﻿ Positive Solutions for Fractional Differential Equations with Multi-Point Boundary Value Problems Journal of Applied Mathematics and Physics, 2014, 2, 108-114 Published Online April 2014 in SciRes. http://www.scirp.org/journal/jamp http://dx.doi.org/10.4236/jamp.2014.25014 How to cite this paper: Zhou, L.N. and Jiang, W.H. (2014) Positive Solutions for Fractional Differential Equations with Multi-Point Boundary Value Problems. Journal of Applied Mathematics and Physics, 2, 108-114. http://dx.doi.org/10.4236/jamp.2014.25014 Positive Solutions for Fractional Differential Equations with Multi-Point Boundary Value Problems Lina Zhou1, Weihua Jiang2 1College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, China 2College of Science, Hebei University of Science and Technology, Shijiazhuang , China Email: lnazhou@163.com, xytzln@gmail.com Received Novemb er 20 13 Abstract In this paper, a fractional multi-point boundary value problem is considered. By using the fixed point index theory and Krein-Rutman theorem, some results on existence are obtained. Keywords Caputo Fractional Derivative, Fractional Integral, Boundary Value Problem, Fixed Point Index Theory 1. Introduction Fractional differential equations have been of great interest recently. This is due to the intensive development of the theory of fractional calculus itself as well as its applications. Apart from diverse areas of mathematics, frac- tional differential equations arise in rheology, dynamical processes in self similar and porous structures, elec- trical networks, visco-elasticity, chemical physics, and many other branches of science. For details, see -. It should be noted that most of papers and books on fractional calculus are devoted to the solvability of linear initial fractional differential equations on terms of special functions. Recently, there are some papers dealing with the existence and multiplicity of solution to the nonlinear fractional differential equations boundary value problems, see -. Zhao  investigated the existence and uniqueness of positive solutions for a local boundary value problem of fractional differential equation. 0()(, ())0 01(0)() 0,(1)() 0Dutf tuttu uuuαβξ γη++= <<′′−= += where α is a real number with 12,0, 1,01αβγξ η<≤≤≤ ≤≤≤, 0Dα+ is the Caputo’s derivative. Inspired by above work, we will consider the fractional boundary value problem L. N. Zhou, W. H. Jiang 109 031()(, ())0 01(0) ()0,(1)()0miiiDutf tuttuuu uαβξγ η+−=+= <<′′−=+ =∑ (1) where α is a real number with 12, 01, 01,1, 2,,3,iimα βγ<≤≤≤≤≤=− 12 301mξη ηη−≤< << <≤,0Dα+ is the Caputo’s derivative. Let 3311(1)(1 ).mmii iiiβγ ηγβξ−−= =∆= ++−∑∑ Now we list some conditions for convenience. (H1) 31(1)(1),1(1) 0miiiαβξγ η−=−−>∆ +−>∑ (H2) : [0,1]f RR++×→ satisfied Carathéodory condition,that is (, )fu⋅ is measurable for each fixed uR+∈ and (,)ft⋅ is continuous for a.e. [0,1]t∈. For any 0r>, there existed 1( )[0,1]tLΦ∈ such that(, )()f tut≤Φ, where [0, ]ur∈; a.e. [0,1]t∈. (H3) 0,0,(,)lLftuL∀>∃ >< where [0, ]ul∈; a.e. [0,1]t∈. 2. Preliminary For the convenience of readers, we provide some background material in this section. {\bf Definition 2.1 The Riemann-Liouville fractional of order α for function y is defined as 1001()( )()()tIy ttsys dsααα−+= −Γ∫ Definition 2.2  The Caputo’s derivative for function y is defined as ()0101 ()() ()()ntnysDytdsntsααα++−=Γ− −∫ Lemma 2.1  Let 0α>, then the fractional differential equation 0()0D utα+= has solutions 2112 3(),,1, 2,,,[]1nniutcct ctctcRinnα−=+ +++∈==+ Lemma 2.2  Let 0α>, then for some ,1, 2,,,[]1ic Rinnα∈= =+ Lemma 2.3 (Krein-Rutman)  Let K be a reproducing cone in a real Banach space X and let :LX X→ be a compact linear operator with ()LKK⊆. ()rLis the spectral radius of L If () 0rL>, then there exists 1\ {0}Kϕ∈ such that 11()L rLϕϕ=. Lemma 2.4  Let X is a Banach space, P be a cone in X and ()PΩ be a bounded open subset inP. Suppose that : ()APPΩ→ is a completely continuous operator. Then the following results hold: (1) If there exists 0\ {0}uP∈ such that 0,( ),0u AuuuPλλ≠+∀∈∂Ω> then the fixed point index ( ,( ),)0iAPPΩ=. (2) If 0 ()P∈Ω and ,( ),1Auu uPλλ≠∀∈ ∂Ω≥, then the fixed point index ( ,( ),)1iAPPΩ=. Take [0,1]XC= with norm [0,1]()max()txt xt∈=, 1[0,1]YL= with norm 110() ()xtxt dt=∫. {|()0,[0,1]}KuX utt=∈ ≥∈ Obviously K is a reproducing cone of X. Lemma 2.5 If [0,1],1 2,yCα∈ <≤ then the unique solution of 031()() 001(0) ()0,(1)()0miiiD utyttuuu uαβξγ η+−=+ =<<′′−=+ =∑ is 10()(, ) (),utGtsys ds=∫where L. N. Zhou, W. H. Jiang , 110 31113211312131[1( )]1()( )() ()1(1 )(1 )(),(1)( )[1( )]1()(1 )( )(1)(1)(()(,)miiimiiimiiimiitts sttsss stttsstGtsααααααβ γηξααβξ ββξ βγη ξααβ γηβξ βξααβξ βγηα−−−=−−−=−−−=−=+−− −+−Γ ∆Γ−+ −++−+−≤≤∆Γ −∆Γ+− −+−+ −∆Γ∆Γ −−++∆Γ=∑∑∑∑11231113211112),11()(1 )( )(1)(1)() ,()1(1 )(1 )(),(1)( )11()(1 )( )(1)(1imiiimiiiss tstts stss stttsss tstts sααααααααξβξ βααβξ βγηξ ηαβξ ββξ βγηξ ηααβξ βαα−−−−−=−−−=−−− ≤≤−+− −+−Γ∆Γ −−++−≤≤ ≤∆Γ−+ −+−+−≤≤ ≤∆Γ −∆Γ−+− −+−Γ∆Γ −−+∑∑3113211)() ,()1(1 )(1 )(),(1)( )mjji ijimjji ijitss stttsss tsαααβξ βγηη ηαβξ ββξ βγηηηαα−−−=−−−−=+−≤≤ ≤∆Γ−+ −+−+−≤≤≤∆Γ −∆Γ∑∑ Proof: The equation 0()() 0D utytα++= has a unique solution 11201()( )()()tuttsysdscc tαα−=− −++Γ∫ where 12,.cc R∈ By 31(0)()0,(1)()0miiiuuu uβξγ η−=′′−=+ =∑, we have 3311 000111[(1)() (1)(1) (1)()]mmiii iiicIyI yIyαα αβγ ηξβξβξγη−−−++ += == ++−+−∆∑∑ 33120 00111[(1)()( )]mmii iiicI yIyIyαααββγηβγξ−−−+++= ==+−∆∑∑ 311 1210 003101[1( )]11()() ()() ()(1)()()()( 1)(1) () ()()imiittimiiittuttsy sdssy s dssy s dstsy s dsα ααηαβ γηβξ βξαα αβξ βγηα−− −−=−−=+− −+=−−+−+ −Γ∆Γ∆Γ −−++−∆Γ∑∫ ∫∫∑∫ The proof is complete. Lemma 2.6 If (H1) hold, then there exist a constant M such that 20( ,)(1),,[0,1]GtsMstsα−≤ ≤−∈ Proof: Obviously ( ,)0,Gts ≥ L. N. Zhou, W. H. Jiang 111 3312 1101 13322 21131[1( )]1(1 )max( ,)()(1)()()( 1)()[1 ]1(1 )(1 )(1 )(1 )()(1)()(1mii miiitimii miiimiitttGtss ssss sαα ααα αβ γηβξ ββξ βξ γηαα αβ γηβξββξβγαα αβγ−−−− −=≤≤ =−−−− −==−=+− −+ −+≤−+− +−∆Γ∆Γ −∆Γ+−+ −+≤−+− +−∆Γ∆Γ −∆Γ+=∑∑∑∑321) (1)(1)(1)(1 )()miiisαηαβξ βγβξ βα−−=+−−++ −+−∆Γ∑∑ Let 3311(1 )(1)(1 )(1 )()mmii iiiMβγ ηαβξβγβξβα−−= =++−−++ −+=∆Γ∑∑.The proof is completed. Define an operator :AK K→ and a linear operator :TX X→ as follows: 10( )( ,)(,())Au tG tsfsusds=∫ 10()(, ) ()TutGts usds=∫ Then the fixed point \ {0}uK∈ of A is the positive solutions of (1) 3. Main Results In order to obtain our main results, we firstly present and prove some lemmas. Lemma 3.1 If (H1)-(H3) hold, then :AK K→ and :TX X→ are completely continuous. Proof: According to the Lebesgue Dominated Convergence Theorem and Lemma 2.6, we have :AK K→ is uniformly bounded and equicontinuous. It follows from Ascoli-Arzela theorem that :AK K→ is com-pletely continuous. By the same method, we can get that :TX X→ is completely continuous also. Lemma3.2 If (H1)-(H2) hold, then () 0rT> (r is the spectral radius of T) Proof: Take () 1ut≡ 10311 1210 00310131()(, ) ()[1( )]11()()(1 )()()(1)(1) ()()[1( )]11()()(1imiittimiiimiiiTu tG tsu sdsttts dss dssdsts dstttα ααηαααβ γηβξ βξαα αβξ βγηαβγη ξβξ βαααα α−− −−=−−=−==+− −+=− −+−+−Γ∆Γ∆Γ −−++−∆Γ+− −+=−++Γ∆Γ∆Γ −∫∑∫ ∫∫∑∫∑31311 (1)) 1()11(1):(1) () (1)miiimiiitlααγηβξ βα ααβξβξ γηα αα−=−=−++− ∆Γ−−≥−+ +=Γ +∆Γ∆Γ +∑∑ 22()(())(),( ())nnTutTTutTll Tutl= ≥≥≥ 11, ()lim0nnnnnTlrT Tl→∞≥ =≥> The proof is completed. By Lemma 2.3, we can get there exists 0\{0}Kϕ∈ such that 00()T rTϕϕ= L. N. Zhou, W. H. Jiang , 112 Define 00 [0,1]\[0,1]\(, )(, )liminfinf,limsup suputE u tEf tuf tuffuu∞→∈ →∞ ∈= =, where [0,1]E⊂, with () 0mE= (()mE is the lebesgue measure of E and the same as follows). Set 1{ |},.()K uKurTρρµ=∈ <= Lemma 3.3 Suppose 0fµ< ≤∞, then there exists 00ρ> such that for 0(0, ]ρρ∈, if ,uAu uKρ≠∀ ∈∂,then ( ,,)0.iAKKρ= Proof: It follows from 0fµ< that there exists 0ε> and 00ρ> such that for a.e. 0[0,1], 0tuρ∈ ≤≤ (, )()f tuuµε≥+ (2) For 00ρρ<≤, assume ,uAu uKρ≠∀ ∈∂ By Lemma 2.4,we need only to prove that 0, ,0u AuuKρλϕ λ≠+∀ ∈∂> where 0 00\ {0},()KTrTϕ ϕϕ∈=. Otherwise, there exists 00,0uKρλ∈∂> such that 0000u Auλϕ= + (3) Then 00000,,uAu uλϕ≥≥by (2), we can get that 100 00( ,)(,())()AuG tsfsusdsTuµε= ≥+∫ (4) Considering 0 00,uλϕ≥we get 00000()Au Tµε λϕλϕ≥+ >. This together with (3) means that 0 002,uλϕ≥ by (4) we get 0 002Auλϕ≥.So 0 003uλϕ≥. Repeating this process, we get that 0 00unλϕ≥, so we have 0 00,.un nλϕ≥→∞ →∞ This is a contradic- tion. It follows from Lemma 2.4 that 0( ,,)0,(0,]iAK Kρρρ= ∈. The proof is completed. Lemma 3.4 Suppose 0fµ∞≤<, then there exists 00r> such that ( ,,)1,riAKK = for each 0rr> Proof: Let 0ε> satisfy fµε∞<−, then there exists 10r> such that 1(,)(),for ,a.e.[0,1]f tuuurtµε≤−> ∈ By (H3),there existed 1( )[0,1]tLΦ∈ such that (, )()f tut≤Φ, where 1[0, ]ur∈; a.e. [0,1]t∈. Thus, for all uR+∈ a.e. [0,1]t∈ (, )()()f tuutµε≤ −+Φ (5) Since 1()rTµ=, 1()ITµε−−− exists. Let 1100(, )(),()ICCGtss dsrTµε µε−= Φ=−−−∫ Take 0rr>, we will show ,Au uλ≠for each ,1ruKλ∈∂ ≥. Otherwise, there exist 00,1ruKλ∈∂ ≥, such that 0 00,Au uλ= This together with (5), implies0 0000()uuAuTu Cλ µε≤= ≤−+ Then 0()()ICTutµε µε−≤−−. So we get 0()().CITu tKµε µε−− ∈−− It follows from 110()()nnnITTµεµε∞−+=−= −−∑ and ()TK K⊂, we get 10() ()ICut Tµε µε−≤−−−. Therefore, we have 00u rr≤≤. This is a contradiction. By Lemma 2.4 (2), we get ( ,,)1,riAKK = for each 0rr>. The proof is completed. Theorem 3.1 Suppose 0fµ< ≤∞ and 0fµ∞≤< then (1) has at least one positive solution. Proof: It follows from 0fµ∞≤< and Lemma 3.4 there exists 0r> such that (,, ) 1riAKK =. By 0fµ< <∞ and Lemma 3.3, we can get there exists 0rρ<< such that either there exists uK∈∂ with or (,,)0uAuiA KKρ= =. In the second case, A has a fixed point uK∈ with urρ≤< by the properties of index. The proof is completed. L. N. Zhou, W. H. Jiang 113 4. Example Let’s consider the following boundary value problem 320()(, ())00111 13(0)() 0,(1)() 05454Dutf tuttuu uu++= <<′′−=+= (6) where ,[0,1] is a irrational number(, )0,[0,1] is a rational numberu ttf tut+∈=∈. Corresponding to the problem (1), we have that31311,,,,24455αξηβγ= = ===. Let {|[0,1] is a rational number}E tt= ∈, then () 0mE=. Obviously, (H1)-(H3) are satisfied. By simple calcula- tion, we get 0,0ff∞=∞=. By Theorem 3.1, we get that (6) has at least one solution. This problem can be not solved by the theorem in . Acknowledgements This work is supported by the Natural Science Foundation of China (11171088), the Natural Youth science Foundation of China (11101118) and the Natural Science Foundation of Hebei Province (A2012205074). References  Kilbas, A.A., Srivastava Hari, M. and Trujillo Juan, J. (2006) Theory and Applications of Fractional Differential Equa-tions. North-Holland Mathematics Studied, Vol. 204. Elsevier Science BV, Amsterdam.  Oldham, K.B. and Spanier, J. (1974) The Fractional Calculus . Academic Press, New York, London.  Ross, B. (1975) The Fractional Calculus and Its Applications. Lecture Notes in Mathematics, Vol. 475, Springer, Ber-lin.  Nonnenmacher, T.F. and Metzler, R. (1995) On the Riemann-Liouvile Fractional Calculus and Some Recent Applica-tions. Fractals, 3, 557-566. http://dx.doi.org/10.1142/S0218348X95000497  Tatom, F.B. (1995) The Relationship between Fractional Calculus and Fractals. Fractals, 3, 217-229. http://dx.doi.org/10.1142/S0218348X95000175  Podlubny, I. (1999) Fractional Differential Equations. Mathematics in Science and Engineering, Vol. 198, Aca-demicPress, NewYork/London/Toronto.  Samko, S.G., Kilbas, A.A. and Marichev, O.I. (1993) Fractional Integrals and Derivatives. (Theory and Applications). Gordon and Breach, Switzerland.  Baleanu, D., Mustafa, O.G. and Agarwal, R.P. (2010) An Existence Result for a Super Linear Fractional Differential Equation. Applied Mathematics Letters, 23, 1129-1132. http://dx.doi.org/10.1016/j.aml.2010.04.049  Baleanu, D., Agarwal, R.P., Mustafa, O.G. and Cosulschi, M. (2011) Asymptotic Integration of Some Nonlinear Dif-ferential Equations with Fractional Time Derivative. Journal of Physics A: Mathematical and Theoretical, 44. http://dx.doi.org/10.1088/1751-8113/44/5/055203  Baleanu, D., Mustafa, O.G. and Aga rwal, R.P. (2010) On the Solution Set for a Class of Sequential Fractional Differ-ential Equations. Journal of Physics A: Mathematical and Theoretical, 43.  Zhao, X.K., Chai, C.W. and Ge, W.G. (2011) Positive Solutions for Fractional Four-Point Boundary Value Problems. Communications in Nonlinear Science and Numerical Simulation, 16, 3665-3672. http://dx.doi.org/10.1016/j.cnsns.2011.01.002  Wang, J.H., Xiang, H.J. and Liu, Z.G. (2010) Positive Solution to Nonzero Boundary Value Problem for a Coupled System of Nonlinear Fractional Differential Equations. International Journal of Differential Equations.  Ahmad, B. and Nieto, J.J. (2009) Existence Results for a Coupled System of Nonlinear Fractional Differential Equa-tions with Three-Point Boundary Conditions. Computers & Mathematics with Applications, 58, 1838-1843. http://dx.doi.org/10.1016/j.camwa.2009.07.091  Zhang, S.Q. (2006) Positive Solutions for Boundary-Value Problem of Fractional Order. Acta Mathematica Scientia, 36, 1-12. L. N. Zhou, W. H. Jiang , 114  Guo, D. and Lakshmikantham, V. (1988) Nonlinear Problem in Abstract Cones. Academic Press, San Diego.  Webb, J.R.L. and Lan, K. Q. (2006) Eigenvalue Criteria for Existence of Multiple Positive Solutions of Nonlinear Boundary Value Problems of Local and Nonlocal Type. Topological Methods in Nonlinear Analysis, Journal of the Juliusz Schauder Center, 27, 91-115.