MATHEMATICAL COMMUNICATIONS 273 Mth. Commun. 21(2016), 273 281 Sturm s theorems for conformble frctionl differentil equtions Michl Pospíšil 1, nd Luci Pospíšilová Škripková2 1 CEITEC Centrl Europen Institute of Technology, Brno University of Technology, Technická 3058/10, 61 600 Brno, Czech Republic 2 Wolkrov 15, 85101 Brtislv, Slovk Republic Received November 26, 2015; ccepted Mrch 21, 2016 Abstrct. In the present pper, we mke use of locl properties of the recently estblished definition of conformble frctionl derivtive. Sturm s seprtion nd Sturm s comprison theorems re proved for differentil equtions involving conformble frctionl derivtive of order 0 < α 1. AMS subject clssifictions: 34A08, 26A33 Key words: Frctionl derivtive, frctionl integrl, frctionl Picone identity 1. Introduction In the lst decdes, the interest in frctionl differentil equtions is rpidly growing, nd clssic frctionl derivtives such s Cputo, Riemnn-Liouville or Hdmrd seemtobewell-developed(seee.g. [8,9]). Onethingthtllthesehveincommonis tht they re defined s integrls with different singulr kernels, i.e., they hve nonlocl structure. Due to this fct, none of the clssic frctionl derivtives stisfies n nlog of integer-orderproduct rule: (fg) = f g+fg for C 1 -functions f, g. On the other hnd, recently introduced definition of the so-clled conformble frctionl derivtive (see Definition 1 below) involves limit insted of n integrl. This locl definition enbles us to prove mny properties nlogous to those of integer-order derivtive (cf. [1, 7]). Nowdys, the Cuchy problems involving conformble frctionl derivtive [4] nd frctionl semigroups [2] re lso investigted. We note tht the notion of conformble frctionl derivtive ws generlized in [5] to time scles. In this pper, we stte nd prove Sturm s theorems (see e.g. [6] for clssic sttements) for differentil equtions with conformble frctionl derivtives. Our results my be used s foundtion for studies of oscilltory properties of conformble frctionl differentil equtions. For the simplicity, we denote by u the Eucliden norm of vector u R n without ny respect to the dimension n N. In the present pper, we lwys ssume 0 < α 1. The pper is orgnized s follows. In the next section, we recll some bsic definitions nd known results tht will be in force when proving the min results. Corresponding uthor. Emil ddresses: michl.pospisil@ceitec.vutbr.cz (M.Pospíšil), luci.skripkov@gmil.com (L.Pospíšilová Škripková) http://www.mthos.hr/mc c 2016 Deprtment of Mthemtics, University of Osijek
274 M.Pospíšil nd L.Pospíšilová Škripková Here we lso prove result on the existence nd uniqueness of solution of n initil vlue problem. Section 3 is devoted to the min results of this pper, nd t the end, we present n exmple compring solutions of frctionl equtions with constnt coefficients. 2. Preliminry results Here we recll bsic notions, nd provide results helpful for the min section. The bsic definition is from [7]. Definition 1. Let 0 < α 1. The conformble frctionl derivtive of function f: [, ) R n is defined s f(t+ε(t ) 1 α ) f(t) D α f(t) = lim, t >, ε 0 ε D α f() = lim t D α f(t). + If D α f(t 0 ) exists nd is finite, we sy tht f is α differentible t t 0. For 2 n N we denote D n αf(t) = D α Dα n 1 f(t). The conformble frctionl integrl is defined s I α f(t) = Whenever = 0, we omit the lower index. f(s) (s ) 1 αds, t. Note tht if f: [, ) R n is differentible t t 0, then D α f(t 0 ) = (t 0 ) 1 α f (t 0 ). We dd severl lemms on properties of conformble frctionl derivtive. The next one is from [7]. Lemm 1. Let f: [, ) R n be continuous function, nd 0 < α 1. Then D α I α f(t) = f(t), t >. It is elementry to prove the next two lemms using the continuity of n α differentible function [7, Theorem 2.1] in the first cse, nd the men vlue theorem for α differentible functions [7, Theorem 2.4] in the second. Lemm 2. Let 0 < α 1, f be differentible t g(t), nd g α differentible t t >. Then D α (f g)(t) = f (g(t)) D α g(t). Lemm 3. If f: (,b) R is α differentible in (,b) nd D α f is positive (negtive) on the whole (,b), then f is incresing (decresing) on (,b). TheinterestedredermightcompreLemm2with thechinrulein[1, Theorem 2.11]. Note tht we immeditely obtin the sttement complementry to Lemm 3: If the function f is α differentible on (, b) nd incresing (decresing) on (, b), then D α f(c) 0 ( D α f(c) 0) for ll c (,b).
Sturm s theorems for conformble FDEs 275 In the present pper, when tlking bout solution of frctionl differentil eqution with the conformble frctionl derivtive we shll lwys hve continuous solution on mind. Next, we present n integrl eqution corresponding to the initil vlue problem for conformble frctionl differentil eqution D α x(t) = f(t,x(t)), t >, x() = x. (1) Lemm 4. Let f C([, ) R n,r n ) be given function. Then the solution x of the initil vlue problem (1) stisfies x(t) = x + f(s,x(s)) (s ) 1 αds, t. (2) Proof. Since f nd x re continuous, I α f(t) exists. Tht mens tht I α D α x(t) exists. Then by [1, Corollry 2.7], x is differentible, nd by [1, Lemm 2.8] I α D α x(t) = x(t) x(). (3) Therefore, pplying the opertor I α on eqution (1) yields for ny t >. Tht ws to be proved. x(t) x() = I α f(t) Note tht the ltter lemm holds for f defined on [,T] B(u,r), where T > nd { } B(u,r) = x C([,T],R n ) mx x(t) u r t [,T] is closed bll in C([,T],R n ). The following result gives sufficient condition for the existence of unique solution. Theorem 1. Let 0 < α 1 nd let f: [,T] B(x,r) R n be given continuous function, where T >. Suppose tht f(t,x) M for ll t [,T], x B(x,r), for some M > 0. Moreover, let f(t, ) be Lipschitz continuous with the constnt L for ll t [,T]. Then there exists unique solution of the initil vlue problem (1) { defined on [,T 1 ] with T 1 = min T,+ ( αr M Proof. Let us introduce the Bnch spce Z = C([,T 1 ],R n ) equipped with the Bielicki norm x Z = mx t [,T1]e Lβt x(t), where Define the opertor F: Z Z s ) 1 α }. ( ) 1 α+le L α β :=. α (Fx)(t) := x + f(s,x(s)) (s ) 1 αds.
276 M.Pospíšil nd L.Pospíšilová Škripková Clerly, F is well defined. Moreover, F mps B(x,r) into itself. Indeed, for ny t [,T 1 ] we hve (Fx)(t) x f(s, x(s)) M(t )α (s ) 1 αds M(T 1 ) α r. α α Next we show tht F is contrction. Let t [,T 1 ] be rbitrry nd fixed, nd x,y B(x,r). Then (Fx)(t) (Fy)(t) L f(s,x(s)) f(s,y(s)) (s ) 1 α ds e Lβs (s ) 1 αds x y Z. Now, we split = +ε + +ε for ε := β 1. If t < +ε, omit the second integrl in the following inequlities for the estimtions to hold. +ε (Fx)(t) (Fy)(t) L (e Lβ(+ε) ds (s ) 1 α + 1 ) ε 1 α e Lβs ds x y Z +ε = L (e Lβ(+ε)εαα + elβt e Lβ(+ε) ) x y Z. Lβε 1 α Hence ( ) Le e Lβt (Fx)(t) (Fy)(t) ε α Lβ( t+ε) + 1 elβ( t+ε) x y Z α βε ( ) Le ε α L α +1 el x y Z = (1 ε α e L ) x y Z. Tking the mximum over ll t [,T 1 ], one obtins Fx Fy Z (1 ε α e L ) x y Z. The sttement follows by the Bnch fixed point theorem nd Lemm 4. The following definition of α Wronskin is from [3], nd will be in force in the next section. Definition 2. Let x, y be given functions α differentible on [,b], α (0,1]. We set W α [x,y](t) := x(t) y(t) D α x(t) D α y(t). 3. Sturm s theorems In this section, we consider the sclr frctionl differentil eqution of second order of the form D 2 αx(t)+( D α x(t))p(t)+x(t)q(t) = 0, t > (4) with continuous functions p, q. Clssiclly [6], two functions x, y continuous on [, b] will be clled linerly dependent if there exist c 1,c 2 R such tht c 1 + c 2 > 0 nd
Sturm s theorems for conformble FDEs 277 c 1 x(t)+c 2 y(t) 0 for ll t [,b]. In the other cse, they re linerly independent. Clerly, if one of two functions is identiclly equl to zero, they cnnot be linerly independent. Using the formul W α [x,y](t) = e 1 p(s) (s ) 1 αds W α [x,y]( 1 ), t (,b) for two solutions, x nd y, of (4) nd some 1 (,b), which follows from [3, Theorem 2.2], we immeditely obtin the next equivlent condition. Lemm 5. Two solutions x, y of eqution (4) defined on (,b) for some < b re linerly independent if nd only if W α [x,y](t) 0 for ll t (,b). One of the min results of this pper follows. It is Sturm s seprtion theorem for frctionl differentil equtions with the conformble derivtive. Theorem 2. Let x, y be linerly independent solutions of (4) defined on (,b) (b is llowed to be + ), p nd q given continuous functions, nd 0 < α 1. Then x hs zero between ny two successive zeros of y. Thus the zeros of x nd y occur lterntely. Proof. Liner independence yields W α [x,y](t) = x(t) D α y(t) ( D α x(t))y(t) 0, t (,b). (5) So W α [x,y] does not chnge the sign over (,b). Suppose tht,t 2 (,b) re two successive zeros of y. Note tht D α y(t i ) 0 for ech i = 1,2. Otherwise, Theorem 1 gives y 0 contrdiction with liner independence. This lso mens tht the zeros of y (nd lso of x) re isolted. Hence, from (5), x( )( D α y( ))x(t 2 ) D α y(t 2 ) > 0. (6) Let us ssume without ny loss of generlity tht D α y( ) > 0. By Lemm 3, y is incresing t. Since y is continuous, nd t 2 is zero of y next to, y is decresing t t 2. By the corollryof Lemm 3, D α y(t 2 ) < 0. Similrly, the cse D α y( ) < 0 gives D α y(t 2 ) > 0. Therefore, ( D α y( )) D α y(t 2 ) < 0, i.e., x( )x(t 2 ) < 0 by (6). The continuity of x yields the existence of t 3 (,t 2 ) such tht x(t 3 ) = 0. Note tht x hs only one zero in (,t 2 ). Indeed, if there were t 4 t 3 in (,t 2 ) such tht x(t 4 ) = 0, then pplying the bove rguments would result in the existence of t 5 (t 3,t 4 ) or t 5 (t 4,t 3 ) such tht y(t 5 ) = 0, i.e. there is nother zero of y between nd t 2, wht is contrdiction. The next result is Sturm s comprison theorem for conformble frctionl differentil equtions. Theorem 3. Let x nd y be nontrivil solutions of the equtions D 2 αx(t)+x(t)r(t) = 0, t >, D 2 α y(t)+y(t)r 1(t) = 0, t >, respectively, where r(t) r 1 (t) for t > re given continuous functions. Then exctly one of the following conditions holds:
278 M.Pospíšil nd L.Pospíšilová Škripková (1) x hs t lest one zero between ny two zeros of y, (2) r(t) = r 1 (t) for ll t >, nd x is constnt multiple of y. Proof. Let us suppose tht (1) does not hold. Let < < t 2 be two consecutive zeros of y. Thus W α [x,y](t i ) = x(t i ) D α y(t i ) for i = 1,2. Without ny loss of generlity, we ssume tht x(t),y(t) > 0 on (,t 2 ) (otherwise, tke x or y). Similrly to the proof of Theorem 2, D α y( ) > 0 > D α y(t 2 ). Therefore, W α [x,y]( ) = x( ) D α y( ) 0, W α [x,y](t 2 ) = x(t 2 ) D α y(t 2 ) 0. For the derivtive, we hve (7) D α W α [x,y](t) = x(t)y(t)(r(t) r 1 (t)) 0, t (,t 2 ). If there is some t 0 (,t 2 ) such tht D α W α [x,y](t 0 ) > 0, then from f( ) = f()+ I α D α f( ), f(t 2 ) = f()+ I α D α f(t 2 ) for continuous function f, nd using (7), we obtin 2 D α W α [x,y](s) 0 W α [x,y](t 2 ) = W α [x,y]( )+ (s ) 1 α ds > 0, wht is contrdiction. Hence, r(t) = r 1 (t) for ll t (,t 2 ). Note tht now x nd y solve the sme eqution. So, Theorem 2 yields tht x nd y relinerly dependent. Obviously, if (2) is vlid, then x cnnot hve zero between two successive zeros of y, i.e., (1) does not hold. The proof is complete. For the finl result of this section, we shll need the frctionl version of the Picone identity. For the simplicity, we omit the rgument t. Lemm 6. Let u, v be nontrivil solutions of the equtions D α (p D α u)+qu = 0 on (, ), D α (p 1 D α v)+q 1 v = 0 on (, ), (8) respectively, where q ndq 1 re given continuousfunctions, p nd p 1 re α differentible. Then ( u D α v (p( D α u)v p 1 u( D α v))) ( = (q 1 q)u 2 +(p p 1 )( D α u) 2 +p 1 D α u ( D α v) u ) 2. (9) v
Sturm s theorems for conformble FDEs 279 Proof. The sttement is obtined by direct differentition of the left-hnd side, pplying the rules for the derivtive of product nd frction of two functions [7] ( ) f D α (fg) = ( D α f)g +f( D α g), D α = ( D α f)g f( D α g) g g 2, nd using (8). The following result is generliztion of Theorem 3. Theorem 4. Let u, v be nontrivil solutions of equtions (8), where 0 < p 1 (t) p(t), q(t) q 1 (t) for t > re given continuous functions, nd p nd p 1 re α differentible. Then between ny two zeros,t 2 > of u, there exists t lest one t 0 [,t 2 ] such tht v(t 0 ) = 0. Proof. Let < < t 2 be two consecutive zeros of u. Let us ssume tht u(t) > 0 for ll t (,t 2 ), nd conversely, tht v(t) > 0 for ll t [,t 2 ] (tke u or v if needed). Then pplying the opertor I α (t 2 ) I α ( ) to the Picone identity (9), we get [ ] t2 u(t) 0 = 2 = v(t) (p(t)( D α u(t))v(t) p 1 (t)u(t)( D α v(t))) 2 + 2 = 2 2 t= (q 1 (t) q(t))u 2 (t) (p(t) p 1 (t))( D α u(t)) 2 (t ) 1 α dt+ (t ) 1 α dt ( p 1 (t) (t ) 1 α D α u(t) ( D α v(t)) u(t) ) 2 dt v(t) ( p 1 (t) (t ) 1 α D α u(t) ( D α v(t)) u(t) ) 2 dt v(t) p 1 (t)v 2 ( ( )) 2 (t) u(t) (t ) 1 α D α dt 0 v(t) with the id of (3). Now, the right-hnd side of the bove inequlity is zero if nd only if ( ) u(t) D α = 0 v(t) for ll t (,t 2 ), i.e., u/v is constnt on (,t 2 ). Then continuity of u yields tht u(t) = 0 for ll t [,t 2 ], which is in contrdiction with the positivity of u on (,t 2 ). Finlly, we present n exmple of equtions with constnt coefficients, illustrting the ppliction of the ltter theorem. Exmple 1. Let us consider the couple of equtions D1 (4 D1 2 2 for R nd prmeter q 1 R. u)+u = 0 on (, ), (10) D1 2 (4 D1 2 v)+q 1v = 0 on (, ) (11)
280 M.Pospíšil nd L.Pospíšilová Škripková It cn be esily verified tht π ) u(t) = sin( 4 + t (12) solves (10) long with u() = 2 2, D1 u() = 2 2 4. Similrly, eqution (11) hs solution ( v(t) = sin c 1 + ) q 1 (t ) (13) stisfying v() = sin(c 1 ), D1 2 v() = q 1 2 cos(c 1). Therefore, if q 1 < 1, function v oscilltes more slowly thn u, nd eventully one of its zeros will not lie between two successive zeros of u. On the other hnd, v oscilltes fster thn u whenever q 1 > 1. This coincides with the sttement of Theorem 4, nd is depicted in Figure 1. Figure 1: For = 0, function u of (12) (solid), v of (13) with c 1 = 0 nd q 1 = 5 8 (dshed), q 1 = 2 (dotted) Acknowledgements This pper ws relised in the CEITEC Centrl Europen Institute of Technology with reserch infrstructure supported by the project CZ.1.05/1.1.00/02.0068 finnced by Europen Regionl Development Fund, nd t the Deprtment of Mthemtics nd Sttistics, Fculty of Science, Msryk University, Brno, Czech Republic. References [1] T. Abdeljwd, On conformble frctionl clculus, J. Comput. Appl. Mth. 279(2015), 57 66. [2] T. Abdeljwd, M. Al Horni, R. Khlil, Conformble frctionl semigroups of opertors, J. Semigroup Theory Appl. 2015(2015), Article ID 7, 9 pges. [3] M. Abu Hmmd, R. Khlil, Abel s formul nd Wronskin for conformble frctionl differentil equtions, Int. J. Differ. Equ. Appl. 13(2014), 177 183.
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