Ageneralizedq-fractional Gronwall inequality and its applications to nonlinear delay q-fractional difference systems

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1 Abdeljwd et l. Journl of Ineulities nd Applictions (206) 206:240 DOI 0.86/s R E S E A R C H Open Access Agenerlized-frctionl Gronwll ineulity nd its pplictions to nonliner dely -frctionl difference systems Thbet Abdeljwd *, Jehd Alzbut nd Dumitru Blenu 2,3 * Correspondence: tbdeljwd@psu.edu.s Deprtment of Mthemtics nd Physicl Sciences, Prince Sultn University P. O. Box 66833, Riydh, 586, Sudi Arbi Full list of uthor informtion is vilble t the end of the rticle Abstrct In thispper, westte ndprove new discrete -frctionl version of the Gronwll ineulity. Bsed on this result, prticulr version expressed by mens of the -Mittg-Leffler function is provided. To pply the proposed results, we prove the uniueness nd obtin n estimte for the solutions of nonliner dely Cputo -frctionl difference system. We exmine our results by providing numericl exmple. MSC: 26A33; 39A Keywords: generlized -frctionl Gronwll ineulity; dely -frctionl difference system; uniueness of solution; estimte for the solution Bckground The study of -difference eutions hs gined intensive interest in the lst yers. It hs been shown tht these types of eutions hve numerous pplictions in diverse fields nd thus hve evolved into multidisciplinry subjects [ 0]. For more detils on -clculus, we refer the reder to the remrkble monogrph []. On the other hnd, the frctionl differentil eutions hve recently received considerble ttention in the lst two decdes. Indeed, mny reserchers hve investigted these types of eutions due to their significnt pplictions in vrious fields of science nd engineering; see for instnce the monogrphs [2 4] nd the references therein. The corresponding theory of frctionl difference eutions is considered to be t its first stges of progress; we suggest [5 26] whose uthors hve tken the led to promote nd develop this theory. The -frctionl clculus nd differentil eutions hve been recently studied in mny ppers; we recommend the monogrph [27] ndthepperscited therein. For the -frctionl difference eutions which serve s bridge between frctionl difference eutions nd -difference eutions there hve ppered some ppers which study the ulittive properties of solutions [5, 27 33]. However, less ttention hs been pid to these types of eutions in the literture. The differentil nd integrl ineulities, which re considered s n effective tools for studying solutions properties, hve lso been under considertion. Due to its benefit in the determintion of uniueness, boundedness nd stbility of solutions, in prticulr, the 206 Abdeljwd et l. This rticle is distributed under the terms of the Cretive Commons Attribution 4.0 Interntionl License ( which permits unrestricted use, distribution, nd reproduction in ny medium, provided you give pproprite credit to the originl uthor(s) nd the source, provide link to the Cretive Commons license, nd indicte if chnges were mde.

2 Abdeljwd et l. Journl of Ineulities nd Applictions (206) 206:240 Pge 2 of 3 Gronwll ineulity hs been min trget for mny reserchers. There hve ppered severl versions for the Gronwll ineulity in the literture; we list here those results which concern with frctionl differentil or difference eutions [34 39]. For 0 < <,wedefinethetimesclet = { n : n Z} {0}, wherez is the set of integers. For = n 0 nd n 0 Z, wedenotet =[, ) = { i : i =0,,2,...}. In[40], which is probbly the first pper in this subject, the current uthors hve estblished discrete -frctionlversionofthegronwllineulity. Indeed, theyobtinedthetheorem given below. Theorem [40] Let α >0,undμ be nonnegtive rel vlued functions such tht 0 μ(t)< t α ( ) α for ll t T (in prticulr if 0 μ(t)< ( ) α ) nd Then u(t) u()+ α u(t)μ(t). u(t) u() Eμ k, k=0 where Eμ k = μk (t ) kα Ɣ (kα+). Bsed on the result of Theorem, the following prticulr estimte which is expressed by mens of the -Mittg-Leffler function ws lso concluded. Theorem 2 [40] Let 0 δ(t)< ( ) for ll t T. If then u(t) u()+ u(t) u()e (t, ), δ(s)u(s) s, where e (t, )= E (, t ) is the nbl -exponentil function for the time scle T. To pply these results, the uthors considered the following discrete -frctionl initil vlue problem: { C αx(t)=f (t, x(t)), 0 < α, T, t T, () x()=γ. Here C α mens the Cputo frctionl difference of order α nd f (t, y) fulfills Lipschitz condition for ll t nd y. Theuniuenessofsolutionsswellsthedependenceonthe initil dt were proved. The purpose of our mnuscript is to extend the results in Theorem nd Theorem 2 nd obtin new discrete -frctionl version of the Gronwll ineulity vlid for nonliner systems contining dely rguments. As n ppliction, we will prove the uniueness nd obtin n estimte forthe solutions of nonlinerdely Cputo -frctionl difference systems. We exmine our results by presenting numericl exmple.

3 Abdeljwd et l. Journl of Ineulities nd Applictions (206) 206:240 Pge 3 of 3 2 Auxiliry ssertions Before strting, we provide some bsic nbl nottions, definitions, nd lemms tht will be used in the seuel. Let f : T R.Wedefinethenbl-derivtive of f by f (t)= f (t) f(t), t T {0}. (2) ( )t The nbl -integrl of f hs the following form: f (s) s =( )t 0 i=0 i f ( t i) (3) nd for 0 T f (s) s = 0 f (s) s 0 f (s) s. The definition of the -fctoril function for n N is given by n (t s) n = ( t i s ). (4) i=0 In the cse α is nonpositive integer, the -fctoril function is defined by (t s) α = tα i=0 s t i s. (5) i+α t Below we present some of the properties of -fctoril functions within the following lemm. Lemm [28] For α, γ, β R, we hve (i) (t s) β+γ =(t s) β (t β s) γ. (ii) (t s) β = β (t s) β. (iii) The nbl -derivtive of the -fctoril function with respect to t is (t s) α = α (t s)α. (iv) The nbl -derivtive of the -fctoril function with respect to s is (t s) α = α (t s)α. For function f : T R, theleft-frctionl integrl α ndstrtingt0< T is defined by α f (t)= of order α 0,, 2,... (t s) α f (s) s, (6)

4 Abdeljwd et l. Journl of Ineulities nd Applictions (206) 206:240 Pge 4 of 3 where Ɣ (α +)= α Ɣ (α), Ɣ () =, α >0. (7) One should note tht the left -frctionl integrl α functions defined on T. mps functions defined on T to Definition [32] Let 0 < α / N. Then the Cputo left -frctionl derivtive of order α of function f defined on T is defined by C α f (t) (n α) n f (t)= Ɣ (n α) (t s) n α n f (s) s, (8) where n =[α]+.inthecseα N, wemywrite C αf (t) n f (t). The (left) Riemnn -frctionl derivtive is defined by ( αf )(t)=( (n α) f )(t). In virtue of [32], the Riemnn nd Cputo -frctionl derivtives re relted by (C α f ) (t)= ( α f ) (t) (t ) α f (). (9) Ɣ ( α) Lemm 2 [32] Let α >0nd f be defined in suitble domin. Thus n α C α f (t)=f(t) (t ) k Ɣ (k +) k f () (0) nd if 0<α we hve k=0 α C α f (t)=f(t) f(). () The following identity plys crucil role in solving the liner -frctionl eutions: α (x )μ = Ɣ (μ +) (x )μ+α (0 < < x < b), (2) Ɣ (α + μ +) where α R + nd μ (, ). The -nlog of the Mittg-Leffler function with double index (α, β)isintroducedin[32]. It ws defined s follows. Definition 2 [32] Forz, z 0 C nd R(α)>0,the-Mittg-Leffler function is defined by E α,β (λ, z z 0 )= k=0 λ k (z z 0) αk Ɣ (αk + β). (3) In the cse β =,weutilize E α (λ, z z 0 ):= E α, (λ, z z 0 ). Exmple [32] Let 0 < α nd consider the left Cputo -frctionl difference eution C α y(t)=λy(t)+f (t), y()= 0, t T. (4)

5 Abdeljwd et l. Journl of Ineulities nd Applictions (206) 206:240 Pge 5 of 3 The solution of (4)is given by y(t)= 0 E α (λ, t )+ (t s) α E α,α ( λ, t α s ) f (s) s. If insted we use the modified -Mittg-Leffler function e α,β (λ, z z 0 )= k=0 λ k (z z 0) αk+(β ), Ɣ (αk + β) then the solution representtion becomes y(t)= 0 e α (λ, t )+ e α,α (λ, t s)f (s) s. Remrk [32] Ifwesetα =,λ =, =0,ndf (t)=0in(4), we obtin -exponentil formul e (t)= t k k=0 Ɣ (k+) on the time scle T,whereƔ (k +)=[k]!=[] [2] [k] with [r] = r.werecllthte (t)=e (( )t), where E (t) denoted specil cse of the bsic hypergeometric series, nmely E (t)= φ 0 (0;, t)= ( n t ) t n =, () n n=0 where () n =( )( 2 ) ( n )denotesthe-pochhmmer symbol. n=0 3 Agenerlized-Gronwll ineulity We stte nd prove the generlized -Gronwll ineulity. Theorem 3 Let α >0, u(t), v(t) be nonnegtive functions nd w(t) be nonnegtive nd nondecresing function for t [, ) = { i : i =0,,2,...}, = n 0 for some n 0 Z such tht w(t) MwhereMisconstnt. If u(t) v(t)+w(t) α u(t), (5) then u(t) v(t)+ Proof Define Bφ(t)=w(t) It follows tht ( w(t)ɣ (α) ) k kα v(t). (6) k= (t s) α φ(s) s, t T. u(t) v(t)+bu(t),

6 Abdeljwd et l. Journl of Ineulities nd Applictions (206) 206:240 Pge 6 of 3 which implies tht u(t) n k=0 Bk v(t)+b n u(t). We clim tht B n u(t) (w(t)) n Ɣ (nα) (t s) nα u(s) s (7) nd B n u(t) 0sn for t T.Itisesytoseetht(7) isvlidforn =. Assume tht it is true for n = k,thtis, B k u(t) If n = k +,then (w(t)) k Ɣ (kα) B k+ u(t)=b ( B k u(t) ) w k+ (t) = w k+ (t) = (w(t)ɣ (α)) k+ Ɣ (kα) = (w(t)ɣ (α)) k+ Ɣ (kα) r (t s) kα u(s) s. () k Ɣ (kα) [ s (t s) α (t s)α r () k Ɣ (kα) (s r) kα u(r) s r (s r)kα u(r) r s ] (t s) α (s r) kα s u(r) r r α (s r)kα u(r) r, where r αu(t)= t r (t s)α u(s) hs been used. It follows from (2)tht B k+ u(t) (w(t)ɣ (α)) k+ Ɣ (kα) = (s r) (k+)α (w(t)) k+ (s r)(k+)α u(r) r. Ɣ ((k +)α) Ɣ (kα) Ɣ ((k +)α) u(r) r Therefore, eution (7) is obtined. Furthermore nd becuse the denomintor goes to infinity fster thn the numertor in the below ineulity, one cn conclude tht B n u(t) (M) n Ɣ (nα) To complete the proof, we let n in (t s) nα u(s) s 0 sn, t [, ). n n u(t) B k v(t)+b n u(t)=v(t)+ B k v(t)+b n u(t) to obtin k=0 u(t) v(t)+ B k v(t). k= k= With the help of the semigroup property α μ = (α+μ) nd the definition of B we get (6). This completestheproof.

7 Abdeljwd et l. Journl of Ineulities nd Applictions (206) 206:240 Pge 7 of 3 The following immedite conseuence of the bove theorem plys key role in our subseuent nlysis. Corollry Under the hypotheses of Theorem 3, ssume further tht v(t) is nondecresing function for t T, then u(t) v(t) E α ( w(t)ɣ (α), t ), t T. (8) Proof From (6) ndthessumptionthtv(t) is nondecresing function for t T,we my write [ u(t) v(t) + k= (w(t)) k Ɣ (kα) (t s) kα s ] or [ u(t) v(t) + kα ( w(t)ɣ (α) ) ] k. k= Then, with the help of (2) it follows tht [ ( u(t) v(t) + w(t)ɣ (α) ) ] k kα k= [ ( = v(t) + w(t)ɣ (α) ) ] k (t )kα Ɣ (kα +) = v(t) k=0 k= (w(t)) k (t ) kα Ɣ (kα +) = v(t) E α ( w(t)ɣ (α), t ). The proof is complete. 4 Applictions to nonliner dely -frctionl difference systems Let R m be the m-dimensionl Eucliden spce nd define I τ = {τ, τ, 2 τ,...,}, N 0 = {0,,2,3,...} nd T τ =[τ, ) = {τ, τ, 2 τ,...} where τ = d T, d N 0 nd I τ = {} with d =0isthenon-delycse.Weobtinourfirstpplictionbyproving the uniueness of the solution for the system: { C αx(t)=a 0x(t)+A x(τt)+f (t, x(t), x(τt)), t [, ), x(t)=ϕ(t), t I τ, where C α denotes the Cputo frctionl difference of order α (0, ), the stte vector x : T τ R m, the constnt mtrices A 0 nd A re of pproprite dimensions, the nonlinerity f : T τ R m R m R m nd the initil function ϕ : I τ R m.let be ny Eucliden norm nd be the mtrix norm induced by this vector. Let D = D(N 0 R m R m, R m ) be the set of ll bounded functions (seuences). Clerly, the spce D is Bnch spce induced by the norm z D := sup t Iτ z(t). (9)

8 Abdeljwd et l. Journl of Ineulities nd Applictions (206) 206:240 Pge 8 of 3 We mke use of the following ssumptions: (H.) f D(T R m R m, R m ) is Lipschitz-type function. Tht is, there exists positive constnt L >0such tht f ( t, x(t), x(τt) ) f ( t, y(t), y(τt) ) L ( x(t) y(t) + x(t τ) y(τt) ), for t [, ). (H.2) There exists positive constnt L 2 such tht f (t, x(t), x(τt)) L 2. The first result in this section provides representtion for the solutions of system (9) tht will be useful in the subseuent nlysis. Theorem 4 x : T τ R m is solution of system (9) if nd only if x(t)=ϕ()+ t s= (t s)α [A 0 x(s)+a x(τs)+f (s, x(s), x(τs))], t [, ), x(t)=ϕ(t), t I τ. (20) Proof For t I τ,itisclerthtx(t)=ϕ(t) is the solution of (9). For t T, we pply α on both sides of eution (20)toobtin α x(t)=ϕ() t α Ɣ ( α) + A 0x(t)+A x(τt)+f ( t, x(t), x(τt) ), where ( α α u)(t) =u(t) hve been used. By using eution (9), we end up with the desired form C α t x(t)=a 0x(t)+A x(τt)+f ( t, x(t), x(τt) ), t [, ). From system (9), we cn see tht x(t)=ϕ(t) fort I τ.fort [, ), we pply α both sides of eution (9)toget α [ C α x(t)] = In view of eution (), one cn esily see tht (t s) α [ A0 x(s)+a x(τs)+f ( s, x(s), x(τs) ) s ]. on x(t)=ϕ()+ t s= (t s) α [ A0 x(s)+a x(τs)+f ( s, x(s), x(τs) )]. Next we stte nd prove the uniueness theorem. Theorem 5 Let condition (H.) hold. If x(t) nd y(t) re two solutions for the system (9), then x(t)=y(t). Proof Let x nd y be two solutions of system (9). Denote z by z(t)=x(t) y(t). Then one cn esily figure out tht z(t) =0fort I τ.thisimpliesthtsystem(9) hsuniue solution for t I τ.

9 Abdeljwd et l. Journl of Ineulities nd Applictions (206) 206:240 Pge 9 of 3 For t T,however,wehve z(t)= (t s) ) α [ A0 z(s)+a z(τs)+f ( s, x(s), x(τs) ) f ( s, y(s), y(τs) )] s. If t I τ = {, τ,...,τ },thenz(τt) = 0. Therefore, z(t)= This implies z(t) = (t s) α [ A0 z(s)+f ( s, x(s), x(τs) ) f ( s, y(s), y(τs) )] s. (2) = A 0 + L (t s) α [ A0 z(s) + f ( s, x(s), x(τs) ) f ( s, y(s), y(τs) ) ] s (t s) α [ A0 z(s) ( + L x(t) y(t) + x(τt) y(τt) )] s (t s) α [( ) ] A0 + L z(s) + Lz(τs) s By pplying the result of Corollry, we hve (t s) α z(s) s. (22) z(t) 0 E α [( A0 + L ) Ɣ (α), t ], (23) which implies tht x(t)=y(t)fort I τ. For t [τ, ),weget z(t)= + It follows tht z(t) (t s) α [ A0 z(s)+f ( s, x(s), x(τs) ) f ( s, y(s), y(τs) )] s t (t s) α A z(τs) s. (24) + A (t s) α [ A0 ( ) ( )] z(s) + f s, x(s), x(τs) f s, y(s), y(τs) s A 0 + L (t s) α z(τs) s Let z(t)=sup θ Iτ z(θt),thenweget z(t) A 0 + L A 0 + A +2L (t s) α z(s) s + A + L (t s) α z(s) s + A + L (t s) α z(τs) s. (t s) α z(s) s (t s) α z(s) s. (25)

10 Abdeljwd et l. Journl of Ineulities nd Applictions (206) 206:240 Pge 0 of 3 By pplying the result of Corollry,weobtin z(t) [( ) z(t) 0 E α A0 + A 0 +2L Ɣ (α), t ]. (26) Hence, we end up with x(t)=y(t)fort T τ =[τ, ). In the following theorem, we provide n estimte for the solution of system (9). Theorem 6 Let condition (H.2) hold. Then the following estimte for the solution x(t) of system (9) is vlid: x(t) [ ϕ + L ] 2 + ϕ ( A 0 + A ) (t ) α [( E α A0 + A ), t ]. (27) Ɣ (α +) Proof For t T =[, ), the solution of system (9)hstheform x(t)=ϕ()+ (t s) α [ A0 x(s)+a x(τs)+f ( s, x(s), x(τs) )] s. (28) It follows tht x(t) ϕ(0) t + (t s) α A 0 x(s)+a x(τs)+f ( s, x(s), x(τs) ) s ϕ + A 0 (t s) α x(s) s + A (t s) α x(τs) s t + (t s) α f ( s, x(s), x(τs) ) s. By the ssumption (H.2), the bove ineulity cn be rewritten s x(t) A 0 + A ϕ + + L 2 [ ] (t s) α sup x(θs) + ϕ s θ I τ (t s) α s (29) = ϕ + L 2 + ϕ ( A 0 + A ) (t ) α Ɣ (α +) + A 0 + A z (t s) α sup x(sθ) s, (30) θ I τ where the power rule (2) hsbeenused.letv(t)= ϕ + L 2+ ϕ ( A 0 + A ) Ɣ (t ) α (α+),thenv is nondecresing function. Therefore, Corollry implies tht [( x(t) supx(θs) v(t) E α A0 + A ), t ]. (3) θ I τ Hence, the solution x of (9) stisfies the estimte x(t) [ ϕ + L ] 2 + ϕ ( A 0 + A ) (t ) α [( E α A0 + A ), t ]. (32) Ɣ (α +) The proof is complete.

11 Abdeljwd et l. Journl of Ineulities nd Applictions (206) 206:240 Pge of 3 Exmple 2 Consider the nonliner dely frctionl difference eution of the form C 2 x(t)=2x(t)+3x(τt) sin x(t)+3sin x(τt), t T =[, ), (33) with the initil function x(t)=cos 2t, t I τ. Clerly, eution (33) is sclr eution nd A 0 =2ndA = 3. The nonlinerity hs the form f (t, x(t), x(τt)) = sin x(t)+3sin x(τt). Therefore, we hve f ( t, x(t), x(τt) ) f ( t, y(t), y(τt) ) = sin x(t)+3sin x(τt)+sin y(t) 3sin y(τt) 3 ( sin x(t) sin y(t) + sin x(τt) sin y(τt) ). Thus, condition (H.) holds with L = 3. By the conseuence of Theorem 4,eution(33) hs uniue solution. Moreover, f ( t, x(t), x(τt) ) = sin x(t)+3sin x(τt) 4, which implies tht condition (H.2) is stisfied with L 2 =4.ByTheorem6, the solution hs the estimte x(t) [ ] 9 + Ɣ ( 3 2 )(t ) 2 (5Ɣ ( 2 ))k 2 (t ) k Ɣ k=0 ( k 2 +). Remrk 2 Thefollowingfeturescnbeconcluded:. The dely term in system (9) cn be considered s function τ : T [ τ, ) R so tht the solution will be defined on the intervl [ τ, ). In this rticle our dely function cts from [, ) to [τ, ) T. 2. Solving eution (33) is not n esy tsk. However, getting bound for the solution could be considered s substntil step forwrd. 3. Clerly, eution (33) cnnotbedelt with using theresults oftheorem nd Theorem 2. Therefore, the results of this pperre essentilly new nd hve their own merits. 5 Conclusion The Gronwll ineulity hs n importnt role in mny differentil nd integrl eutions. The recent yers hve witnessed the ppernce of n incresing number of generlized Gronwll ineulities which hve been ddressed to overcome difficulties encountered in differentil eutions. To the best of the uthors knowledge, however, there is no pper tht hs delt with generlized -frctionl Gronwll ineulity. In this pper, we extend our previous work nd estblish new generlized version of discrete -frctionl Gronwll ineulity. The new estblished Gronwll ineulity is designed to del with dely -frctionl difference systems. Therefore, we set n initil vlue problem involving nonliner dely Cputo -frctionl difference system. We proved the uniueness nd found n estimte for the solutions of this problem.

12 Abdeljwd et l. Journl of Ineulities nd Applictions (206) 206:240 Pge 2 of 3 Competing interests The uthors declre tht they hve no competing interests. Authors contributions All uthors contributed eully to the writing of this pper. All uthors red nd pproved the finl mnuscript. Author detils Deprtment of Mthemtics nd Physicl Sciences, Prince Sultn University P. O. Box 66833, Riydh, 586, Sudi Arbi. 2 Deprtment of Mthemtics, Çnky University, Blgt, Ankr 06530, Turkey. 3 Institute of Spce Sciences, Mgurele, Romni. Received: 25 August 206 Accepted: 20 September 206 References. Finkelstein, R, Mrcus, E: Trnsformtion theory of the -oscilltor. J. Mth. Phys. 36(6), (995) 2. Finkelstein, RJ: The -Coulomb problem. J. Mth. Phys. 37(6), (996) 3. Florenini, R, Vinet, L: Automorphisms of the -oscilltor lgebr nd bsic orthogonl polynomils. Phys. Lett. A 80(6), (993) 4. Florenini, R, Vinet, L: Symmetries of the -difference het eution. Lett. Mth. Phys. 32(), (994) 5. Florenini, R, Vinet, L: Quntum symmetries of -difference eutions. J. Mth. Phys. 36(6), (995) 6. Freund, PGO, Zbrodin, AV: The spectrl problem for the -Knizhnik-Zmolodchikov eution nd continuous -Jcobi polynomils. Commun. Mth. Phys. 73(), 7-42 (995) 7. Mrin, M: On existence nd uniueness in thermoelsticity of micropolr bodies. C. R. Acd. Sci. Pris, Ser. II 32(2), (995) 8. Mrin, M, Mrinescu, C: Thermoelsticity of initilly stressed bodies, symptotic euiprtition of energies. Int. J. Eng. Sci. 36(), (998) 9. Hn, G-N, Zeng, J: On -seuence tht generlizes the medin Genocchi numbers. Ann. Sci. Mth. Qué. 23(), (999) 0. Mrin, M: Lgrnge identity method for microstretch thermoelstic mterils. J. Mth. Anl. Appl. 363(), (200). Ernst, T: A Comprehensive Tretment of -Clculus. Birkhäuser, Bsel (202) 2. Smko, SG, Kilbs, AA, Mrichev, OI: Frctionl Integrls nd Derivtives: Theory nd Applictions. Gordon & Brech, Yverdon (993) 3. Podlubny, I: Frctionl Differentil Eutions. Mthemtics in Science nd Engineering, vol. 98. Acdemic Press, Sn Diego (999) 4. Kilbs, AA, Srivstv, HM, Trujillo, JJ: Theory nd Applictions of Frctionl Differentil Eutions. North-Hollnd Mthemtics Studies, vol Elsevier, Amsterdm (2006) 5. Jrd, F, Abdeljwd, T, Blenu, D: Stbility of -frctionl non-utonomous systems. Nonliner Anl., Rel World Appl. 4(), (203) 6. Abdeljwd, T, Jrd, F, Blenu, D: A semigroup-like property for discrete Mittg-Leffler functions. Adv. Differ. Eu. 202, Article ID 72 (202) 7. Bstos, NRO, Ferreir, RAC, Torres, DFM: Necessry optimlity conditions for frctionl difference problems of the clculus of vritions. Discrete Contin. Dyn. Syst., Ser. A 29(2), (20) 8. Bstos, NRO, Ferreir, RAC, Torres, DFM: Discrete time vritionl problems. Signl Process. 9(3), (20) 9. Atici, FM, Şengül, S: Modeling with frctionl difference eutions. J. Mth. Anl. Appl. 369(),-9 (200) 20. Atici, FM, Eloe, PW: A trnsform method in discrete frctionl clculus. Int. J. Difference Eu. 2(2),65-76 (2007) 2. Atici, FM, Eloe, PW: Initil vlue problems in discrete frctionl clculus. Proc. Am. Mth. Soc. 37(3), (2009) 22. Goodrich, CS: Continuity of solutions to discrete frctionl initil vlue problems. Comput. Mth. Appl. 59(), (200) 23. Goodrich, CS: Solutions to discrete right-focl frctionl boundry vlue problem. Int. J. Difference Eu. 5(2), (200) 24. Anstssiou, GA: Nbl discrete frctionl clculus nd nbl ineulities. Mth. Comput. Model. 5(5-6), (200) 25. Cheng, J-F, Chu, Y-M: Frctionl difference eutions with rel vrible. Abstr. Appl. Anl. 202, Article ID98529 (202) 26. Cheng, JF, Wu, GC: Solutions of frctionl difference eutions of order (2, ). Act Mth. Sin. 55(3), (202) 27. Annby, MH, Mnsour, ZS: -frctionl Clculus nd Eutions. Lecture Notes in Mthemtics, vol Springer, Heidelberg (202) 28. Atici, FM, Eloe, PW: Frctionl -clculus on time scle. J. Nonliner Mth. Phys. 4(3), (2007) 29. Rjkovic, PM, Mrinkovi, SD, Stnkovi, MS: Frctionl integrls nd derivtives in -clculus. Appl. Anl. Discrete Mth. (), (2007) 30. Mnsour, ZSI: Liner seuentil -difference eutions of frctionl order. Frct. Clc. Appl. Anl. 2(2), (2009) 3. Zho, Y, Chen, H, Zhng, Q: Existence results for frctionl -difference eutions with nonlocl -integrl boundry conditions. Adv. Differ. Eu. 203, Article ID 48 (203) 32. Abdeljwd, T, Blenu, D: Cputo -frctionl initil vlue problems nd -nlogue Mittg-Leffler function. Commun. Nonliner Sci. Numer. Simul. 6(2), (20) 33. Abdeljwd, T, Benli, B, Blenu, D: A generlized -Mittg-Leffler function by Cputo frctionl liner eutions. Abstr. Appl. Anl. 202,Article ID (202) 34. Ye, H, Go, J, Ding, Y: A generlized Gronwll ineulity nd its ppliction to frctionl differentil eution. J. Mth. Anl. Appl. 328(2), (2007) 35. M, Q-H, Pecri, J: Some new explicit bounds for wekly singulr integrl ineulities with pplictions to frctionl differentil nd integrl eutions. J. Mth. Anl. Appl. 34(2), (2008)

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