Existence of Solutions for Nonlocal Boundary Value Problems of Higher-Order Nonlinear Fractional Differential Equations

Hindawi Publishing Co po a ion Abs ac and Applied Analysis Volume 2009, A icle ID 494720, 9pages doi:10.1155/2009/494720 Resea ch A icle Exis ence o Solu ions o Nonlocal Bounda y Value P oblems o Highe -O de Nonlinea F ac ional Di e en ial Equa ions Bashi Ahmad1and Juan J. Nie o2 1Depa men o Ma hema ics, Facul y o Science, King Abdulaziz Uni e si y, P.O. Box 80203, Jeddah 21589, Saudi A abia 2Depa amen o de An´ alisis Ma em´ a ico, Facul ad de Ma em´ a icas, Uni e sidad de San iago de Compos ela, 15782 San iago de Compos ela, Spain Co espondence should be add essed o Bashi Ahmad, bashi [email p o ec ed] Recei ed 19 Feb ua y 2009; Accep ed 27 Ap il 2009 Recommended by Paul Eloe We s udy some exis ence esul s in a Banach space o a nonlocal bounda y alue p oblem in ol ing a nonlinea diffe en ial equa ion o ac ional o de qgi en by cDqx   , x , 0< <1, q∈m−1,m,m∈N,m≥2, x00, x00,x 00,...,xm−200, x1αxη. Ou esul s a e based on he con ac ion mapping p inciple and K asnoselskii’s ixed poin heo em. Copy igh q2009 B. Ahmad and J. J. Nie o. This is an open access a icle dis ibu ed unde he C ea i e Commons A ibu ion License, which pe mi s un es ic ed use, dis ibu ion, and ep oduc ion in any medium, p o ided he o iginal wo k is p ope ly ci ed. 1. In oduc ion F ac ional diffe en ial equa ions in ol e de i a i es o ac ional o de . They a ise in many enginee ing and scien i ic disciplines such as he ma hema ical modeling o sys ems and p ocesses in he ields o physics, chemis y, ae odynamics, elec o-dynamics o complex medium, and polyme heology. In consequence, he subjec o ac ional diffe en ial equa ions is gaining much impo ance and a en ion. Fo examples and de ails, see 1–17 and he e e ences he ein. Howe e , he heo y o bounda y alue p oblems o nonlinea ac ional diffe en ial equa ions is s ill in he ini ial s ages and many aspec s o his heo y need o be explo ed. The subjec o mul ipoin nonlocal bounda y alue p oblems, ini ia ed by Ilin and Moisee 18,19, has been add essed by many au ho s, o ins ance, 20–26. The mul ipoin bounda y condi ions appea in ce ain p oblems o he modynamics, elas ici y, and wa e p opaga ion, see 27and he e e ences he ein. The mul ipoin bounda y condi ions may be unde s ood in he sense ha he con olle s a he end poin s dissipa e o add ene gy acco ding o censo s loca ed a in e media e posi ions. 2 Abs ac and Applied Analysis Fo m∈N,m ≥2,and q∈m−1,m,we conside he ollowing nonlinea ac ional diffe en ial equa ion o o de qwi h nonlocal bounda y condi ions: cDqx   ,  ,0< <1, x00,x 00,x 00,...,x m−200,x 1αxη, 0<η<1,αη m−1/ 1,α∈R, 1.1 whe e cDis he Capu o ac ional de i a i e and :0,1×X→Xis con inuous. He e, X, ·is a Banach space and CC0,1,Xdeno es he Banach space o all con inuous unc ions om 0,1→Xendowed wi h a opology o uni o m con e gence wi h he no m deno ed by ·. By a solu ion o 1.1, we mean a unc ion x∈Co class Cm0,1which sa is ies he nonlocal ac ional bounda y alue p oblem 1.1. 2. P elimina ies Le us ecall some basic de ini ions 12,15,17on ac ional calculus. De ini ion 2.1. Fo a unc ion g:0,∞→R, he Capu o de i a i e o ac ional o de qis de ined as cDqg 1 Γn−q 0  −sn−q−1gnsds, n −1<q<n, nq1,2.1 whe e qdeno es he in ege pa o he eal numbe q. De ini ion 2.2. The Riemann-Liou ille ac ional in eg al o o de qis de ined as Iqg 1 Γq 0 gs  −s1−qds, q > 0,2.2 p o ided ha he in eg al exis s. De ini ion 2.3. The Riemann-Liou ille ac ional de i a i e o o de q o a unc ion g is de ined by Dqg 1 Γn−qd d n 0 gs  −sq−n1ds, n q1,2.3 p o ided he igh hand side is poin wise de ined on 0,∞. We ema k ha he Capu o de i a i e becomes he con en ional n h de i a i e o he unc ion as q→nand he ini ial condi ions o ac ional diffe en ial equa ions e ain he same o m as ha o o dina y diffe en ial equa ions wi h in ege -o de de i a i es. On he o he hand, he Riemann-Liou ille ac ional de i a i e could ha dly p oduce he physical in e p e a ion o he ini ial condi ions equi ed o he ini ial alue p oblems Abs ac and Applied Analysis 3 in ol ing ac ional diffe en ial equa ions  he same applies o he bounda y alue p oblems o ac ional diffe en ial equa ions. Mo eo e , he Capu o de i a i e o a cons an is ze o while he Riemann-Liou ille ac ional de i a i e o a cons an is nonze o. Fo mo e de ails, see 17. Lemma 2.4 see 28.Fo q>0, he gene al solu ion o he ac ional diffe en ial equa ion cDqx 0is gi en by x c0c1 c2 2···cn−1 n−1,2.4 whe e ci∈R,i0,1,2,...,n−1(nq1). In iew o Lemma 2.4, i ollows ha IqcDqx x c0c1 c2 2···cn−1 n−1,2.5 o some ci∈R,i0,1,2,...,n−1nq1. Now, we s a e a known esul due o K asnoselskii 29which is needed o p o e he exis ence o a leas one solu ion o 1.1. Theo em 2.5. Le Mbe a closed con ex and nonemp y subse o a Banach space X. Le A, B be he ope a o s such ha iAx By ∈Mwhene e x, y ∈M, iiAis compac and con inuous, iiiBis a con ac ion mapping. Then he e exis s z∈Msuch ha zAz Bz. To s udy he nonlinea p oblem 1.1, we i s conside he associa ed linea p oblem and ob ain i s solu ion. Lemma 2.6. Fo a gi en σ∈C0,1, he unique solu ion o he bounda y alue p oblem, cDqx σ ,0< <1,q∈m−1,m ,m∈N,m≥2, x00,x 00,x 00,...,x m−200,x 1αxη, 0<η<1,αη m−1/ 1,α∈R, 2.6 is gi en by x  0  −sq−1 Γqσsds − m−1 1−αηm−11 0 1−sq−1 Γqσsds −αη 0η−sq−1 Γqσsds. 2.7 4 Abs ac and Applied Analysis P oo . Using 2.5, we ha e x  0  −sq−1 Γqσsds −c0−c1 −c2 2−···−cm−1 m−1,2.8 whe e c0,c 1,c 2,...,c m−1∈Ra e a bi a y cons an s. In iew o he ela ions cDqIqx x  and IqIpx Iqpx  o q, p > 0,x∈L0,1,we ob ain x  0  −sq−2 Γq−1σsds −c1−2c2 −···−m−1cm−1 m−2, x  0  −sq−3 Γq−2σsds −2c2−···−m−1m−2cm−1 m−3,.... 2.9 Applying he bounda y condi ions o 2.6,we ind ha c00,c 10,...,c m−20,and cm−11 1−αηm−11 0 1−sq−1 Γqσsds −αη 0η−sq−1 Γqσsds.2.10 Subs i u ing he alues o c0,c 1,...,c m−1in 2.8,weob ain x  0  −sq−1 Γqσsds − m−1 1−αηm−11 0 1−sq−1 Γqσsds −αη 0η−sq−1 Γqσsds. 2.11 This comple es he p oo . 3. Main Resul s Fo he o hcoming analysis, we need he ollowing assump ions: A1  , x−  , y≤Lx−y, o all ∈0,1,x,y∈X; A2  , x≤μ , o all  , x∈0,1×X, μ ∈L10,1,R . In ela ion o he nonlocal p oblem 1.1, we de ine he cons an s: Λ L Γq1λ, λ L1|α|ηq Γq11−αηm−1 .3.1 Theo em 3.1. Assume ha :0,1×X→Xis a join ly con inuous unc ion and sa is ies he assump ion A1.Then he bounda y alue p oblem 1.1has a unique solu ion p o ided Λ<1,whe e Λis gi en by 3.1. Abs ac and Applied Analysis 5 P oo . De ine :C→Cby x  0  −sq−1 Γq s, xsds − m−1 1−αηm−1 ×1 0 1−sq−1 Γq s, xsds −αη 0η−sq−1 Γq s, xsds, ∈0,1. 3.2 Le us se sup ∈0,1  , 0M, and choose ≥M 1−βΓq111|α|ηq 1−αηm−1,3.3 whe e βis such ha Λ≤β<1.Now we show ha B ⊂B ,whe e B {x∈C:x≤ }. Fo x∈B ,we ha e x ≤ 0  −sq−1 Γq  s, xs ds  m−1 1−αηm−11 0 1−sq−1 Γq  s, xs ds |α|η 0η−sq−1 Γq s, xsds ≤ 0  −sq−1 Γq  s, xs − s, 0   s, 0 ds  m−1 1−αηm−11 0 1−sq−1 Γq s, xs − s, 0 s, 0ds |α|η 0η−sq−1 Γq  s, xs − s, 0  s, 0ds ≤L M 0  −sq−1 Γqds  m−1 1−αηm−11 0 1−sq−1 Γqds |α|η 0η−sq−1 Γqds ≤L Γq1L1|α|ηq Γq11−αηm−1 M Γq111|α|ηq 1−αηm−1 Λ M Γq111|α|ηq 1−αηm−1 ≤Λ1−β ≤ . 3.4 6 Abs ac and Applied Analysis Now, o x, y ∈Cand o each ∈0,1,we ob ain x −y  ≤ 0  −sq−1 Γq s, xs − s, ysds  m−1 1−αηm−11 0 1−sq−1 Γq s, xs − s, ysds |α|η 0η−sq−1 Γq s, xs − s, ysds ≤L x−y  0  −sq−1 Γqds  m−1 1−αηm−11 0 1−sq−1 Γqds |α|η 0η−sq−1 Γqds L x−y  q Γq1 m−1 1−αηm−11|α|ηq Γq1 ≤L x−y  1 Γq111|α|ηq 1−αηm−1 Λ  x−y . 3.5 Clea ly Λdepends on he pa ame e s q, m,α,η,Lin ol ed in he p oblem. As Λ<1, he e o e, is a con ac ion. Thus, he conclusion o he heo em ollows by he con ac ion mapping p inciple. Theo em 3.2. Le :0,1×X→Xbe a join ly con inuous unc ion mapping bounded subse s o 0,1×Xin o ela i ely compac subse s o X. Fu he , he assump ions A1−A2hold wi h λ<1, whe e λis gi en by 3.1. Then he bounda y alue p oblem 1.1has a leas one solu ion on 0,1. P oo . Le us ix ≥ μ L1 Γq11|α|ηq−1 1−αηm−1,3.6 and conside B {x∈C:x≤ }.We de ine he ope a o s Φand Ψon B as Φx 1 Γq 0  −sq−1 s, xsds, Ψx − m−1 1−αηm−11 0 1−sq−1 Γq s, xsds −αη 0η−sq−1 Γq s, xsds. 3.7 Abs ac and Applied Analysis 7 Fo x, y ∈B ,we ind ha  ΦxΨy ≤ μ L1 Γq11|α|ηq−1 1−αηm−1≤ . 3.8 Thus, ΦxΨy∈B .I ollows om he assump ion A1 ha Ψis a con ac ion mapping o λ<1.Con inui y o implies ha he ope a o Φis con inuous. Also, Φis uni o mly bounded on B as Φx≤ μ L1 Γq.3.9 To show ha he ope a o Φis compac , we use he classical A zela-Ascoli heo em. Le Abe a bounded subse o C.We ha e o show ha ΦAis equicon inuous and o each , he se ΦA is ela i ely compac in X. In iew o A1,A2,we de ine sup ,x∈0,1×B   , x max,and consequen ly we ha e Φx 1−Φx 2      1 Γq 1 0 2−sq−1− 1−sq−1 s, xsds  2 1  2−sq−1 s, xsds     ≤ max Γq12 2− 1q q 1− q 2, 3.10 which is independen o x. Thus, Φis equicon inuous. Using he ac ha maps bounded subse s in o ela i ely compac subse s, we ha e ha ΦA is ela i ely compac in X o e e y . The e o e, Φis ela i ely compac on B .Hence, By A zela-Ascoli heo em, Φ is compac on B .Thus all he assump ions o Theo em 2.5 a e sa is ied and he conclusion o Theo em 2.5 implies ha he bounda y alue p oblem 1.1has a leas one solu ion on 0,1. Example 3.3. Conside he ollowing bounda y alue p oblem: cDqx 1  72 x 1x,2<q≤3, ∈0,1, x00,x 00,x 1x1 2. 3.11 He e, m3,  , x   1/ 72x/1x,α 1,η 1/2.As   , x−  , y≤ 1/49x−y, he e o e, A1is sa is ied wi h L1/49.Fu he , Λ L Γq111|α|ηq |1−ηm−1|1 49Γq114 311 2q<1,2<q≤3.3.12 Thus, by Theo em 3.1, he bounda y alue p oblem 3.11has a unique solu ion on 0,1. 8 Abs ac and Applied Analysis Acknowledgmen s The au ho s hank he e iewe s o hei use ul commen s. The esea ch o J. J. Nie o has been pa ially suppo ed by Minis e io de Educacion y Ciencia and FEDER, p ojec MTM2007- 61724, and by Xun a de Galicia and FEDER, p ojec PGIDIT06PXIB207023PR. Re e ences 1B. Ahmad and J. J. Nie o, “Exis ence esul s o a coupled sys em o nonlinea unc ional diffe en ial equa ion wi h h ee-poin bounda y alue p oblem,” p ep in . 2B. Ahmad and J. J. 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