Available a hp://pvamu.edu/aam Appl. Appl. Mah. ISSN: 1932-9466 Vol. 9, Issue 1 June 2014, pp. 416 427 Applicaions and Applied Mahemaics: An Inernaional Journal AAM Exisence of Soluions for Muli-Poins Fracional Evoluion Equaions Soumia Belarbi and Zoubir Dahmani USTHB, Algiers, Algeria soumia-mah@homail.fr & Laboraory LPAM, Faculy SEI UMAB, Universiy of Mosaganem, Algeria zzdahmani@yahoo.fr Received: March 2, 2013; Acceped: February 6, 2014 Absrac In his paper we sudy an impulsive fracional evoluion equaion wih nonlinear boundary condiions. Sufficien condiions for he exisence and uniqueness of soluions are esablished. To illusrae our resuls, an example is presened. Keywords: Capuo fracional derivaive; fixed poin; fracional inegral. MSC 2010 No.: 26A33; 34A12 1. Inroducion During he las decades, he heory of fracional differenial equaions has araced many auhors since i is much richer han he heory of differenial equaions of ineger order Aangana and Secer, 2013, Balachandran e al., 2011, Benchohra e al., 2008, Bonila e al., 2007, Fu, 2013, Kosmaov, 2009, Mirshafaei and Toroqi, 2012, Wu and Baleanu, 2013. This fracional heory has many applicaions in physics, chemisry, biology, blood flow problems, signal and image processing, biophysics, aerodynamics, see for insance Aangana and Alabaraoye, 2013, 416
AAM: Inern. J., Vol. 9, Issue 1 June 2014 417 Baneanu e al., 2012, Bonila e al., 2007, He, 1999, He, 1998, Luchko e al., 2010, Samko e al., 1993 and he reference herein. Moreover, he fracional impulsive differenial equaions have played a very imporan role in modern applied mahemaical models of real processes arising in phenomena sudied in physics, populaion dynamics, opimal conrol, ec. Ahmed, 2007, Lakshmikanham e al., 1989, Samolenko and Peresyuk, 1995. This impulsive heory has been addressed by several researchers: in Benchohra and Slimani, 2009, Benchohra e al. esablished sufficien condiions for he exisence of soluions for some iniial value problems for impulsive fracional differenial equaions involving he Capuo fracional derivaive. In Wang e al., 2010, J.R. Wang e al. sudied nonlocal impulsive problems for fracional differenial equaions wih ime-varying generaing operaors in Banach spaces. In Zhang e al., 2012, he auhors invesigaed he exisence of soluions for nonlinear impulsive fracional differenial equaions of order α 2, 3] wih nonlocal boundary condiions. Ergoren and Kilicman, 2012 esablished some sufficien condiions for he exisence resuls for impulsive nonlinear fracional differenial equaions wih closed boundary condiions. Balachandran e al. Balachandran e al., 2011 discussed he exisence of soluions of firs order nonlinear impulsive fracional inegro-differenial equaions in Banach spaces, while Mallika Arjunane e al. Arjunana e al., 2012 invesigaed he sudy of exisence resuls for impulsive differenial equaions wih nonlocal condiions via measures of non-compacness. In Dahmani and Belarbi, 2013, he auhors sudied an impulsive problem using a bounded linear operaor and some lipschizian funcions. Oher research papers relaed o he fracional impulsive problems can be found in Ahmad and Sivasundaram, 2009, Dabas and Gauam, 2013, Maho e al., 2013, Nieo and ORegan, 2009. In his paper, we are concerned wih he exisence of soluions for he following nonlinear impulsive fracional differenial equaion wih nonlinear boundary condiions: D α x = f, x,xα 1,..., x α n, i, J, 0 < α < 1, x =i = I i x i, i = 1, 2,..., m, x 0 = g x s 0,..., x s r, where D α is he Capuo derivaive, J = [0, b], 0 = s 0 < s 1 <... < s r = b, and 0 < 1 < 2 <... < i <... < m < m+1 = b are consans for r, m N, and x =i = x + i x i, such ha x + i and x i represen he righ-hand limi and lef-hand limi of x a = i respecively, f is an impulsive Carahéodory funcion, α k,...n C 1 J, J, g and I i i=1,...m are appropriae funcions ha will be specified laer. The res of he paper is organized as follows: In Secion 2, some preliminaries are presened. Secion 3 is devoed o he sudy of he exisence and he uniqueness of soluions for he impulsive fracional problem 1.1. A he end, an illusraive example is discussed and a conclusion is given. 1.1 2. Preliminaries In his secion, we inroduce some preliminary facs which are used hroughou his paper Goreno and Mainardi, 1997, Kilbas e al., 2006, Podlubny, 1999, Samko e al., 1993.
418 S. Belarbi & Z. Dahmani Definiion 1: A real valued funcion f, > 0 is said o be in he space C µ, µ R if here exiss a real number p > µ such ha f = p f 1, where f 1 C[0,. Definiion 2: A funcion f, > 0 is said o be in he space C n µ, n N, if f n C µ. Definiion 3: The Riemann-Liouville fracional inegral operaor of order α 0, for a funcion f C µ, µ 1, is defined as J α f = 1 0 τα 1 fτdτ; α > 0, > 0, 2.1 J 0 f = f. Definiion 4: The fracional derivaive of f C n 1 in he sense of Capuo is defined as: D α f = { 1 Γn α 0 τn α 1 f n τdτ, n 1 < α < n, n N, d n d n f, α = n. In order o define soluions of 1.1, we will consider he following Banach space: Le PC J, R = { x : J R, is coninuous a i, x is lef coninuous a = i, and has righ hand limis a i, i = 1, 2,..., m endowed wih he norm x PC = sup J x. We also give he following auxiliary resul Kilbas e al., 2006: Lemma 1: For α > 0, he general soluion of he problem D α x = 0 is given by 2.2 }, 2.3 x = c 0 + c 1 + c 2 2 +... + c n 1 n 1, where c i are arbirary real consans for i = 0, 1, 2,..., n, n 1 = [α]. We also prove he following lemma: Lemma 2: Le 0 < α < 1 and le F, κ PC n+1 J, R. A soluion of he problem D α x = F,, κ; J { 1,..., m }, 0 < α < 1, x =i = I i x i, i = 1, 2,..., m, is given by x = x 0 = g x s 0,..., x s r g x s 0,..., x s r + 1 0 τα 1 F τ, κdτ, [0, 1 ], g x s 0,..., x s r + 1 i j j=1 j 1 j τ α 1 F τ, κdτ 2.4 2.5 + 1 τ α 1 F τ, κdτ + i j=1 I jx j, [ i, i+1 ], i = 1,...m.
AAM: Inern. J., Vol. 9, Issue 1 June 2014 419 Proof: Assume x saisfies 2.4. If [0, 1 ], hen D α x = F, κ. Hence by Lemma 5, i holds x = g x s 0,..., x s r + 1 If [ 1, 2 ], hen using Lemma 5 again, we ge 0 τ α 1 F τ, κdτ. x = x + 1 1 + τ α 1 F τ, κ dτ 1 = x =1 +x 1 1 + τ α 1 F τ, κdτ 1 = I 1 x 1 + g x s0,..., x s r + 1 1 1 τ α 1 F τ, κ dτ 0 + 1 τ α 1 F τ, κdτ. 1 If [ 2, 3 ], hen by he same lemma, we have x = x + 1 2 + τ α 1 F τ, κ dτ 2 = x =2 +x 1 2 + τ α 1 F τ, κ dτ 2 = I 2 x 2 + I1 x 1 + g x s0,..., x s r + 1 1 τ α 1 F τ, κ dτ 0 + 1 2 2 τ α 1 F τ, κdτ + 1 τ α 1 F τ, κdτ. 1 2 By repeaing he same procedure for [ i, i+1 ], i = 1,..m, we obain he second quaniy in 2.5. The proof of Lemma 6 is complee. To end his secion, we give he following assumpions: H 1 : The nonlinear funcion f : J PC n+1 J, R PCJ, R is an h υ impulsive Carahéodory funcion and here exis consans β, I, and θ k,..n such ha for each k = 1, 2,..., n f, x, x 1,..., x n f, y, y 1,..., y n β [ x y + x 1 y 1 +... + x n y n ]; x, y, x k, y k PCJ, R, k = 1,..n, x k y k θ k x y ; x, y, x k, y k PCJ, R, and I := max J h υ. H 2 : The funcions I i : PCJ, R PCJ, R are coninuous for every i = 1,...m and here exis consans ϖ i i=1,...m, such ha 1 I i x I i y ϖ i x y ; I i 0 ω; x, y PCJ, R, i = 1,..m,
420 S. Belarbi & Z. Dahmani H 3 : The funcion g : PC r+1 J, R PCJ, R is coninuous and here exis wo posiive consans ϱ and ˆϱ, such ha for each, x, y PCJ, R, s l J, l = 0,..., r we have [gx s 0,..., x s r ] [gy s 0,..., y s r ] ϱ x y, [gx s 0,..., x s r ] ˆϱ. H 4 : There exiss a posiive consan ρ such ha m + 1 γ [β n + 1 ρ + I] + ρ m ϖ i + mω + ˆϱ ρ; γ = i=1 b α Γα + 1. 3. Main Resuls In his secion, we will derive some exisence and uniqueness resuls concerning he soluion for he sysem 1.1 under he assumpions H j j=1,4 : Theorem 1: If he hypoheses H j j=1,4 and 0 Λ := m + 1 γβ 1 + θ k + m ϖ i + ϱ < 1 3.1 i=1 are saisfied, hen he problem 1.1 has a unique soluion on J. Proof: The hypohesis H 4 allows us o consider he se B ρ = {x PCJ, R : x ρ}. On B ρ we define an operaor T : PCJ, B ρ PCJ, B ρ by Tx = 1 i i τ α 1 f τ, x τ,xα 1 τ,..., x α n τdτ + 1 + i τ α 1 f τ, x τ,xα 1 τ,..., x α n τdτ I i x i + g x s 0,..., x s r. We shall prove ha he operaor T has a unique fixed poin. The proof will be given in wo seps: Sep1: We show ha TB ρ B ρ. Le x B ρ, hen we have: Tx 1 + 1 + i 3.2 i τ α 1 f τ, x τ,xα 1 τ,..., x α n τ dτ 3.3 i τ α 1 f τ, x τ,xα 1 τ,..., x α n τ dτ I i x i + g x s 0,..., x s r.
AAM: Inern. J., Vol. 9, Issue 1 June 2014 421 Consequenly, Tx 1 i [ i τ α 1 f τ, x τ,xα 1 τ,..., x α n τ f τ, 0, 0,..., 0 dτ] 3.4 + 1 i i τ α 1 f τ, 0, 0,..., 0 dτ 1 + 1 i [ τ α 1, f τ, x τ,xα 1 τ,..., x α n τ f τ, 0, 0,..., 0 dτ] + 1 τ α 1 f τ, 0, 0,..., 0 dτ + I i x i I i 0 + I i 0 + g x s 0,..., x s r. By H 2 and H 3, and using he fac ha f is an h υ impulsive carahéodory funcion, we can wrie Tx m + 1 γβ n + 1ρ + m + 1 γ h υ + ρ m ϖ i + mω + ˆϱ. 3.5 i=1 Since h υ I, J, hen we obain m Tx m + 1 γ [β n + 1ρ + I] + ρ ϖ i + mω + ˆϱ. 3.6 i=1 And, by H 4, we have Tx ρ, 3.7 which implies ha TB ρ B ρ. Sep2: Now we prove ha T is a conracion mapping on B ρ : le x and y B ρ, hen for any
422 S. Belarbi & Z. Dahmani J, we have 1 Tx Ty [ i i τ α 1 f τ, x τ,xα 1 τ,..., x α n τdτ + 1 τ α 1 f τ, x τ,xα 1 τ,..., x α n τdτ 3.8 + 1 [ I i x i + g x s 0,..., x s r ] i i τ α 1 f τ, y τ,y α 1 τ,..., y α n τdτ Hence, + 1 τ α 1 f τ, y τ,y α 1 τ,..., y α n τ dτ + Tx Ty m + 1 γ Then by 3.1, we have I i y i + g y s 0,..., y s r ]. [ β x y + ] m θ k x y + ϖ i + ϱ x y. i=1 3.9 Tx Ty Λ x y. 3.10 By 3.1, we can sae ha T is a conracion mapping on B ρ. Combining he Seps1-2, ogeher wih he Banach fixed poin heorem, we conclude ha here exiss a unique fixed poin x PCJ, B ρ such ha Tx = x. Theorem 7 is hus proved. Using he Krasnoselskii s fixed poin heorem in Krasnoselskii, 1964, we prove he following resul. Theorem 2: Suppose ha he hypoheses H j j=1,4 are saisfied. If m + 1γβ hen he problem 1.1 has a soluion on J. 1 + θ k + ϱ < 1, 3.11
AAM: Inern. J., Vol. 9, Issue 1 June 2014 423 Proof: On B ρ, we define he operaors R and S by he following expression Rx = 1 i 1 i τ α 1 f τ, x τ,xα 1 τ,..., x α n τ dτ + 1 τ α 1 f τ, x τ,xα 1 τ,..., x α n τdτ 3.12 and +g x s 0,..., x s r Sx = Le x, y B ρ. Then, for any J, we have Tha is, Rx + Sy 1 I i x i. 3.13 Rx + Sy Rx + Sx. 3.14 i i τ α 1 f τ, x τ,xα 1 τ,..., x α n τ dτ + 1 τ α 1 f τ, x τ,xα 1 τ,..., x α n τ dτ + By H 1, H 2 and H 3, i follows ha Using H 4, we obain Hence, I i y i + g x s 0,..., x s r. Rx + Sy m + 1γ [β n + 1 ρ + I] + ρ. 3.15 m ϖ i + mω + ˆϱ. 3.16 i=1 Rx + Sy ρ. 3.17 Rx + Sy B ρ. Le us now prove he conracion of R : We have Rx Ry = 1 i 1 i τ α 1 f τ, x τ,xα 1 τ,..., x α n τdτ + 1 τ α 1 f τ, x τ,xα 1 τ,..., x α n τdτ + g x s 0,..., x s r 1 i i τ α 1 f τ, y τ,y α 1 τ,..., y α n τdτ + 1 i τ α 1 f τ, y τ,y α 1 τ,..., y α n τdτ + g y s 0,..., y s r. 3.18
424 S. Belarbi & Z. Dahmani Wih he same argumens as before, we ge Rx Ry m + 1 γβ x y + θ k x y + ϱ x y, By 3.1 we can sae ha R is a conracion on B ρ. Now, we shall prove ha he operaor S is compleely coninuous from B ρ o B ρ. Since I i C J, R, hen S is coninuous on B ρ. Now, we prove ha S is relaively compac as well as equiconinuous on PCJ, R for every J. To prove he compacness of S, we shall prove ha SB ρ PCJ, R is equiconinuous and SB ρ is precompac for any ρ > 0, J. Le x B ρ and + h J, hen we can wrie Sx + h Sx I i x i I i x i. 3.20 +h The quaniy 3.20 is independen of x, hus S is equiconinous and as h 0 he righ hand side of he above inequaliy ends o zero, so S B ρ is relaively compac, and S is compac. Finally by Krasnosellkii heorem, here exiss a fixed poin x. in B ρ such ha Tx = x, and his poin x. is a soluion of 1.1. This ends he proof of Theorem 8. Example 1: Consider he following fracional differenial equaion: D α x = exp x+ 0 B 25+exp@1+x+ x 2 k+1 x 1 C 2 k+1 A J = [0, 1], 1, k = 1,..., n, 0 < α < 1, 2 ] 1 x =i = b i x i, b i 2, 1, i = 1, 2,..., m, 3.21 x 0 = 1 3 x ξ + cosx0 n a sinxη, ξ, η [0, 1], a R +.
AAM: Inern. J., Vol. 9, Issue 1 June 2014 425 Corresponding o 1.1, we have f, x,xα k = α k = I is easy o see ha and 2 k+1, I i x i = b i x i, b i exp x + x 2 k+1 25 + exp 1 + x + x, 2 k+1 ] 1 2, 1. s 0 = 0, s 1 = ξ, s 2 = η, g x s 0,xs 1, x s 2 = 1 3 x ξ + cos x 0 n a sinxη. Now, for x, y PC [0, 1] ; R, we have f, x,xα k f, y,y α k exp x + x 2 exp k+1 y + y 2 k+1 25 + exp 1 + x + x 25 + exp 1 + y + y, 2 k+1 2 k+1 exp x + x y + y 2 k+1 2 k+1 25 + exp 1 + x + x 1 + y + y, 2 k+1 2 k+1 exp 25 + exp 1 + x + x 1 + y + 2 k+1 x y + x y, 2 k+1 2 k+1 exp x y 25 + exp 2 k+1 2 k+1 1 x y + x y, 26 2 k+1 2 k+1 + x y, y 2 k+1 Furher we can easily show ha he condiions H i i=1,4 are saisfied and i is possible o choose β, ω, ϖ, n and ϱ in such a way ha he consan Λ < 1. Hence, by Theorem 7, he sysem 3.21 has a unique soluion defined on [0, 1].
426 S. Belarbi & Z. Dahmani 4. Conclusion In his paper, we have sudied an impulsive fracional differenial equaion. Using Banach fixed poin heorem, we have esablished new sufficien condiions for he exisence of a unique soluion for he problem 1.1. To illusrae his resul, we have presened an example. Anoher resul for he exisence of a leas a soluion for 1.1 is also given using Krasnosellkii heorem. REFERENCES Ahmad, B. and Sivasundaram, S. 2009. Exisence resuls for nonlinear impulsive hybrid boundary value problems involving fracional differenial equaions. Nonlinear Anal., 3:251 258. Ahmed, N. 2007. Opimal feedback conrol for impulsive sysems on he space of finiely addiive measures. Publ. Mah. Debrecen., 70:371 393. Arjunana, M. M., Kavihab, V., and Selvic, S. 2012. Exisence resuls for impulsive differenial equaions wih nonlocal condiions via measures of noncompacness. J. Nonlinear Sci. Appl., 5:195 205. Aangana, A. and Alabaraoye, E. 2013. Solving a sysem of fracional parial differenial equaions arising in he model of hiv infecion of cd4+ cells and aracor one-dimensional keller-segel equaions. Advances in Difference Equaions, 201394. Aangana, A. and Secer, A. 2013. A noe on fracional order derivaives and able of fracional derivaives of some special funcions. Absrac and Applied Analysis, 2013279681. Balachandran, K., S.Kiruhika, and Trujillo, J. 2011. Exisence resuls for fracional impulsive inegrodifferenial equaions in banach spaces. Communicaions in Nonlinear Science and Numerical Simulaion, 164:1970 1977. Baneanu, D., Diehelm, K., Scalas, E., and Trujillo, J. 2012. Fcacional calculus: Models and numirical mehods, Vol. 3 of Series on Complexiy, Nonlineariy and Chaos. World Scienific, Hackensack, NJ, USA. Benchohra, M., Henderson, J., Nouyas, S., and Ouahab, A. 2008. Exisence resuls for fracional order funcional differenial equaions wih in nie delay. J. Mah. Anal. Appl., 338:1340 1350. Benchohra, M. and Slimani, B. 2009. Exisence and uniqueness of soluions o impulsive fracional differenial equaions. Elecronic Journal of Differenial Equaions, 200910:1 11. Bonila, B., Rivero, M., Rodriquez-Germa, L., and Trujilio, J. 2007. Fracional differenial equaions as alernaive models o nonlinear differenial equaions. Appl. Mah. Compu., 187:79 88. Dabas, J. and Gauam, G. 2013. Impulsive neural fracional inegro-differenial equaions wih sae dependen delays and inegral condiion. Elecronic Journal of Differenial Equaions, 2013273:1 13. Dahmani, Z. and Belarbi, S. 2013. New resuls for fracional evoluion equaions using banach fixed poin heorem. In. J. Nonlinear Anal. Appl., 41:40 48. Ergoren, H. and Kilicman, A. 2012. Some exisence resuls for impulsive nonlinear fracional
AAM: Inern. J., Vol. 9, Issue 1 June 2014 427 differenial equaions wih closed boundary condiions. Absrac and Applied Analysis, 201238. Fu, X. 2013. Exisence resuls for fracional differenial equaions wih hree-poin boundary condiions. Advances in Difference Equaions, 2013257:1 15. Goreno, R. and Mainardi, F. 1997. Fracional calculus: inegral and differenial equaions of fracional order. Springer Verlag, Wien., pages 223 276. He, J. 1998. Approximae analyical soluion for seepage flow wih fracional derivaives in porous media. Compuer Mehods in Applied Mechanics and Engineering, 1671-2:57 68. He, J. 1999. Some applicaions of nonlinear fracional differenial equaions and heir approximaions. Bull. Sci. Technol., 152:86 90. Kilbas, A. A., Srivasava, H. M., and Trujillo, J. J. 2006. Theory and Applicaions of Fracional Differenial Equaions, Norh-Holland Mahemaics Sudies. 204. Elsevier Science B.V., Amserdam. Kosmaov, N. 2009. Inegral equaions and iniial value problems for nonlinear differenial equaions of fracional order. Nonlinear Analysis, 70:2521 2529. Krasnoselskii, M. 1964. Posiive Soluions of Operaor Equaions. Nordho Groningen Neherland. Lakshmikanham, V., Bainov, D., and Simeonov, P. 1989. Theory of Impulsive Differenial Equaions. World Scienific, Singapore. Luchko, Y., Rivero, M., Trujillo, J., and Velasco, M. 2010. Fracional models, nonlocaliy and complex sysems. Comp. Mah. Appl., 59:1048 1056. Maho, L., Abbas, S., and Favini, A. 2013. Analysis of capuo impulsive fracional order differenial equaions wih applicaions. Inernaional Journal of Differenial Equaions, 2013707457. Mirshafaei, S. and Toroqi, E. 2012. An approximae soluion of he mahieu fracional equaion by using he generalized differenial ransform mehod. Applicaions and Applied Mahemaics, 71:374 384. Nieo, J. and ORegan, D. 2009. Variaional approach o impulsive differenial equaions. Nonlinear Anal., 10:680 690. Podlubny, I. 1999. Fracional Differenial Equaions. Academic Press, San Diego. Samko, S. G., Kilbas, A. A., and Marichev, O. I. 1993. Fracional Inegrals and Derivaives, Theory and Applicaions. Gordon and Breach, Yverdon, Swizerland. Samolenko, A. and Peresyuk, N. 1995. Impulsive Differenial Equaions. World Scienic, Singapore. Wang, J., Yang, Y., and Wei, W. 2010. Nonlocal impulsive problems for fracional differenial equaions wih ime-varying generaing operaos in banach space. Opuscula Mahemaica, 303:361 381. Wu, G. and Baleanu, D. 2013. Variaional ieraion mehod for fracional calculus, a universal approach by laplace ransform. Advances in Difference Equaions, 201318. Zhang, L., Wang, G., and Song, G. 2012. Exisence of soluions for nonlinear impulsive fracional differenial equaions of order α 2, 3] wih nonlocal boundary condiions. Absrac and Applied Analysis, 2012717235.