T in ( 00,00). (Here is ny convenient vector norm.) Thus, 8^ (in) is Fréchet spce. In wht follows we suppose tht there exist continuous functions /* : [, 00) x R > R+, I < *' <», such tht (3) /<(*,6,...,Ç«) /?(*, 161,..., Çn ), l< <n, for ll (*,&,..., n) [, 00) x Rn. THEOREM l. Suppose tht f*(t, ui,..., un), 1 such tht / oo = 0 (4) J f:(t,pi,...,pn)dt<^-^pi, Let Ci,..., cn be constnts such tht (5) M < j^, 1 n(to) sucn tht (6) l*»(*) - ctl ^ -<-zp, 1 + o _0 < t < 00, 1 t0, nd (8) lim îi(t) = Ci, 1 < i < n. t ^oo PROOF. Let S be the closed convex subset of 8 (io) consisting of functions X such tht Q (9) x (í) - c < Pi, -00 < t < 00, 1 t0, Ci-f fi(s;x)ds, t<t0, Becuse of (4) nd (10), S*~(S) C S. Lebesgue's dominted convergence theorem implies tht if {Yj} is sequence in S nd K- Y s j -* 00, then {S^Yj}

3 FUNCTIONAL DIFFERENTIAL EQUATIONS 1093 converges to STY uniformly on ( 00,00), which is more thn sufficient to show tht S7~ is continuous. Differentiting (11) shows tht "-0! ( U, ts'ni 0 ' with pproprite one-sided interprettions t in- Now the Arzel theorem implies tht Sf(S) hs compct closure. Hence, S?~ stisfies the hypotheses of the Schuder- Tychonoff fixed point theorem on S, nd so there exists n X in S such tht X = S^X. It is esy to see tht X stisfies (6), (7) nd (8). This completes the proof. The following theorem is "locl" ner infinity. THEOREM 2. Suppose tht f*(t,ui,...,un), 1 0. Let,.,cn be given constnts function X in Wn(to) such tht \xi(t) ci\ <, -co < t < 00, 1 0 such tht nd 0 so tht fjtn 'to Ci\ 0. Then there exists Then pply Theorem 1 with pi = = pn p nd 6 = e/(p e). DEFINITION 2. Suppose tht the functions f*(t,ui,...,un) in (3) re ll nondecresing in ui,...,un nd the functions u_1f*(t,u,...,u) re monotonie in u for ech t >. Then we sy tht (1) is purely superliner if (13) lim u~~lf*(t,u,...,u) 0, t >, 1 <» < n, u to+ or purely subliner if (14) lim u'1f*(t,u,...,u)=0, t>, u >oo According to this defintion, (2) is purely superliner if 7 j > 1 for 1 < i, j < n, or purely subliner if 0 < m < 1 for 1 < i,j < n. We now consider globl existence theorems which gurntee the existence of solutions of (1) on given intervl [in, 00). THEOREM 3. Suppose tht (1) is purely superliner nd tht (12) holds for some p > 0. Let to > nd 6 > 0 be given. Then there is number po > 0 such tht if (15) ci -ÍT ' 1^í^'ft'

4 1094 W. F. TRENCH AND TAKASI KUSANO then there is function X in Wn(to) which stisfies (7), (8), nd the inequlities Û \xi(t) - Ci\ < - po, -oo < t < oo, 1 < t < n. 1 + V PROOF. From (13) nd the Lebesgue dominted convergence theorem, there is sufficiently smll constnt po > 0 such tht (16) J f*(t,p0,...,po)dt< p0. Now pply Theorem 1 with pi = = pn = po to obtin the stted conclusion. THEOREM 4. Let (1) be purely subliner nd suppose tht (12) holds for some p > 0. Suppose tht to > nd ci,..., cn re rbitrry constnts. Then there is function X in Wn(to) which stisfies (7) nd (8). PROOF. Let 9 > 0 be rbitrry. Then choose po > 0 so lrge tht (15) nd (16) both hold. (We cn stisfy (16) with lrge po becuse of (14).) Now pply Theorem 1 with pi = = pn = po- DEFINITION 3. Suppose tht in (1) the functions /, : [,oo) x [R\{0}]" R, 1 < i < n, re continuous. Then the system (1) is sid to be purely singulr if there exist continuous functions f*(t,ui,...,un), 1 < i < n, on [,co) x (0,oo)n which re nonincresing in ech u3, 1 < j < n, nd stisfy (3) for ll (t, i,..., n) [,oo)x[r\{0}]". The system (2) is purely singulr if 7^ < 0 for ll 1 < i, j < n. THEOREM 5. Suppose tht (1) is purely singulr nd tht IJ f * (t, Pi,---,pn) dt < 00, l<i<n, for some positive numbers pi,..., pn Let to > be given. Then there is number 6 in (0,1) with the following property: ifci,...,cn re ny constnts such tht (17) \ct\>j^, l<i<n, then there exists function X in Wn(to) which stisfies (7), (8) nd the inequlities \xi(t) - Ci\ < - -pi, -co < t < 00, 1 Pi, -co < t < 00, 1 <i <n. Let S*~ be s in the proof of Theorem 1 (cf. (11)). Becuse of (19) nd the pure singulrity of (1), it is strightforwrd to verify tht if S is s in (18), then f~(s) C ^ if f*(t,pi,...,pn)dt<y jjpi, /OO ß A These inequlities cn be forced to hold by choosing 6 sufficiently close to 1. The rest of the proof is the sme s tht of Theorem 1. When specilized to the system (2), the bove theorems yield the following results.

5 Corollry FUNCTIONAL DIFFERENTIAL EQUATIONS 1095 l. Suppose tht (20) / J \i:j(t)\dt < oo, 1 < i,j < n. (i) Ifiij > 0, 1 < i,j < n, then there exists solution X of (2) which is defined in some neighborhood of oo nd stisfies (8) for ny given constnts ci,...,c. (ii) 7/0 < 7 j; < 1, 1 < i,j < n, then there exists solution X of (2) which is defined on given intervl [to, oo) nd stisfies (8) for ny given constnts C\,...,cn. (iii) If 7,j > 1, 1 < i,j < n, then there exists solution X of (2) which is defined on given intervl [to, oo) nd stisfies (8), provided tht ci,..., c re sufficiently smll. (iv) If iij < 0, 1 < i,j < n, then there exists solution X of (2) which is defined on given intervl [to, oo) nd stisfies (8), provided tht cj,..., cn re sufficiently lrge. It is lso possible to obtin some results if (1) is singulr with respect to some vribles nd nonsingulr with respect to others. We illustrte this for the system (2). THEOREM 6. Suppose tht A nd B re nonempty subsets of {1,2,...,n} such tht A n B = 0 nd A U B = {1,2,..., n}. Let (21) m >0, l<i<n, if je A (i.e., (2) is nonsingulr with respect to x for i e A) nd (22) 7 J <0, 1 <i<n, ifjeb (i.e., (2) is singulr with respect to x for i e B). Suppose tht (20) holds nd tht there re constnts pi (i A) nd to > such tht 3t/. /oo o \ij(t)\dt<pl, iea. Let Ci,...,cn be given constnts. Then there exists function X in ^«(io) which stisfies (7) nd (8), provided tht c is sufficiently smll for i in A, or sufficiently lrge for i in B. PROOF. Choose positive constnts r (j B) nd sufficiently lrge so tht (24) YpT \ij(t)\dt+yr]" [lj(t)[dt<t -Pi, iea. 3 A Jt jeb J*o l + (This is possible becuse of (22) nd (23).) Now pick ß in (0,1) so tht roo / oo /-oo (25) YpT M*) d*+x>7'/ \i0(t)\dt < j^-ñn, ieb. Let j A Jt j B /*0 l P (26) M<7T-> l + *' A, nd

6 1096 W. F. TRENCH AND TAKASI KUSANO Let S be the subset of Wn(to) such tht ct (28) \xi(t) - CÁ <-pi, -oo < t < oo, iea, l + nd (29) \xi(t) -Ci\< ri, -oo < t < oo, ie B. Define the mpping ET : S ^i(ín) by (11) with n fi(t; X) = Y ij{t)[xj(gij(t))[^ sgnfrjignit))], 3 = 1 Since (26) nd (28) imply tht \xi(t)\ < Pi, co < t < co, iea, while (27) nd (29) imply tht \xi(t)\ > fi, -co < t < co, i e B, (21), (22), (24) nd (25) imply tht S^(S) C S. The rest of the proof is like tht of Theorem 1. If the nonsingulr prt of (2) is purely subliner or purely superliner, then it is lwys possible to stisfy (23) with ny í0 > o by choosing pi (i e A) ppropritely. This implies the following globl result. COROLLARY 2. Let A nd B be prtition of {1,2,...,n}, s in Theorem 6, nd suppose tht (20) nd (22) hold. Let to be given, to >. (i) Suppose tht 0 < 7«< 1, 1 1, 1 < 1' < n, if j e A. Then there is function X in fén(to) which stisfies (7) nd (8), provided tht c (i e A) re sufficiently smll nd \ci\ (i e B) re sufficiently lrge. REFERENCES 1. I. Foltynsk nd J. Werbowski, On the oscilltory behvior 0/ solutions o systems o differentil equtions with deviting rguments, Colloq. Mth. Soc. János Bolyi 30: Qulittive Theory 0} Differentil Equtions (M. Frks, ed.), Vol. 1, Szeged, 1981, pp Y. Kitmur nd T. Kusno, Oscilltion nd clss oj nonliner differentil systems with generl deviting rguments, Nonliner Anl. 2 (1978), P. MruSik, Oscilltory properties o solutions 0/ nonliner differentil systems with deviting rguments, Czechoslovk Mth. J. 36 (1986), V. Sed, On nonliner differentil systems with deviting rguments, Czechoslovk Mth. J. 36 (1986), V. N. Shevelo, Oscilltion 0/solutions o differentil equtions with deviting rgument, Nukov Dumk, Kiev, (Russin)

7 FUNCTIONAL DIFFERENTIAL EQUATIONS V. N. Shevelo, N. V. Vrch nd A. G. Gritsi, Oscilltory properties o solutions of systems of differentil equtions with deviting rgument, Preprint 85.10, Institute of Mthemtics, Ukrinin Acdemy of Sciences, Kiev, (Russin) 7. W. F. Trench, Asymptotics o differentil systems with deviting rguments, Proc. Amer. Mth. Soc. 92 (1984), Deprtment of Mthemtics, Trinity University, 715 Stdium Drive, Sn Antonio, Texs Deprtment of Mthemtics, Fculty of Science, Hiroshim University, Hiroshim, Jpn

KRASNOSEL SKII TYPE FIXED POINT THEOREM FOR NONLINEAR EXPANSION

Fixed Point Theory, 13(2012), No. 1, 285-291 http://www.mth.ubbcluj.ro/ nodecj/sfptcj.html KRASNOSEL SKII TYPE FIXED POINT THEOREM FOR NONLINEAR EXPANSION FULI WANG AND FENG WANG School of Mthemtics nd