Approximating solutions of nonlinear periodic boundary value problems with maxima

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1 Dhage, Coget Mathematics 216, 3: COMPUAIONAL SCIENCE RESEARCH ARICLE Approximatig solutios of oliear periodic boudary value problems with maxima Bapurao C. Dhage 1 * Received: 14 February 216 Accepted: 14 Jue 216 First Published: 3 Jue 216 Correspodig author: Bapurao C. Dhage, Kasubai, Gurukul Coloy, Dist. Latur, Ahmedpur , Maharashtra, Idia bcdhage@gmail.com Reviewig editor: Shaoyog Lai, Southwester Uiversity of Fiace ad Ecoomics, Chia Additioal iformatio is available at the ed of the article Abstract: I this paper we study a periodic boudary value problem of first order oliear differetial equatios with maxima ad discuss the existece ad approximatio of the solutios. he mai result relies o the Dhage iteratio method embodied i a recet hybrid fixed poit theorem of Dhage 214 i a partially ordered ormed liear space. At the ed, we give a example to illustrate the applicability of the abstract results to some cocrete periodic boudary value problems of oliear differetial equatios. Subjects: Advaced Mathematics; Aalysis - Mathematics; Applied Mathematics; Differetial Equatios; Mathematics & Statistics; No-Liear Systems; Operator heory; Sciece Keywor: differetial equatios with maxima; Dhage iteratio method; approximatio of solutios AMS Subject Classificatios: 34A12; 34A45; 47H7; 47H1 1. Itroductio he study of fixed poit theorems for the cotractio mappigs i partially ordered metric spaces is iitiated by Ra ad Reurigs 24 which is further cotiued by Nieto ad Rodriguez-Lopez 25 ad applied to periodic boudary value problems of oliear first order ordiary differetial equatios for provig the existece results uder certai mootoic coditios. Similarly, the study of hybrid fixed poit theorems i a partially ordered metric space is iitiated by Dhage 213, 214, 215a with applicatios to oliear differetial ad itegral equatio uder weaker mixed coditios of oliearities. See Dhage 215b, 215c ad the refereces therei. I this paper we ivestigate the existece of approximate solutios of certai hybrid differetial equatios with maxima usig the Dhage iteratio method embodied i a hybrid fixed poit theorem. We claim that the results of this paper are ew to the theory of oliear differetial equatios with maxima. ABOU HE AUHOR he author of this paper is a oliear aalyst always egaged i the ivestigatio of ew techique or method for studyig the oliear equatios i differet abstract spaces uder differet situatios. He has also a kee iterest i the study of umerical or approximate solutios of oliear differetial ad itegral equatios. He deals with ad claims that the periodic BVPs of first ad secod order ordiary oliear differetial equatios a easily be aalyzed uder miimum atural coditios for umerical or approximate ad other qualitative aspects of the solutios. PUBLIC INERES SAEMEN May of the atural ad physical processes of the uiverse are govered by periodic boudary value problems of oliear differetial equatios. herefore, the results of this paper are useful to scietists ad mathematicias to describe some uiversal pheomea i a scietific maer for better coclusio. Agai the Dhage iteratio priciple which has bee used i this paper is a powerful techique for oliear equatios ad so other researchers of dyamic systems ad applicatios may explore this ew techique to tackle differet oliear problems for qualitative aalysis ad applicatios. 216 he Authors. his ope access article is distributed uder a Creative Commos Attributio CC-BY 4. licese. Page 1 of 12

2 Dhage, Coget Mathematics 216, 3: Give a closed ad bouded iterval J =[, ] of the real lie R for some >, we cosider the followig hybrid differetial equatio i short HDE of first order periodic boudary value problems, x t =f t, xt + g t, max ξ t, x =x, 1.1 for all t J =[, ] ad f, g : J R R are cotiuous fuctios. By a solutio of Equatio 1.1 we mea a fuctio x C 1 J, R that satisfies Equatio 1.1, where C 1 J, R is the space of cotiuously differetiable real-valued fuctios defied o J. Differetial equatios with maxima are ofte met i the applicatios, for istace i the theory of automatic cotrol. Numerous results o existece ad uiqueess, asymptotic stability as well as umerical solutios for such equatios have bee obtaied. o ame a few, we refer the reader to Baiov & Hristova, 211; Otrocol, 214 ad the refereces therei. he PBVP s of oliear first order ordiary differetial equatios have also bee a topic of great iterest sice log time. he HDE 1.1 is a liear perturbatio of first type of the PBVP of first order oliear differetial equatios. he details of differet types of perturbatio appears i Dhage 21. he special cases of the HDE 1.1 are { x t =f t, xt, t J, x =x, ad x t =g t, max xξ, t J, ξ t x =x he HDE 1.2 has already bee discussed i the literature for differet aspects of the solutios usig usual Picard as well as Dhage iteratio method. See Zeidler 1986 ad Dhage ad Dhage 215a. Similarly, the HDE 1.3 has bee studied earlier usig Picard method. See Baiov, ad Hristova 211 ad the refereces therei. Very recetly, Dhage ad Octrocol 216 have iitiated the study of iitial value problems of first order ordiary oliear differetial equatios via ew Dhage iteratio method, however to the best of author s kowledge the HDE 1.3 is ot discussed via Dhage iteratio method. herefore, the HDE 1.1 is ew to the literature i the set up of Dhage iteratio method. I this paper we discuss the HDE 1.1 for existece ad approximatio of the solutios via a ew approach based upo Dhage iteratio method which iclude the existece ad approximatio results for the HDEs 1.2 ad 1.3 as special cases which are agai ew to the theory of differetial equatios. I the followig sectio we give some prelimiaries ad the key tool that will be used for provig the mai result of this paper. 2. Prelimiaries hroughout this paper, uless otherwise metioed, let E,, deote a partially ordered ormed liear space. wo elemets x ad y i E are said to be comparable if either the relatio x or y x hol. A o-empty subset C of E is called a chai or totally ordered if all the elemets of C are comparable. It is kow that E is regular if {x } is a odecreasig resp. oicreasig sequece i E such that x x as, the x x resp. x x for all N. he coditios guarateeig the regularity of E may be foud i Heikkilä ad Lakshmikatham 1994 ad the refereces therei. We eed the followig defiitios see Dhage, 213, 214 ad the refereces therei i what follows. Page 2 of 12

3 Dhage, Coget Mathematics 216, 3: Defiitio 2.1 A mappig : E E is called isotoe or mootoe odecreasig if it preserves the order relatio, that is, if x y implies x y for all x, y E. Similarly, is called mootoe oicreasig if x y implies x y for all x, y E. Fially, is called mootoic or simply mootoe if it is either mootoe odecreasig or mootoe oicreasig o E. Defiitio 2.2 A mappig : E E is called partially cotiuous at a poit a E if for ε > there exists a δ > such that x a <ε wheever x is comparable to a ad x a <δ. called partially cotiuous o E if it is partially cotiuous at every poit of it. It is clear that if is partially cotiuous o E, the it is cotiuous o every chai C cotaied i E. Defiitio 2.3 A o-empty subset S of the partially ordered Baach space E is called partially bouded if every chai C i S is bouded. A operator o a partially ormed liear space E ito itself is called partially bouded if E is a partially bouded subset of E. is called uiformly partially bouded if all chais C i E are bouded by a uique costat. Defiitio 2.4 A o-empty subset S of the partially ordered Baach space E is called partially compact if every chai C i S is a relatively compact subset of E. A mappig : E E is called partially compact if E is a partially relatively compact subset of E. is called uiformly partially compact if is a uiformly partially bouded ad partially compact operator o E. is called partially totally bouded if for ay bouded subset S of E, S is a partially relatively compact subset of E. If is partially cotiuous ad partially totally bouded, the it is called partially completely cotiuous o E. Remark 2.1 Suppose that is a odecreasig operator o E ito itself. he is a partially bouded or partially compact if C is a bouded or relatively compact subset of E for each chai C i E. Defiitio 2.5 he order relatio ad the metric d o a o-empty set E are said to be compatible if {x } is a mootoe sequece, that is, mootoe odecreasig or mootoe oicreasig sequece i E ad if a subsequece {x k } of {x } coverges to x implies that the origial sequece {x } coverges to x. Similarly, give a partially ordered ormed liear space E,,, the order relatio ad the orm are said to be compatible if ad the metric d defied through the orm are compatible. A subset S of E is called Jahavi if the order relatio ad the metric d or the orm are compatible i it. I particular, if S = E, the E is called a Jahavi metric or Jahavi Baach space. Clearly, the set R of real umbers with usual order relatio ad the orm defied by the absolute value fuctio has this property. Similarly, the fiite dimesioal Euclidea space R with usual compoetwise order relatio ad the stadard orm possesses the compatibility property ad so is a Jahavi Baach space. Defiitio 2.6 A upper semi-cotiuous ad mootoe odecreasig fuctio ψ: R + R + is called a -fuctio provided ψ =. A operator : E E is called partially oliear -cotractio if there exists a -fuctio ψ such that x y ψ x y 2.1 for all comparable elemets x, y E, where <ψr < r for r >. I particular, if ψr =kr, k >, is called a partial Lipschitz operator with a Lischitz costat k ad moreover, if < k < 1, is called a partial liear cotractio o E with a cotractio costat k. he Dhage iteratio method embodied i the followig applicable hybrid fixed poit theorem of Dhage 214 i a partially ordered ormed liear space is used as a key tool for our work cotaied i this paper. he details of a Dhage iteratio method is give i Dhage 215b, 215c, Dhage, Dhage, ad Graef 216 ad the refereces therei. Page 3 of 12

4 Dhage, Coget Mathematics 216, 3: heorem 2.1 Dhage, 214 Let E,, be a regular partially ordered complete ormed liear space such that every compact chai C of E is Jahavi. Let, : E E be two odecreasig operators such that a is partially bouded ad partially oliear -cotractio, b is partially cotiuous ad partially compact, ad c there exists a elemet x E such that x x + x or x x + x.he the operator equatio x + x = x has a solutio x i E ad the sequece {x } of successive iteratios defied by x +1 = x + x, =,1,..., coverges mootoically to x. Remark 2.2 he coditio that every compact chai of E is Jahavi hol if every partially compact subset of E possesses the compatibility property with respect to the order relatio ad the orm i it. Remark 2.3 We remark that hypothesis a of heorem 2.1 implies that the operator is partially cotiuous ad cosequetly both the operators ad i the theorem are partially cotiuous o E. he regularity of E i above heorem 2.1 may be replaced with a stroger cotiuity coditio of the operators ad o E which is a result proved i Dhage 213, Mai results I this sectio, we prove a existece ad approximatio result for the HDE 1.1 o a closed ad bouded iterval J =[, ] uder mixed partial Lipschitz ad partial compactess type coditios o the oliearities ivolved i it. We place the HDE 1.1 i the fuctio space CJ, R of cotiuous real-valued fuctios defied o J. We defie a orm ad the order relatio i CJ, R by x = sup xt t J 3.1 ad x y xt yt for all t J. 3.2 Clearly, CJ, R is a Baach space with respect to above supremum orm ad also partially ordered w.r.t. the above partially order relatio. It is kow that the partially ordered Baach space CJ, R is regular ad lattice so that every pair of elemets of E has a lower ad a upper boud i it. he followig useful lemma cocerig the Jahavi subsets of CJ, R follows immediately form the Arzelá-Ascoli theorem for compactess. Lemma 3.1 Let CJ, R,, be a partially ordered Baach space with the orm ad the order relatio defied by 3.1 ad 3.2 respectively. he every partially compact subset of CJ, R is Jahavi. Proof he proof of the lemma is well-kow ad appears i the works of Dhage 215b, 215c, Dhage ad Dhage 214, Dhage et al. 216 ad so we omit the details. he followig useful lemma is obvious ad may be foud i Dhage 28 ad Nieto Lemma 3.2 For ay fuctio σ L 1 J, R, x is a solutio to the differetial equatio x t+λxt =σt, t J, x =x, } 3.3 if ad oly if it is a solutio of the itegral equatio xt = G λ t, s σs 3.4 Page 4 of 12

5 Dhage, Coget Mathematics 216, 3: where, G λ t, s = { e λs λt+λ e λ 1 e λs λt e λ 1, if s t,, if t < s. 3.5 Notice that the Gree s fuctio G λ is cotiuous ad oegative o J J ad therefore, the umber K λ : = max { G λ t, s : t, s [, ]} exists for all λ R +. For the sake of coveiece, we write G λ t, s =Gt, s ad K λ = K. Aother useful result for establishig the mai result is as follows. Lemma 3.3 If there exists a differetiable fuctio u CJ, R such that u t+λut σt, t J, u u, } 3.6 for all t J, where λ R, λ > ad σ L 1 J, R, the ut Gt, s σs, for all t J, where Gt, s is a Gree s fuctio give by Proof Suppose that the fuctio u CJ, R satisfies the iequalities give i 3.6. Multiplyig the first iequality i 3.6 by e λt, e λt ut e λt σt. A direct itegratio of above iequality from to t yiel t e λt ut u+ e λs σs, Now u u, so oe has for all t J. herefore, i particular, e λ u u+ e λs σs ue λ ue λ. 3.1 From 3.9 ad 3.1 it follows that e λ u u+ e λs σs which further yiel e λs u σs. e λ 1 Substitutig 3.12 i 3.8 we obtai ut Gt, sσs, Page 5 of 12

6 Dhage, Coget Mathematics 216, 3: for all t J. his completes the proof. We eed the followig defiitio i what follows. Defiitio 3.1 A fuctio u C 1 J, R is said to be a lower solutio of the Equatio 1.1 if it satisfies u t f t, ut + g t, max ξ t, u u, for all t J. Similarly, a differetiable fuctio v C 1 J, R is called a upper solutio of the HDE 1.1 if the above iequality is satisfied with reverse sig. We cosider the followig set of assumptios i what follows: H 1 here exist costats λ >, μ> with λ μ such that [f t, x+λx] [f t, y+λy] μx y for all t J ad x, y R, x y. Moreover, λ K < 1. H 2 here exists a costat M g > such that gt, x M g, for all t J, x R; H 3 gt, x is odecreasig i x for each t J. H 4 HDE 1.1 has a lower solutio u C 1 J, R. Now we cosider the followig HDE x t+λxt x = f t, xt + g t, max ξ t, = x, 3.13 for all t J =[, ], where f t, x =f t, x+λx, λ>. Remark 3.1 A fuctio u C 1 J, R is a solutio of the HDE 3.13 if ad oly if it is a solutio of the HDE 1.1 defied o J. We also cosider the followig hypothesis i what follows. H 5 here exists a costat M f > such that f t, x M f for all t J ad x R. Lemma 3.4 Suppose that the hypotheses H 2, H 3 ad H 5 hold. he a fuctio x CJ, R is a solutio of the HDE 3.3 if ad oly if it is a solutio of the oliear hybrid itegral equatio i short HIE xt = Gt, s f s, xs + for all t J. Gt, sg s, max xξ, ξ s 3.14 heorem 3.1 Suppose that hypotheses H 1 H 5 hold. he the HDE 1.1 has a solutio x defied o J ad the sequece {x } of successive approximatios defied by Page 6 of 12

7 Dhage, Coget Mathematics 216, 3: x = u, x +1 t = Gt, s f s, x s + Gt, sg s, max x ξ ξ s, for all t J, coverges mootoically to x Proof Set E = CJ, R. he, i view of Lemma 3.1, every compact chai C i E possesses the compatibility property with respect to the orm ad the order relatio so that every compact chai C is Jahavi i E. Defie two operators ad o E by xt = Gt, s f s, xs, t J, ad xt = Gt, sg s, max xξ, t J. ξ s 3.16 From the cotiuity of the itegral, it follows that ad defie the operators, : E E. Applyig Lemma 3.4, the HDE 1.1 is equivalet to the operator equatio xt+ xt =xt, t J Now, we show that the operators ad satisfy all the coditios of heorem 2.1 i a series of followig steps. Step I: ad are odecreasig o E. Let x, y E be such that x y. he by hypothesis H 1, we get xt = Gt, s f s, xs Gt, s f s, ys = yt, for all t J. Next, we show that the operator is also odecreasig o E. Let x, y E be such that x y. he xt yt for all t J. Sice y is cotiuous o [, t], there exists a ξ [a, t] such that yξ =max yξ. a ξ t By defiitio of, oe has xξ yξ. Cosequetly, we obtai max a ξ t xξ =xξ yξ =max a ξ t yξ. Now, usig hypothesis H 3, it ca be show that the operator is also odecreasig o E. Step II: is partially bouded ad partially cotractio o E. Let x E be arbitrary. he by H 5 we have Page 7 of 12

8 Dhage, Coget Mathematics 216, 3: xt Gt, s s, xs f M f M f K, Gt, s for all t J. akig the supremum over t, we obtai xt M f K ad so, is a bouded operator o E. his implies that is partially bouded o E. Let x, y E be such that x y. he by H 1 we have xt yt ] Gt, s [ f s, xs f s, ys Gt, sμ xs ys Gt, sλ xs ys λ Gt, s x y λ K x y, for all t J. akig the supremum over t, we obtai x y L x y for all x, y E with x y, where L = λ K < 1. Hece is a partially cotractio o E ad which also implies that is partially cotiuous o E. Step III: is partially cotiuous o E. Let {x } N be a sequece i a chai C such that x x, for all N. he lim x t =lim Gt, sg s, max x ξ ξ s [ ] = Gt, s lim g s, max x ξ ξ s = Gt, sg s, max x ξ ξ s = xt, for all t J. his shows that x coverges mootooically to x poitwise o J. Now we show that { x } N is a equicotiuous sequece of fuctios i E. Let t 1, t 2 J with t 1 < t 2. We have Bx t x t 2 1 = Gt 2, sg s, max x ξ ξ s Gt 1, sg s, max x ξ ξ s Gt, s Gt, s 2 1 g s, max x ξ ξ s M g Gt 2, s Gt 1, s as t 2 t 1, Page 8 of 12

9 Dhage, Coget Mathematics 216, 3: uiformly for all N. his shows that the covergece x x is uiform ad hece is partially cotiuous o E. Step IV: is partially compact operator o E. Let C be a arbitrary chai i E. We show that C is uiformly bouded ad equicotiuous set i E. First we show that C is uiformly bouded. Let y C be ay elemet. he there is a elemet x C such that y = x. By hypothesis H 2 yt = xt = Gt, sg s, max xξ ξ s Gt, s g s, max xξ ξ s KM g = r, for all t J. akig the supremum over t we obtai y x r, for all y C. Hece C is uiformly bouded subset of E. Next we show that C is a equicotiuous set i E. Let t 1, t 2 J, with t 1 < t 2. he, for ay y C, oe has yt yt 2 1 = xt xt 2 1 = Gt 2, sg s, max xξ ξ s Gt 1, sg s, max xξ ξ s Gt, s Gt, s 2 1 g s, max xξ ξ s M g Gt 2, s Gt 1, s as t 1 t 2 uiformly for all y C. his shows that C is a equicotiuous subset of E. So C is a uiformly bouded ad equicotiuous set of fuctios i E. Hece it is compact i view of Arzelá-Ascoli theorem. Cosequetly : E E is a partially compact operator of E ito itself. Step V: u satisfies the iequality u u + u. By hypothesis H 4 the Equatio 1.1 has a lower solutio u defied o J. he we have u t u f t, ut + g t, max ξ t, t J, u A direct applicatio of lemma 3.3 yiel that ut Gt, s f s, us + Gt, sg s, max uξ, ξ s 3.19 for t J. From defiitios of the operators ad it follows that ut ut+ ut, for all t J. Hece u u + u. hus ad satisfy all the coditios of heorem 2.1 ad we apply it to coclude that the operator equatio x + x = x has a solutio. Cosequetly the itegral equatio ad the Equatio 1.1 has a solutio x defied o J. Furthermore, the sequece {x } = of successive approximatios defied by 3.5 coverges mootoically to x. his completes the proof. Page 9 of 12

10 Dhage, Coget Mathematics 216, 3: Remark 3.2 he coclusio of heorem 3.1 also r s true if we replace the hypothesis H 4 with the followig oe. H 4 he HDE 1.1 has a upper solutio v C1 J, R. Remark 3.3 We ote that if the PBVP 1.1 has a lower solutio u as well as a upper solutio v such that u v, the uder the give coditios of heorem 3.1 it has correspodig solutios x ad x ad these solutios satisfy x x. Hece they are the miimal ad maximal solutios of the PBVP 1.1 i the vector segmet [u, v] of the Baach space E = C 1 J, R, where the vector segmet [u, v] is a set i C 1 J, R defied by [u, v] ={x C 1 J, R u x v}. his is because the order relatio defied by 3.2 is equivalet to the order relatio defied by the order coe ={x CJ, R x θ} which is a closed set i CJ, R. I the followig we illustrate our hypotheses ad the mai abstract result for the validity of coclusio. Example 3.1 We cosider the followig HDE x t =arcta xt xt+tah max ξ t xξ, t J =[, 1], x =x Here f t, x =arcta xt xt ad gt, x =tah x. he fuctios f ad g are cotiuous o J R. Next, we have 1 arcta xt arcta yt x y, ξ for all x, y R, x >ξ>y. herefore λ = 1 > 1 = μ. Hece the fuctio f satisfies the hypothesis ξ 2 +1 H 1. Moreover, the fuctio f t, x =arcta xt is bouded o J R with boud M f = π 2, so that the hypothesis H 5 is satisfied. he fuctio g is bouded o J R by M g = 1, so H 2 hol. he fuctio gt, x is icreasig i x for each t J, so the hypothesis H 3 is satisfied. he HDE 3.2 has a lower 1 solutio ut = 2 Gt, s, t [, 1], where Gt, s is a Gree s fuctio associated with the homogeeous PBVP x t+xt =, t J, x =x1, give by } 3.21 Gt, s = { e s t+1 e 1 e s t e 1, if s t 1,, if t < s 1. Fially, λ K = sup t,s J Gt, s < 1. hus all the hypotheses of heorem 3.1 are satisfied ad hece the HDE 3.11 has a solutio x defied o J ad the sequece {x } = defied by 1 x = 2 Gt, s, 1 x +1 t = Gt, s arcta x s + 1 Gt, s tah max x ξ ξ s Page 1 of 12

11 Dhage, Coget Mathematics 216, 3: for each t J, coverges mootoically to x. Remark 3.4 Fially while cocludig this paper, we metio that the study of this paper may be exteded with appropriate modificatios to the oliear hybrid differetial equatio with maxima, x t =f t, xt, max xξ + g t, xt, max xξ, a ξ t a ξ t x =x, 3.22 for all t J =[a, b], where f, g: J R R R are cotiuous fuctios. Whe g, the differetial Equatio 3.12 reduces to the oliear differetial equatios with maxima, x t =f t, xt, max xξ, t J, a ξ t x =x, 3.23 which is ew ad could be studied for existece ad uiqueess theorem via Picard iteratios uder strog Lipschitz coditio. herefore, the obtaied results for differetial Equatio 3.12 with maxima via Dhage iteratio method will iclude the existece ad approximatio results for the differetial equatio with maxima 3.13 uder weak partial Lipschitz coditio. Ackowledgemets he author is thakful to the referee for poitig out some misprits/correctios for the improvemet of this paper. Fudig he author received o direct fudig for this research. Author details Bapurao C. Dhage 1 bcdhage@gmail.com 1 Kasubai, Gurukul Coloy, Dist. Latur, Ahmedpur , Maharashtra, Idia. Citatio iformatio Cite this article as: Approximatig solutios of oliear periodic boudary value problems with maxima, Bapurao C. Dhage, Coget Mathematics 216, 3: Refereces Baiov, D. D., & Hristova, S Differetial equatios with maxima. New York, NY: Chapma & Hall/CRC Pure ad Applied Mathematics. Dhage, B. C. 28. Periodic boudary value problems of first order Carathéodory ad discotiuous differetial equatios. Noliear Fuctioal Aalysis ad its Applicatios, 13, Dhage, B. C. 21. Quadratic perturbatios of periodic boudary value problems of secod order ordiary differetial equatios. Differetial Equatios & Applicatios, 2, Dhage, B. C Hybrid fixed poit theory i partially ordered ormed liear spaces ad applicatios to fractioal itegral equatios. Differetial Equatios & Applicatios, 5, Dhage, B. C Partially codesig mappigs i ordered ormed liear spaces ad applicatios to fuctioal itegral equatios. amkag Joural of Mathematics, 45, Dhage, B. C. 215a. Noliear D-set-cotractio mappigs i partially ordered ormed liear spaces ad applicatios to fuctioal hybrid itegral equatios. Malaya Joural of Matematik, 3, Dhage, B. C. 215b. Operator theoretic techiques i the theory of oliear hybrid differetial equatios. Noliear Aalysis Forum, 2, Dhage, B. C. 215c. A ew mootoe iteratio priciple i the theory of oliear first order itegro-differetial equatios. Noliear Studies, 22, Dhage, B. C., & Dhage, S. B Approximatig solutios of oliear first order ordiary differetial equatios. GJMS Special Issue for Recet Advaces i Mathematical Scieces ad Applicatios-13, GJMS, 2, Dhage, B. C., & Dhage, S. B. 215a. Approximatig solutios of oliear PBVPs of hybrid differetial equatios via hybrid fixed poit theory. Idia Joural of Mathematics, 57, Dhage, B. C., & Dhage, S. B. 215b. Approximatig positive solutios of PBVPs of oliear first order ordiary quadratic differetial equatios. Applied Mathematics Letters, 46, Dhage, B. C., Dhage, S. B., & Graef, J. R Dhage iteratio method for iitial value problems for oliear first order hybrid itegrodifferetial equatios. he Joural of Fixed Poit heory ad Applicatios, 18, doi:1.17/s Dhage, B. C., & Octrocol, D Dhage iteratio method for approximatig solutios of oliear first order ordiary differetial equatios with maxima. Fixed Poit heory, 172. Heikkilä, S., & Lakshmikatham, V Mootoe iterative techiques for discotiuous oliear differetial equatios. New York, NY: Marcel Dekker. Nieto, J. J Basic theory for oresoace impulsive periodic problems of first order. he Joural of Mathematical Aalysis ad Applicatios, 25, Nieto, J. J., & Rodriguez-Lopez, R. 25. Cotractive mappigs theorems i partially ordered sets ad applicatios to ordiary differetial equatios. Order, 22, Otrocol, D Systems of fuctioal differetial equatios with maxima, of mixed type. Electroic Joural of Qualitative heory of Differetial Equatios, 2145, 1 9. PetruŞel, A., & Rus, I. A. 26. Fixed poit theorems i ordered L-spaces. Proceedigs of the America Mathematical Society, 134, Ra, A. C. M., & Reurigs, M. C. 24. A fixed poit theorem i partially ordered sets ad some applicatios to matrix equatios. Proceedigs of the America Mathematical Society, 132, Zeidler, E Noliear fuctioal aalysis ad its applicatios, Vol. I: Fixed poit theorems. New York, NY: Spriger. Page 11 of 12

12 Dhage, Coget Mathematics 216, 3: he Authors. his ope access article is distributed uder a Creative Commos Attributio CC-BY 4. licese. You are free to: Share copy ad redistribute the material i ay medium or format Adapt remix, trasform, ad build upo the material for ay purpose, eve commercially. he licesor caot revoke these freedoms as log as you follow the licese terms. Uder the followig terms: Attributio You must give appropriate credit, provide a lik to the licese, ad idicate if chages were made. You may do so i ay reasoable maer, but ot i ay way that suggests the licesor edorses you or your use. No additioal restrictios You may ot apply legal terms or techological measures that legally restrict others from doig aythig the licese permits. Coget Mathematics ISSN: is published by Coget OA, part of aylor & Fracis Group. Publishig with Coget OA esures: Immediate, uiversal access to your article o publicatio High visibility ad discoverability via the Coget OA website as well as aylor & Fracis Olie Dowload ad citatio statistics for your article Rapid olie publicatio Iput from, ad dialog with, expert editors ad editorial boar Retetio of full copyright of your article Guarateed legacy preservatio of your article Discouts ad waivers for authors i developig regios Submit your mauscript to a Coget OA joural at Page 12 of 12