For the solutions of hyperbolic partial difference equation 

D2/1,2 y(m,n) = a(m,n) y(m,n) 

satisfying some boundary conditions an analytical formula is presented. The solutions are then studied in relation to their oscillatory properties

A. Musielak

J. Popenda

Hrčak - Portal of scientific journals of Croatia

GLASNIK MATEMATICKIVol. 33(53) (1998).209-221ON THE HYPERBOLIC PARTIAL DIFFERENCEEQUATIONS AND THEIR OSCILLATORY PROPERTIESA. Musielak andJ. Popenda,Poznan,PolandAbstract. For thesolutionshyperbolicpartialdifferenceequationD71,2y(m,n) =a(m,n)y(m,n)satisfyingsomeboundaryconditionsananalyticalformulais presented.The solutionsarethenstudiedin relationto theiroscillatoryproperties.In thisnoteweshallstudythelinearpartialdifferenceequationsof theformD/1,2y(m, n) =a(m,n)y(m,n), (m,n) E N5 (E)whereNo ={O,1,... } is thesetof nonnegativeintegers.In Section1a closedanalyticalformulafor thesolutionof (E) satisfyingsomeboundaryconditionsis given. In Section2 we shallstudyexistenceof oscillatorysolutions,by usingthemethodof separationof variables.We shallconsiderrealvaluedsequencesof two independentvariables,thatisthefunctionsy : N6 --+ R.For thesequencey ={y(m,n)}:=o:o we definepartialdifferenceoperatorsof thefirstorder:D/1 y(m,n) =y(m+1,n) - y(m,n), m,n E NoD/2y(m, n) =y(m,n +1) - y(m,n), m,n E No;andof thesecondorderIt is evidenthatD/1,2y(m, n) =D/2,1y(m,n) =y(m+1,n+ 1)-y(m+ 1,n)-y(m, n+ l)+y(m, n).Mathematicssubjectclassification(1991): 39AIO.Keywordsandphrases: hyperbolicpartialdifferenceequation,boundaryvalueproblem,oscillation.210 A. MUSIELAK AND J. POPENDA1. Explicitformof solutionsof theequation(E)In thisSectionwe introducesomesumoperators,similarto thoseusedin [2]'whichallowustopresenteverysolutionof (E) in anexplicitform.For anytwosequencesx : N6 -+ R, W : No -+ R wedefine,L,x(i,})oW(i)= L (IIx(ik,ik))W(i,)iE)l,m m~il>i2...>ir~)l k=1jEv,n n~h>h···>jr~vfor /1, m,v,n E No, 1 ~ r ~ min(m- /1 +1,n - v +1),m ~ /1, n ~ v;L,x(i,}) 0 w(i) =0iE)l,mjEv,nfor r =0, or r >min(m- /1 +1,n - v +1),orm </1 or J1 <v.For exampleL1 x(i,})ow(i)=x(2, 6)w(2)+x(2,7)w(2)+x(3,6)w(3)+x(3,7)w(3)iE2,5jE6,7+x(4,6)w(4)+x(4,7)w(4)+x(5,6)w(5)+x(5,7)w(5),L2x(i,})oW(i)=x(3, 7)x(2,6)w(2)+x(4,7)x(2,6)w(2)+x(5,7)x(2,6)w(2)iE2,5jE6,7+x(5,7)x(3,6)w(3)+x(4,7)x(3,6)w(3)+x(5,7)x(4,6)w(4),L3 x(i,})ow(i)=O.iE2,5jE6,7Thefollowingpropertiesof theoperatorL:,canbeobservediE)l,mjEK,KnL1 x(i,}) 0 w(i) =LX(K,})W(K),iEK,K j=vjEv,nmL1 x(i,}) 0 w(i) =Lx(i, K)W(i),i=)lhenceiEJl,mjEv,n+lmL1 x(i,}) 0 w(i) +LX(s, n +l)w(s) = L1 x(i,}) 0 w(i),S=)liE)l,mjEv,niE)l,m+ljEv,nnL1 x(i,}) 0 w(i) +Lx(m +1,s)w(m+1) = L1 x(i,}) 0 w(i);s=viE)l,mjEv,nfurthermoreON THE HYPERBOLIC PARTIAL DIFFERENCE EQUATIONS AND THEIR OSCILLATORY PROPERTIES 211m i]-1 i'_I-1L,x(i,j)OW(i) = L (Lx(iI,jdLx(iz,h) ... L X(i"j,)W(i,))iE/l,m n~h>h-··>k~vil=/l h=/l i,=/ljEv,nandm n m nL LX(p, q) L'-1 x(i,j) 0 w(i) = L L x(p, q) L'-1 x(i,j) 0 w(i)P=/lq=v iE/l,p-l p=/l+1q=v+l iE/l,p-ljEv,q-l jEv,q-l=L, x(i,j) 0 w(i),iE/l,mjEv,nniE/l,m-ljEv,q-liE/l,m-ljEv,nL, x(i,j) 0 w(i) = L, x(i,j) 0 w(i) + L x(m,q) L'-1 x(i,j) 0 w(i)q=v+,-1iE/l,mjEv,nfor r ~ 2.Similarpropertiesarepossessedby theoperatoriE/l,mjEv,n,L,x(i,j)ow(j)= L (IIx(ik,jk))w(j,).m~il>h...>i"~/lk=1n~h>h···>j,~vFor w(n) == Ion No weshallwriteiEJl,mjEv,niE/l,mjEv,nL x(i,j)ow(j) =L x(i,j)ow(i) =L x(i,j) = L (IT x(ik,A))., , ,m~il >h...>i,~/lk=1n~h>h···>j,~viEJl,mjEv,nFromtheaboveweobtain:(PI)THEOREM 1. Let a : N6 -+ R, and cp,1jI : No -+ R befunctionssuchthatcp(O)=1jI(0).Thenthereexistsa uniquesolutionof theproblem{ D71,zy(m,n) =a(m,n)y(m,n)y(m,0) =cp(m)y(O,n) =1jI(n)for m,n E No. Thissolutioncanbepresentedin theformm-l n-ly(m,n) =cp(m)+L L, a(i,j) 0 cp(i)+1jI(n)+L L, a(i,j) 0 1jI(j),=1 -- ,=1 --iEl,m-l iEO,m-ljEO,n-l jEl,n-lP+ (cp(O)a(O,0) - cp(O))L L, a(i,j) +(cp(O)a(O,0) - cp(O)) (1),=1 --iEl,m-ljEl,n-l212 A. MUSIELAK AND J. POPENDAfor all m,11E N, wherep =minim - 1,11- 1}.Proof Equation(E) canbewrittenin thefollowingequivalentform:y(m + 1,11+ 1)=y(m + 1,11) + y(m, 11+ 1) - y(m, 11)+ a(m, Il)y(m, 11). (2)Assumingthat(1)holdsforallm E {1,2, ... , mI},11E {1,2, ... , 1lI}wecanprove(1) inductivelyfrom (2), first for m =m] + 1 andsuccessively11= 1,2, ... , 11];thenfor 11=III + 1andsuccessivelym=1,2, ... , m], andfinallyform =ml + 1,11=III+1. Theproofis rathertechnicalsotoelucidatewepresentshortpartof it.Let (1) holdsfory(m, 1),thatism-]y(m, 1) =qJ(m) +L Lr a(i,j) 0 qJU) + 1JI(1) + (qJ(O)a(O,0) - qJ(O))r=1 --iEI,m-ljEO,Om-l= qJ(m) +La(i, 0) 0 qJU) + 1JI( 1)+ (qJ(O)a(O,0) - qJ(O)).1=1Hencefory(m +1,1)wegetfrom(2) by initialconditionsy(m+ 1, l)=y(m+ 1,O)+y(m, l)-y(m, O)+a(m, O)y(m, 0)m-l=qJ(m+ 1)+qJ(m)+ LaU, 0) 0 qJ(i)+ 1JI( 1)+(qJ(O)a(O,0)-qJ(O))i=1-qJ(m)+a(m,O)qJ(m)m=qJ(m+ 1)+La(i, 0) 0 qJU)+ 1JI( 1)+(qJ(O)a(O,O)-qJ(O)).i=1Thatis (1) validfory(m + 1,1). Similarly(1) canbeverifiedfory( 1,11).Assumethat(1) holdsfory(m + 1,k), y(m, k + 1),y(m, k), wherem >k. WeON THE HYPERBOLIC PARTIAL DIFFERENCE EQUATIONS AND THEIR OSCILLATORY PROPERTIES 213provethat(1)holdsfory(m +1,k +1). By (2) andinductiveassumptionweobtainy(m+1, k+1)=y(m+1, k)+y(m, k+1)-y(m, k)+a(m, k)y(m, k){ k k-I= cp(m+1)+L L, a(i,j)ocp(i)+If/(k)+ L L, a(i,j)olf/(j)r=1 - ,=1 -iEI,m iEO,mjEO,k-1 jEI,k-1+(cp(O)a(O,O)-cp(O)) ~ L, a(i,j)+(cp(O)a(O, O)-cp(O)) },=1 iEI,mjEI,k-1+{cp(m)+I: L, a(i,j)ocp(i)+If/(k+1)+ tL, a(i,j)olf/(j)r=1 -- r=1 --iEI,m-1 iEO,m-1- -jEO,k jEI,kk+(cp(O)a(O,O)-cp(O)) L L, a(i,j)+(cp(O)a(O, O)-cp(O)) }r=1 --iEI,m-1jEI,k{ k k-I- cp(m)+L L, a(i,j)ocp(i)+If/(k)+ L L, a(i,j)olf/(j),=1 -- r=1 --iEI,m-1 iEO,m-1jEO,k-1 jEI,k-1+(cp(O)a(O,O)-cp(O)) ~ L, a(i,j)+(cp(O)a(O, O)-cp(O)) }r=1 --iEI,m-1jEI,k-1{ k k-I+a(m, k) cp(m)+L L, a(i,j)ocp(i)+If/(k)+ L L, a(i,j)olf/(j),=1 -- ,=1 --iEI,m-1 iEO,m-1jEO,k-1 jEI,k-1+(cp(O)a(O,O)-cp(O)) ~ L, a(i,j)+(cp(O)a(O, O)-cp(O)) }.,=1 --iEI,m-1jEI,k-1Noticethatsuitableupperlimitsof summationshavebeenchangendbecauseof theconditionm >k. Weshallconsiderthesum214 A. MUSIELAK AND 1.POPENDAk m-I kL Lr a(i,j)ocp(i) +L Lr a(i,j)ocp(i) - L Lr a(i,j)ocp(i)r=1 - r=1 -- r=1 --iEI,m iEI,m-1 iEI,m-1jEO,k-1 jEO,k jEO,k-1k+a(l1l, k)cp(l1l) +a(l1l, k) L Lr a(i,j)ocp(i)r=1 --iEI,m-1jEO,k-1m-I k=L Lr a(i,j)ocp(i) +a(l1l, k)cp(l1l) +a(l1l, k) L Lr a(i,j)ocp(i)r=1 -- r=I --iEI,m-1 iEI,m-1jEO,k jEO,k-1+ LI a(i,j)ocp(i) + Lz a(i,j)ocp(i) +... + Lk a(i,j)ocp(i)iEI,mjEO,k-1iEI,mjEO,k-1iEI,mjEO,k-1- '" a(i,j)ocp(i) - '" a(i,j)ocp(i) - ... - '" a(i,j)ocp(i)L.." I L.." z L.." kiEI,m-1jEO,k-1iEI,m-1jEO,k-1iEI,m-1jEO,k-1m-I k=L Lr a(i,j)ocp(i) +a(l1l, k) cp(Ill) +a(m, k) L Lr a(i,j)ocp(i)r=1 --iEI,m-1jEO,kk-I k-I+L a(l1l, s)cp(s) +L a (Ill, s)s=O s=1r=1 --iEl,m-1jEO,k-1'" a(i,j)ocp(i) +...L.." IiEI,m-1jEO,s-1k-I+ L a(m, S) Lk-I a(i,j)ocp(i)s=k-IiEI,m-1jEO,s-Im-I k=L Lr a(i,j)ocp(i) +L a (Ill, s)cp(s)r=1 iEI,m-1 s=ojEO,kiEI,m-1jEO,s-1iEI,m-1jEO,k-1k-I+a(l1l, k) LI a(i,j)ocp(i) +L a(l1l, s) LI a(i,j)ocp(i) +...s=1k-I+a(l1l, k) '" a(i,j)ocp(i) + '" a(l1l, s) '" a(i,j)ocp(i)L.."k-I L.." L.."k-ls=k-IiEI,m-1jEO,k-1iEI,m-1jEO,s-1+a(l1l, k) Lk a(i,j)ocp(i)iEI,m-1jEO,k-1ON THE HYPERBOLIC PARTIAL DIFFERENCE EQUATIONS AND THEIR OSCILLATORY PROPERTIES 215m-I k k=L Lr a(i,j)ocp(i)+L a(l1l,s)cp(s)+L a(111, s) LI a(i,j)ocp(i)r=1 iEI,m-1 s=o s=1 iEI,m-1jEO,k jEO,s- IiEI,m-IjEO,k-IiEl,m-IjEO,s-Ik+... + '" a(l1l,s) '" a(i,j)ocp(i)+a(l1l,k) '" a(i,j)ocp(i)~ ~k-I ~ks=k-IiEI,m-IjEO,kiEl,m-IjEO,kk= L1 a(i,j)ocp(i)+L a(m,s)cp(s)+ L2 a(i,j)ocp(i)s=OiEI,m-IjEO,kiEI,m-IjEO,s-Ik+La(l1l, s) L1 a(i,j)ocp(i) +... + Lk a(i,j)ocp(i)s=IiEI,m-IjEO,s-1k+ L a(l1l,s) Lk-I a(i,j)ocp(i)s=k-I+a(l1l,k) Lk a(i,j)ocp(i)( +Lk+I a(i,j)ocp(i) if 111>k +1)iEI,m-I iEI,m-IjEO,k-I jEO,k=L1a(i,j)oCP(i)+L2 a(i,j)ocp(i)+...iEl,mjEO,kiEI,mjEO,k+'" a(i,j)ocp(i)+'" a(i,j)ocp(i)~k ~k+IiEI,mjEO,kiEI,mjEO,kk+I m=L Lr a(i,j)ocp(i) =L Lr a(i,j)ocp(i).r=I - r=I -~I~ ~I~jEO,k jEO,k216In similarwaywecanproveA. MUSIELAK AND J. POPENDAk-I k k-IL L, a(i,j) 0 l/fV) +L L, a(i,j) 0 l/fV) - L L, a(i,j) 0 l/fV)r=I - ,=1 -- ,=1 --iEO,m iEO,m-1 iEO,m-1jEI,k-1 jEI,k jEI,k-1k-I k+ a(m, k)l/f(k) +a(m, k) L L, a(i,j) 0 l/fV) =L L,a(i,j) 0 l/fV),,=1 -- ,=1 -iEO,m-1 iEO,mjEI,k-1 jEI,kandk-I k(cp(O)a(O,O)-cp(O)) L L, a(i,j)+(cp(O)a(O, O)-cp(O)) L L, a(i,j)r=1 - ,=1 --iEI,m iE1,m-1jEI,k-1 jEI,kk ~I- (cp(O)a(O,O)-cp(O)) L L, a(i,j)+a(m, k) (cp(O)a(O,O)-cp(O)) L L, a(i,j)r=1iEI,m-1 ,=1iEI,m-1jEI,k-1 jEI,k-1k+a(m, k) (cp(O)a(O,O)-cp(O)) =(cp(O)a(O,O)-cp(O)) L L a(i,j).,,=1 -iE1,mjE1,kSinceweconsiderthecasem>k, it sufficesto replaceupperlimitsof summationm(because,forexamplein thesumL L, a( i,j) termswithr >k areequaltozero),r=1 -iEI,mjEI,kand(2) holdsfory(m +1,k +1).In thefollowingresultweshallconsideraproblemwhichwill beusefulin thenextsection.THEOREM 2. Let a : N5 ~ Rand cp,y : No ~ R be functions such thatcp(O)=y(O). Then thereexistsa unique solution of theproblemform,n E No.{ D71,2y(m,n) =a(m, n)y(m, n)y(m,O) =cp(m)y(n, n) =y(n)(P2)ON THE HYPERBOLIC PARTIAL DIFFERENCE EQUATIONS AND THEIR OSCILLATORY PROPERTIES 217Proof It sufficestotakein Theorem1n-l n-llfI(n) =y(n) - cp(n) - L L, a(i,j)ocp(i) - L L, a(i,j)olflU),=1 -- ,=1 --iEl,n-l iEO,n-ljEO,n-l jEl,n-ln-l- (cp(O)a(O,0) - cp(O))L L, a(i,j) - (cp(O)a(O,0) - cp(O)),=1 --iEl,n-ljEl,n-Ifor n;;:::1, lfI(O) =cp(O),andnotethatfrom(1) wegety(n, n) =y(n).A similartheoremcanbeformulatedfor theproblem{ D71,zy(m,n) =a(m, n)y(m, n)y(O, n) =lfI(n)y(m, m) =y(m)for m,n E No.2. Oscillatory properties(P3)Definition. A sequencey ={y(m, n)}:=o':o is saidto be nonoscillatory(inrelationto0) if thereexistpositiveintegers/1, v suchthaty(m, n) >0 (positivesequence) for all m;;:::/1,n ;;:::vory(m, n) <0 (negativesequence) for all m;;:::/1,n ;;:::v.Otherwisethesequencey is calledoscillatory.Howeverthisdefinitioncallseventuallyzerosequenceasoscillatory,weexcludethistypeof sequencesfromourconsiderations.If thesequencey is nonoscillatorythenit is nonoscillatoryfor eachsectionalongeacharbitrarybutfixedm =m,m ;;:::/1 aswell asarbitrarybutfixedn =n,n ;;:::v.Moreover,all thesesectionsareof thesamesign(positiveor negative).A necessaryandsufficientconditionforoscillationcanbeformulatedasfollows.THEOREM 3. A sequencey ={y(m, n)}:=0':0 of real numbers is oscillatoryif and only if there exist increasing to infinity sequencesTn ={mt}~1 and n ={nk}~1such that {y(mk,nk)}~1 is an oscillatory sequence.Example 1. ConsidertheequationD/1,zy(m, n) =y(m, n).218 A. MUSIELAK AND J. POPENDAOneof theoscillatorysolutionsof thisequation,namelysatisfyingy(m,0) =1 for m E No,andy(m,m) =(_I)m for mE No,canbepresentedin theformn m jny(m,n) =(-It+2:{(-lt-k2: 2:k=I jn=n+ljn-l=nFor examplejn-k+22: I} for m ~ n,n E N.jn-k+!=n-k+23 m jS-ky(m,3) =(-l)3+2:{(-1)3-k2:'" 2: l}k=I h=4 j4-k=5-km m h m h h=(_1)3+(-1)22:1+(_1)12:2:1+(-1)°2:2:2: 1.h=4 h=4h=3 h=4h=3h=2Assumingthatt{(-lt-kttk=1 In=n+lln-1=n=t{(_1)n-kt tk=1 In=n+lln-1=n s=m(P5)ntheaboveformularemainstruefor m ::::;n, if furthermorewe takeI: ...=0 fork=1n =0 alsofory(m,0),m E No.FromTheorem2 it followsimmediately:THEOREM4. Let a : N5 ~ R. For anyfunction({J: No ~ R thereexistsafamilyof oscillatorysolutionsof theproblem{ D71,2y(m,n) =a(m,n)y(m,n) (P4)y(m,O) =({J(m)form,n E NoProof To getthedesiredresultwe cantakein theTheorem2 anyarbitraryoscillatorysequencelfI.In factwecanobtainthesameresultasin theTheorem4 for theproblem{ D71,2y(m,n) =a(m,n)y(m,n)y(m,O) =((J(m)y(O, a(n)) =y(m)wherea : No ~ No is strictlyincreasingandsuchthatcard{No\ a(No)} =No.ON THE HYPERBOLIC PARTIAL DIFFERENCE EQUATIONS AND THEIR OSCILLATORY PROPERTIES 219THEOREM 5. Let a : N5 -+R+. and cp,1/1 : No -+R+ befunctions such thatcp(O)=1/1(0). Then thesolution of theproblem (PI) ispositive on N5'In whatfollows we shall studysolutionsof theequation(E) of a particulartype,namelythosewhich canbe presentedas a productof two functionsof oneindependentvariable. We usethemethodof separationof variables(see[1] fordetails)andexaminethesolutionsthusobtained.THEOREM 6. Let a, f3 : No -+R. Then thereexistsolutions of the equationD/1,2y(m, n) =a(m)f3(n)y(m, n), (m, n) E N5 (El)which can be written aswhere c is any real nonzero constant.Proof Supposethat the solutionof (El) can be presentedas the producty(m,n) =u(m)v(n) for all (l1l,n) E N5' ThenD/1,2Y(I1l,n) = (Llu(m)) (Llv(n)) ,whereLlu(l1l)=u(11l+1) - u(I1l). Assumingthatthesuitableoperationsareadmis-sible,wegetLlu(l1l) = f3(n)v(n) (=c).a(l1l)u(l1l) Llv(n)Fromthisweobtaintwofirstorderdifferenceequationsin onevariable{ Llu(l1l)=ca(l1l)u(I1l), 111E Nov n) (l/c)f3(n)v(n), II E No.Thesolutionsof theseequationsarem-Iu(l1l) =u(O) II(1 +ca(i)), 111E Noi=Oandn-Iv(n) =v(O)II(1+f3U)/c), n E No.j=OThereforem-I n-Iu(l1l)v(n) =(u(O) II (1+ca(i)))(v(O) II (1+f3U)/c))i=O j=Om-I n-I=y(O, 0)(II (1+ca(i)))(II (1+f3U)/c))i=O j=Oshouldbesolutionof (El). By directsubstitutionto (El) wecheckvalidityof thisstatement.From the formula (3) we can deduceexistenceof solutionsof (El) whichpossessomeinterestingproperties.For examplein relationtooscillationweget220 A. MUSIELAK AND J. POPENDATHEOREM 7. Let a,f3 : No -7 R. The equation (El) hasan oscillatory solutionin theform given by theformula (3) ifand only if the sequencea or f3possessesasubsequencebounded awayfrom zero.Proof Necessity.Supposefor contraryboth hm a(m) =0 and lim f3(n) =m---+(X) n---+CX)O.Hencefor anyc thereexistm(c) EN andn(c) EN suchthat1+ ca(lIl) >0 forall m ~ m(c) and1+f3(n)/c >0 for alln ~ n(c). So for all m ~ m(c), n ~ n(c)thereissgn{y(o, 0)(1](1+ca(i))) (}] (1+f3U)/c)) }m(e)-l n(e)-l=sgn{y(o,o)( TI (1+ca(i)))( II(1+f3U)/c))}.Sufficiency.Supposethatthereexistsasequence{md~landaconstantE <0-1suchthata(mk) <E for all kEN. Thenfor c >- wehaveE1+ ca(lIlk) <1+ CE <0 for all kEN.Thereforethesequence{1X(1 + ca(i)) }~=1isoscillatory,andhencethesamebe-haviourcharacterisesthesequences{y(O, O)(1X (l+ca(i))) Cu(1+f3U)/c)) }~=lfor anyfixedbutarbitraryv E N. HencethesolutionY(IIl, n) givenby theformula(3) is oscillatory.Similarproofscanbemadein othercasesthatis: a(lIlk) >E >0,f3(nk) <E <0, f3(nk) > E >O.Example 2. Let in (El) a(m) =a (constant),f3(n) =1. Thenthesolutionsof(El) givenby (3) arey(m, n) =y(O, 0)(1 +ca)m(l+l/ct.Let in (El) a(m) = _1_, f3(n) _1_ Then Y(IIl, n) =y(O, 0)mrf(1 +m+1 n +1 i=OC ) n-l ( 1)i+l Xl 1+ cU+l) .Sincemrr-l ( c) mrr-l 1 . 1 1(c+m);=0 l+ i+l = ;=0 i+l (I + 1+ c) =m! r(c)n-l n-Irr(1+ -.1-) =rr _1_U+ 1+ 1)=l r(n+ l/c)j=O cU+l) j=O J+l c n! r(1/c)thenfor c f. -k, kEN,1for c f. k' kEN,(m n) = (0 O)-.L l r(111+ c)r(n + l/c)y, y, m! n! r(c)r(I/c)ON THE HYPERBOLIC PARTIAL DIFFERENCE EQUATIONS AND THEIR OSCILLATORY PROPERTIES 221for (m, n) E N2, e '" 0, e '" -k, e '" -11k,kEN.For e = -k, kEN thereis y(m, n) =0 for m ~ k,similarlyfor e =-11k,kEN wegety(m,n) =0 for n ~ k.Let in (El) a(m) =m, f3(n) =n. Thenthesolutiongivenby (3) for e '" 0,e '" -k,e '" -i,kEN is of theform( ) _ (0 ) m-nr(m - 1+l/e)r(n - 1+c)y m,n -y ,0 e r(l/e)r(e) for m ~ 1, n ~ 1.Furthermorey(m,n) =0 for n ~ k +1in thecasee =-k, kEN andy(m,n) =0for m ~ k +1in thecasee =-1 Ik,kEN.REFERENCES[1J R. P. Agarwal, DifferenceEquationsand Inequalities,TheoryMethodsand Applications,MarcelDekkerInc., New York, 1992.[2J J. Popenda,On thediscreteanalogyofGronwall-Wendroffinequality,Demonstr.Math. 18,4(1985),1083-1103.(ReceivedJune 24, 1997)(RevisedApril2, 1998)Instituteof MathematicsPOVlan Universityof TechnologyPiotrowo 3a60-965POVlanPoland

On the hyperbolic partial difference equations and their oscillatory properties

Abstract

Similar works

Full text

Available Versions

Hrčak - Portal of scientific journals of Croatia