Keywords: fractional calculus; weighted Cauchy-type problem; stability

Transcription

1 ISSN (rin), (online) Inernaional Journal of Nonlinear Science Vol.5(28) No.3, soluion of Weiged Caucy-ye Prolem of a Diffre-inegral Funcional Equaion A. M. A. El-Sayed, S. A. Ad El-Salam Faculy of Science, Alexandria Universiy, Alexandria, Egy (Received 29 Ocoer 27, acceed 23 Aril 28) Asrac: Te oic of fracional calculus, (inegraion and differeniaion of fracional-order) is a one of e singular inegral and inegro-differenial oeraors, is enjoying ineres among maemaicians, ysiciss and engineers ( see []-[2], [5]-[] and [2]-[3] and e references erein ). In is work, we are concerned wi a nonlinear weiged Caucy ye rolem ( For e earlier work and alicaion see, for examle, [6] and [8]) of a diffre-inegral funcional equaion of fracional order. We will rove some local and gloal exisence eorems for is rolem, also we will sudy e uniqueness and sailiy of is soluion. Keywor: fracional calculus; weiged Caucy-ye rolem; sailiy Inroducion In an earlier work e auor (see [8]) sudy e weiged Caucy-ye rolem: { D α u() = f(, u), >, α u() = =. were e funcion f(, u) is assumed o e coninuous on R R, f(, u) µ e σ ψ() u m, µ, m >, σ >, ψ() is a coninuous funcion on R. Here, we deal wi e nonlinear funcional weiged Caucy-ye rolem: { D α u() = f(, u(φ())), α u() = =. () We invesigae e eavior of soluions for rolem () wi cerain nonlineariies, using e equivalence of e fracional diffre-inegral rolem wi e corresonding Volerra inegral equaion, we rove e exisence of -soluion suc a e funcion f saisfies e Caraeodory condiions and e grow condiion f(, u) a() k u, for eac (, ), u R, were a(.) (, ) and k e a consan. Moreover, we will sudy e uniqueness and e sailiy of e soluion. Te lan of our aer is as follows. In e nex secion, we reare some maerial needed o rove our resuls. Secion 3 is devoed o our main resuls on e exisence of some local and gloal soluion. Secion 4 is devoed o e uniqueness of e soluion. In e las secion, we rove e sailiy of e soluion. Corresonding auor. address: amasayed@omail.com address: srnamed@makoo.com Coyrig c World Academic Press, World Academic Union IJNS /55

2 282 Inernaional Journal of Nonlinear Science,Vol.5(28),No.3, Preliminaries e (I) e e class of eesgue inegrale funcions on e inerval I = [a, ], were a < < and le Γ(.) e e gamma funcion. Also le (I) e e sace of e funcions wi ( ) inegrale ower wi e norm u = u() d,. Recall a e oeraor T is comac if i is coninuous and mas ounded ses ino relaively comac ones. Te se of all comac oeraors from e susace U X ino e Banac sace X is denoed y C(U, X). Moreover, we se B r = {u (I) : u < r, r > }. Definiion 2. Te fracional inegral of e funcion f(.) (I) of order β R is defined y (see [] - [4]) Ia β ( s) β f() = f(s), Γ(β) were (see []) I β I γ f() = I βγ f(), β, γ >. a Definiion 2.2 Te Riemann-iouville fracional-order derivaive of f() of order α (, ) is defined as (see [] - [4]) Da α f() = d d I a α f(), [a, ]. Now, le us recall some resuls wic will e needed in e sequel. Teorem 2. (Roe Fixed Poin Teorem) [3] e U e an oen and ounded suse of a Banac sace E, le T C(Ū, E). Ten T as a fixed oin if e following condiion ol T ( U) Ū. Teorem 2.2 (Nonlinear alernaive of aray-scauder ye) [3] e U e an oen suse of a convex se D in a Banac sace E. Assume U and T C(Ū, E). Ten eier (A) T as a fixed oin in Ū, or (A2) ere exiss γ (, ) and x U suc a x = γ T x. Teorem 2.3 (Kolmogorov comacness crierion) [4] e Ω (, ), <. If (i) Ω is ounded in (, ) and (ii) x x as uniformly wi resec o x Ω, en Ω is relaively comac in (, ), were 3 Exisence of soluion x () = x(s). We egin is secion y roving e equivalence of rolem () wi e corresonding Volerra inegral equaion: u() = α ( s) α f(s, u(φ(s))), (, ). (2) Indeed: e u() e a soluion of (2), mulily o sides of (2) y α, we ge α u() = α I α f(, u(φ())), IJNS for conriuion: edior@nonlinearscience.org.uk

3 A. M. A. El-Sayed, S. A. Ad El-Salam: - soluion of Weiged Caucy-ye Prolem of 283 wic gives α u() = =. Now, oeraing y I α on o sides of (2), en I α u() = I f(, u(φ())). Differeniaing o sides we ge D α u() = f(, u(φ())). Conversely, le u() e a soluion of (), inegrae o sides, en I α u() I α u() = = I f(, u(φ())). Oeraing y I α on o sides of e las equaion, en Iu() I α C = I α f(, u(φ())), differeniae o sides, en u() C α = I α f(, u(φ())), from e iniial condiion, we find a C =, en we oain (2), i.e. Prolem () and equaion (2) are equivalen o eac oer. Now define e oeraor T as T u() = α ( s) α f(s, u(φ(s))), (, ). To solve equaion (2) i is necessary o find a fixed oin of e oeraor T. Now, we resen our main resul y roving some local and gloal exisence eorems for e diffre-inegral weiged Caucy-ye rolem () in. To faciliae our discussion, le us firs sae e following assumions: (i) f : (, ) R R e a funcion wi e following roeries: () for eac (, ), f(,.) is coninuous, (2) for eac u R, f(., u) is measurale, (3) for eac (, ), u R, f(, u) saisfies e grow condiion f(, u) a() k u, were a(.) (, ) and k e a consan. (ii) φ : (, ) (, ) is nondecreasing and ere is a consan M > suc a φ M a.e. on (, ). Now, for e local exisence of e soluions we ave e following eorem: Teorem 3. e e assumions (i) and (ii) are saisfied. If k < M Γ( α) and < α, (3) en e diffre-inegral weiged Caucy-ye rolem () as a soluion u B r, were r (α ) Γ(α) a k M Γ(α). IJNS omeage:://

4 284 Inernaional Journal of Nonlinear Science,Vol.5(28),No.3, Proof: e u e an arirary elemen in B r. Ten from e assumions (i) - (ii), we ave T u = = { { { } T u() d } α d (α ) (α ) (α ) (α ) } { ( s) α { { ( s) α { { ( s) α (α ) Γ( α) (α ) Γ( α) (α ) Γ( α) } f(s, u(φ(s))) d } f(s, u(φ(s))) } d } ( a(s) k u(φ(s)) ) } d α { d ( a(s) k u(φ(s)) ) ( a k { M a { a { φ() k M φ() k } M u. } } ) u(φ(s)) φ (s) u(x) dx } Te las esimae sows a e oeraor T mas ino iself. Now, le u B r, a is, u = r, en e las inequaliy imlies T u (α ) { a km } Γ( α) r Ten T ( B r ) B r (closure of B r ) if T u (α ) Γ( α) { a km r } r, wic imlies a Terefore (α ) Γ( α) r (α ) { a km r } Γ(α) a k M Γ(α) Using inequaliy (3) we deduce a r >. Moreover, we ave. r. f = ( ( ) f(s, u(φ(s))) ) ( a(s) k u(φ(s)) ) a k M u. Tis esimaion sows a f in (, ). Furer, f is coninuous in u (assumion ) and I α mas (, ) coninuously ino iself, I α f(, u(φ())) is coninuous in u. Since u is an arirary elemen in B r, T mas B r coninuously ino (, ). IJNS for conriuion: edior@nonlinearscience.org.uk

5 A. M. A. El-Sayed, S. A. Ad El-Salam: - soluion of Weiged Caucy-ye Prolem of 285 Now, we will sow a T is comac, o acieve is goal we will aly Teorem 2.3. So, le Ω e a ounded suse of B r. Ten T (Ω) is ounded in (, ), i.e. condiion (i) of Teorem 2.3 is saisfied. I remains o sow a (T u) T u in (, ) as, uniformly wi resec o T u T Ω. We ave e following esimaion: (T u) T u = = { { { { { } (T u) () (T u)() d ( } (T u)(s) (T u)() d ) (T u)(s) (T u)() } s α α d } d } I α f(s, u(φ(s))) I α f(, u(φ())) d. Since f (, ) we ge a I α f(.) (, ). Moreover α (, ). So, we ave (see [5]) and s α α I α f(s, u(φ(s))) I α f(, u(φ())) for a.e. (, ). Terefore, y Teorem 2.3, we ave a T (Ω) is relaively comac, a is, T is a comac oeraor. Terefore, Teorem 2. wi U = B r and E = (, ) imlies a T as a fixed oin. Tis comlee e roof. Now for more gloal soluion of e diffre-inegral weiged Caucy-ye rolem (), consider e following assumion: (iii) Assume a every soluion u(.) (, ) o e equaion ( u() = γ α ( s) α ) f(s, u(φ(s))) saisfies u r (r is arirary u fixed). a.e. on (, ), < α < Teorem 3.2 e e condiions (i) - (iii) e saisfied, en e diffre-inegral weiged Caucy-ye rolem () as a leas one soluion u (, ). Proof: e u e an arirary elemen in e oen se B r = {u : u < r, r > }. Ten from e assumions (i) - (ii), we ave T u (α ) { a k } Γ( α) M u. Te aove inequaliy means a e oeraor T mas B r ino. Moreover, we ave f a k M u. Tis esimaion sows a f in (, ). As a consequence of Teorem 3. we ge a T mas B r coninuously ino (, ) and T is comac. Se U = B r and D = E = (, ), en in e view of assumion (iii) e condiion A2 of Teorem 2.2 does no old. Terefore, Teorem 2.2 imlies a T as a fixed oin. Tis comlee e roof. IJNS omeage:://

6 286 Inernaional Journal of Nonlinear Science,Vol.5(28),No.3, Uniqueness of e soluion For e uniqueness of e soluion we ave e following eorem: Teorem 4. e e assumions of Teorem 3. e saisfied, u insead of assumion (i) consider e following condiions: f(, u) f(, v) u v and f(, ) a(), en e diffre-inegral weiged Caucy-ye rolem () as a unique soluion. Proof: e u () and u 2 () e any wo soluions of equaion (2), en Terefore u 2 () u () ( s) α = {f(s, u 2 (φ(s))) f(s, u (φ(s)))} ( ( s) α u 2 (φ(s)) u (φ(s)) ). ( u 2 () u () d ( s) α u 2 (φ(s)) u (φ(s)) ) d, { ( ( s) α } u 2 u u 2 (φ(s)) u (φ(s)) ) d α ( ) d u 2 (φ(s)) u (φ(s)) Γ( α) Γ( α) ( ( M u 2 (φ(s)) u (φ(s)) φ (s) M φ() φ() M Γ( α) u 2 u. u 2 (x) u (x) dx ) ) 5 Sailiy Now we sudy e sailiy of e diffre-inegral weiged Caucy-ye rolem (). Teorem 5. e e assumions of Teorem 4. e saisfied, en e soluion of e diffre-inegral weiged Caucy-ye rolem () is uniformly sale. Proof: e u() e a soluion of u() = α ( s) α f(s, u(φ(s))), IJNS for conriuion: edior@nonlinearscience.org.uk

7 A. M. A. El-Sayed, S. A. Ad El-Salam: - soluion of Weiged Caucy-ye Prolem of 287 and le ũ() e a soluion of e aove equaion suc a α ũ() = =, en ( u() ũ() = ( ) α ( s) α u() ũ() = ( ) α ) u() ũ() d u ũ { { { ( s) α ( ) α d {f(s, u(φ(s))) f(s, ũ(φ(s)))}, {f(s, u(φ(s))) f(s, ũ(φ(s)))}, } ( s) α {f(s, u(φ(s))) f(s, ũ(φ(s)))} (α ) (α ) } } d ( ) u ũ M Γ( α) u ũ { ( ( s) α } f(s, u(φ(s))) f(s, ũ(φ(s))) ) d (α ) { ( ( s) α = (α ) α d ( } u(φ(s)) ũ(φ(s)) ) d ) u(φ(s)) ũ(φ(s)) (α ) ( u(φ(s)) ũ(φ(s)) φ (s) Γ( α) M ( (α ) φ() Γ( α) M (α ) (α ), ( M Γ( α) φ() M Γ( α) u ũ, ) (α ). ) u(x) ũ(x) dx ) Terefore, if < δ(ε), en u ũ < ε. Now from e equivalence we ge a e soluion of e diffre-inegral weiged Caucy-ye rolem () is uniformly sale. References [] Ad El-Salam, S. A. and El-Sayed, A. M. A: On e sailiy of some fracional-order nonauonomous sysems. EJQTDE. 6 :-4(27) IJNS omeage:://

8 288 Inernaional Journal of Nonlinear Science,Vol.5(28),No.3, [2] Amed, E., El-Sayed, A.M.A. and El-Saka, H.A.A: Equilirium oins, sailiy and numerical soluions of fracional-order redaor-rey and raies models. Journal of Maemaical Analysis and Alicaions. 325 : (27) [3] Deimling, K: Nonlinear Funcional Analysis. Sringer-Verlag (985) [4] Dugundji, J. Granas, A: Fixed Poin eory. Monografie Maemayczne, PWN, Warsaw (982) [5] El-Sayed, A.M.A. and Gaafar, F.M: Fracional calculus and some inermediae ysical rocesses Al. Ma. and Comue Vol. 44 (23) [6] El-Sayed, A. M. A. and Ad El-Salam, S. A: Weiged Caucy-ye rolem of a funcional differinegral equaion, EJQTDE. 3 : -9(27) [7] El-Sayed, A.M.A, El-Mesiry, A. E. M and El-Saka H. A. A: On e fracional-order logisic equaion J. Al. Ma. eers. 2 :87-823(27) [8] Furai, K. M. and Taar N. E: Power-ye esimaes for a nonlinear fracional differenial equaion. nonlinear analysis. 62: 25-36(25) [9] Gorenflo, R. and Mainardy, F: Fracional calculus: Inegral and Differenial Equaions od Fracional Order. Vol. 378 of CISM courses and lecures, Sringer-Verlag, Berlin (997) [] Hilfer, R: Alicaion of fracional calculus in ysics. World Scienific, Singaore (2) [] Miller, K. S. and Ross, B: An Inroducion o e Fracional Calculus and Fracional Differenial Equaions. Jon Wiley, New York (993) [2] Podluny, I. and E-Sayed, A. M. A: On wo definiions of fracional calculus, Prerin UEF 3-96 (ISBN ), Slovak Academy of Science- Insiue of Exerimenal ys. (996) [3] Podluny, I: Fracional Differenial Equaions.Acad. ress, San Diego-New York-ondon (999) [4] Samko, S., Kilas, A. and Maricev, O. : Fracional Inegrals and Derivaives. Gordon and Breac Science Puliser (993) [5] Swarz, C: Measure, Inegraion and Funcion saces. World Scienific, Singaore (994) IJNS for conriuion: edior@nonlinearscience.org.uk