Existence Theorem for Abstract Measure Delay Integro-Differential Equations

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1 Applied ad Computatioal Mathematics 5; 44: 5-3 Published olie Jue 4, 5 doi:.648/j.acm.544. IN: Prit; IN: Olie istece Theorem for Abstract Measure Delay Itegro-Differetial quatios.. Bellale,. B. Birajdar, D.. Palimkar 3 Mathematics Research Cetre, Dayaad ciece College, Maharashtra, Idia Departmet of Mathematics, Bidve gieerig College, Latur, Maharashtra, Idia 3 Departmet of Mathematics, Vasatrao Naik College, Naded, Maharashtra, Idia mail address: sidhesh.bellale@gmail.com.. Bellale, lsbidave@gmail.com. B. Birajdar, dspalimkar@rediffmail.com D.. Palimkar To cite this article:.. Bellale,. B. Birajdar, D.. Palimkar. istece Theorem for Abstract Measure Delay Itegro-Differetial quatios. Applied ad Computatioal Mathematics. Vol. 4, No. 4, 5, pp doi:.648/j.acm.544. Abstract: I this paper, we have proved the eistece ad uiqueess results for a abstract measure delay itegro-differetial equatio by usig Leray-chauder oliear alterative uder certai Caratheodory coditios. The various aspects of the solutios of the abstract measure itegro-differetial equatios have bee studied i the literature usig the various fied poit techiques such as chauder,s fied poit priciple ad Baach cotractio mappig pricipal etc. I this paper we have proved eistece ad uiqueess coditio for Abstract Measure delay itegro-differetial equatios. Keywords: Time cale, Abstract Measure Itegro-Differetial quatio, Abstract Measure Delay Itegro-Differetial quatio, istece Theorem ad termal olutios. Itroductio The cocept of stability has bee widely used by may scietists uder various model formulatios. Absolute stability was origially formulated by Lur e ad Postikov ad is coected betwee with egieerig ad mathematical cosideratios. From a mathematical poit of view, oe arrives at this cocept from of cotiuity. I our study we shall be cocered with a view of such physical pheomea. Fuctioal itegro-differetial equatios with delay is a hereditary system i which the rate of charge or the derivative of the ukow fuctio or set fuctio depeds upo the past history. The fuctioal itegro differetial equatios of eutral type is a hereditary system i which the derivative of the ukow fuctio is determied by the values of a state variable as well as the derivative of the state variable over some past iterval i the phase space. Although the geeral theory ad the basic result for itegro-differetial equatios have ow bee thoroughly ivestigated, the study of fuctioal itegro-differetial equatios has ot bee complete yet. I recet years, this has bee a icreasig iterest for such equatios amog the mathematicias all over the word. The study of abstract measure itegro-differetial equatios is iitiated ad developed at legth i a series of papers by Dhage [,,3]. The study of abstract measure delay differetial equatios was iitiated by Joshi [9,], hedge ad Joshi [4] ad Bellale [4,5,6]. Usig the approach of above metioed papers, i this paper, we prove the eistece ad stability results for a abstract measure delay itegro-differetial equatios.. Prelimiaries Let R deote the real lie, R a uclidea space with respect to the orm defied by For {,..., } = ma.,..., =, R. Let X be a real Baach space with ay coveiet orm. For ay two poits, y i X, the segmets y i X is defied by {, r } y = X = + r y. Let ad y be two fied poits i X, such that y, where is the ero vector of X. Let be a poit of

2 6.. Bellale et al.: istece Theorem for Abstract Measure Delay Itegro-Differetial quatios X, such that. For this ad y, defie the sets ad as follows. For, y { : } { : } = r < r < r r = <, we write or < > if y y. Let the positive umber y be deoted by. For each, >, let deote that elemet of y which <, =. Note that, ad are ot the same poits uless = ad =. Let M deote the σ -algebra of all subsets of X so that X, M becomes a measurable space. Let ca X, M by p = p X. Where p is a total variatio measure of p ad is give by For all i X p p X = i.3 i with X =, = θ for i j. It i= i i i is kow that ca X, M is a Baach space with respect to the orm defied by.. Let be a σ fiite measure o X ad let p ca X, M. We say p is absolutely cotiuous with respect to the measure if = implies p = for all M p <<.. I this case, we write For a fied X, let M be the smallest σ -algebra o, cotaiig { } ad the sets, y. Let X be such that > ad let M deote the σ -algebra of all sets cotaiig M ad the sets of the form for. We defie the set B H ad C H by BH = { R < H}, { } B = p ca, M q + C < H, H Where H > is give large eough ad C >. Fially let, L, R deote the space of all itegrable oegative real valued fuctios h o with the orm L, defied by h = L h d 3. tatemet of the Problem Let be a σ fiite real measure o X. Give a, p AC' X M with p << cosider the followig abstract measure delay itegro differetiate equatio ivolvig the delay, a. e.{ } o o,,, t t dp f t p k t p d d d = 3. p q M, = dp Where q is a give kow vector measure, is Rado d Nikodym derivative of p with respect to ad the fuctio F : R R R is such that,,, t t t f t p k t p d is itegrable for each p AC, M '. Defiitio 3. :- Give a iitial real measure q o M, a vector p AC', M > is said to be a solutio of delay. if =, p q M p << o Remark 3. :- The delay itegro-differetial equatio 3. is equivalet to the abstract measure delay itegro-differetial equatio. ad, t,, t ;, t { p = f t p k t p d d M = q ; M A solutio p of delay itegro-differetial equatio 3. o will be deoted by p, q We apply the chauder s fied poit theorem foe formulatig the mai eistece result for the delay itegro-differetial equatio 3.. Before statig this result, we recall defiitio. Defiitio 3.. :- A operator Q o a Baach space X ito it self is called compact if for ay bouded subset of of X. Q X is a

3 Applied ad Computatioal Mathematics 5; 44: relatively compact subset of X. Q is called totally bouded if Q is a totally bouded subset of X for each bouded subset of X. If Q is cotiuous ad totally bouded, the it is called completely cotiuous o X. very compact operator is totally bouded, but the coverse may ot be true however both otios coicide o bouded subset of X. Theorem 3. :- Let be a o-empty, closed cove ad bouded subset of the Baach space X ad let Q: be a cotiuous ad compact operator. The the operator equatio Q = has a solutios. Now we shall prove the mai eistece theorem for the delay 3. uder suitable coditios of the fuctio R. 4. istece ad Uiqueess Theorem Defiitio 4.:- A fuctio β : R R R is said to satisfy coditios of Caratheodory or simply it is Caratheodory if β, y, y, R R, is measurable for each ad y, β, y, is cotiuous for almost everywhere o. Defiitio 4.:- A Caratheodory fuctio f is called L Caratheodory if For each give real umber p > there eists a fuctio, hp L R such that,, p..[ ] β y h ae, for all y, R with y p, p. Defiitio 4.3 :- A fuctio : R R β R is called L R Caratheodory if there eists a fuctio h L, R that.,,..[ ] β y h ae for all y, R Cosider the followig set of assumptios. A { } =, A For ay δ >, such the σ -algebra M is compact with respect to topology geerated by the pseudo-metric defied by Tp d =,, M A q is cotiuous o pseudo-metric d defied i B A 3 The fuctio,, M with respect to the f y is L R Caratheodory. A 4 The fuctio f is cotiuous ad there eist fuctios + l, l L R such,,,, f y f y l y y + l for all y, y R Theorem 4.:- uppose the assumptio B B The for a give iitial measure admits a solutios p o, q o Proof:-Let { }, hold. 4 q CH, the delay 3. for some. r r > be a decreasig sequece of real umbers such that r as as > >... > >... roo r r lim = The we have { r } There fore, these eists a real umber r ad a poit = r such that Where m w d < H q hk L m =. This is possible by virtue of A ad the positiveess of. How i the Baach space, subset by { AC M we defie a, = p AC M p = q 4. if M ad p k} Where, the costat k is give by k = q + h m 4.3 k L From 4. ad 4.3 it follows that p < H for all p. Defie a operator T from ito, f t, p t, k t, p t d d,if M, t = q, if M AC M by 4.4 We shall show that the operator T satisfies all the coditios of theorem 3. o. tep I:- We show that + cotiuously maps ito itself. First show that the operator T maps ito itself. Let p t be arbitrary ad let M. The there are sets F M ad G M, G such that = FUG. The

4 8.. Bellale et al.: istece Theorem for Abstract Measure Delay Itegro-Differetial quatios Tp q F + f t, p t, k t, p t d d q + hk t d d q h d q hk m t L L + + G for all M. From the defiitio of the orm i Tp q + h m = K. AC, M, oe has k L This show that T maps ito itself. Now we show that T is p be a sequece of vector measures cotiuous o. Let { } i covergig to vector measure p, that is, lim p p =. The for η >, by hypothesis A 4, these eists a δ > such that Therefore, fore ay M, η m,,, t,,, t p p < δ f p k t p d f p k t p d <,,, t,,, t Tp Tp t t p k t p d F t p k t p d d d t t η d d η < m p p < δ. This shows that T is a cotiuous operator o. tep II:- Net, we show that T is a uiformly bouded Tp ad set i, AC M. Now as i step I, it is proved that T is a subset of ad hece it is uiformly bouded set i AC, M. Now by the defiitio of the map T are has f t, p t, k t, p t d d,if M, = t q, if M Further we show that T is a equi - cotiuous set i Ac, M. Let, M the there are sets F, F M ad G, G M with G, G, ad Fi Gi = φ, i =,. We kow that the set idetities Therefore, we have G = G G G G G = G G G G = q F q F Tp Tp t t t t t + t t, p, k t, p d d G G f t, p, k t, p d d G G t ice f, y, is L - Caratheodory, we have that G G q F q F Tp Tp + f, p, k, p d d r G G q F q F + q F q F + G G r L h d d h d Assume that d, =. The we have ad cosequetly F F ad G G. From the cotiuity of o M it follows that q F q F Tp Tp + G G h d as r L This show that T is a equi-cotiuous set o AC, M, thus T s is uiformly bouded ad equi- cotiuous set i ', topology o, AC M so it is compact i the orm AC M. Now a applicatio of Arela

5 Applied ad Computatioal Mathematics 5; 44: Ascoli Theorem yields that T is a compact subset of AC, M. As a result, T is a cotiuous ad compact operator o esseces, a applicatio of theorem 3. yields that the operator equatio p = Tp has a solutio is. As a result, the delay 3. has a solutio o. This completes the proof. 5. tesio of tability A solutio of the delay 3. so obtaied ca be eteded to the larger segmet, wheever { } =. A eistece result i this coditio is. Theorem 5.. Uder the hypothesis of theorem 4. let p, q be solutio of the delay 4.3. o. The the larger segmet if { } = Proof :- Defiitio 5.:- We cosider the followig assumptios. H The fuctio f : B R is -itegrable ad, f = for all for some >. H Give, δ > there eists a > such that,,,, δ f y f y y + y For all y, y,, R with y ε, y ε, ε, ε Theorem 5.:- Let the assumptios. ε > ad a fied umber H H hold. The for each b,, there eists a uique solutio p, q of the delay 3. satisfyig p ε wheever q bε. b Proof:- Let δ m =. m The correspodig this δ there is a umber ε > such that =, where,,,, f y f y δ y + y 5. For all y, y,, R y ε, ε, ε, y ε With Defie a subset AC, M by ε of Let p ε. { p AC M p ε} ε, = 5. Usig H ad t H we obtai f, p, k t, p d δε 5.3 st Defie a operator T from ε ito, AC M by Tp t, t,, t t p k t p d d d if M, t = q ; if M 5.4 Now if M, the there are two disjoit sets F ad G i M such that Hece for M, from 5. ad 5.3 it follows that t t G = F G, F M, G.,,, Tp q + f t p k t p d d d q + δ d d q + m δ ε bε + b ε = ε b For all p ε, where q bε ad δ = m Therefore Tp ε for all p ε o ε. Let p, p ε. The by G. This shows that T maps e ito itself. Net, we show that T is a cotractio operator H,, t,, t, t,, t Tp Tp t p k t p d d d f t p k t p d d d

6 3.. Bellale et al.: istece Theorem for Abstract Measure Delay Itegro-Differetial quatios,,,,,, t t t t f t p k t p f t p k t p d d d t t, t, + δ p p k t p d k t p t d d d δ p p d d δ m p p b p p 5.5 For all M,. This further implies that. Where, b Tp Tp α p p α = <. This shows that T is a cotractio operator o ε with the cotractio costats α. Therefore, by a applicatio of cotractio mappig priciple, there is a uique solutio p q of the delay 3. satisfyig p ε wheever q bε. This completes the proof. ample 5.:- Let X = R the Lebesgue measure o = [ ] > ad q, [,] R,, the delay AMIGD =. Cosider ad Here dp = 6 d t p d p = q,, = for, we observe that, [, ] p = p = q = If [, ] the we have t p = q + 6p ds dt = + 6 s ds dt s [, ] t t 3 7 = + t dt = + 3 t = I this way the solutio p for the liear delay AMIGD 5.6 ca be foud recursively o [, ]. Ackowledgmets The authors are thakful to the referee for their valuable suggestios. The author.. Bellale is also thakful to UGC, New Delhi MRP-Major, R.N.4-75, for their fiacial support. Refereces [] B. C. Dhage O abstract measure itegro-differetial equatios, J. Math. Phy. ci. 986, [] B. C. Dhage O system of abstract measure itegro differetial iequatios ad applicatios, Bull. Ist. Math. Acad. iica8 989, [3] B. C. Dhage Mied mootoicity theorems for a system of abstract of measure delay itegro differetial equatios, A. tit. Uiv. Al. I. Cua IasiXLII [4] B. C. Dhage ad.. Bellale, Abstract measure itegro differetial equatios, Global Jour Math. Aal. 7, 9 8. [5] B. C. Dhage ad.. Bellale, istece theorem for perturbd abstract measure differetial equatios, Noliear Aalysis, 79,e39-e38 [6].. Bellale, Hybrid Fied Poit Theorem For Abstract Measure Differetial quatio, World Academy Of ciece, gieerig ad Techology, INe-3778 [7] B. C. Dhage, D. N. Chate ad. K. Ntouyas, Abstract measure differetial equatios, Dyamic ystems & Appl [8] J. Dagudji ad A. Graas, Fied poit Theory, Moograhie Math. PNW. Warsaw 98. [9]. R. Joshi, A system of abstract measure delay differetial equatios, J. Math. Phy. ci , []. R. Joshi ad. G. Deo, O abstract measure delay differetial equatios, A. tit. Uiv. Al I. CuaIassiXXVI 98, [] W. Rudi, Real ad Comple, Aalysis, McGraw, Hill Ic.New York, 966. [] R. R. harma, A abstract measure differetial equatio, Proc, Amer, Math. oc [3] R. R. harma, A measure differetial iequality with applicatios, Proc. Amer. Math. oc

7 Applied ad Computatioal Mathematics 5; 44: [4] G. R. hedge ad. R. Joshi, Abstract measure differetial iequalities ad applicatios, Acta Math Hug , [5] D. R. mart, Fied poit Theorems, Cambridge Uive. Press, Cambridge 974.