Approximating positive solutions of nonlinear first order ordinary quadratic differential equations

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1 Dhage & Dhage, Cogen Mahemaics (25, 2: 2367 hp://dx.doi.org/.8/ APPLIED & INTERDISCIPLINARY MATHEMATICS RESEARCH ARTICLE Approximaing posiive soluions of nonlinear firs order ordinary quadraic differenial equaions Bapurao C. Dhage * and Shyam B. Dhage Received: 4 November 24 Acceped: 9 February 25 Published: 2 March 25 *Corresponding auhor: Bapurao C. Dhage, Kasubai, Gurukul Colony, Ahmedpur, Maharashra, India bcdhage@gmail.com Reviewing edior: Kok Lay Teo, Curin Universiy, Ausralia Addiional informaion is available a he end of he aricle Absrac: In his paper, he auhors prove he exisence as well as approximaions of he posiive soluions for an iniial value problem of firs-order ordinary nonlinear quadraic differenial equaions. An algorihm for he soluions is developed and i is shown ha he sequence of successive approximaions converges monoonically o he posiive soluion of relaed quadraic differenial equaions under some suiable mixed hybrid condiions. We base our resuls on he Dhage ieraion mehod embodied in a recen hybrid fixed-poin heorem of Dhage (24 in parially ordered normed linear spaces. An example is also provided o illusrae he absrac heory developed in he paper. Subjecs: Advanced Mahemaics; Analysis - Mahemaics; Differenial Equaions; Mahemaics & Saisics; Operaor Theory; Science Keywords: quadraic differenial equaion; iniial value problem; Dhage ieraion mehod; approximae posiive soluion AMS subjec classificaions: 34A2; 34A38. Inroducion Given a closed and bounded inerval J =[, a], of he real line R for some, a R wih, a >, consider he iniial value problem (in shor IVP of firs-order ordinary nonlinear quadraic differenial equaion, (in shor HDE d x( d f (,x( λ x( = g(, x(, J, f (,x( x( =x R, } (. ABOUT THE AUTHORS The key research projec of he auhors of he paper is o prove exisence and find he algorihms for differen nonlinear equaions ha arise in mahemaical analysis and allied areas of mahemaics via newly developed Dhage ieraion mehod. The quadraic differenial equaions form an imporan class in he heory of differenial equaions. In he presen paper, i is shown ha he new mehod is also applicable o such ype of nonlinear quadraic differenial equaions for proving he exisence as well as approximaions of he soluions under mixed monoonic and geomeric condiions. PUBLIC INTEREST STATEMENT I is known ha many of he naural, physical, biological, and social processes or phenomena are governed by mahemaical models of nonlinear differenial equaions. So if a person is engaged in he sudy of such complex universal phenomena and no having he knowledge of sophisicaed nonlinear analysis of his paper, hen one may convinced he use of he resuls of his paper, in paricular when one comes across a cerain dynamic process which is based on a mahemaical model of quadraic differenial equaions. In such siuaions, he applicaion of he resuls of he presen paper yields numerical concree soluions under some suiable naural condiions hereby which i is possible o improve he siuaion for beer desired goals. 25 The Auhor(s. This open access aricle is disribued under a Creaive Commons Aribuion (CC-BY 4. license. Page of

2 Dhage & Dhage, Cogen Mahemaics (25, 2: 2367 hp://dx.doi.org/.8/ for λ R, λ >, where f : J R R {} and g : J R R are coninuous funcions. By a soluion of he QDE (., we mean a funcion x C (J, R ha saisfies (i x is a coninuously differeniable funcion for each x R, and f (, x (ii x saisfies he equaions in (. on J,where C(J, R is he space of coninuously differeniable real-valued funcions defined on J. The QDE (. wih λ = is well known in he lieraure and is a hybrid differenial equaion wih a quadraic perurbaion of second ype. Such differenial equaions can be ackled wih he use of hybrid fixed-poin heory (cf. Dhage 999; 23; 24a. The special cases of QDE (. have been discussed a lengh for exisence as well as oher aspecs of he soluions under some srong Lipschiz and compacness-ype condiions which do no yield any algorihm o deermine he numerical soluions. See Dhage and Regan (2, Dhage and Lakshmikanham (2 and he references herein. Very recenly, he sudy of approximaion of he soluions for he hybrid differenial equaions is iniiaed in Dhage, Dhage, and Nouyas (24 via hybrid fixed-poin heory. Therefore, i is of ineres and new o discuss he approximaions of soluions for he QDE (. along he similar lines. This is he main moivaion of he presen paper and i is proved ha he exisence of he soluions may be proved via an algorihm based on successive approximaions under weaker parial coninuiy and parial compacness-ype condiions. 2. Auxiliary resuls Unless oherwise menioned, hroughou his paper ha follows, le E denoes a parially ordered real-normed linear space wih an order relaion and he norm. I is known ha E is regular if {x n is a nondecreasing (resp. nonincreasing sequence in E such ha x n x as n, hen x n x (resp. x n x for all n N. Clearly, he parially ordered Banach space, C(J, R is regular and he condiions guaraneeing he regulariy of any parially ordered normed linear space E may be found in Nieo and Lopez (25 and Heikkilä and Lakshmikanham (994 and he references herein. We need he following definiions in he sequel. Definiion 2. A mapping : E E is called isoone or nondecreasing if i preserves he order relaion, ha is if x y implies x y for all x, y E. Definiion 2.2 (Dhage, 2 A mapping : E E is called parially coninuous a a poin a E if for ε> here exiss a δ > such ha x a <ε whenever x is comparable o a and x a <δ. called parially coninuous on E if i is parially coninuous a every poin of i. I is clear ha if is parially coninuous on E, hen i is coninuous on every chain C conained in E. Definiion 2.3 A mapping : E E is called parially bounded if T(C is bounded for every chain C in E. is called uniformly parially bounded if all chains (C in E are bounded by a unique consan. is called bounded if T(E is a bounded subse of E. Definiion 2.4 A mapping : E E is called parially compac if (C is a relaively compac subse of E for all oally ordered ses or chains C in E. is called uniformly parially compac if (C is a uniformly parially bounded and parially compac on E. is called parially oally bounded if for any oally ordered and bounded subse C of E, (C is a relaively compac subse of E. If is parially coninuous and parially oally bounded, hen i is called parially compleely coninuous on E. Definiion 2.5 (Dhage, 29 The order relaion and he meric d on a nonempy se E are said o be compaible if {x n is a monoone, ha is, monoone nondecreasing or monoone nonincreasing sequence in E and if a subsequence {x nk of {x n converges o x implies ha he whole Page 2 of

3 Dhage & Dhage, Cogen Mahemaics (25, 2: 2367 hp://dx.doi.org/.8/ sequence {x n converges o x. Similarly, given a parially ordered normed linear space (E,,, he order relaion and he norm are said o be compaible if and he meric d defined hrough he norm are compaible. Clearly, he se R of real numbers wih usual order relaion and he norm defined by he absolue value funcion has his propery. Similarly, he finie-dimensional Euclidean space R n wih usual componenwise order relaion and he sandard norm possesses he compaibiliy propery. Definiion 2.6 (Dhage, 2 An upper semi-coninuous and nondecreasing funcion ψ : R R is called a -funcion, provided ψ( =. Le (E,, be a parially ordered normed linear space. A mapping : E E is called parially nonlinear -Lipschiz if here exiss a -funcion ψ : R R such ha x y ψ( x y (2. for all comparable elemens x, y E. If ψ(r =kr, k >, hen is called a parially Lipschiz wih a Lipschiz consan k. and Le (E,, be a parially ordered normed linear algebra. Denoe E = { x E x θ, where θ is he zero elemen of E } ={E E uv E for all u, v E }. (2.2 The elemens of he se are called he posiive vecors in E. The following lemma follows immediaely from he definiion of he se, which is ofenimes used in he hybrid fixed-poin heory of Banach algebras and applicaions o nonlinear differenial and inegral equaions. Lemma 2. (Dhage, 999 If u, u 2, v, v 2 are such ha u v and u 2 v 2, hen u u 2 v v 2. Definiion 2.7 R (. An operaor : E E is said o be posiive if he range R( of is such ha The Dhage ieraion principle or mehod (in shor DIP or DIM developed in Dhage (2; 23; 24a may be rephrased as monoonic convergence of he sequence of successive approximaions o he soluions of a nonlinear equaion beginning wih a lower or an upper soluion of he equaion as is iniial or firs approximaion and which forms a useful ool in he subjec of exisence heory of nonlinear analysis. The Dhage ieraion mehod is differen from oher ieraions mehods and embodied in he following applicable hybrid fixed-poin heorem of Dhage (24b, which is he key ool for our work conained in he presen paper. A few oher hybrid fixed-poin heorems conaining he Dhage ieraion principle appear in Dhage (2; 23; 24a; 24b. ( Theorem 2. Le E,, be a regular parially ordered complee normed linear algebra such ha he order relaion and he norm in E are compaible in every compac chain of E. Le, : E be wo nondecreasing operaors such ha (a is parially bounded and parially nonlinear -Lipschiz wih -funcion ψ, (b is parially coninuous and uniformly parially compac, (c Mψ (r < r, r >, where M = sup{ (C : C ch (E}, and (d here exiss an elemen x X such ha x x x or x x x. Page 3 of

4 Dhage & Dhage, Cogen Mahemaics (25, 2: 2367 hp://dx.doi.org/.8/ Then he operaor equaion x x = x has a posiive soluion x in E and he sequence {x n } of successive ieraions defined by x n = x n x n, n =,, ; converges monoonically o x. Remark 2. The compaibiliy of he order relaion and he norm in every compac chain of E is held if every parially compac subse S of E possesses he compaibiliy propery wih respec o and. This simple fac is used o prove he desired characerizaion of he posiive soluion of he QDE (. defined on J. 3. Main resuls The QDE (. is considered in he funcion space C(J, R of coninuous real-valued funcions defined on J. We define a norm and he order relaion in C(J, R by (2.3 x = sup x( J (3. and x y x( y( (3.2 for all J, respecively. Clearly, C(J, R is a Banach algebra wih respec o above supremum norm and is also parially ordered w.r.. he above parially order relaion. I is known ha he parially ordered Banach algebra C(J, R has some nice properies w.r.. he above order relaion in i. The following lemma follows by an applicaion of Arzelá Ascoli heorem. Lemma 3. Le ( C(J, R,, be a parially ordered Banach space wih he norm and he order relaion defined by (3. and (3.2, respecively. Then, and are compaible in every parially compac subse of C(J, R. Proof The proof of he lemma is given in Dhage and Dhage (in press. Since i is no well known, we give he deails of proof for he sake of compleeness. Le S be a parially compac subse of C(J, R and le {x n } be a monoone nondecreasing sequence of poins in S. Then, we have x ( x 2 ( x n ( (ND for each J. Suppose ha a subsequence {x nk } of {x n } is convergen and converges o a poin x in S. Then he subsequence {x nk (} of he monoone real sequence {x n (} is convergen. By monoone characerizaion, he whole sequence {x n (} is convergen and converges o a poin x( in R for each J. This shows ha he sequence {x n (} converges poinwise in S. To show he convergence is uniform, i is enough o show ha he sequence {x n (} is equiconinuous. Since S is parially compac, every chain or oally ordered se and consequenly {x n } is an equiconinuous sequence by Arzelá Ascoli heorem. Hence {x n } is convergen and converges uniformly o x. As a resul, and are compaible in S. This complees he proof. We need he following definiion in wha follows. Definiion 3. A funcion u C (J, R is said o be a lower soluion of he QDE (. if he funcion u( is coninuously differeniable and saisfies f (, u( Page 4 of

5 Dhage & Dhage, Cogen Mahemaics (25, 2: 2367 hp://dx.doi.org/.8/ d u( d f (,u( λ u( g(, u(, f (,u( u( x } for all J. Similarly, a funcion v C (J, R is said o be an upper soluion of he QDE (. if i saisfies he above propery and inequaliies wih reverse sign. We consider he following se of assumpions in wha follows: (A The map x x is injecion for each J. f (, x (A f defines a funcion f :J R R. (A 2 There exiss a consan M f > such ha < f (, x M f for all J and x R. (A 3 There exiss a -funcion φ, such ha f (, x f (, y φ(x y, for all J and x, y R, x y. (B g defines a funcion g:j R R. (B 2 There exiss a consan M g > such ha g(, x M g for all J and x R. (B 3 g(, x is nondecreasing in x for all J. (B 4 The QDE (. has a lower soluion u C (J, R. Remark 3. Noice ha Hypohesis (A holds in paricular if he funcion x x is increasing in f (,x R for each J. Lemma 3.2 Suppose ha hypohesis (A holds. Then a funcion x C(J, R is a soluion of he QDE (., if and only if i is a soluion of he nonlinear quadraic inegral equaion (in shor QIE, x( = [ f (, x( ] ( ce λ (3.3 e λ( s g(s, x(s ds for all J, where c = x e λ. Theorem 3. Assume ha hypoheses (A (A 3 and (B (B 4 hold. Furhermore, assume ha ( x M a g φ(r < r, r <, (3.4 hen he QDE (. has a posiive soluion x defined on J and he sequence {x n } n= of successive approximaions defined by x n ( = [ f (, x n ( ] ( ce λ for R, where x = u, converges monoonically o x. Proof Se E = C(J, R Then, by Lemma 3., every compac chain in E possesses he compaibiliy propery wih respec o he norm and he order relaion in E. Define wo operaors and on E by e λ( s g(s, x n (s ds (3.5 Page 5 of

6 Dhage & Dhage, Cogen Mahemaics (25, 2: 2367 hp://dx.doi.org/.8/ x( =f (, x(, J, (3.6 and ce λ x( = e λ( s g(s, x(s ds, J. From he coninuiy of he inegral, i follows ha and define he maps, : E E. Now by Lemma 3.2, he QDE (. is equivalen o he operaor equaion (3.7 x( x( =x(, J. (3.8 We shall show ha he operaors and saisfy all he condiions of Theorem 2.. This is achieved in he series of following seps. Sep I: and are nondecreasing on E. Le x, y E be such ha x y. Then by hypohesis (A 3, we obain x( =f (, x( f (, y( = y( for all J. This shows ha is nondecreasing operaor on E ino E. Similarly using hypohesis (B 3, i is shown ha he operaor is also nondecreasing on E ino iself. Thus, and are nondecreasing posiive operaors on E ino iself. Sep II: is parially bounded and parially -Lipschiz on E. Le x E be arbirary. Then by (A 2, x( f (, x( M f for all J. Taking supremum over, we obain x M f and so, is bounded. This furher implies ha is parially bounded on E. Nex, le x, y E be such ha x y. Then, x( y( = f (, x( f (, y( φ(x( y( φ( x y for all J. Taking supremum over, we obain x y φ( x y for all x, y E, x y. Hence, is a parial nonlinear D-LIpschiz on E which furher implies ha is a parially coninuous on E. Sep III: is parially coninuous on E. Le {x n be a sequence in a chain C of E such ha x n x for all n N. Then, by dominaed convergence heorem, we have n lim x ce λ n n ( =lim n lim = ce λ = ce λ e λ( s e λ( s g(s, x n (s ds [ ] lim g(s, x (s n n ds e λ( s g(s, x(s ds = x( Page 6 of

7 Dhage & Dhage, Cogen Mahemaics (25, 2: 2367 hp://dx.doi.org/.8/ for all J. This shows ha x n converges monoonically o x poinwise on J. Nex, we will show ha { x n is an equiconinuous sequence of funcions in E. Le, 2 J wih < 2. Then ce λ x n ( 2 x n ( ce λ2 e λ( s g(s, x n (s ds e λ( 2 s g(s, x n (s ds e λ( 2 2 s g(s, x n (s ds e λ( 2 s g(s, x n (s ds ce λ ce λ2 e λ( s e λ( 2 s g(s, x n (s ds g(s, x n (s ds 2 ce λ a ce λ2 M g e λ( s e λ( 2 s ds M g 2 as 2 uniformly for all n N. This shows ha he convergence x n x is uniform and hence is parially coninuous on E. Sep IV: is uniformly parially compac operaor on E. Le C be an arbirary chain in E. We show ha (C is a uniformly bounded and equiconinuous se in E. Firs, we show ha (C is uniformly bounded. Le y (C be any elemen. Then here is an elemen x C, such ha y = x. Now, by hypohesis (B 2, ce λ y( e λ( s g(s, x(s ds ce λ e λ( s g(s, x(s ds x a g(s, x(s ds x M g a = M for all J. Taking supremum over, we obain y = x M for all y (C. Hence, (C is a uniformly bounded subse of E. Moreover, (C M for all chains C in E. Hence, is a uniformly parially bounded operaor on E. Nex, we will show ha (C is an equiconinuous se in E. Le, 2 J wih < 2. Then, for any y (C, one has Page 7 of

8 Dhage & Dhage, Cogen Mahemaics (25, 2: 2367 hp://dx.doi.org/.8/ y( 2 y( = x( 2 x( ce λ ce λ2 e λ( s g(s, x(s ds e λ( 2 s g(s, x(s ds e λ( 2 2 s g(s, x(s ds e λ( 2 s g(s, x(s ds ce λ ce λ2 e λ( s e λ( 2 s g(s, x(s ds g(s, x(s ds 2 ce λ a ce λ2 M g e λ( s e λ( 2 s ds M g 2 as 2 uniformly for all y (C. Hence (C is an equiconinuous subse of E. Now, (C is a uniformly bounded and equiconinuous se of funcions in E, so i is compac. Consequenly, is a uniformly parially compac operaor on E ino iself. Sep V: u saisfies he operaor inequaliy u u u. By hypohesis (B 4, he QDE (. has a lower soluion u defined on J. Then, we have d u( d f (,u( λ u( g(, u(, f (, u( u( x for all J. Muliplying he above inequaliy (3.9 by he inegraing facor e λ, we obain ( e λ u( e λ g(, u( f (, u( for all J. A direc inegraion of (3. from o yields u( [ f (, u( ] ( ce λ } e λ( s g(s, u(s ds (3.9 (3. (3. for all J. From definiions of he operaors and, i follows ha u( u( u(, for all J. Hence u u u. Sep VI: -funcion φ saisfies he growh condiion Mψ (r < r, r >. Finally, he -funcion φ of he operaor saisfies he inequaliy given in hypohesis (d of Theorem 2.. Now from he esimae given in Sep IV, i follows ha ( x Mψ (r M a g φ(r < r for all r >. Page 8 of

9 Dhage & Dhage, Cogen Mahemaics (25, 2: 2367 hp://dx.doi.org/.8/ Thus, and saisfy all he condiions of Theorem 2. and we apply i o conclude ha he operaor equaion x x = x has a soluion. Consequenly he inegral Equaion 3.3 and he QDE (. has a soluion x defined on J. Furhermore, he sequence {x n } n= of successive approximaions defined by (3.5 converges monoonically o x. This complees he proof. Remark 3.2 The conclusion of Theorem 3. also remains rue if we replace he hypohesis (B 4 wih he following: (B 4 The QDE (. has an upper soluion v C (J, R. The proof under his new hypohesis is similar o he proof of Theorem 3. wih appropriae modificaions. Example 3. Given a closed and bounded inerval J =[, ], consider he IVP of QDE, d x( = d f (,x( 4 2 anh x(, J, x( = R } (3.2 where he funcions f, g : J R R are defined as f (, x =, if x, x, if < x < 3, 4, if, x 3 and g(, x = 2 anh x 4 Clearly, he funcions f and g are coninuous on J R ino R. The funcion f saisfies he hypohesis (A 3 wih φ(r =r. To see his, we have f (, x f (, y x y for all x, y R, x y. Therefore, φ(r =r. Moreover, he funcion f (, x is posiive and bounded on J R wih bound M f = 4 and so he hypohesis (A 2 is saisfied. Again, since g is posiive and bounded on J R by M g = 3 4, he hypohesis (B 2 holds. Furhermore, g(, x is nondecreasing in x for all J, and hus hypohesis (B 3 is saisfied. Also, condiion (3.4 of Theorem 3. is held. Finally, he QDE (3.2 has a lower soluion u( = defined on J, hus all hypoheses of Theorem 3. are saisfied. Hence, we apply Theorem 3. and conclude ha he QDE (3.2 has a soluion x defined on J 4 and he sequence {x n } n= defined by x n ( = 4 [ f (, xn ( ] ( [ 2 an hxn (s ] ds for all J, where x = u, converges monoonically o x. (3.3 Page 9 of

10 Dhage & Dhage, Cogen Mahemaics (25, 2: 2367 hp://dx.doi.org/.8/ Funding The auhors received no direc funding for his research. Auhor deails Bapurao C. Dhage Shyam B. Dhage Kasubai, Gurukul Colony, Ahmedpur, Maharashra, India. Ciaion informaion Cie his aricle as: Approximaing posiive soluions of nonlinear firs order ordinary quadraic differenial equaions, Bapurao C. Dhage & Shyam B. Dhage, Cogen Mahemaics (25, 2: References Dhage, B. C. (999. Fixed poin heorems in ordered Banach algebras and applicaions. PanAmerican Mahemaical Journal, 9, Dhage, B. C. (29. Local asympoic araciviy for nonlinear quadraic funcional inegral equaions. Nonlinear Analysis, 7, Dhage, B. C. (2. Quadraic perurbaions of periodic boundary value problems of second order ordinary differenial equaions. Differenial Equaions and Applicaions, 2, Dhage, B. C. (23. Hybrid fixed poin heory in parially ordered normed linear spaces and applicaions o fracional inegral equaions. Differenial Equaions and Applicaions, 5, Dhage, B. C. (24a. Global araciviy resuls for comparable soluions of nonlinear hybrid fracional inegral equaions. Differenial Equaions and Applicaions, 6, Dhage, B. C. (24b. Parially condensing mappings in parially ordered normed linear spaces and applicaions o funcional inegral equaions. Tamkang Journal of Mahemaics, 45, Dhage, B. C., & Dhage, S. B. (in press. Approximaing soluions of nonlinear firs order ordinary differenial equaions. Global Journal of Mahemaical Sciences, 3. Dhage, B. C., Dhage, S. B., & Nouyas, S. K. (24. Approximaing soluions of nonlinear hybrid differenial equaions. Applied Mahemaics Leers, 34, Dhage, B. C., & Lakshmikanham, V. (2. Basic resuls on hybrid differenial equaions. Nonlinear Analysis: Hybrid Sysems, 4, Dhage, B. C., & Regan, D. O. (2. A fixed poin heorem in Banach Algebras wih applicaions o funcional inegral equaions. Funcional Differenial Equaions, 7, Heikkilä, S., & Lakshmikanham, V. (994. Monoone ieraive echniques for disconinuous nonlinear differenial equaions. New York, NY: Marcel Dekker. Nieo, J. J., & Rodriguez-Lopez, R. (25. Conracive mappings heorems in parially ordered ses and applicaions o ordinary differenial equaions. Order, 22, The Auhor(s. This open access aricle is disribued under a Creaive Commons Aribuion (CC-BY 4. license. You are free o: Share copy and redisribue he maerial in any medium or forma Adap remix, ransform, and build upon he maerial for any purpose, even commercially. The licensor canno revoke hese freedoms as long as you follow he license erms. Under he following erms: Aribuion You mus give appropriae credi, provide a link o he license, and indicae if changes were made. You may do so in any reasonable manner, bu no in any way ha suggess he licensor endorses you or your use. No addiional resricions You may no apply legal erms or echnological measures ha legally resric ohers from doing anyhing he license permis. Cogen Mahemaics (ISSN: is published by Cogen OA, par of Taylor & Francis Group. Publishing wih Cogen OA ensures: Immediae, universal access o your aricle on publicaion High visibiliy and discoverabiliy via he Cogen OA websie as well as Taylor & Francis Online Download and ciaion saisics for your aricle Rapid online publicaion Inpu from, and dialog wih, exper ediors and ediorial boards Reenion of full copyrigh of your aricle Guaraneed legacy preservaion of your aricle Discouns and waivers for auhors in developing regions Submi your manuscrip o a Cogen OA journal a Page of