Ioan A. Rus

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208 Radu Precup 6 RI,VUE D'ÂNALYSE NUMÉRIQUE ET DD THÉORIE DE L'APPROXIM.{TIONTome XXVI, NN 1-2,1997, pp. 209-213THEoREM 4.3. Suppose E is a Hilbert space. Let Ai E -> 2E be a hyper-accretíve map satisfying (4.2), (4.3) and (4.6). Then the sequenc. (*,),r,,0 - K,' <l"l< n'1- InA GENERAL FLNCTIONAL INEQUALITYAND ITS APPLICATIONSstronglyconyergestoasolution x eK of 0 e A(x), and r < lrl < n.For other applications of the continuation principles to the theory of nonlinearmaps of monotone type we refer to [6] and [7].IOAN A. RUStREFERENCES 1, INTtrIODUCTIO}"I1. F. E. Browcler, Nonexpansive nonlinear operators in a Bønach space, Proc. Acad. Nat. Sci.usA s4 (1965),1041 7044.2. M. G. Cranclall and A. Pazy, Semigroups of nonlinear contractions and dissipative seß,J. Funct. Anal. 3 (1969),316-418.3. K. Deimling, Nonlínear'Functional Analyis, Springor Vorlag, Berlin, 1985.4. K. Deimling, Positivefixed poínts of weakly inward øøp¡ Nonlinoar Analysis 12 (1988),223-226.5. K. Deimling ancl H. S. Chuan, Fixed poinß of weakly ínward maps in conical shells, NonlinearAnaþsis 12 (1988), 22'7-230.6. R. Procup, Generalized topological transversality and mappings of monotone typq SllttdiaUniv. Baboç-Bolyai (Mathematica) 35, 2 (1990), 44-50.7, R. Precup, Generalized topological transversality and existence theorerns, LibertasMathomatica 11 (1997), 65-19.8. R. l'recup, On somefxecl point theorems' of Deimling, Nonlinear Analysis 23 (1994), 1315-1320.9. R. Procup, On the continuation principlefor nonanpansive maps:, St:udia Univ. Babeç-Bolyai(Mathomatica) 41, 3 (1,996) (in print).In [10] we studiecl the inequalities involving Picard operators, The mainresult in that paper is the following abstract Gronwall theorem:THEonEM L Let (X,¿,<) be aru ordered metric space. Let A',X -+ X besuch th,at:(í) A is monotonícalþ increasing,Receivecl }lay 15, 1996 Facuhy of Mathematics and Computer Science" B a b eS - B o ly ai " Univers' i ty1, M. Kogãlniceanu St.3400 Cltg'-NapocaRomania(ii) A ß a Pí.card operator (O^ = {.r^})Then:(o) x<A(x) impties x3x*n;(b) * > A(x) impties x2x*n.Frorn this abstract theie resülts alargeclass of integral inequalities (Gronwtrll,Bellmnnn, Giuliano,Ifarlamorr, Willet, Wong, Beesack, Wenclorfl etc.) and somo -abstract inequalities (see, for instance, [j] and [10]) follow.The present paper rleals with a now abstract functionai inequality, Somoapplications to tl-re intogral inequalitv, whicb rlo not involvc Picard operators, arc given.2. IIUISC'i'IO¡{AI, IFIE(}T]AI-I'I'IÐSOur main result is thc lbllowing'lFrEORnu 2, LeÍ F:la,b)x C(¡a,b),R*) + I. be afunctional(i) I(.,x) eCl(¡a,bl,R,), fo,'all x eC(la,b),R*);I4/e supposef,hat1 991 AMS Subject Classifìcation: 4'7LI19, 3'7 A402 Functional Inequality 2tl210 Ioarr r\. Rus(ä) there axr.srs g € C(la,ól * &) such. that(r(',x))'(r)1"=", r¡ == 0(lfor all t ela,b'l;(äi) thefunction þ(t,.) is monotonically increasing for all t ela,b);(iv) (r(.,r))'(l) . ö(/,x(t)) for all tela,b], and all monotonicallvincreasing x e C(la,¿1, &);(v) there exists a e R*, such that F(a, x) = a, for aII x e C(lo,bl, n.).Let'.(a) x e C(la,bl, R.) be a solution of the inequality*(t) < F(t,x), for atl t elo,b);(b) y be the maximal solution of thefollowing Cauclty problemy'(t)= ö(r,y(r¡), vt ela,bf,Y(o) = o'Thenx<y.Proof,Lat u(t) = F(t,r), From (iii) we haveu'(t):(r(.,x))'(r) . (r(. ,"))'Q)1"=,r,) < (r(', ,))'U)1"=,r,1.FIenceu'(t) < 0(r, r(r¡¡,u(a) = ç1'By the theorern of differential inequaiities ([3], [9], [11]) it follows thattt. 1!,i.e.,x1v.Remarlc 1. We can lake la,bl or la,+ *1, instead of [a, b], in Theorem 2'RemarkZ,If(r(','))'(t)1,=,r,1 < Þ(t,x(t)), t > n,then the conclusion of Theorom 2 followsRemarlc3.If(r'(., x))'(r) < ö(r, ^1r¡, ,(s(t)), t > ofor the solutions x of x . F(t,x), where g(r) > a, for all t )- ct, an:d for thefo llowing Cauchy problemy'(t) = 0(¿, .y).Y(') = owe have a theorem of differential inequalities, then the conclusion of Theorem 2follows,3. SOME INTEGRAI, INEQUALITIESFrom Theorem2 and Remarks I-,2 and 3 on this theorem, we haveTHBoRBIT¡ 3 (see t6l, t8l). Let a eR*,P and q eC(1a,.+"o['n.). 4x e C(la,+ *[,&) is such that,(t) < o2 + zl"lz(')'(")* a(')nf'þ)]o',for all t ) a, thenüC . [" . fut'lu,],*n fir(,)o'Proof. We haver(t,x)..= d.2 , zl,lt!)r(')+ a(')n1'[]]a',Jx( ),tandThe Cauchy problemhas aunique solution0(¿, y) = zp(t)y(t) + zq\rÇoy' (t) = zp(t)y(t) + zqorÇo, t ) at(a) = oz;(r) = {[" . J"n(,f,]"*n J'r1,¡4,]2212 Ioan A. Rus 4- TuBoREv_ 4 (see [7]). Let u,Þ e À, antl p,q eC(la,+-['n,)' ryx e C(la,* -1, 4) is a solution of the inequalityFunctional Inoquality 273F(r, x)= c( * fr(')'(')a, + fø(s)r(g('))¿'and0(r, y(r¡) = p(t)y(t) + q(t)y(g(t))The proof follows from Remark 3,Remarlc4' Theolem l and rheorem 2 irnprove' unif,i and extend the resultsof Gronwall (1919), Chaplighin(1919),Be1lmam(1943), Giuliano(1946),Harlamov( 1 g 5 5), Willet-Wong ( 1 9 6 5), B eesack (19 69, I97 5),'Wendorff, Li, Bìhari, 1'urinici,Lungu, Lakshrnikantham, Leela, Young, Bainov, ou-lang, Defermos, Pachpatte,and of many others (see [2], [3], U0l, [12])'REFERENCE S5,"(/) <Proof. 'We haveF(t, x) =+ fz(,)'(,þ']þ + J'ø(,),(,)0,1,þfor all t ela,+ *1, thenx3y,wherey is a unique solution of theþllowing Cauchy problemy' (t) = I'q (,)+ pp(r )] /r ) *loØ f,n.),rs + q(r ) f p1 "¡c"lr' ( r )Y(o) = o'0(r,y(r¡)= (r( ,x))'(r)1"=rr,l = f.',q(r)+ pp(¡)]y(¿)+ [r(,)1,ø(,)0 , * qùf,pþto,]r'(r)It is easy to observe that F and þ satisff the conditions of Theorem 2. ApplyrngTheorem 2with the above F and 0, we obtain the conclusion of Theorem 4'THEoREM 5 (see lI)). Let ct,cx e R.,P,Q eC(la,+ *[,&) andg ee C(la,+ *[, [4, + *l). Let x e C(la,+ ""[, &) be a solution of the inequalilt'(¿) < o * l,nß),(s)ds + fø(f"(s(,))a,(If g ís monolonically i.ncreasíng and g(t) < t, for o\t t =la, + *l-, therr1. S. M. Aziz ancl A. FI. Nasr, Asymptotic behøviour and oscillations of second order dffirentialequations vlith deviating argument, J. Ma . Anal. Appl. 197 (1996),448458.2. V. iakshmikantharn ancl S. léoh, Dffirentiøl and Integral Inequalíties, Acaclomic Pross,New York, 1969.3. A. A. Marlynyuk, V. Lakshmikantham and S. Leela, Motion Støbility: The Method of InÍegralIneclualities, Kiev, 1989.4. J. Norbnry ancl A. M. Shtart, Volterrø integral ecluatíon and o new Gronwall ineclualíly, ProcRoyal Soc. Eclinburglr 106 Ä (1 987)' 361-313'5. B. G. pachpatte, On ã certain inequality arising in the theory of differential ecluations,J. Math. A,nalysis and Appl. 182 (1994), 143-t5'l '6. B. On,tome-iew inequalitíes related to certain ìnequalilies in the tlteory oJuations, J. Math. Anal. Appl. 189 (1995), 128-144'j. B. 01 a'new inequalie suggested by the s'tudy of certain epidemic tnodels,J. Math. A,nal. Appl. 19s (1995), 638-644.g. p. \/. IrI. pang ancl^R. P. Agarwail On an integral ineEtality and its cliscretu analogue,J. Math. Anal. Appl. 194 (1995), 569-51'7,9. L. C. piccinini, G. -Sìarnpacchia alid G. Vidossich, Ordinar.y Differenr'ial Lquations in R",Springer Vorlag, Berlin, 1984.t0. I. A. -Rus,"¡icarct Operaørc anct Applicøtions, Serninar on Fixecl Point Theory, Prcprint No. .3, 19961it. I. a. ¡¡tts, Ecua¡ii ctiferenliale, Lcualíi integrale Si sisteme dinarnice, "Transilvania" Press,Cluj-Napoca, 1996.12. p. Vei Eecka, tlpptications tlu calcul r)ilférenllel, Presses Universitaires do !'ranco, 1985.c(+ Jn('),("r'l[o Jø(,)'(,)4,]+and. x3y,wlterey is a unique solution of the"following Cauclty probletny' (t) = p(t)y(t) + q(t)y(g(t)),,' oY(o) = o'Proof. 'We haveReceivecl May 15, 1996 Faculty of Mathematics and Conrputer Sciettce" B ab e,1 -B olYa i " Universi lY1, M. Kogãlniceanu SL3400 CtLtj-NaPoca' llomania

A general functional inequality and its applications

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A general functional inequality and its applications

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