COMPARISON PRINCIPLES FOR DIFFERENTIAL EQUATIONS INVOLVING CAPUTO FRACTIONAL DERIVATIVE WITH MITTAG-LEFFLER NON-SINGULAR KERNEL

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1 Electronic Journl of Differentil Equtions, Vol (2018, No. 36, pp ISSN: URL: or COMPARISON PRINCIPLES FOR DIFFERENTIAL EQUATIONS INVOLVING CAPUTO FRACTIONAL DERIVATIVE WITH MITTAG-LEFFLER NON-SINGULAR KERNEL MOHAMMED AL-REFAI Communicted by Mokhtr Kirne Abstrct. In this rticle we study liner nd nonliner differentil equtions involving the Cputo frctionl derivtive with Mittg-Leffler non-singulr kernel of order 0 < < 1. We first obtin new estimte of the frctionl derivtive of function t its extreme points nd derive necessry condition for the existence of solution to the liner frctionl eqution. The condition obtined determines the initil condition of the ssocited frctionl initilvlue problem. Then we derive comprison principles for the liner frctionl equtions, nd pply these principles for obtining norm estimtes of solutions nd to obtin uniqueness results. We lso derive lower nd upper bounds of solutions. The pplicbility of the new results is illustrted through severl exmples. 1. Introduction Frctionl differentil equtions hve been implemented to model vrious problems in severl fields, [15, 19, 20, 21]. The non-loclity of the frctionl derivtive mkes frctionl models more prcticl thn the usul ones, especilly for systems which involve memory. In recent yers there re gret interests to develop new types of non-locl frctionl derivtive with non-singulr kernel, see [11, 13]. The ide is to hve more types of non-locl frctionl derivtives, nd it is the role of ppliction tht will determine which frctionl model is pproprite. The theory of frctionl models is effected by the type of the frctionl derivtive. Therefore, severl ppers hve been devoted recently to study the new types of frctionl derivtives nd their pplictions, see [2, 3, 7] for the Cputo-Fbrizio frctionl derivtive nd [4, 12, 14, 16, 22, 23] for the Abdon-Blenu frctionl derivtive. In this rticle, we nlyze the solutions of clss of frctionl differentil equtions involving the Cputo frctionl derivtive with Mittg-Leffler non-singulr kernel of order 0 < < 1. To the best of our knowledge this is the first theoreticl study of frctionl differentil equtions with frctionl derivtive of non-singulr kernel. We strt with the definition nd min properties of the nonlocl frctionl derivtive with Mittg-Leffler non-singulr kernel. For more detils the reder is referred to [11, 12, 1] Mthemtics Subject Clssifiction. 34A08, 35B50, 26A33. Key words nd phrses. Frctionl differentil equtions; mximum principle. c 2018 Texs Stte University. Submitted October 14, Published Jnury 29,

2 2 M. AL-REFAI EJDE-2018/36 Definition 1.1. Let f H 1 (, b, < b, (0, 1, the left Cputo frctionl derivtive with Mittg-Leffler non-singulr kernel is defined by ( ABC D f(t = B( 1 t E [ 1 (t s] f (sds. (1.1 where B( > 0 is normliztion function stisfying B(0 = B(1 = 1, nd E [s] is the well known Mittg-Leffler function. The derivtive is known in the literture by the Abdon-Blenu frctionl derivtive. Definition 1.2. Let f H 1 (, b, < b, (0, 1, the left Riemnn-Liouville frctionl derivtive with Mittg-Leffler non-singulr kernel is defined by ( ABR D f(t = B( d 1 dt t E [ The ssocited frctionl integrl is defined by 1 (t s] f(sds. (1.2 ( AB I f(t = 1 B( f(t + B( ( I f(t, (1.3 where ( I f(t is the left Riemnn-Liouville frctionl integrl of order > 0 defined by ( I f(t = 1 t (t s 1 f(sds. Γ( The following sttements hold: ( ABC 0D f(t = ( ABR 0D f(t B( 1 f(0e [ 1 t ], (1.4 ( ABR D AB I f(t = f(t, (1.5 ( AB I ABR D f(t = f(t. (1.6 The rest of the pper is orgnized s follows. In Section 2, we present new estimte of the frctionl derivtive of function t its extreme points. In Section 3, we develop new comprison principles for liner frctionl equtions nd obtin norm bound to their solutions. We lso, obtin the solution for clss of liner equtions in closed form, nd present necessry condition for the existence of their solutions. In Section 4, we consider nonliner frctionl equtions. We obtin uniqueness result nd derive upper nd lower bounds to the solution of the problem. Finlly we present some exmples to illustrte the pplicbility of the obtined results. 2. Estimtes of frctionl derivtives t extreme points We strt with estimting the frctionl derivtive of function t its extreme points, this result is nlogous to the ones obtined in [5] for the Cputo nd Riemnn-Liouville frctionl derivtives. The pplicbility of these results were indicted in ([6]-[10] by estblishing new comprison principles nd studying vrious frctionl diffusion models. Therefore, the current result cn be used to study frctionl diffusion models involving the Cputo nd Riemnn-Liouville frctionl derivtives with Mittg-Leffler non-singulr kernel, nd we leve this for future work.

3 EJDE-2018/36 COMPARISON PRINCIPLES 3 Lemm 2.1. Let function f H 1 (, b ttin its mximum t point t 0 [, b] nd 0 < < 1. Then ( ABC D f(t 0 B( 1 E [ 1 (t 0 ](f(t 0 f( 0. (2.1 Proof. We define the uxiliry function g(t = f(t 0 f(t, t [, b]. Then it follows tht g(t 0, on [, b], g(t 0 = g (t 0 = 0 nd ( ABC D g(t = ( ABC D f(t. Since g H 1 (, b, then g is integrble nd integrting by prts with yields u = E [ 1 (t 0 s ], dv = g (sds, t0 ( ABC D g(t 0 = B( E [ 1 1 (t 0 s ]g (s ds = B( ( E [ 1 1 (t 0 s ]g(s t0 t0 d ds E [ 1 (t 0 s ]g(sds = B( ( E [0]g(t 0 E [ 1 1 (t 0 ]g( t0 d ds E [ 1 (t 0 s ]g(sds = B( ( E [ 1 1 (t 0 ]g( t0 d ds E [ 1 (t 0 s ]g(sds. (2.2 We recll tht for 0 < < 1, see [17], we hve E [ t ] = 0 e rt K (rdr, where Thus, K (r = 1 π r 1 sin(π r 2 + 2r cos(π + 1 > 0. d ds E [ 1 (t 0 s ] = d [ ( ds E ] ( 1 1/ (t 0 s = d e r( 1 1/ (t d 0 s K (rdr = ds 0 0 ds e r( 1 1/ (t 0 s K (rdr = ( 1 1/ re r( 1 1/ (t 0 s K (rdr > 0, 0 (2.3

4 4 M. AL-REFAI EJDE-2018/36 which together with g(t 0 on [, b], will led to the integrl in (2.2 is nonnegtive. We recll here tht E [t] > 0, 0 < < 1, see [18], nd thus ( ABC D g(t 0 B( ( E [ 1 1 (t 0 ]g( = B( 1 E [ (2.4 1 (t 0 ](f(t 0 f( 0. The lst inequlity yields ( ABC D f(t 0 B( 1 E [ 1 (t 0 ](f(t 0 f( 0, which proves the result. By pplying nlogous steps for f we hve the following result. Lemm 2.2. Let function f H 1 (, b ttin its minimum t point t 0 [, b] nd 0 < < 1. Then ( ABC D f(t 0 B( 1 E [ 1 t 0](f(t 0 f( 0. (2.5 Lemm 2.3. Let function f H 1 (, b then it holds tht ( ABC D f( = 0, 0 < < 1. (2.6 Proof. Becuse E [ 1 (t s] is continuous on [, b], then it is in L2 [, b]. Applying the Cuchy-Schwrtz inequlity we hve ( ABC D f(t 2 B2 ( t ( (1 2 E [ 2 t ( 2 1 (t s ] ds f (s ds. (2.7 Since f H 1 (, b then f is squre integrble nd it holds tht ( f (s 2 ds = 0. The result is obtined s the first integrl in (2.7 is bounded. 3. Liner equtions We implement the results in Section 1 to obtin new comprison principles for the liner frctionl differentil equtions of order 0 < < 1, nd to derive necessry condition for the existence of their solutions. We then use these principles to obtin norm bound of the solution. We lso present the solution of certin liner eqution by the Lplce trnsform. Lemm 3.1 (Comprison Principle-1. Let function u H 1 (, b C[, b] stisfies the frctionl inequlity P (u = ( ABC D u(t + p(tu(t 0, t >, 0 < < 1, (3.1 where p(t 0 is continuous on [, b] nd p( 0. Then u(t 0, t. Proof. Since u H 1 (, b then by Lemm 2.3 we hve ( ABC D u( = 0. By the continuity of the solution, the frctionl inequlity (3.1 yields p(u( 0, nd hence u( 0. Assume by contrdiction tht the result is not true, becuse u is continuous on [, b] then u ttins bsolute mximum t t 0 with u(t 0 > 0. Since u( 0, then t 0 >. Applying the result of Lemm 2.1 we hve ( ABC D u(t 0 B( 1 E [ 1 (t 0 ](u(t 0 u( > 0.

5 EJDE-2018/36 COMPARISON PRINCIPLES 5 We hve ( ABC D u(t 0 + p(t 0 u(t 0 ( ABC D u(t 0 > 0, which contrdicts the frctionl inequlity (3.1, nd completes the proof. Corollry 3.2 (Comprison Principle-2. Let u 1, u 2 H 1 (, b C[, b] be the solutions of ( ABC D u 1 (t + p(tu 1 (t = g 1 (t, t >, 0 < < 1, ( ABC D u 2 (t + p(tu 2 (t = g 2 (t, t >, 0 < < 1, where p(t 0, g 1 (t, g 2 (t re continuous on [, b] nd p( 0. If g 1 (t g 2 (t, then u 1 (t u 2 (t, t. Proof. Let z = u 1 u 2, then P (z = ( ABC D z(t + p(tz(t = g 1 (t g 2 (t 0, t >, 0 < < 1. (3.2 By Lemm 3.1 we hve z(t 0, nd hence the result follows. Lemm 3.3. Let u H 1 (, b be the solution of ( ABC D u(t + p(tu(t = g(t, t >, 0 < < 1, (3.3 where p(t > 0 is continuous on [, b]. Then it holds tht u [,b] = mx u(t M = mx { g(t t [,b] t [,b] p(t }. Proof. We hve M g(t p(t, or Mp(t g(t for t [, b]. Let v 1 = u M, then P (v 1 = ( ABC D v 1 (t + p(tv 1 (t = ( ABC D u(t + p(tu(t p(tm = g(t p(tm g(t p(tm 0. Thus by Lemm 3.1 we hve v 1 = u M 0, which implies Anlogously, let v 2 = M u, then it holds tht P (v 2 = ( ABC D v 2 (t + p(tv 2 (t u M. (3.4 = ( ABC D u(t p(tu(t p(tm = g(t p(tm g(t g(t 0. Thus by Lemm 3.1 we hve v 2 = u M 0, thus u M. (3.5 By combining (3.4 nd (3.5 we hve u(t M, t [, b] nd hence the result follows. Lemm 3.4. The frctionl initil vlue problem ( ABC D u(t = λu + f(t, t > 0, 0 < < 1, (3.6 u(0 = u 0. (3.7 hs the unique solution 1 ( u(t = B(u 0 E [ωt ] + (1 (g(t f (t + f(0g(t, (3.8 B( λ(1

6 6 M. AL-REFAI EJDE-2018/36 in the functionl spce H 1 (0, b C[0, b], if nd only if, λu 0 + f(0 = 0, where λ ω = B( λ(1, nd t 1 g(t = E [ωt ] + 1 Γ( E [wt ]. Proof. Since u H 1 (0, b we hve ( ABC D u(0 = 0. Thus, necessry condition for the existence of solution to (3.6 is tht λu 0 + f(0 = 0. (3.9 Applying the Lplce trnsform to (3.6 nd using the fct tht we hve ( ABC 0D u(t = B( 1 E [ 1 t ] u (t, λl(u + L(f(t = B( ( 1 L E [ 1 t ] u (t. Applying the convolution result of the Lplce trnsform nd leds to Direct clcultions led to L(E [ 1 t ] = s 1 s + 1 s 1 λl(u + L(f(t = B( 1 s + s 1 B(u 0 L(u = B( λ(1 s ω + 1, 1 s < 1, 1 1 s + B( λ(1 (sl(u u(0. ( s ω L(f(t, (3.11 where ω = Let then λ B( λ(1. Thus, u(t = B(u 0 B( λ(1 L 1( s 1 s ω 1 + B( λ(1 L 1( s + 1 s ω L(f(t B(u 0 = B( λ(1 E [ωt ] 1 + B( λ(1 L 1( s + 1 s ω L(f(t. G(s = 1 s + 1 s s ω = s 1 s ω t 1 s s 1 s ω, g(t = L 1( G(s = E [ωt] + 1 Γ( E [ωt ]., (3.12

7 EJDE-2018/36 COMPARISON PRINCIPLES 7 Applying the convolution result we hve L 1( s + 1 s ω L(f(t = L 1( G(ssL(f(t = L 1( G(s[sL(f f(0 + f(0] = L 1( G(s[L(f + f(0] = L 1( G(sL(f + f(0g(s = g(t f (t + f(0g(t. The result follows by substituting (3.13 in (3.12. (3.13 Corollry 3.5. The frctionl differentil eqution ( ABC 0D u(t = λu, t > 0, 0 < < 1, (3.14 hs only the trivil solution u = 0, in the functionl spce H 1 (0, b C[0, b]. Proof. Applying Lemm 3.4 with f(t = 0, yields u(t = 1 B( λ(1 B(u 0E [ωt ]. The necessry condition for the existence of solution yields tht u 0 = 0, nd hence the result. 4. Nonliner equtions In this section we pply the obtined comprison principles to estblish uniqueness result for nonliner frctionl differentil eqution nd to estimte its solution. Lemm 4.1. Consider the nonliner frctionl differentil eqution ( ABC D u(t = f(t, u, t >, 0 < < 1, (4.1 where f(t, u is smooth function. If f(t, u is non-incresing with respect to u then the bove eqution hs t most one solution u H 1 (, b. Proof. Let u 1, u 2 H 1 (, b be two solutions of the bove eqution nd let z = u 1 u 2. Then ( ABC D z(t = f(t, u 1 f(t, u 2. Applying the men vlue theorem we hve for some u between u 1 nd u 2. Thus, f(t, u 1 f(t, u 2 = f u (u (u 1 u 2, ( ABC D z(t f u (u z = 0. (4.2 Since f u (u > 0, then z(t 0, by Lemm 3.1. Also,(4.2 holds true for z nd thus z 0, by virtue of Lemm 3.1. Thus, z = 0 which proves tht u 1 = u 2.

8 8 M. AL-REFAI EJDE-2018/36 Lemm 4.2. Consider the nonliner frctionl differentil eqution ( ABC D u(t = f(t, u, t >, 0 < < 1, (4.3 where f(t, u is smooth function. Assume tht λ 2 u + h 2 (t f(t, u λ 1 u + h 1 (t, where λ 1, λ 2 < 0. Let v 1 nd v 2 be the solutions of for ll t (, b, u H 1 (, b, ( ABC D v 1 (t = λ 1 v 1 + h 1 (t, t >, 0 < < 1, (4.4 nd ( ABC D v 2 (t = λ 2 v 2 + h 2 (t, t >, 0 < < 1. (4.5 Then v 2 (t u(t v 1 (t, t. Proof. We shll prove tht u(t v 1 (t nd by pplying nlogous steps one cn show tht v 2 (t u(t. By subtrcting (4.4 from (4.3 we hve ( ABC D (u v 1 (t = f(t, u λ 1 v 1 h 1 (t λ 1 u + h 1 (t λ 1 v 1 h 1 (t = λ 1 (u v 1. Let z = u v 1. Then ( ABC D z(t λ 1 z(t 0. Since λ 1 > 0, it follows tht z 0, by Lemm 3.1, which completes the proof. We now present some exmples to illustrte the obtined results. Exmple 4.3. Consider the nonliner frctionl initil vlue problem ( ABC 0D u(t = e u 2, t > 0, 0 < < 1, u(0 = ln(2. Since e u 2 u 1, letting v be the solution of (4.6 ( ABC 0D v(t = v 1, t > 0, 0 < < 1, (4.7 we hve v(t u(t by Lemm 4.2. The solution of (4.7 is given by (3.8 with λ = 1, nd f(t = 1. Thus, u(t v(t 1 ( = B(E [wt ] + (1 (E [wt ] + t 1 B( Γ( E [wt ], where ω = B(+1. We recll tht (4.7 hs solution only if v(0 = 1. Exmple 4.4. Consider the nonliner frctionl initil vlue problem ( ABC 0D u(t = e u 1 2 u2, t > 0, 0 < < 1, u(0 = u 0, (4.8 where u 0 is the unique solution of e u0 = 1 2 u2 0. By the Tylor series expnsion of f(u = e u, one cn esily show tht e u 1 2 u2 1 u. Let v be the solution of ( ABC 0D v(t = v + 1, t > 0, 0 < < 1, (4.9 then v(t u(t by virtue of Lemm 4.2. The solution of (4.9 is given by (3.8 with λ = 1, nd f(t = 1. Thus, u(t v(t

9 EJDE-2018/36 COMPARISON PRINCIPLES 9 = 1 ( B(E [wt ] + (1 (E [wt ] + B( t 1 Γ( E [wt ], where ω = B(+1. We recll tht (4.9 hs solution only if v(0 = 1. Moreover, pplying the result of Lemm 3.3 we hve v 1, nd hence u 1. Exmple 4.5. Consider the nonliner frctionl initil vlue problem ( ABC 0D u(t = e u (3 + cos(u + 4e t, t > 0, 0 < < 1, u(0 = 0. (4.10 Let h(u = e u (3 + cos(u, since h (u = e u ( sin(u 0, by the Tylor series expnsion method one cn esily show tht h(u h(0 + h (0u = 4 4u. Let v be the solution of ( ABC 0D v(t = 4v 4 + 4e t, t > 0, 0 < < 1, (4.11 v(0 = 0, then v(t u(t by Lemm 4.2. The solution of (4.11 is given by (3.8 where λ = 4, nd f(t = 4 + 4e t. Applying Lemm 3.3 we hve 4 + 4e t u v = 1 e t, t > 0. 4 Acknowledgments. The uthor cknowledges support from the United Arb emirtes University under the Fund No. 31S239-UPAR( References [1] T. Abdeljwd, D. Blenu; Integrtion by prts nd its pplictions of new nonlocl frctionl derivtive with Mittg-Leffler nonsingulr kernel, J. Nonliner Sci. Appl., 10 (2017, [2] T. Abdeljwd, D. Blenu; On frctionl derivtives with exponentil kernel nd their discrete versions, Journl of Reports in Mthemticl Physics, (2017. [3] T. Abdeljwd, D. Blenu; Monotonicity results for frctionl difference opertors with discrete exponentil kernels, Advnces in Difference Equtions ( :78 DOI /s [4] T. Abdeljwd, D. Blenu; Discrete frctionl differences with nonsigulr discrete Mittg- Leffler kernels, Advnces in Difference Equtions ( :232, DOI /s [5] M. Al-Refi; On the frctionl derivtive t extreme points, Electron. J. of Qulittive Theory of Diff. Eqn., 2012, no. 55 (2012, 1-5. [6] M. Al-Refi; Bsic results on nonliner eigenvlue problems of frctionl order, Electron. Journl of Differentil Equtions, 2012 (2012, [7] M. Al-Refi, T. Abdeljwd; Anlysis of the frctionl diffusion equtions with frctionl derivtive of non-singulr kernel, Advnces in Difference Equtions, ( : [8] M. Al-Refi, Yu. Luchko; Mximum principles for the frctionl diffusion equtions with the Riemnn-Liouville frctionl derivtive nd their pplictions. Frct. Clc. Appl. Anl., 17(2014, [9] M. Al-Refi, Yu. Luchko; Mximum principle for the multi-term time-frctionl diffusion equtions with the Riemnn-Liouville frctionl derivtives, Journl of Applied Mthemtics nd Computtion, 257 (2015, [10] M. Al-Refi, Yu. Luchko; Anlysis of frctionl diffusion equtions of distributed order: Mximum principles nd its pplictions, Anlysis 2015, DOI: /nly [11] A. Atngn, D. Blenu; New frctionl derivtives with non-locl nd non-singulr kernel: theory nd pplictions to het trnsfer model, Therm. Sci., 20 (2016, [12] A. Atngn, I. Koc; Chos in simple nonliner system with Atngn-Blenu derivtives with frctionl order, Chos, Solitons nd Frctls, 89 (2016,

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