Journal of Applied Mathematics and Physics, 2014, 2, 108114 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 MultiPoint Boundary Value Problems. Journal of Applied Mathematics and Physics, 2, 108114. http://dx.doi.org/10.4236/jamp.2014.25014 Positive Solutions for Fractional Differential Equations with MultiPoint 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 multipoint boundary value problem is considered. By using the fixed point index theory and KreinRutman 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, viscoelasticity, chemical physics, and many other branches of science. For details, see [1][7]. 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 [8][14]. Zhao [11] investigated the existence and uniqueness of positive solutions for a local boundary value problem of fractional differential equation. 0()(, ())0 01 (0)() 0,(1)() 0 Dutf tutt u uuu α βξ γη + += << ′′ −= += where is a real number with , is the Caputo’s derivative. Inspired by above work, we will consider the fractional boundary value problem
L. N. Zhou, W. H. Jiang 0 3 1 ()(, ())0 01 (0) ()0,(1)()0 m ii i Dutf tutt uuu u α βξγ η + − = += << ′′ −=+ = ∑ (1) where is a real number with 12, 01, 01,1, 2,,3, iim α βγ <≤≤≤≤≤=− , is the Caputo’s derivative. Let 33 11 (1)(1 ). mm ii i ii βγ ηγβξ −− = = ∆= ++− ∑∑ Now we list some conditions for convenience. (H1) 3 1 (1)(1),1(1) 0 m ii i αβξγ η − = −−>∆ +−> ∑ (H2) satisfied Carathéodory condition,that is is measurable for each fixed and is continuous for a.e. . For any , there existed such that , where ; a.e. . (H3) where ; a.e. . 2. Preliminary For the convenience of readers, we provide some background material in this section. {\bf Definition 2.1[7] The RiemannLiouville fractional of order for function y is defined as 1 00 1 ()( )() () t Iy ttsys ds αα α − += − Γ∫ Definition 2.2 [7] The Caputo’s derivative for function y is defined as () 01 0 1 () () () () n t n ys Dytds nts α α α ++− =Γ− − ∫ Lemma 2.1 [14] Let , then the fractional differential equation has solutions 21 12 3 (),,1, 2,,,[]1 n ni utcct ctctcRinn α − =+ +++∈==+ Lemma 2.2 [14] Let , then for some ,1, 2,,,[]1 i c Rinn α ∈= =+ Lemma 2.3 (KreinRutman) [15] Let be a reproducing cone in a real Banach space and let be a compact linear operator with . is the spectral radius of If , then there exists such that . Lemma 2.4 [16] Let is a Banach space, be a cone in and be a bounded open subset in . Suppose that is a completely continuous operator. Then the following results hold: (1) If there exists such that then the fixed point index . (2) If and , then the fixed point index . Take with norm , with norm . Obviously is a reproducing cone of . Lemma 2.5 If then the unique solution of 0 3 1 ()() 001 (0) ()0,(1)()0 m ii i D utytt uuu u α βξγ η + − = + =<< ′′ −=+ = ∑ is where
L. N. Zhou, W. H. Jiang , 3 11 1 3 21 1 3 12 1 3 1 [1( )] 1()( ) () () 1(1 ) (1 )(), (1)( ) [1( )]1 ()(1 ) ( )(1) (1)( () (,) m ii i m ii i m ii i m i i t ts s tt sss st tt ss t Gts αα αα αα β γη ξ αα βξ ββξ βγη ξ αα β γηβξ β ξ αα βξ βγη α − −− = − −− = − −− = − = +− − −+− Γ ∆Γ −+ −+ +−+−≤≤ ∆Γ −∆Γ +− −+ −+ − ∆Γ∆Γ − −+ +∆Γ = ∑ ∑ ∑ ∑1 12 311 1 3 21 1 1 12 ), 11 ()(1 ) ( )(1) (1)() , () 1(1 ) (1 )(), (1)( ) 11 ()(1 ) ( )(1) (1 i m ii i m ii i ss ts t ts s tss st tt sss ts t ts s α αα α αα αα ξ βξ β αα βξ βγηξ η α βξ ββξ βγηξ η αα βξ β αα − −− −− = − −− = −− − ≤≤ −+ − −+− Γ∆Γ − −+ +−≤≤ ≤ ∆Γ −+ −+ −+−≤≤ ≤ ∆Γ −∆Γ −+ − −+− Γ∆Γ − − + ∑ ∑ 311 3 21 1 )() , () 1(1 ) (1 )(), (1)( ) m jji i ji m jji i ji tss st tt sss ts α αα βξ βγηη η α βξ ββξ βγηηη αα −− − = − −− − = + −≤≤ ≤ ∆Γ −+ −+ −+−≤≤≤ ∆Γ −∆Γ ∑ ∑ Proof: The equation has a unique solution 112 0 1 ()( )() () t uttsysdscc t α α − =− −++ Γ∫ where By 3 1 (0)()0,(1)()0 m ii i uuu u βξγ η − = ′′ −=+ = ∑ , we have 33 1 1 000 11 1[(1)() (1)(1) (1)()] mm iii i ii cIyI yIy αα α βγ ηξβξβξγη −− − ++ + = = = ++−+− ∆ ∑∑ 33 1 20 00 11 1[(1)()( )] mm ii i ii cI yIyIy ααα ββγηβγξ −− − +++ = = =+− ∆ ∑∑ 3 1 1 12 1 0 00 31 0 1 [1( )] 11 ()() ()() ()(1)() ()()( 1) (1) () () () i m ii tt i m ii i tt uttsy sdssy s dssy s ds tsy s ds α αα ηα β γηβξ β ξ αα α βξ βγη α − − −− = −− = +− −+ =−−+−+ − Γ∆Γ∆Γ − −+ +− ∆Γ ∑ ∫ ∫∫ ∑∫ The proof is complete. Lemma 2.6 If (H1) hold, then there exist a constant such that 2 0( ,)(1),,[0,1]GtsMsts α − ≤ ≤−∈ Proof: Obviously
L. N. Zhou, W. H. Jiang 3 3 12 1 1 01 1 3 3 22 2 1 1 3 1 [1( )]1(1 ) max( ,)()(1)() ()( 1)() [1 ]1(1 ) (1 )(1 )(1 ) ()(1)() (1 m ii m iii ti m ii m ii i m i i ttt Gtss ss ss s αα α αα α β γηβξ ββξ β ξ γη αα α β γηβξββξβγ αα α βγ − − −− − = ≤≤ = − − −− − = = − = +− −+ −+ ≤−+− +− ∆Γ∆Γ −∆Γ +−+ −+ ≤−+− +− ∆Γ∆Γ −∆Γ + = ∑∑ ∑∑ 3 2 1 ) (1)(1)(1) (1 ) () m ii i s α ηαβξ βγβξ β α − − = +−−++ −+ − ∆Γ ∑∑ Let 33 11 (1 )(1)(1 )(1 ) () mm ii i ii M βγ ηαβξβγβξβ α −− = = ++−−++ −+ =∆Γ ∑∑ .The proof is completed. Define an operator and a linear operator as follows: 1 0 ( )( ,)(,())Au tG tsfsusds= ∫ Then the fixed point of 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 and are completely continuous. Proof: According to the Lebesgue Dominated Convergence Theorem and Lemma 2.6, we have is uniformly bounded and equicontinuous. It follows from AscoliArzela theorem that is com pletely continuous. By the same method, we can get that is completely continuous also. Lemma3.2 If (H1)(H2) hold, then ( is the spectral radius of ) Proof: Take 1 0 3 1 1 12 1 0 00 31 0 1 3 1 ()(, ) () [1( )] 11 ()()(1 ) ()()(1) (1) () () [1( )] 11 ()()(1 i m ii tt i m ii i m ii i Tu tG tsu sds tt ts dss dssds ts ds t tt α αα ηα αα β γηβξ β ξ αα α βξ βγη α βγη ξβξ β αααα α − − −− = −− = − = = +− −+ =− −+−+− Γ∆Γ∆Γ − −+ +− ∆Γ +− −+ =−++ Γ∆Γ∆Γ − ∫ ∑ ∫ ∫∫ ∑∫ ∑ 3 1 3 1 1 (1) ) 1() 11(1): (1) () (1) m ii i m ii i t l α α γη βξ β α αα βξβξ γη α αα − = − = −+ + − ∆Γ −− ≥−+ += Γ +∆Γ∆Γ + ∑ ∑ 22 ()(())(),( ()) nn TutTTutTll Tutl= ≥≥≥ 11 , ()lim0 nn nn n TlrT Tl →∞ ≥ =≥> The proof is completed. By Lemma 2.3, we can get there exists such that
L. N. Zhou, W. H. Jiang , Define 00 [0,1]\[0,1]\ (, )(, ) liminfinf,limsup sup utE u tE f tuf tu ff uu ∞ →∈ →∞ ∈ = = , where , with ( is the lebesgue measure of and the same as follows). Set Lemma 3.3 Suppose , then there exists such that for , if ,then Proof: It follows from that there exists and such that for a.e. (2) For , assume By Lemma 2.4,we need only to prove that where . Otherwise, there exists such that (3) Then by (2), we can get that 1 00 0 0( ,)(,())() AuG tsfsusdsTu µε = ≥+ ∫ (4) Considering we get . This together with (3) means that by (4) we get .So . Repeating this process, we get that , so we have This is a contradic tion. It follows from Lemma 2.4 that . The proof is completed. Lemma 3.4 Suppose , then there exists such that for each Proof: Let satisfy , then there exists such that 1 (,)(),for ,a.e.[0,1]f tuuurt µε ≤−> ∈ By (H3),there existed such that , where ; a.e. . Thus, for all a.e. (5) Since , exists. Let 11 0 0 (, )(),() IC CGtss dsrT µε µε − = Φ=− −− ∫ Take , we will show for each . Otherwise, there exist , such that This together with (5), implies 0 0000 ()uuAuTu C λ µε ≤= ≤−+ Then . So we get It follows from 11 0 ()() nn n ITT µε µε ∞ −+ = −= − − ∑ and , we get . Therefore, we have . This is a contradiction. By Lemma 2.4 (2), we get for each . The proof is completed. Theorem 3.1 Suppose and then (1) has at least one positive solution. Proof: It follows from and Lemma 3.4 there exists such that . By and Lemma 3.3, we can get there exists such that either there exists with . In the second case, has a fixed point with by the properties of index. The proof is completed.
L. N. Zhou, W. H. Jiang 4. Example Let’s consider the following boundary value problem 3 2 0 ()(, ())001 11 13 (0)() 0,(1)() 0 5454 Dutf tutt uu uu + += << ′′ −=+= (6) where ,[0,1] is a irrational number (, )0,[0,1] is a rational number u tt f tut +∈ =∈ . Corresponding to the problem (1), we have that 31311 ,,,, 24455 αξηβγ = = === . Let {[0,1] is a rational number}E tt= ∈ , then . Obviously, (H1)(H3) are satisfied. By simple calcula tion, we get . By Theorem 3.1, we get that (6) has at least one solution. This problem can be not solved by the theorem in [11]. 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 [1] Kilbas, A.A., Srivastava Hari, M. and Trujillo Juan, J. (2006) Theory and Applications of Fractional Differential Equa tions. NorthHolland Mathematics Studied, Vol. 204. Elsevier Science BV, Amsterdam. [2] Oldham, K.B. and Spanier, J. (1974) The Fractional Calculus . Academic Press, New York, London. [3] Ross, B. (1975) The Fractional Calculus and Its Applications. Lecture Notes in Mathematics, Vol. 475, Springer, Ber lin. [4] Nonnenmacher, T.F. and Metzler, R. (1995) On the RiemannLiouvile Fractional Calculus and Some Recent Applica tions. Fractals, 3, 557566. http://dx.doi.org/10.1142/S0218348X95000497 [5] Tatom, F.B. (1995) The Relationship between Fractional Calculus and Fractals. Fractals, 3, 217229. http://dx.doi.org/10.1142/S0218348X95000175 [6] Podlubny, I. (1999) Fractional Differential Equations. Mathematics in Science and Engineering, Vol. 198, Aca demicPress, NewYork/London/Toronto. [7] Samko, S.G., Kilbas, A.A. and Marichev, O.I. (1993) Fractional Integrals and Derivatives. (Theory and Applications). Gordon and Breach, Switzerland. [8] Baleanu, D., Mustafa, O.G. and Agarwal, R.P. (2010) An Existence Result for a Super Linear Fractional Differential Equation. Applied Mathematics Letters, 23, 11291132. http://dx.doi.org/10.1016/j.aml.2010.04.049 [9] 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/10.1088/17518113/44/5/055203 [10] 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. [11] Zhao, X.K., Chai, C.W. and Ge, W.G. (2011) Positive Solutions for Fractional FourPoint Boundary Value Problems. Communications in Nonlinear Science and Numerical Simulation, 16, 36653672. http://dx.doi.org/10.1016/j.cnsns.2011.01.002 [12] 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. [13] Ahmad, B. and Nieto, J.J. (2009) Existence Results for a Coupled System of Nonlinear Fractional Differential Equa tions with ThreePoint Boundary Conditions. Computers & Mathematics with Applications, 58, 18381843. http://dx.doi.org/10.1016/j.camwa.2009.07.091 [14] Zhang, S.Q. (2006) Positive Solutions for BoundaryValue Problem of Fractional Order. Acta Mathematica Scientia, 36, 112.
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