Research Article Existence and Uniqueness of Positive and Nondecreasing Solutions for a Class of Singular Fractional Boundary Value Problems

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1 Hindawi Publishing Corporaion Boundary Value Problems Volume 29, Aricle ID 42131, 1 pages doi:1.1155/29/42131 Research Aricle Exisence and Uniqueness of Posiive and Nondecreasing Soluions for a Class of Singular Fracional Boundary Value Problems J. Caballero Mena, J. Harjani, and K. Sadarangani Deparameno de Maemáicas, Universidad de Las Palmas de Gran Canaria, Campus de Tafira Baja, 3517 Las Palmas de Gran Canaria, Spain Correspondence should be addressed o K. Sadarangani, ksadaran@dma.ulpgc.es Received 24 April 29; Acceped 14 June 29 Recommended by Juan José Nieo We esablish he exisence and uniqueness of a posiive and nondecreasing soluion o a singular boundary value problem of a class of nonlinear fracional differenial equaion. Our analysis relies on a fixed poin heorem in parially ordered ses. Copyrigh q 29 J. Caballero Mena e al. This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. 1. Inroducion Many papers and books on fracional differenial equaions have appeared recenly. Mos of hem are devoed o he solvabiliy of he linear fracional equaion in erms of a special funcion see, e.g., 1, 2 and o problems of analyiciy in he complex domain 3. Moreover, Delbosco and Rodino 4 considered he exisence of a soluion for he nonlinear fracional differenial equaion D α u f, u, where <α<1andf :,a R R, <a is a given coninuous funcion in,a R. They obained resuls for soluions by using he Schauder fixed poin heorem and he Banach conracion principle. Recenly, Zhang 5 considered he exisence of posiive soluion for equaion D α u f, u, where <α<1 and f :, 1,, is a given coninuous funcion by using he sub- and supersoluion mehods. In his paper, we discuss he exisence and uniqueness of a posiive and nondecreasing soluion o boundary-value problem of he nonlinear fracional differenial equaion D α u f, u, <<1, u u 1 u, 1.1

2 2 Boundary Value Problems where 2 <α 3, D α is he Capuo s differeniaion and f :, 1,, wih lim f, i.e., f is singular a. Noe ha his problem was considered in 6 where he auhors proved he exisence of one posiive soluion for 1.1 by using Krasnoselskii s fixed poin heorem and nonlinear alernaive of Leray-Schauder ype in a cone and assuming cerain hypoheses on he funcion f.in6 he uniqueness of he soluion is no reaed. In his paper we will prove he exisence and uniqueness of a posiive and nondecreasing soluion for he problem 1.1 by using a fixed poin heorem in parially ordered ses. Exisence of fixed poin in parially ordered ses has been considered recenly in This work is inspired in he papers 6, 8. For exisence heorems for fracional differenial equaion and applicaions, we refer o he survey 13. Concerning he definiions and basic properies we refer he reader o 14. Recenly, some exisence resuls for fracional boundary value problem have appeared in he lieraure see, e.g., Preliminaries and Previous Resuls For he convenience of he reader, we presen here some noaions and lemmas ha will be used in he proofs of our main resuls. Definiion 2.1. The Riemman-Liouville fracional inegral of order α> of a funcion f :, R is given by I α 1 f Γα s α 1 fsds 2.1 provided ha he righ-hand side is poinwise defined on,. Definiion 2.2. The Capuo fracional derivaive of order α > of a coninuous funcion f :, R is given by D α f 1 Γn α f n s ds, α n1 s 2.2 where n 1 <α n, provided ha he righ-hand side is poinwise defined on,. The following lemmas appear in 14. Lemma 2.3. Le n 1 <α n, u C n, 1. Then I α Dα u u c 1 c 2 c n n 1, 2.3 where c i R, i 1, 2,...,n.

3 Boundary Value Problems 3 Lemma 2.4. The relaion I α Iβ ϕ I αβ ϕ 2.4 is valid when Re β>, Reα β >, ϕx L 1,b. The following lemmas appear in 6. Lemma 2.5. Givenf C, 1 and 2 <α 3, he unique soluion of D α u f, <<1, u u 1 u, 2.5 is given by u G, sfsds, 2.6 where α 11 s α 2 s α 1, s 1, G, s Γα 1 s α 2 Γα 1, s Remark 2.6. Noe ha G, s > for / andg,s see 6. Lemma 2.7. Le < σ < 1, 2 < α 3 and F :, 1 R is a coninuous funcion wih lim F. Suppose ha σ F is a coninuous funcion on, 1. Then he funcion defined by H G, sfsds 2.8 is coninuous on [,1, where G, s is he Green funcion defined in Lemma 2.5. Now, we presen some resuls abou he fixed poin heorems which we will use laer. These resuls appear in 8. Theorem 2.8. Le X, be a parially ordered se and suppose ha here exiss a meric d in X such ha X, d is a complee meric space. Assume ha X saisfies he following condiion: if {x n } is a non decreasing sequence in X such ha x n x hen x n x for all n N. LeT : X X be a nondecreasing mapping such ha d ( Tx,Ty ) d ( x, y ) ψ ( d ( x, y )), for x y, 2.9

4 4 Boundary Value Problems where ψ :,, is coninuous and nondecreasing funcion such ha ψ is posiive in,, ψ and lim ψ. If here exiss x X wih x Tx hen T has a fixed poin. If we consider ha X, saisfies he following condiion: for x, y X here exiss z X which is comparable o x and y, 2.1 hen we have he following heorem 8. Theorem 2.9. Adding condiion 2.1 o he hypoheses of Theorem 2.8 one obains uniqueness of he fixed poin of f. In our consideraions, we will work in he Banach space C, 1 {x :, 1 R, coninuous} wih he sandard norm x max 1 x. Noe ha his space can be equipped wih a parial order given by x, y C, 1, x y x y, for, In 1 i is proved ha C, 1, wih he classic meric given by d ( x, y ) { } max x y saisfies condiion 2 of Theorem 2.8. Moreover, for x, y C, 1, as he funcion max{x, y} is coninuous in, 1, C, 1, saisfies condiion Main Resul Theorem 3.1. Le < σ < 1, 2 < α 3, f :, 1,, is coninuous and lim f,, σ f, y is a coninuous funcion on, 1,. Assume ha here exiss <λ Γα σ/γ1 σ such ha for x, y, wih y x and, 1 σ( f (, y ) f, x ) λ ln ( y x 1 ) 3.1 Then one s problem 1.1 has an unique nonnegaive soluion. Proof. Consider he cone P {u C, 1 : u }. 3.2 Noe ha, as P is a closed se of C, 1, P is a complee meric space.

5 Boundary Value Problems 5 Now, for u P we define he operaor T by Tu G, sfs, usds. 3.3 By Lemma 2.7, Tu C, 1. Moreover, aking ino accoun Remark 2.6 and as σ f, y for, y, 1, by hypohesis, we ge Tu G, ss σ s σ fs, usds. 3.4 Hence, TP P. In wha follows we check ha hypoheses in Theorems 2.8 and 2.9 are saisfied. Firsly, he operaor T is nondecreasing since, by hypohesis, for u v Tu G, sfs, usds G, ss σ s σ fs, usds G, ss σ s σ fs, vsds Tv. 3.5 Besides, for u v dtu,tv max Tu Tv,1 maxtu Tv max,1,1 max,1 max,1 [ [ [ G, ss σ s σ( fs, us fs, vs ) ds G, ss σ λ lnus vs 1ds G, s ( fs, us fs, vs ) ds 3.6 As he funcion ϕx lnx 1 is nondecreasing hen, for u v, lnus vs 1 ln u v 1 3.7

6 6 Boundary Value Problems and from las inequaliy we ge dtu,tv max,1 [ G, ss σ λ lnus vs 1ds λ ln u v 1 max,1 G, ss σ ds λ ln u v 1 [ α 11 s α 2 s α 1 max s σ ds,1 Γα λ ln u v 1 [ α 11 s α 2 max s σ ds,1 Γα 1 s α 2 s σ ds Γα 1 1 s α 2 Γα 1 s σ ds λ ln u v 1 [ α 11 s α 2 max s σ ds,1 Γα [ λ ln u v 1 max,1 λ ln u v 1 Γα 1 max,1 [ 1 s α 2 s σ ds Γα 1 1 s α 2 s σ ds 1 s α 2 s σ ds Γα 1 1 s α 2 s σ ds Γα λ ln u v 1 Γα 1 λ ln u v 1 Γα 1 λ ln u v 1 Γα 1 λ ln u v 1 1 s α 2 s σ ds β1 σ, α 1 Γ1 σ Γα 1 Γα σ Γ1 σ Γα σ Γ1 σ λ ln u v 1 Γα σ Γ1 σ Γα σ ln u v 1 u v u v ln u v 1. Pu ψx x lnx1. Obviously, ψ :,, is coninuous, nondecreasing, posiive in,, ψ and lim x ψx. Thus, for u v dtu,tv du, v ψdu, v. 3.9

7 Boundary Value Problems 7 Finally, ake ino accoun ha for he zero funcion, T, by Theorem 2.8 our problem 1.1 has a leas one nonnegaive soluion. Moreover, his soluion is unique since P, saisfies condiion 2.1see commens a he beginning of his secion and Theorem 2.9. Remark 3.2. In 6, lemma 3.2 i is proved ha T : P P is compleely coninuous and Schauder fixed poin heorem gives us he exisence of a soluion o our problem 1.1. In he sequel we presen an example which illusraes Theorem 3.1. Example 3.3. Consider he fracional differenial equaion his example is inspired in 6 D 5/2 u 1/22 ln2 u, <<1 u u 1 u 3.1 In his case, f, u 1/2 2 ln2 u/ for, u, 1,. Noehaf is coninuous in, 1, and lim f,. Moreover, for u v and, 1 we have ( ( 1 ) 2 ( ln2 u 1 2 ) ln2 v 2 2) 3.11 because gx lnx 2 is nondecreasing on,,and ( ( 1 2 ) 2 ( ln2 u 1 2 ) ln2 v 2) ( 1 2 ln2 u ln2 v 2) ( 1 ) 2 [ ( ) 2 u ln 2 2 v ( 1 2 ln 2) ( ) 2 v u v 2 v 3.12 ( ) 1 2 ln1 u v. 2 Noe ha Γα σ/γ1 σ Γ5/2 1/2/Γ1 1/2 Γ2/Γ1/2 1/ π 1/4. Theorem 3.1 give us ha our fracional differenial 3.1 has an unique nonnegaive soluion. This example give us uniqueness of he soluion for he fracional differenial equaion appearing in 6 in he paricular case σ 1/2 andα 5/2 Remark 3.4. Noe ha our Theorem 3.1 works if he condiion 3.1 is changed by, for x, y, wih y x and, 1 σ( f (, y ) f, x ) λ ψ ( y x ) 3.13

8 8 Boundary Value Problems where ψ :,, is coninuous and ϕx x ψx saisfies a ϕ :,, and nondecreasing; b ϕ ; c ϕ is posiive in, ; d lim x ϕx. Examples of such funcions are ψx arcgx and ψx x/1 x. Remark 3.5. Noe ha he Green funcion G, s is sricly increasing in he firs variable in he inerval, 1. In fac, for s fixed we have he following cases Case 1. For 1, 2 s and 1 < 2 as, in his case, G, s 1 sα 2 Γα I is rivial ha G 1,s 11 s α 2 Γα 1 < 21 s α 2 Γα 1 G 2,s Case 2. For 1 s 2 and 1 < 2, we have G 2,s G 1,s [ α 12 1 s α 2 Γα 2 s α 1 Γα [ 1 1 s α 2 Γα 1 21 s α s α 2 Γα 1 > s α 2 Γα 1 2 s α 1 Γα 1 2 s α 1 Γα s α 2 Γα 1 Now, s and 1 s 2 s hen 2 s 2 s α 2. Γα s α 2 Γα 1 > 2 s 2 s α Γα 1 Hence, aking ino accoun he las inequaliy and 3.16, weobaing 1,s <G 2,s. Case 3. For s 1, 2 and 1 < 2 < 1, we have G α 11 sα 2 α 11 s α 2 Γα α 1 ( 1 s α 2 s α 2), Γα 3.18

9 Boundary Value Problems 9 and, as 1 s α 2 > s α 2 for, 1, i can be deduced ha G/ > and consequenly, G 2,s >G 1,s. This complees he proof. Remark 3.5 gives us he following heorem which is a beer resul han ha 6, Theorem 3.3 because he soluion of our problem 1.1 is posiive in, 1 and sricly increasing. Theorem 3.6. Under assumpions of Theorem 3.1, our problem 1.1 has a unique nonnegaive and sricly increasing soluion. Proof. By Theorem 3.1 we obain ha he problem 1.1 has an unique soluion u C, 1 wih u. Now, we will prove ha his soluion is a sricly increasing funcion. Le us ake 2, 1, 1 wih 1 < 2, hen u 2 u 1 Tu 2 Tu 1 G 2,s G 1,sfs, usds Taking ino accoun Remark 3.4 and he fac ha f, we ge u 2 u 1. Now, if we suppose ha u 2 u 1 hen G 2,s G 1,sfs, usds and as, G 2,s G 1,s > we deduce ha fs, us a.e. On he oher hand, if fs, us a.e. hen u G, sfs, usds for, Now, as lim f,, hen for M> here exiss δ> such ha for s, 1 wih <s<δwe ge fs, >M. Observe ha,δ {s, 1 : fs, us >M}, consequenly, δ μ,δ μ ({ s, 1 : fs, us >M }) 3.21 and his conradics ha fs, us a.e. Thus, u 2 u 1 > for 2, 1, 1 wih 2 > 1. Finally, as u G,sfs, usds we have ha <u for /. Acknowledgmen This research was parially suppored by Miniserio de Educación y Ciencia Projec MTM 27/6576. References 1 L. M. B. C. Campos, On he soluion of some simple fracional differenial equaions, Inernaional Journal of Mahemaics and Mahemaical Sciences, vol. 13, no. 3, pp , K. S. Miller and B. Ross, An Inroducion o he Fracional Calculus and Fracional Differenial Equaions, A Wiley-Inerscience Publicaion, John Wiley & Sons, New York, NY, USA, 1993.

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