The Journal of Nonlinear Science and Applications

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1 J. Noliear Sci. Appl. 2 (29), o. 2, Te Joural of Noliear Sciece ad Applicaios p:// POSITIVE SOLUTIONS FOR A CLASS OF SINGULAR TWO POINT BOUNDARY VALUE PROBLEMS RAHMAT ALI KHAN, AND NASEER AHMAD ASIF 2 Commuicaed by J. J. Nieo Absrac. Exisece of posiive soluio for a class of sigular boudary value problems of e ype x () = f(, x(), x ()), (, ) x() =, x() =, is esablised. Te olieariy f C((, ) (, ) (, ), (, )) is allowed o cage sig ad is sigular a =, = ad/or x =. A example is icluded o sow e applicabiliy of our resul.. Irocio ad prelimiaries Sigular boudary value problems arise i various fields of Maemaics ad Pysics suc as uclear pysics, boudary layer eory, oliear opics, gas dyamics, ec, [, 5, 9, 2, 4, 7, 8, 9]. For more deails o sigular BVPs ad rece developmes, we refer e readers o e rece moograp by R. P. Agarwal ad D. O Rega [4] ad [6, 8, 9, 5]. I is paper, we cosider a class of secod order sigular boudary value problems of e ype x () = f(, x(), x ()), (, ), x() =, x() =, (.) Dae: Received: 2 December 28; Revised: 25 February Maemaics Subjec Classificaio. Primary 34A45; Secodary 34B5. Key words ad prases. Posiive soluios, Sigular differeial equaios, Diricle boudary codiios. Correspodig auor. 26

2 SINGULAR BVPS 27 were f C((, ) (, ) (, ), (, )) ad may be sigular a =, = ad/or x = ad is allowed o cage sig. We esablis exisece of posiive soluios for e BVP (.) uder a weaker ypoesis o f. Recely, exisece of posiive soluios for sigular boudary value problems, i e case we e olieariy f is idepede of e derivaive erm, as bee sudied by may auors, [3,,, 3, 2]. I ese papers, e olieariy is assumed o be o-egaive ad eier be subliear or superliear. Hece, e resuls of ese papers would be applicable o a limied class of boudary values problems. Moreover, mos of oliear problems from e applied scieces, e olieariy explicily depeds o e derivaive erm, for example, e differeial equaio x () = λ ( 2 x() ), [, ), x() ogeer wi some suiable boudary codiios, a explicily depeds o e derivaive, arises i e boudary layer eory i fluid mecaics [9]. Furer, e olieariy f does o saisfy e subliear ad superliear codiios i mos cases ad may cage sig. Hece, e sudy of boudary value problems wiou e above meioed resricios is of grea imporace. Recely, exisece eory for posiive soluios of wo poi boudary value problems wiou e firs derivaive erm is sudied i [6, 9, 2]. Ispired by e above papers, e aim of e prese paper is o improve ad geeralize e resul sudied i [2] o e case we e olieariy f explicily depeds o e derivaive erm x. We sudy e problem uder muc weaker ypoesis o f. We iclude a example o sow e applicabiliy of our resul. Trougou is paper, we assume a e followig codiio olds. (A ) ere exis k C((, ), (, )) ad a decreasig F C((, ), (, )) suc a Te codiio R F (u) ( )k()d < ad F (u) =. = implies a we ca coose R > suc a { /2 } F (u) > max sk(s)ds, ( s)k(s)ds. (.2) /2 For fixed {3, 4, 5,...}, le M = max{f ( )k() : [, ]} ad coose C > 2MR. For u C[, ] we wrie u = max{ u() : [, ]} ad for u C [, ], we wrie u = max{ x, γ 3 x } were γ = 2 for 3. Clearly, C [, ] wi e orm. is a Baac space. Te oly codiio we are imposig o e olieariy f is e followig: (A 2 ) f(, x(), x ()) k()f (x()) o (, ) (, R] [ C, C].

3 28 KHAN, ASIF For fixed {3, 4, 5,...}, cosider e BVPs x () = f(, x(), x ()), [, ], x( ) =, x( ) =. (.3) We wrie (.3) as a equivale iegral equaio were x() = + / G (, s) = / G (, s)f(s, x(s), x (s))ds, [, ], (.4) { (s )( ), if s < 2 ( )( s), if s, is e Gree s fucio for e correspodig omogeeous problem x () =, [, ] x( ) =, x( ) =. (.5) Noice a G (, s) o (, ) (, ) ad G (, s) G (s, s), (, ). Moreover, max [,] max [,] / / / / G (, s)ds = G (, s)ds = / / / / 2. Mai resuls G (s, s)ds = γ2 6, G (, s) ds = γ 2. Teorem 2.. Assume a (A ) ad (A 2 ) old. Te boudary value problem (.3) as a soluio x C [, ] suc a x() < R ad x () < C for [, ]. Proof. Defie reracios q : (, ) [ C, C] by q(v) = max{ C, mi{v, C}} ad p : (, ) [, R] by p (x()) = max{, mi{x(), R}}. Clearly, q, p are coiuous ad q(v) = v for v C, p(v) = v for v R. Cosider e modified BVP x () = F (, x(), x ()), [, ], x( ) =, x( ) =, (2.) were F (, x(), x ()) = f(, p (x()), q(x ())). Clearly, F is coiuous, bouded ad oegaive o [, ] R R. Furer, ay soluio x C [, ] of (2.) suc a x() < R, x () < C, [, ], (2.2) is a soluio of (.3). Obviously, x() o [, ] as F.

4 SINGULAR BVPS 29 We wrie e BVP (2.) as a equivale iegral equaio of e ype x() = / + G (, s)f (s, x(s), x (s))ds (2.3) / ad defie a operaor T : C [, ] C [, ] by (T x)() = / + G (, s)f (s, x(s), x (s))ds. (2.4) / By a soluio of (2.) we mea a soluio of e operaor equaio (I T )(x) =, a is, a fixed poi of T. We sow a T as a fixed poi x C [, ]. Clearly, T is coiuous ad compleely coiuous as F is coiuous ad bouded. Coose R > max{r, + M γ 2 }, were 3 6 M = max{f(, x, x ) : [, ], x [, R], x [ C, C]} ad defie a ope, bouded ad covex se For x Ω R, we ave I follows a T x + max [,] Ω R = {x C [, ] : x < R}. 3 + M / / / / G (, s)f (s, x(s), x (s))ds G (s, s) ds = 3 + M γ 2, 6 / (T x) G max [,] / (, s)f (s, x(s), x (s))ds M γ 2. T x = max{ T x, γ 3 (T x) } 3 + M γ 2 6 < R for every x Ω R. Hece, T (Ω R) Ω R. Cosequely, by Scauder s fixed poi eorem, e BVP (2.) as a soluio i Ω R. Now, we sow a ay soluio x of (2.) mus saisfies (2.2). Firsly, we sow a x < R o [, ]. Assume a is is o rue ad x() R for some [, ]. Le ξ = mi{ [, ] : x() = R}. We discuss differe cases: Case : If ξ /2, sice x( ) = < R, ere exis subiervals, say [ξ 2i, ξ 2i ] [, ξ], i =, 2, 3,, m suc a (): ξ =, ξ 2m = ξ, ξ 2i < ξ 2i for i =, 2, 3,, m, (2) ξ 2i ξ 2i+, x(ξ 2i ) = x(ξ 2i+ ) ad x (ξ 2i ) = for i =, 2, 3,, m, (3) x () for [ξ 2i, ξ 2i ], i =, 2, 3,, m.

5 3 KHAN, ASIF For [ξ 2i, ξ 2i ], i =, 2, 3,, m, usig (A 2 ) ad e fac a x() [, R], we ave x () = F (, x(), x ()) = f(, p (x()), q(x ())) k()f (x()), [ξ 2i, ξ 2i ]. (2.5) Iegraig (2.5) from o ξ 2i, usig (2) ad e decreasig propery of F, we obai x () F (x()) k(s)ds, [ξ 2i, ξ 2i ], i =, 2, 3,, m, wic implies a x () F (x()) Iegraig (2.6) from ξ 2i o ξ 2i, we ave ξ 2i wic ca be wrie as x(ξ2i ) F (u) x(ξ 2i ) k(s)ds, [ξ 2i, ξ 2i ], i =, 2, 3,, m. (2.6) x () F (x()) d ξ 2i k(s)dsd, ξ 2i sk(s)ds, i =, 2, 3,, m. (2.7) Summig from i = o m ad usig (2) (x(ξ 2i ) = x(ξ 2i+ )), we obai Leig, we ave R / R F (u) F (u) /2 /2 sk(s)ds. sk(s)ds, a coradicio o (.2). Case 2: Le ξ /2 ad η = max{ [, ] : x() = R}. Sice x( ) = 2 < R, ere exis subiervals [η 2i, η 2i ] [, ], i =, 2, 3,, 2 m suc a (4) η =, η 2m = η, η 2i < η 2i for i =, 2, 3,, m, (5) η 2i+ η 2i, x(η 2i ) = x(η 2i+ ), x (η 2i ) = for i =, 2, 3,, m ad (6) x () for [η 2i, η 2i ], i =, 2, 3,, m. Iegraig (2.5) from η 2i o, usig (5), e decreasig propery of F ad e iegraig from η 2i o η 2i, we obai x(η2i ) x(η 2i ) F (u) η2i η 2i ( s)k(s)ds, i =, 2, 3,, m. (2.8) Summig (2.8) from i = o i = m ad usig (5) (x(η 2i ) = x(η 2i+ )), we obai R / F (u) /2 ( s)k(s)ds.

6 Leig limi, we ge R F (u) ( s)k(s)ds /2 SINGULAR BVPS 3 wic is a coradicio o (.2). Now we sow a ay soluio x() of (2.) mus saisfies x () C, [, ]. From e boudary codiios, x( ) = ad x( ) =, i follows a ere exis p (, ) suc a x (p) =. Suppose ere exis (, ) suc a x ( ) > C. As i e firs par of is eorem, coose [ξ 2i, ξ 2i ] [, ] suc a x () o [ξ 2i, ξ 2i ] ad x (ξ 2i ) =, i =, 2, 3,. Hece, ere exis some i suc a [ξ 2i, ξ 2i ]. Le C = max{x () : [ξ 2i, ξ 2i ]} = x (ξ ). Clearly, C C ad i view of (A 2 ), we ave Hece, x () = f(, x(), q(x ())) k()f (x()) M. Iegraig from ξ o ξ 2i, we ave implies a x ()x () Mx (), [ξ 2i, ξ 2i ]. ξ C x ()x ()d M ξ x ()d, vdv MR C 2MR, wic coradic e defiiio of C. Hece, x () C, [, ]. Similarly, we ca sow a x () C, [, ]. Teorem 2.2. Assume a (A ) ad (A 2 ) old. Te, e boudary value problem (.) as a posiive soluio x. Proof. By Teorem 2., ay soluio x of (.3) saisfies x () R, x () C for [, ], = 3, 4, 5,... Hece, for eac (, /2), ere exis a aural umber m {3, 4, 5, } suc a x () > for all [, ] ad m. Cosider e iegral equaio, x () = x ( ) x () ( ) + x () + + (s )( ) f(s, x (s), x (s))ds ( )( s) f(s, x (s), x (s))ds, [, ]. (2.9)

7 32 KHAN, ASIF Differeiaig wi respec o, we obai x () = x ( ) x () + (s )f(s, x (s), x (s))ds ( s)f(s, x (s), x (s))ds. For ay, 2 [, ], we ave x ( 2 ) x 2 ( ) = (s )f(s, x (s), x (s))ds ( s)f(s, x (s), x (s))ds (s )f(s, x (s), x (s))ds (2.) ( s)f(s, x (s), x (s))ds = 2 2 (s )f(s, x (s), x (s))ds + ( s)f(s, x (s), x (s))ds L 2, were L = max{f(, u, v) : (, u, v) [, ] [, R] [ C, C]}. Tus e sequeces {x } ad {x } are uiformly bouded ad equicoiuous. By Arzelà- Ascoli eorem, ere exis a subsequece {x k } of {x } covergig uiformly o [, ] suc a lim x k k () = x(), lim k x k () = x (), were x C [, ]. Takig lim, we ave lim x k k () = x() o (, ). Furer, x > o (, ). Leig lim k, (2.2) ad (2.) yield x( ) x() x() = ( ) + x() + + ( )( s)f(s, x(s), x (s))ds, (s )( )f(s, x(s), x (s))ds x () = x( ) x() + (s )f(s, x(s), x (s))ds ( s)f(s, x(s), x (s))ds.

8 Hece, SINGULAR BVPS 33 x () = ( )f(, x(), x ()) ( )f(, x(), x ()) = f(, x(), x ()) x () =f(, x(), x ()), (, ) (2.) wic implies a x saisfies e differeial equaio (.). Moreover, ( ) ( ) x() = lim x = lim k x k k k = lim =, k k k ad ( x() = lim x ) ( = lim k x k k k ) = lim =, k k k wic implies a x also saisfies e boudary codiios ad ece is a soluio of (.). Example 2.3. Cosider e boudary value problem x x () + 5 () = ; (, ) ( )(x()) 2 x() =, x() =. (2.2) Coose k() = ad F (u) = 9. Clearly F is decreasig ad =. ( ) u 2 F (u) Sice 2 Hece, from e relaio R k()d = l 2, 2 ( )k()d = l 2. > l 2, we ave R > ( + 27 l F (u) 2)3. Also, M = max{f ( )k() : [, ]} = max{ 92 ( ) : [, ]} = 9. Hece C = 3. Moreover, x () + 5 µ ( ) ν (x()) 2 = f(, x(), x ()) k()f (x()) for x () [ 3, 3], (, ). By Teorem 2.2, e problem (2.2) as a soluio x suc a < x() ( + 27 l 2) 3, x () 3, (, ). Ackowledgemes: Researc of R. A. Ka is suppored by HEC, Pakisa, Projec 2-3(5)/PDFP/HEC/28/.

9 34 KHAN, ASIF Refereces [] G. A. Afrouzi, M. Kalegy Mogaddam, J. Moammadpour ad M. Zamei, O e posiive ad egaive soluios of Laplacia BVP wi Neuma boudary codiios, J. Noliear Sci. Appl., 2 (29), [2] R.P. Agarwal ad D. O Rega, Twi soluios o sigular Diricle problems, J. Ma. Aal. Appl., 24 (999), [3] R.P. Agarwal ad D. O Rega, A upper ad lower soluio approac for a geeralized Tomas-Fermi eory of eural aoms, Maemaical Problems i Egieerig, 8 (22), [4] R. P. Agarwal ad D. O Rega, Sigular differeial ad iegral equaios wi applicaios, Kluwer Academic Publisers, Lodo, 23. [5] R. P. Agarwal, D. O Rega ad P. K. Palamides, Te geeralized Tomas-Fermi sigular boudary value problems for eural aoms, Ma. Meods i e Appl. Sci., 29 (26), [6] M. va de Berg, P. Gilkey ad R. Seeley, Hea coe asympoics wi sigular iiial emperaure disribuios, J. Fuc. Aal., 254 (28), [7] A. Callegari ad A. Nacma, A oliear sigular boudary value problem i e eory of pseudoplasic fluids, SIAM J. Appl. Ma., 38 (98), [8] J. Cu ad D. Fraco, No-collisio periodic soluios of secod order sigular dyamical sysems, J. Ma. Aal. Appl., 344 (28), [9] J. Cu ad J.J. Nieo, Impulsive periodic soluios of firs-order sigular differeial equaios, Bull. Lodo Ma. Soc., 4 (28), [] X. Du ad Z. Zao, Exisece ad uiqueess of posiive soluios o a class of sigular m-poi boudary value problems, Appl. Ma. Compu., 98 (28), [] M. Fega ad W. Gea, Posiive soluios for a class of m-poi sigular boudary value problems, Ma. Compu. Modellig, 46 (27), [2] J. Jaus ad J. Myjak, A geeralized Emde-Fowler equaio wi a egaive expoe, Noliear Aal., 23 (994), [3] R. A. Ka, Posiive soluios of four poi sigular boudary value problems, Appl. Ma. Compu., 2 (28), [4] S. K. Nouyas ad P. K. Palamides, O Surm-Liouville ad Tomas-Fermi Sigular Boudary Value Problems, Noliear Oscillaios, 4 (2), [5] A. Orpel, O e exisece of bouded posiive soluios for a class of sigular BVPs, Noliear Aal., 69 (28), [6] D. O Rega, Sigular differeial equaios wi liear ad oliear boudary codiios, Compuers Ma. Appl., 35 (998), [7] J. V. Si, A sigular oliear differeial equaio arisig i e Homa flow, J. Ma. Aal. Appl., 22 (997) [8] E. Soewoo, K. Vajravelu ad R. N. Moapara, Exisece ad ouiqueess of soluios of a sigular oliear boudary layer problem, J. Ma. Aal. Appl., 59 (99), [9] G.C. Yag, Exisece of soluios o e ird-order oliear differeial equaios arisig i boudary layer eory, Appl. Ma. Le., 6 (23), [2] G.C. Yag, Posiive soluios of sigular Diricle boudary value problems wi sigcagig olieariies, Comp. Ma. Appl., 5 (26), , [2] Z. Yog, Posiive soluios of sigular subliear Emde-Fower boudary value problems, J. Ma. Aal. Appl., 85 (994),

10 SINGULAR BVPS 35 Cere for Advaced Maemaics ad Pysics, Naioal Uiversiy of Scieces ad Tecology, Campus of College of Elecrical ad Mecaical Egieerig, Pesawar Road, Rawalpidi, Pakisa address: rama 2 Cere for Advaced Maemaics ad Pysics, Naioal Uiversiy of Scieces ad Tecology, Campus of College of Elecrical ad Mecaical Egieerig, Pesawar Road, Rawalpidi, Pakisa address: aseerasif@yaoo.com