On two general nonlocal differential equations problems of fractional orders

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1 Malaya Journal of Maemaik, Vol. 6, No. 3, , 28 ps://doi.org/.26637/mjm63/3 On wo general nonlocal differenial equaions problems of fracional orders Abd El-Salam S. A. * and Gaafar F. M.2 Absrac In is paper, we prove some local and global exisence eorems for a fracional orders differenial equaions wi nonlocal condiions, also e uniqueness of e soluion will be sudied. Keywords Fracional calculus; fracional order differenial equaions wi nonlocal condiions. AMS Subjec Classificaion 26A33, 3E25, 34A2, 34A34, 34A37, 37C25, 45J5.,2 Faculy of Science, Damanour Universiy, Damanour, Egyp. *Corresponding auor: srnamed@sci.dmu.edu.eg; 2 famagaafar@sci.dmu.edu.eg Aricle Hisory: Received 2 February 28; Acceped 26 May 28 Conens Inroducion Preliminaries Main Resuls Uniqueness of e soluion c 28 MJM. were e funcion f saisfies Caraeodory condiions and e grow condiion. And, in [3], e auors proved by using e Banac conracion fixed poin eorem, e exisence of a unique soluion of e fracional-order differenial equaion: CD α x() c() f (x()) b(), References wi e nonlocal condiion:. Inroducion m In is work, we consider an arbirary (fracional) orders differenial equaion of e form: du f (, Dα u()), α (, ) d (.) wi e nonlocal condiions I α u() η I α u(), η (, ) x() ak x(k ) x, k were x ℜ and < < 2 < < m <, and ak 6 for all k, 2,, m. (Were C Dα is e Capuo derivaive). Also, e nonlocal problems is sudied in [5] - [7]. (.2) 2. Preliminaries or α u() η α u(), η (, ) (.3) Te nonlocal problems ave been inensively sudied by many auors, for insance in [4], e auors proved e exisence of L -soluion of e nonlocal boundary value problem Dβ u() f (, u(φ ())), β (, 2), (, ), I γ u(), γ (, ], αu(η) u(), < η <, < αη β <. Define L (I) as e class of Lebesgue inegrable funcions on e inerval I [a, b], were a < b < and le Γ(.) be e gamma funcion. Le C(U, X) be Te se of all compac operaors from e subspace U X ino e Banac space X and le Br {u L (I) : u < r, r > }. Definiion. Te fracional inegral of e funcion f (.) L (I) of order β R is defined by (see [8] - []) Iaβ f () ( s)β a Γ(β ) f (s) ds.

2 On wo general nonlocal differenial equaions problems of fracional orders 479/482 Definiion.2 Te Riemann-Liouville fracional-order derivaive of f () of order α (, ) is defined as (see [8] - []) Dαa d α f () I f (), [a, b]. d a Differeniaing bo sides we ge Dα u() u α I α f (, Dα u()). Γ( α) In is paper, we prove e exisence of L -soluions for problems (.) - (.2) and (.) - (.3). Also, we will sudy e uniqueness of e soluion. Take y() Dα u(), we ge (3.) Conversely, operae by I α on bo sides of (3.3), and differeniae wice we obain (.). Now, le us sae e eorems wic will be needed in e paper. Now define e operaor T as Teorem 2.. (Roe Fixed Poin Teorem) [] Ty() Le U be an open and bounded subse of a Banac space E, le T C(U, E). Ten T as a fixed poin if e following condiion olds u α Γ( α) ( s) α Γ( α) f (s, y(s)) ds, (, ). To solve equaion (3.), we mus prove a e operaor T as a fixed poin. T ( U) U. Consider e following assumpions: Teorem 2.2. (Nonlinear alernaive of Laray-Scauder ype) [] Le U be an open subse of a convex se D in a Banac space E. Assume U and T C(U, E). Ten eier (a) f : (, ) R R be a funcion wi e following properies: (i) for eac (, ), f (,.) is coninuous, (A) T as a fixed poin in U, or (A2) ere exiss γ (, ) and x U suc a x γ T x. (ii) for eac y R, f (., y) is measurable, (iii) ere exis wo real funcions a(), b() suc a Teorem 2.3. (Kolmogorov compacness crierion) [2] Le Ω L p (, ), p <. If f (, y) a() b() y, for eac (, ), y R, (i) Ω is bounded in L p (, ) and (ii) x x as uniformly wi respec o x Ω, en Ω is relaively compac in L p (, ), were x () were a(.) L (, ) and b(.) is measurable and bounded. x(s) ds. Now, for e local exisence of e soluions we ave e following eorem: 3. Main Resuls Teorem 3.. Firsly, we will prove e equivalence of equaion (.) wi e corresponding Volerra inegral equaion: u α y() Γ( α) ( s) α Γ( α) If assumpions (i) - (iii) are saisfied, suc a f (s, y(s)) ds, (, ). sup b() <, (3.) Indeed: inegrae bo sides of (.), we ge u() u I f (, Dα u()), en e fracional order inegral equaion (3.) as a soluion y Br, were (3.2) Now, operaing by I α on bo sides of (3.2), en I α u() I α u I 2 α f (, Dα u()). (3.4) r (3.3) 479 u sup b() a.

3 On wo general nonlocal differenial equaions problems of fracional orders 48/482 I α maps L (, ) coninuously ino iself, en I α f (, y()) is coninuous in y. Since y is an arbirary elemen in Br, en T maps Br ino L (, ) coninuously. Now, we will sow a T is compac, by using Teorem 2.3. So, le Ω be a bounded subse of Br. Ten T (Ω) is bounded in L (, ), i.e. condiion (i) of Teorem 2.3 is saisfied. I remains o sow a (Ty) Ty in L (, ) wen, uniformly. Proof. Le u be an arbirary elemen in Br. Ten from e assumpions (i) - (iii), we ave Z Ty Ty() d Z u α d Γ( α) ( s) α f (s, y(s)) ds d Γ( α) Z α u (Ty) Ty Z Z ( s) α d f (s, y(s)) ds Z Z I α f (s, y(s)) u u s α α ds Γ( α) Γ( α) and I α f (s, y(s)) I α f (, y()) ds for a.e. (, ). Terefore, by Teorem 2.3, we ave a T (Ω) is relaively compac, a is, T is a compac operaor. Terefore, Teorem 2. wi U Br and E L (, ) implies a T as a fixed poin. Tis complees e proof. Now, for e exisence of global soluion, we will prove e following eorem : a. Teorem 3.2. Le e condiions (i) - (iii) be saisfied in addiion o e following condiion: From inequaliy (3.4) we deduce a r >. Also, since f u α ds d Γ( α) Since f L (, ), en I α f (.) L (, ). Moreover, since α L (, ). Ten, we ave (see [2]) Terefore u s α Γ( α) I α f (, y()) ds d. u a sup b() r. r Z Z Ten T ( Br ) B r (closure of Br ) if Γ(2 α) sup b() Γ(2 α) Z Z u Ty a sup b() r. u Z erefore e operaor T maps L ino iself. Now, le y Br, a is, y r, en e las inequaliy implies r (Ty) () (Ty)() d s Γ( α) u Z ( s) α a(s) b(s) y(s) ds s u Z ( s) α a(s) b(s) y(s) ds u Z a(s) b(s) y(s) ds u a sup b() y. (Ty)(s)ds (Ty)() d Z (Ty)(s) (Ty)() ds d Z (b) Assume a every soluion y(.) L (, ) o e equaion uo ( s) α α y() γ f (s, y(s)) ds Γ( α) Γ( α) f (s, y(s)) ds a(s) b(s) y(s) ds a sup b() y. a.e. on (, ), < α < Ten f in L (, ). Furer, from (assumpion (i)) f is coninuous in y and since saisfies y 6 r (r is arbirary bu fixed). 48

4 On wo general nonlocal differenial equaions problems of fracional orders 48/482 Ten e fracional order inegral equaion (3.) as a leas one soluion y L (, ). Proof. Le y be an arbirary elemen in e open se Br {y : y < r, r > }. Ten from e assumpions (i) - (iii), we ave Ty u a sup b() y. wic implies a y () y2 (). Now for e exisence and uniqueness of e soluion of problems (.) - (.2) and (.) - (.3), we ave e following wo eorems: Teorem 4.2. Te above inequaliy means a e operaor T maps Br ino L. Moreover, we ave If e assumpions of eorem 4. are saisfied, en problem (.) - (.2) as a unique soluion. Proof. Since f a sup b() y. Tis esimaion sows a f in L (, ). Ten from Teorem 3. we ge a T maps Br ino L (, ) coninuously, and e operaor T is compac. Se U Br and D E L (, ), en from assumpion (b), we find a condiion A2 of Teorem 2.2 does no old. Terefore, Teorem 2.2 implies a T as a fixed poin. Tis complees e proof. u() u I f (, y()) from (3.2), en from condiions (.2), we ge Z u (η α ) ( s)α f (s, y(s)) ds Z η (η s)α f (s, y(s)) ds, Z u 4. Uniqueness of e soluion Teorem 4.. G(η, s) f (s, y(s)) ds, were If e funcion f : (, ) R R saisfy assumpion (ii) of Teorem 3. and saisfy e following assumpion f (, y) f (, z) L y z, G(η, s) (4.) ( s)α (η s)α ηα s η, ( s)α ηα η s. Terefore, en e fracional order inegral equaion (3.) as a unique soluion. Proof. From assumpion (4.), we ge Z u() G(η, s) f (s, y(s)) ds I f (, y()), wic complees e proof. f (, y) f (, ) L y, Teorem 4.3. bu since If e assumpions of eorem 4. are saisfied, en problem (.) - (.3) as a soluion. Proof. Since f (, y) f (, ) f (, y) f (, ) L y, erefore u() u I f (, y()) f (, y) f (, ) L y, en from condiions (.3), we ge i.e. assumpions (i) and (iii) of eorem 3. are saisfied. Now, le y () and y2 () be any wo soluions of equaion (3.), en y2 () y () L ( s) α Γ( α) y2 () y () d L y2 y L u y2 (s) y (s) ds. Γ( α) Z Z ( s) α s f (s, y(s))ds Z η η α f (s, y(s))ds, Z Z ( s) α Z u (η α ) Γ( α) G(η, s) f (s, y(s)) ds, were Terefore Z from (3.2), y2 (s) y (s) dsd, d y2 (s) y (s) ds G(η, s) s η, η α η s. Terefore, Z u() G(η, s) f (s, y(s)) ds I f (, y()), L y2 y. wic complees e proof. 48

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