Method of upper lower solutions for nonlinear system of fractional differential equations and applications

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1 Malaya Journal of Maemak, Vol. 6, No. 3, , 218 hps://do.org/ /mjm63/1 Mehod of upper lower soluons for nonlnear sysem of fraconal dfferenal equaons and applcaons D.B. Dhagude1 *, N.B. Jadhav2 and J.A. Nanware3 Absrac Our am s o develop he mehod of upper lower soluons and apply o prove exsence and unqueness of soluon of perodc boundary value problems for a sysem of fraconal dfferenal equaons nvolvng a Remann - Louvlle fraconal dervaves. Keywords Perodc boundary value problems, Sysem of fraconal dfferenal equaons, Remann-Louvlle fraconal dervaves, Upper and lower soluons, Exsence and unqueness resuls. AMS Subjec Classfcaon 26A33;34A8,34B15,34B99. 1 Deparmen of Mahemacs, Dr.Babasaheb Ambedkar Marahwada Unversy, Aurangabad , Inda. of Mahemacs, Yeshwanrao Mahavdyalaya, Tuljapur , Inda. 3 Deparmen of Mahemacs, Shrkrashana Mahavdyalaya, Gunjo , Inda. *Correspondng auhor: 1 dnyanraja@gmal.com, 2 narsngjadhav4@gmal.com, 3 jagskmg91@redffmal.com Arcle Hsory: Receved 24 January 218; Acceped 21 Aprl Deparmen Conens 1 Inroducon Upper Lower Soluons Man Resuls Acknowledgemen References Inroducon Now-a-days he heory of fraconal dfferenal equaons have been occupyng an mporance place n scence and echnology.fraconal dfferenal equaons have been wdely used for modelng varous processes n physcs, chemsry,bology, aerodynamcs of complex medum,polymer rheology, hermoelascy and conrol of dynamcal sysems(see [3, 5, 17] and he references heren). Recenly, many researchers have gven aenon o he exsence and unqueness of soluon of he nal value problems [8, 15], perodc boundary value problems [2, 14, 18], problems wh negral boundary condons [4, 7, 1 13] and wh nonlocal negral boundary condons [1] for fraconal dfferenal equaons. I s well known ha he mehod of upper and lower soluons [6, 16]coupled wh c 218 MJM. s assocaed monoone eraon scheme s an neresng, consrucve and powerful mechansm whch offers exsence and unqueness resuls for nonlnear problems n a closed se. Recenly,We e.al.[19] proved exsence and unqueness of he soluon of perodc boundary value problem for a fraconal dfferenal equaon,usng he mehod of upper lower soluons and s assocaed monoone eraons.in hs paper, we exend hese resuls for nonlnear sysem of RemannLouvlle fraconal dfferenal equaons,by removng he bounded demand of f (, u()) n [9]. We organze he paper as follows: In Secon 2,we consder he perodc boundary value problem for nonlnear sysem of Remann - Louvlle fraconal dfferenal equaons and nroduce he noon of upper lower soluon. Exsence and unqueness resuls of perodc boundary value problem for sysem of nonlnear fraconal dfferenal equaons nvolvng Remann-Louvlle fraconal dervaves are proved n he las secon. 2. Upper Lower Soluons In hs secon,we consder he perodc boundary value problems for a sysem of nonlnear Remann - Louvlle fraconal dfferenal equaons and nroduce he noon of upper

2 Mehod of upper lower soluons for nonlnear sysem of fraconal dfferenal equaons and applcaons 468/472 and lower soluons.consder he followng sysem of nonlnear Remann-Louvlle fraconal dfferenal equaons D u1 () = f1, u1 (), u2 (), (, T ], < 1, D u2 () = f2, u1 (), u2 (), (2.1) I s called perodc boundary value problems(pbvp) for he sysem of nonlnear Remann-Louvlle fraconal dfferenal equaons. Assume ha J = [, T ] R s a compac nerval and f (, u1 (), u2 ()) C([, T ] R2, R), = 1, 2. Furher assume ha u1 and u2 are measurable Lebesgue funcons.e.u1, u2 L1 (, T ). Suppose wh perodc boundary condons u1 () = = u1 () =T, u2 () = = u2 () =T. (2.2) C([, T ]) = u : u ()s connuous on [, T ], u C = max u (), = 1, 2. [,T ] C ([, T ]) = u C[, T ] : u () C([, T ]), u C = u C. Assume ha upper and lower soluons sasfy he followng order relaon (v1, v2 ) (w1, w2 ), (, T ] : v () = w () =, = 1, 2. (2.3) Now we defne he order nerval or ( funconal nerval )secor as follows: Defnon 2.1. The order nerval n a space C ([, T ]) L1 (, T ) s denoed by S and s defned as S = (u1, u2 ) C ([, T ]) L1 (, T ) : v1 (), v2 () u1 (), u2 () w1 (), w2 (), (, T ]; v () = u () = w () =. In he followng,we defne quasmonooncy and Lpschz condon of funcon f (, u1, u2 ), = 1, 2 as follows. Defnon 2.2. A funcon f (, u1, u2 ) C(J R2, R), = 1, 2 s sad o be quasmonoone nondecreasng nonncreasng f for each, u v and u j = v j, 6= j, hen f (, u1, u2 ) f (, v1, v2 ) f (, u1, u2 ) f (,, v1, v2 ). Defnon 2.3. Le f (, u1, u2 ) : [, T ] R2 R be a real valued connuous funcon. We say ha f (, u1, u2 ) sasfes one sded Lpschz condon f here exss M such ha f1 (, u1, u2 ) f1 (, u 1, u2 ) M1 (u1 u 1 ) for v1 u 1 u1 w1, f2 (, u1, u2 ) f2 (, u1, u 2 ) M2 (u2 u 2 ) for v2 u 2 u2 w2. (2.4) Furher o ensure he unqueness of soluon of PBVP (2.1)-(2.2),we assume ha here exss N such ha f1 (, u1, u2 ) f1 (, u 1, u2 ) N1 (u1 u 1 ) for v1 u 1 u1 w1, f2 (, u1, u2 ) f2 (, u1, u 2 ) N2 (u2 u 2 ) for v2 u 2 u2 w2. (2.5) From condons (2.4) and (2.5), we conclude ha funcon f=( f1, f2 ) sasfes Lpschz condon f (, u1, u2 ) f (, u 1, u 2 ) K ( u1 u 1 u2 u 2 ), (2.6) wh M = N = K. Now we consder he followng resuls of he lnear PBVP for a fraconal dfferenal equaon whch are man ngred468 ens n he proof of our exsence and unqueness resuls of soluon of he PBVP (2.1)-(2.2). Lemma 2.4. [19] The lnear perodc boundary value prob-

3 Mehod of upper lower soluons for nonlnear sysem of fraconal dfferenal equaons and applcaons 469/472 as n on (, T ], where he funcons (v1 (), v2 ()) and (w1 (), w2 ()) are mnmal and maxmal soluons on S for he PBVP (2.1) (2.2) and sasfy he monoone propery lem D u() Mu() = σ (), u() = = u() =T, v1 v11... vn1... v1 w1... wn1... w11 w1, where M > s a consan and σ C[, T ] has he followng negral represenaon of he soluon u= T Γ() 1 E, ( M ) [1 Γ()]E, ( MT ) ( ) v2 v12... vn2... v2 w2... wn2... w12 w2. (3.1) Also,f he one sded Lpschz condon (2.5) holds,hen he PBVP (2.1) (2.2) has unque soluon on S. (2.7) (T s) 1 E, M(T s) σ (s)ds M( s) σ (s)ds, ( s) 1 E, Proof:Consder PBVP for sysem of lnear fraconal dfferenal equaons (2.8) D u1 () M1 u1 = f1 (, η1, η2 ) M1 η1 k where E, () = k= Γ((k1)) s he Mag-Leffler funcon (see [5]). = σ1 (, η1, η2 ), (, T ), Lemma 2.5. [19] If u() C ([, T ]) L1 (, T ) and sasfes he relaons u() = = u1 () = = (3.2) u1 () =T, D u2 () M2 u2 = f2 (, η1, η2 ) M2 η2 = σ2 (, η1, η2 ), (, T ), D u() Mu(), (, T ), u() =T, where M > s a consan,hen u(), (, T ]. 3. Man Resuls In hs secon we develop mehod of upper lower soluons and consruc wo monoone convergen sequences,whch converge monooncally from above and below o maxmal and mnmal soluons respecvely. As an applcaon of hs mehod,exsence and unqueness resuls for he PBVP (2.1) (2.2) are proved when he funcons f1 (, u1, u2 ) and f2 (, u1, u2 ) are quasmonoone nonncreasng as well as quasmonoone nondecreasng u2 () = = (3.3) u2 () =T, for any (η1, η2 ) S. Clearly,lnear problems (3.2) and (3.3) have exacly one soluon u1 () and u2 () C ([, T ]) L1 (, T ) respecvely, follows from Lemma 2.1 and whose negral represenaon s as n (2.7). Now defne A[η1, µ] = u1 () as follows: u1 = T Γ() 1 E, ( M1 ) [1 Γ()]E, ( M1 T ) ( ) Theorem 3.1. Suppose ha (T s) 1 E, M1 (T s) σ1 ds ( s) 1 E, M1 ( s) σ1 ds. (3.4) Also we defne and A[η2, µ] = u2 (), as follows: () v = (v1, v2 ) and w = (w1, w2 ) C ([, T ]) L1 (, T ) are lower and upper soluons of he PBVP (2.1) (2.2), such ha order relaon (2.3) holds, u2 = T Γ() 1 E, ( M2 ) [1 Γ()]E, ( M2 T ) () funcon f (, u1, u2 ) C([, T ] R2, R) sasfes one-sded Lpschz condon (2.4), ( ) () funcons f1 and f2 are quasmonoone nondecreasng. (T s) 1 E, M2 (T s) σ2 ds ( s) 1 E, M2 ( s) σ2 ds. (3.5) Then here exs monoone sequences An operaor A s from [v (), w ()] no C ([, T ]) L1 (, T ) {vn1 (), vn2 ()}, {wn1 (), wn2 ()} C ([, T ]) L1 (, T ) such and η s soluon of he PBVP (2.1)- (2.2) ff η = A[η, µ]. ha Now we prove {vn1 (), vn2 ()} (v1, v2 ) and {wn1 (), wn2 ()} (w1, w2 ), (I)v () A[v (), w ()] and (II) If v η µ w w () A[w (), v ()], = 1, 2, (3.6) hen A[η, µ] A[η, µ ], = 1, 2. (3.7) 469

4 Mehod of upper lower soluons for nonlnear sysem of fraconal dfferenal equaons and applcaons 47/472 and boundary condons To prove (I), se A[v (), w ()] = v1 () where v1 () = 1 (v1 (), v12 ()). Noe ha v11 () and v12 () are he unque soluons of lnear PBVP (3.2) and (3.3) respecvely. The funcon v () s a lower soluon of he PBVP (2.1) (2.2). Se p () = v () v1 () wh η = v (). We observe ha p () = = u () = v () = = u =T v () =T = p () =T. Applyng Lemma 2.2,we ge p () mples ha u () v (). Hence A[η, µ] A[η, µ ] Thus he operaor A possess he monoone propery on [v (), w ()]. Defne he sequences {vn } and {wn } by vn = A[vn 1, wn 1 ] and wn = A[wn 1, vn 1 ]. Usng (3.6) and (3.7),we oban D p () = D v () D v1 (), f (, v1 (), v2 ()) f (, v1 (), v2 ()) M (v () v1 ()), M (v () v1 ()), v v1... vn... wn... w1 w, D p () M p (), = 1, 2. (3.8) Le P = {vn : n = 1, 2,...} and Q = {wn : n = 1, 2,...}. We show ha he ses P and Q are relavely compac n C ([, T ]) L1 (, T ). For any η S and by defnon of lower and upper soluon along wh one sded Lpschz condon,we have and boundary condons p () = = v () = v1 () = = v () =T v1 () =T D v M v f (, v1, v2 ) M v = p () =T. f (, η1, η2 ) M η Usng Lemma 2.2,we ge p () mples ha v () v1 () = A[v (), w ()]. To prove ha w () A[w (), v ()]; we se A[w (), v ()] = w1 () where w1 () = (w11 (), w12 ()). Noe ha w11 () and w12 ()) are he unque soluons of lnear PBVP (3.2) and (3.3) respecvely. The funcon w () s an upper soluon of he PBVP (2.1) (2.2). Defne p () = w () w1 () wh η = w (). We observe ha D p () = D f (, w1, w2 ) M w D w M w. Le P = {vn : n = 1, 2,...}, = 1, 2 and S C ([, T ]) L1 (, T ) are bounded ses.furhermore,he se {σ (, η1, η2 ) = f (, η1, η2 ) M η η S} s also a bounded se.hence here exs consans B, = 1, 2 such ha σ (, vn ) = max σ (, vn ) T w () D w1 (), (3.9) B σ (, vn ) B, (, T ] (3.1) D p () M p (), On he oher hand {vn () n = 1, 2,...}, = 1, 2 sasfy and boundary condons ( M ) p () = = w () = w1 () = vn () = Γ()u e = w () =T w1 () =T ( M ( s)) e σ (vn 1 )(s)ds (3.11) where ( M ) = 1 E, ( M ) e = p () =T. Usng Lemma 2.2,we ge p () mples ha u = w () w1 () = A[w (), v ()]. Now,we prove (II).The operaor A s monoone.le η = (η1, η2 ) and µ = (µ1, µ2 ) n [v (), w ()] be such ha η µ. Suppose ha A[η, µ] = u = (u1, u2 ) and A[η, µ ] = v = (v1, v2 ). Consder p () = u () v () and observe ha R ( M ( s)) T σ (vn 1 )(s)ds [1 Γ()]E, ( M T ) e From he condon (3.9),we have, (, T ], η, η P 1 2 σ (, η1, η2 ) B u B T Γ(2)[1 Γ()E, ( M T )] Whou lose of generaly,we assume ha and for ε > here exs δ = δ (ε) when 1 2 < δ,and snce E, () C[, T ], we have D p () = D u () D v () = f (, η1, η2 ) f (, µ1, µ2 ) M (η u ) M (µ v ) ε, 3Γ()max{ u, B /M } εγ(2) (2 1 ) < (3.12) 6B Γ() E, ( M1 ) E, ( M2 ) < M (η u ) M (µ v ) M (µ η ) M (u () v ()), D p () M p (), 47

5 Mehod of upper lower soluons for nonlnear sysem of fraconal dfferenal equaons and applcaons 471/472 (2.1)-(2.2). By he assumpon of he funcon f1 and Lemma 2.1,v1 () s he classcal soluon of he PBVP (2.1)-(2.2). Ths proves ha he lower sequence {vn1 ()} converges o a soluon v1 () of problem (2.1)-(2.2). Furher,we know From above equaons,we oban 1 vn (1 ) 2 vn (2 ) ( M 1 ) = Γ()u [1 e (3.13) ( M 2 ) 2 e ] σ2 (, η1, η2 ) = f2 (, η1, η2 ) M2 η2 ( M ) [1 e 1 σ (vn 1 )(1 ) ( M 1 ) 2 e Clearly, he funcon σ2 s connuous and monoone nondecreasng and monoone convergence of {vn2 ()} o v2 () as n on (, T ], mples ha σ2 (vn2 )() converges o σ2 (v2 )(), (, T ]. Le n n (3.11) and apply he domnaed convergence heorem,we observe ha v2 () sasfes he negral equaon σ (vn 1 )(2 )] Γ() u E, ( M1 ) E, ( M2 ) LΓ() E, ( M1 ) E, ( M2 ) M 2B Γ() (2 1 ) Γ(2) < ε. v2 (3.14) = N2 Ths mples ha P s equ-connuous and by he AscolArzela heorem, we conclude ha P s relavely compac se of C ([, T ]) L1 (, T ). Smlarly,we can show ha Q s relavely compac se of C ([, T ]) L1 (, T ).Therefore,he sequences {vn1, vn2 } and {wn1, wn2 } converge unformly o (v1, v2 ) and (w1, w2 ) on [, T ] respecvely.we have pon wse lms {vn1 (), vn2 ()} (v1, v2 ) and {wn1 (), wn2 ()} (w1, w2 ) as n on (, T ]. Moreover, by (3.8), he lm funcons sasfy he followng monoone propery v1 v11... vn1... v1 w1... wn1... w11 w1 v2 v12... vn2... v2 w2... wn2... w12 w2 (3.15) Now,we prove ha (v1, v2 ) and (w1, w2 ) are soluons of PBVP (2.1) - (2.2).We know σ1 (, η1, η2 ) = f1 (, η1, η2 ) M1 η1 Clearly, he funcon σ1 s connuous and monoone nondecreasng and monoone convergence of {vn1 ()} o v1 () as n on (, T ] mples ha σ1 (vn1 )() converges o σ1 (v1 )(), (, T ]. Le n n (3.11) and apply he domnaed convergence heorem,we observe ha v1 () sasfes he negral equaon v1 () = N1 (T s) 1 E, M1 (T s) σ1 (v1 )(s)ds ( s) 1 E, M2 ( s) σ2 (v2 )(s)ds, (3.17) T Γ() 1 E ( M ) 2 where N2 = [1 Γ()]E,, ( M2 T ). We conclude ha v2 () s an negral represenaon of he soluon o problem (3.3),.e.v2 () s an negral represenaon of he soluon o problem (2.1)-(2.2).By he assumpon of he funcon f2 and lemma 2.1,v2 () s he classcal soluon of he PBVP (2.1)-(2.2).Ths proves ha he lower sequence {vn2 ()} converges o a soluon v2 () of he problem (2.1)-(2.2). Smlarly,we can prove ha upper sequence {wn1, wn2 } converge unformly o a soluon (w1, w2 ) of perodc boundary value problems (2.1) - (2.2) and sasfes he relaon v1 () w1 () and v2 () w2 () (, T ]. I follows ha relaons (3.1) hold as well as (v1, v2 ) and (w1, w2 ) are mnmal and maxmal soluons of he PBVP (2.1)-(2.2) on he order nerval S respecvely. Fnally, f condon (2.9) holds, hen v () = w (), = 1, 2 s a unque soluon of he PBVP (2.1)-(2.2). I s suffcen o prove v () w (), (, T ], snce we have v () w (). We observe ha he funcon u () = v () w () sasfes he relaons D u () M1 u () = [ f (, w1, w2 ) f (, v, v2 )] M1 (v () w ()) u () = = u () =T, = 1, 2, (, T ] Then Lemma 2.3 mples ha u (), (, T ], whch proves v () w (), (, T ] and hence we oban ha v () = w () s a unque soluon of he PBVP (2.1)-(2.2). Ths complees he proof. Corollary 3.2. Assume ha ( s) 1 E, M1 ( s) σ1 (v1 )(s)ds, (3.16) T Γ() 1 E (T s) 1 E, M2 (T s) σ2 (v2 )ds ( M ) 1 where N1 = [1 Γ()]E,, ( M1 T ). We conclude ha v1 () s an negral represenaon of he soluon o problem (3.2),.e.v1 () s an negral represenaon of he soluon o problem 471 () v = (v1, v2 ) and w = (w1, w2 ) C ([, T ]) L1 (, T ) are lower and upper soluons of he PBVP (2.1) (2.2), such ha order relaon (2.3) holds, () funcon f (, u1, u2 ) C([, T ) R2, R) sasfes Lpschz condon (2.6),

6 Mehod of upper lower soluons for nonlnear sysem of fraconal dfferenal equaons and applcaons 472/472 () funcons f1 and f2 are quasmonoone nondecreasng. Then he PBVP (2.1) (2.2) has unque soluon n he order nerval. [11] Proof. Observe ha K (u u ) f (, u1, u2 ) f (, u 1, u 2 ) (3.18) K (u u ), (3.19) [12] for v u u w, whch follows from (2.6).e.Lpschz condons (2.4) and (2.5) hold wh K = M. Then he Theorem 3.1 mples ha he problem (2.1)-(2.2) has one and only one soluon n he ordered nerval. [13] 4. Acknowledgemen The frs auhor s graeful o he UGC,New Delh for award of Emerus FellowshpNo.F.6-6/215-17/EMERITUS OBC-7176/(SA-II) References [1] [2] [3] [4] [5] [6] [7] [8] [9] [1] [14] [15] R. Chaudhary and D. N. Pandey, Exsence resuls for nonlnear fraconal dfferenal equaon wh nonlocal negral boundary condons, Malaya J. Ma., 4(3)(216), D.B. Dhagude,J.A.Nanware and V.R.Nkam,Monoone echnque for weakly coupled sysem of Capuo fraconal dfferenal equaons wh perodc boundary condons,dynamcs of Connuous, Dscree and Impulsve Sysems,Seres A: Mahemacal Analyss, 19(212), R. Hlfer, Applcaons of Fraconal Calculus n Physcs, World Scenfc, Sngapore, 2. N.B.Jadhav and J.A.Nanware, Inegral boundary value problem for sysem of nonlnear fraconal dfferenal equaons, Bull. Marahwada Mah. Soc., 18(2)(217), A.A.Klbas,H.M.Srvasava and J.J.Trujllo, Theory and Applcaons of Fraconal Dfferenal Equaons, Elsever, Amserdam, 26. G.S.Ladde,V.Lakshmkanham and A.S.Vasala, Monoone Ierave Technques for Nonlnear Dfferenal Equaons, Pman Pub.Co.Boson, X. Lu, M. Ja and B. Wu, Exsence and unqueness of soluon for fraconal dfferenal equaons wh negral boundary condons,ejqtde, 69(29),1 1. F.A.McRae, Monoone erave echnque and exsence resuls for fraconal dfferenal Equaons,Nonlnear Analyss:TMA, 71(12)(29), Mohammed Belmekk,J.J.Neo and Rosana Rodrı guezlo pez,exsence of perodc soluons for a nonlnear fraconal dfferenal equaon, Boundary Value Problems, 29(29) Ar. ID J.A.Nanware and D.B.Dhagude, Monoone erave scheme for sysem of Remann-Louvlle fraconal 472 [16] [17] [18] [19] dfferenal equaons wh negral boundary condons,mah. Modellng Scence Compuaon, SprngerVerlag, 283(212), J.A.Nanware and D.B.Dhagude, Exsence and unqueness Of soluon of Remann-Louvlle fraconal dfferenal equaons wh negral boundary condons, Iner. J.Nonl. Sc., 14(4)(212), J.A.Nanware,N.B.Jadhav and D.B.Dhagude, Monoone erave echnque for fne sysem of Remann-Louvlle fraconal dfferenal equaons wh negral boundary condons,inerna. Conf. Mah. Sc., June 214, J.A.Nanware and D.B.Dhagude, Exsence and unqueness of soluons of dfferenal equaons of fraconal order wh negral boundary condons,j. Nonln Sc.Appl., 7(214), J.A.Nanware and D.B. Dhagude,Monoone echnque for fne weakly coupled sysem of Capuo fraconal dfferenal equaons wh perodc boundary condons,dyn.con., Ds. Impul. Sys.Seres A: Mah. Anal., 22(1)(215), J.A.Nanware, N.B.Jadhav and D.B.Dhagude, Inal value problems for fraconal dfferenal equaons nvolvng Remann-Louvlle dervave, Malaya J. Ma., 5(2)(217), C.V.Pao, Nonlnear Parabolc and Ellpc Equaons, New York,Plenum Press,1992. I. Podlubny, Fraconal Dfferenal Equaons,Academc Press, San Dego, J. Vasundhara Dev, Generalzed monoone mehod for perodc boundary value problems of Capuo fraconal dfferenal equaons, Comm. Appl.Anal., 12(4)(28), Z.We, W. Dong and J. Che, Perodc boundary value problems for fraconal dfferenal equaons nvolvng a Remann-Louvlle fraconal dervave, Nonlnear Analyss,73(21), ????????? ISSN(P): Malaya Journal of Maemak ISSN(O): ?????????