A Caputo Boundary Value Problem in Nabla Fractional Calculus

Transcription

1 University of Nebrsk - Lincoln DigitlCommons@University of Nebrsk - Lincoln Disserttions, Theses, nd Student Reserch Ppers in Mthemtics Mthemtics, Deprtment of A Cputo Boundry Vlue Problem in Nbl Frctionl Clculus Juli St. Gor University of Nebrsk-Lincoln, jstgor@gmil.com Follow this nd dditionl works t: Prt of the Mthemtics Commons St. Gor, Juli, "A Cputo Boundry Vlue Problem in Nbl Frctionl Clculus" (2016). Disserttions, Theses, nd Student Reserch Ppers in Mthemtics This Article is brought to you for free nd open ccess by the Mthemtics, Deprtment of t DigitlCommons@University of Nebrsk - Lincoln. It hs been ccepted for inclusion in Disserttions, Theses, nd Student Reserch Ppers in Mthemtics by n uthorized dministrtor of DigitlCommons@University of Nebrsk - Lincoln.

2 A CAPUTO BOUNDARY VALUE PROBLEM IN NABLA FRACTIONAL CALCULUS by Juli St. Gor A DISSERTATION Presented to the Fculty of The Grdute College t the University of Nebrsk In Prtil Fulfilment of Requirements For the Degree of Doctor of Philosophy Mjor: Mthemtics Under the Supervision of Professor Alln C. Peterson Lincoln, Nebrsk June, 2016

3 A CAPUTO BOUNDARY VALUE PROBLEM IN NABLA FRACTIONAL CALCULUS Juli St. Gor, Ph.D. University of Nebrsk, 2016 Adviser: Alln C. Peterson Boundry vlue problems hve long been of interest in the continuous differentil equtions context. However, with the dvent of new res like Nbl Frctionl Clculus, we my consider such problems in new contexts. In this work, we will consider severl right focl boundry vlue problems, involving Cputo frctionl difference opertor, in the Nbl Frctionl Clculus context. Properties of the Green s functions for ech of these boundry vlue problems will be investigted nd, in the cse of prticulr boundry vlue problem, used to estblish the existence of positive solutions to nonliner version of the boundry vlue problem.

4 iii GRANT INFORMATION This work ws supported in prt by the U.S. Deprtment of Eduction grnt P200A (GAANN, Grdute Assistnce in Ares of Ntionl Need).

5 iv ACKNOWLEDGMENTS First, I d like to thnk my dvisor, Dr. Alln C. Peterson, for ll of his help over the yers. He hs provided incredible guidnce, both s I trnsitioned into the re of Discrete Frctionl Clculus nd s I completed the reserch contined in the present work. The fntstic experience I hd in the REU t the University of Nebrsk- Lincoln under Dr. Peterson convinced me to ttend UNL to pursue my Ph.D. Lter fter tking the course in Ordinry Differentil Equtions from him erly in my grdute school creer, I felt further encourged to continue reserch with him. I hve been incredibly lucky to hve him s my Ph.D. dvisor. Mny thnks, s well, to the professors, grdute students, nd stff in the University of Nebrsk-Lincoln Mthemtics Deprtment. Together these individuls hve creted highly supportive environment for which I m grteful. In prticulr, I d like to thnk Dr. Lynn Erbe for his support, especilly during reserch seminr nd in his Ordinry Differentil Equtions course. Specil thnks is due to Christin Edholm for her mthemticl nd morl support both in nd outside the clssroom; grdute school would not hve been the sme without her. Thnks lso to Jmes Crrher for his morl support. For helpful reserch discussions nd feedbck, thnk you to Kevin Ahrendt. Lstly, thnk you to my committee reders Dr. Richrd Rebrber nd Dr. Petronel Rdu for your helpful suggestions nd dvice, nd thnk you to my other committee members Dr. Brin Hrbourne nd Dr. Vinodchndrn Vriym. These individuls, long with mny others in the UNL Mthemtics Deprtment, hve provided immesurble help over the lst few yers, nd I could never hope to dequtely thnk them for ll they hve done for me.

6 v DEDICATION To Edwrd nd Rebecc St. Gor

7 vi Tble of Contents 1 Introduction 1 2 Bckground Discrete Nbl Opertors The Rising Function nd Tylor Monomils Frctionl Sums nd Differences Cputo Frctionl Difference A Nonliner Right Focl Boundry Vlue Problem Involving Cputo Opertor with 1 < ν Introduction Uniqueness of Solutions The Green s Function Definition nd Bsic Properties of the Green s Function Bounds on the Green s Function The Nonliner Cse Existence of Positive Solutions Conditions for the Existence of Multiple Positive Solutions Future Work

8 vii 4 A Right Focl Boundry Vlue Problem Involving Cputo Opertor for Lrger Vlues of ν Introduction Further Results for 1 < ν Bounds on the Green s Function for 2 < ν Generliztion of the Boundry Vlue Problem Future Work A Right Focl Boundry Vlue Problem Involving the Opertor ν Introduction The Green s Function Bounds on the Green s Function Future Work Bibliogrphy 99

9 viii List of Figures 3.1 A solution to the homogenous BVP (3.6) A grph of G(t, 5) for 1 t

10 1 Chpter 1 Introduction Two wys of pproching Discrete Frctionl Clculus re extensively introduced in Goodrich nd Peterson [31] nd differ from the strt in the wy the difference opertor is defined. Delt Frctionl Clculus mkes use of the forwrd difference opertor, for f : N R, defined by f(t) : f(t + 1) f(t), for t N where N : {, + 1, + 2,...}, wheres Nbl Frctionl Clculus mkes use of the nbl (bckwrd) difference opertor, for f : N R, defined by f(t) : f(t) f(t 1), where t N +1. These initil differences give rise to differing definitions of the integrl nd frctionl derivtive. This thesis will focus on the Nbl Frctionl Clculus cse, which, of the two res, is the reltively newer focus mong reserchers nd thus hs been less extensively elborted. However, few fetures of the most prominent opertors in Nbl Frctionl Clculus indicte not only distinctions in behvior but lso some

11 2 properties which pper prticulrly dvntgeous. In Nbl Frctionl Clculus, the frctionl nbl sum, which is relted to the continuous integrl, is defined s follows for f : N +1 R nd µ R + : µ f(t) : t H µ 1 (t, ρ(s))f(s) s for t N, where by convention µ f() 0. In the bove, H ν (t, s) is nottion for Tylor monomils in this context. Notice tht the domin of the function f nd of µ f(t) differ only by single vlue; tht is, the domin is shifted to the left by 1. Hence, the domins before nd fter the ppliction of the frctionl sum re quite similr. This stte of ffirs differs mrkedly from the frctionl sum opertor in the Delt Frctionl Clculus context. In the Delt Frctionl Clculus context, the ppliction of the frctionl sum opertor my result in the domin of the result being shifted by non-integer vlue, resulting in domin tht differs entirely from tht of the originl function. Such n opertor in the Delt Frctionl Clculus still holds much mthemticl mening nd usefulness. However, the reltive consistency in domins before nd fter the ppliction of such frctionl opertors in Nbl Frctionl Clculus is certinly plesnt. Some ppers in the field of discrete nbl frctionl clculus include [13], [15], [2], nd the references therein. Nbl frctionl clculus is considered in the context of the more generl re of time scles by Anstssiou in [4] nd [7] nd by Anderson in [8]. Some ppers in the field of delt frctionl clculus include [14], [16], [29], [30]. In [17], Atici nd Şengül use frctionl opertors in the context of delt frctionl clculus to model tumor growth. In [39], Boguo, Erbe nd Peterson discuss some reltionships between symptotic behvior of nbl nd delt frctionl difference equtions. Before discussing the prticulr focus of this work, it is importnt to refer to the

12 3 prominent field of frctionl clculus in the context of stndrd clculus. This field hs long history nd extensive reserch tht cn only be touched on here. A few of the mny instnces of reserch in frctionl differentil equtions include [11], [20] which considers boundry vlue problem involving Cputo opertor, [47], nd [44]. Finlly Oldhm nd Spnier survey nd discuss the mny pplictions of frctionl clculus in [45]. The prticulr focus will be on vrious boundry vlue problems in the context of nbl frctionl clculus. In these problems one looks for functions which stisfy difference eqution on given domin s well s severl conditions on the boundries of the domin. Additionlly ll boundry vlue problems in this thesis involve frctionl Cputo difference opertor, denoted by ν for ν > 0. The Cputo opertor hs few useful properties tht distinguish it from the Riemnn-Liouville opertor, including the fct tht ν C 0 for constnt C nd ν 1, which does not necessrily hold for the Riemnn-Liouville opertor. The ppers [5] nd [6] by Anstssiou introduce the Cputo opertor in the nbl frctionl clculus context nd shows some other properties in nbl frctionl clculus. There re lso monotonicity results relted to the Cputo opertor, s discussed by B. Ji, etl. [40], s well s in [31] nd [28]. The Riemnn-Liouville opertor, briefly introduced in Chpter 2, is lso frequently used in reserch nd its properties re discussed t length in [28]. Similr boundry vlue problems to the one considered in this pper hve been investigted by Holm [37], whose problem involved Riemnn-Liouville frctionl opertor in the context of the Delt Frctionl Clculus, nd by Erbe nd Peterson [27], whose problem involved whole order differences in the context of Time Scles. A liner boundry vlue problem similr to the one in Chpter 5 ws investigted by Ahrendt et l. [3] in the Nbl Frctionl Clculus context nd involved Cputo opertor.

13 4 Boundry vlue problems re lso considered in Discrete Frctionl Clculus contexts by Atici nd Eloe [16], Goodrich [32], Awsthi [19], nd Brckins [23]. A boundry vlue problem involving whole order delt nd nbl opertor on discrete domin ws investigted by [24]. Within the considertion of ech boundry vlue problem, I mke use of nd investigte Green s functions. Green s functions re specific functions typiclly unique to ech boundry vlue problem. They led directly to solutions in the cse of liner boundry vlue problems nd cn be useful tools in showing the existence of solutions to nonliner boundry vlue problems. In the ppers by Holm [37] nd Erbe nd Peterson [27], positive solutions to the nonliner cse were sought using Green s Functions nd fixed point theorem from Krsnosel skiĭ [42] nd Deimling [26]. The fixed point theorem stted s norm type ws shown by Guo in [34] nd [33]. In the present work, boundry vlue problem we consider is ν x(t) h(t, x(t 1)), t Nb +1 x( 1) 0, (1.1) x(b) 0, where 1 < ν 2, h : N b +1 R + R +, b Z, b 1, nd where the solutions x re defined on N b 1. The Guo-Krsnosel skiĭ fixed point theorem is frequently used theorem within the lrger context of cone theory. Kwong discusses Krsnosel skiĭ s theorem s well s its connection to other fixed-point theorems in [43]. The lrger context of cone theory nd its pplictions to nonliner problems is more fully elborted in the text by Guo nd Lkshmiknthm [35]. Also, fixed-point theory is surveyed by Agrwl, Meehn, nd O Regn in [1] nd by Zeidler in [48].

14 5 The re of nbl frctionl clculus nd discrete frctionl clculus, especilly s presented in this work, my be seen within the context of time scles. The re of time scles works with functions whose domins re ny nonempty closed subsets of the rel numbers nd ws ingurted by Hilger [36]. Hence, both stndrd clculus nd discrete clculus my be viewed s contined within the lrger context of time scles. The re of time scles hs been extensively surveyed by Bohner nd Peterson in [21] nd [22]. Boundry vlue problems with whole order opertors hve been considered in the context of time scles in [27], [10], [12], [25] to nme few. After n overview of bckground nottion nd theorems in Chpter 2, we will focus on boundry vlue problems. In Chpter 3, we will consider the BVP (1.1). The Green s function for this BVP is found, bounds re estblished for this Green s function, nd the Guo-Krsnosel skiĭ fixed point theorem is used to show existence of positive solutions in some cses nd existence of multiple positive solutions in others. In Chpter 4, we consider the BVP ν t Nb +1 k x( 1) 0, 0 k N 2 (1.2) N 1 x(b) 0, where 1 < ν, h : N b +1 R +, b Z, b N 1, nd where the solutions x re defined on N b N+1. We first expnd our knowledge of the BVP (1.2) for the cse where 1 < ν 2. We then continue on to consider the Green s function of the BVP (1.2) for 2 < ν 3 nd estblish bounds on the Green s function. In so doing, we show tht the Green s function for the 2 < ν 3 cse differs mrkedly in its behvior from tht of 1 < ν 2. Lstly, we consider generliztion of the BVP (1.2). In

15 6 Chpter 5, for 0 < ν 1, we consider the BVP ν x(t) h(t), t Nb +2 x() x(b) 0,. (1.3) where, b re positive integers such tht b 2 nd h : N b +2 R. In prticulr, we estblish severl bounds on the Green s function for the BVP (1.3) nd compre the reveled behvior to the behvior of the Green s function for n nlogous BVP in the continuous differentil equtions context.

16 7 Chpter 2 Bckground In this chpter we will introduce relevnt definitions nd theorems from the re of Nbl Frctionl Clculus. 2.1 Discrete Nbl Opertors In this section we will ddress the nottion for the domins of functions in the re of Nbl Frctionl Clculus. Additionlly we will introduce some bsic opertors in this context long with some of their properties. All definitions nd theorems found in this chpter my be found in [31], long with more generl introduction to the re of Nbl Frctionl Clculus. Definition 2.1. Let R nd b Z +. Then N : {, + 1, + 2, }

17 8 nd N b : {, + 1, + 2,, b}. The nottion in Definition 2.1 shows the typicl nottion for domins of functions in the re of Nbl Frctionl Clculus. Definition 2.2 shows dditionl nottion commonly used in this re. Definition 2.2 (Bckwrd Jump Opertor). We define the bckwrd jump opertor, ρ : N +1 N, by ρ(t) t 1. If we consider the derivtive, we notice tht the stndrd definition of the derivtive will not work when considering functions with discrete domins becuse in order to tke derivtive t point, the function must be defined on some open intervl round the point. However, we wish to define n nlogue of this opertor for discrete functions. While the definition must be different from the definition given in stndrd clculus, we still wnt the opertor we use to relte to the ide of slope. Since there re no points in the domin rbitrrily close to ny other points, the next best option is to use the slope of the line between two djcent points s our derivtive. Hence, we define the nbl opertor, or bckwrds difference opertor, for this context. Definition 2.3 (Nbl Opertor). For n rbitrry f : N R, we define the nbl opertor,, by ( f)(t) : f(t) f(t 1), t N +1.

18 9 Note dditionlly tht 0 is the identity, tht is it is defined by 0 f(t) f(t). For N N, we define N by N f(t) : ( N 1 f(t)), for t N +N. The theorem below gives severl bsic properties of the nbl opertor, including two properties nlogous to the product rule nd quotient rule from stndrd clculus. Theorem 2.4. Assume f, g : N R nd α, β R. Then for t N +1, (i.) α 0; (ii.) αf(t) α f(t); (iii.) (f(t) + g(t)) f(t) + g(t); (iv.) if α 0, then α t+β α 1 α αt+β ; (v.) (f(t)g(t)) f(ρ(t)) g(t) + f(t)g(t); (vi.) ( ) f(t) g(t) f(t) f(t) g(t), if g(t) 0, t N g(t) g(t)g(ρ(t)) +1. In ddition to n opertor similr to the derivtive, we lso need n opertor similr to the integrl from stndrd clculus. Of course, s before, the definition must differ from the stndrd definition in this discrete context. Definition 2.5 (Discrete Nbl Integrl). Assume f : N R nd c, d N. Then d c d tc+1 f(t), if d > c f(t) t : 0, if d c.

19 10 Notice tht Definition 2.5 is essentilly right Riemnn sum, involving rectngles of width one. The theorem below shows some bsic properties of the discrete nbl integrl. Theorem 2.6. Assume f, g : N R, b, c, d N, b < c < d, nd α R. Then (i.) c b αf(t) t α c b f(t) t; (ii.) c b (f(t) + g(t)) t c b f(t) t + c b g(t) t; (iii.) f(t) t 0; b (iv.) d b f(t) t c b f(t) t + d c f(t) t; (v.) c b f(t) t c b f(t) t; (vi.) if F (t) : t b f(s) s, for t Nc b, then F (t) f(t), t Nc b+1 ; (vii.) if f(t) g(t) for t {b + 1, b + 2,, c}, then c b f(t) t c b g(t) t. 2.2 The Rising Function nd Tylor Monomils Among the properties of the nbl opertor listed in Theorem 2.4, notice tht there is no property similr to the power rule listed. In order to hve property nlogous to the power rule, s stted in Theorem 2.8, we must define the rising function. Definition 2.7 (Rising Function). Assume n is positive integer nd t R. Then we define the rising function, t n, by t n : t(t + 1) (t + n 1).

20 11 Theorem 2.8 (Nbl Power Rules). For n N, α R, (t + α) n n(t + α) n 1, for t R. Tylor monomils will be used in Section 2.3 to define frctionl sums nd differences nd re often used throughout this work. Definition 2.9 (Nbl Tylor Monomils). We define the nbl Tylor monomils, H n (t, ), n N 0 by H 0 (t, ) 1, t N nd H n (t, ) (t )n, n! for t N n+1 nd n N. 2.3 Frctionl Sums nd Differences In this section we introduce more definitions nd theorems in order to generlize the nbl opertor nd the discrete nbl integrl. In order to mke these generliztions, we use the Gmm function, which is defined to be Γ(z) : 0 e t t z 1 dt, where z is such tht the rel prt of z is positive. By integrtion by prts, one cn show tht Γ(z + 1) zγ(z). (2.1)

21 12 It follows tht Γ(N + 1) N Γ(N) for N positive integer. Hence Γ(N + 1) N! for N N. The definition of Γ(z) is extended, using (2.1), to ll complex z such tht z 0, 1, 2, 3,.... Additionlly, lim z n Γ(z) for n 0, 1, 2, 3,.... Below, we use the Gmm function to generlize the rising function. Note tht for positive integer N, t N (t+n+1)!. With this in mind, we define the generliztion of (t+1)! the rising function s follows. Definition 2.10 ((Generlized) Rising Function). The (generlized) rising function is defined by t r Γ(t + r) Γ(t) (2.2) for those vlues of t nd r such tht the right hnd side of eqution (2.2) mkes sense. We lso use the convention tht if t is nonpositive integer, but t + r is not nonpositive integer then t r : 0. Note tht t 0 1 for t 0, 1, 2,... Theorem 2.11 (Generlized Nbl Power Rules). The formuls (t + α) r r(t + α) r 1, (2.3) nd (α t) r r(α ρ(t)) r 1, (2.4) hold for those vlues of t, r nd α so tht the expressions (2.3) nd (2.4) mkes sense. Additionlly we define the nlogue of the Tylor monomil, (t )n n!, from stndrd clculus nd describe some of its properties.

22 13 Definition Let µ 1, 2 3,, then we define the µ-th order nbl frctionl Tylor monomil, H µ (t, ), by H µ (t, ) (t )µ Γ(µ + 1) (2.5) whenever the right hnd side of the eqution (2.5) mkes sense. Theorem The following hold whenever the expressions below re well-defined: (i.) H µ (, ) 0; (ii.) H µ (t, ) H µ 1 (t, ); (iii.) t H µ(s, ) s H µ+1 (t, ); (iv.) t H µ(t, ρ(s)) s H µ+1 (t, ); (v.) for k N, H k (t, ) 0, t N. Next we define the nbl frctionl sum, denoted ν for R nd ν 0, which is n extension of the discrete nbl integrl. The opertor is defined so tht 1 f(t) t f(t) t. The definition is dditionlly motivted by the fct tht t τ1 τn 1 f(τ n ) τ n τ 2 τ 1 t H n 1 (t, ρ(s))f(s) s. Definition 2.14 (Nbl Frctionl Sum). Let f N +1 R be given nd µ R +, then µ f(t) : t H µ 1 (t, ρ(s))f(s) s for t N, where by convention µ () 0.

23 14 The following theorem gives us n importnt property of the nbl frctionl sum. Theorem Let ν R + nd µ R such tht µ nd ν + µ re not negtive integers. Then we hve tht ν H µ (t, ) H µ+ν (t, ) for t N. This property is nlogous to the fct tht t (τ ) n dτ (t )n+1 n! (n+1)! in stndrd clculus. Next we use the nbl frctionl sum in the definition of the nbl frctionl difference. Definition 2.16 (Nbl Frctionl Difference). Let f : N +1 R, ν R +, nd choose N so tht N 1 < ν N. Then we define the ν-th order nbl frctionl difference, ν f(t) by ν f(t) N (N ν) f(t) for t N +N. Note tht for N N, N f(t) N (N N) f(t) N f(t) for t N +N. 2.4 Cputo Frctionl Difference Definition 2.16 represents the Riemnn-Liouville definition of the frctionl difference. However, one my lso define the Cputo frctionl difference by chnging the order of the opertors in Definition 2.16.

24 15 Definition 2.17 (Cputo Frctionl Difference). Assume f : N N+1 R nd µ > 0. Then the µ-th Cputo nbl frctionl difference of f is defined by µ f(t) (N µ) N f(t) for t N +1, where N µ. Note tht for constnt C, µ C (N µ) N C (N µ) 0 0. Such property is resonble for n opertor extending the difference opertor. Yet this property does not hold for the Riemnn-Liouville opertor defined in Definition This work is primrily concerned with the Cputo Frctionl Difference s the extension of the nbl difference opertor. We describe bsic property of this opertor below from [31, Theorem on p. 229]. Theorem Assume µ > 0 nd N µ. Then the nbl Tylor monomils, H k (t, ), 0 k N 1, re N linerly independent solutions of µ x 0 on N N+1.

25 16 Chpter 3 A Nonliner Right Focl Boundry Vlue Problem Involving Cputo Opertor with 1 < ν Introduction In this chpter we will consider the nonliner right focl boundry vlue problem (3.1) in the context of Nbl Frctionl Clculus, s shown below ν x(t) h(t, x(t 1)), t Nb +1 x( 1) 0, (3.1) x(b) 0, where 1 < ν 2, h : N b +1 R + R +, b N 1, nd where the solutions x re defined on N b 1. Recll tht the Cputo difference opertor is denoted by ν for ν > 0.

26 17 The eventul gol of this chpter is to estblish the existence of positive solutions nd, in some cses, even of multiple positive solutions to the boundry vlue problem (3.1). However, this chpter will begin by considering liner version of the BVP (3.1) for ny ν > 1. This more generl liner BVP is ν t Nb +1 k x( 1) A k, 0 k N 2 (3.2) N 1 x(b) B, where h : N b +1 R, ν > 1, N ν, A k R for 0 k N 2, B R, b Z, b N 1, nd where the solutions x re defined on N b N+1. In Section 3.2, the uniqueness of solutions will be estblished under certin conditions for the BVP (3.2). In Section 3.3, we will consider the Green s function for the BVP ν t Nb +1 k x( 1) 0, 0 k N 2 (3.3) N 1 x(b) 0, where ν > 1, N ν, b Z, b N 1, nd where the solutions x re defined on N b N+1. The Green s function is useful in clculting solutions to the BVP (3.2). In prticulr, the Green s function is defined so tht x(t) G(t, s)h(s) s for t N b N+1 solves the BVP (3.2). In Subsection 3.3.2, the Green s function will be considered only for the cse where 1 < ν 2. Bounds will be estblished on the Green s function in this context in Subsection so tht these bounds my then be used to solve the nonliner BVP (3.1). In Section 3.4, existence of positive solutions

27 18 to the BVP (3.1) will be estblished by finding fixed point for the opertor Ax(t) G(t, s)h(s, x(s 1)) s for x in cone in n pproprite Bnch spce nd t N b 1. The generl definition of cone is given in Definition This fixed point will be found by using the Guo- Krsnosel skiĭ theorem. In Subsection 3.4.2, conditions re estblished under which multiple solutions to the BVP (3.1) my be found. Finlly, directions for future work will be discussed in Section 3.5. Note tht mny of the results in this chpter re included in [46]. 3.2 Uniqueness of Solutions First consider the following nth order initil vlue problem: ν x(t) h(t), t N +1 k x() c k, 0 k N 1. where, ν R, ν > 0, N : ν, c k R for 0 k N 1, nd h : N +1 R. By the theorem below, the solution to the bove IVP is defined on N N+1. By [31, Theorem on p. 230], we hve the following theorem: Theorem 3.1. The unique solution to the IVP is given by x(t) N 1 k0 H k (t, )c k + ν h(t) for t N N+1, where by convention ν h(t) 0 for N + 1 t.

28 19 Noting tht ν h(t) t H ν 1(t, ρ(s))h(s) s for t N +1, we define the Cuchy function for the IVP. Definition 3.2. The Cuchy function for ν x(t) 0 is defined to be x(t, s) : H ν 1 (t, ρ(s)) for those vlues of t, s, nd ν such tht the right side mkes sense. Then by Theorem 3.1, we hve the following vrition of constnts formul. Theorem 3.3 (Vrition of Constnts). For h : N +1 R nd for t N N+1, the solution to the initil vlue problem ν y(t) h(t), t N +1 k y() 0, 0 k N 1. is given by y(t) t x(t, s)h(s) s. In Section 3.3 the Cuchy function will be used in the definition of the Green s function. The proofs of the two theorems tht follow re similr to proofs in [31] nd [3]. Theorem 3.4 (Generl Solution of the Homogeneous Eqution). Let, ν R, ν > 0, nd N ν. Assume x 1, x 2,..., x N re N linerly independent solutions of ν x 0 on N N+1. Then the generl solution to ν x 0 is given by x(t) c 1 x 1 (t) + + c N x N (t), t N N+1

29 20 where c 1, c 2,..., c N R re rbitrry constnts. Proof. Let x 1, x 2,..., x N be linerly independent solutions of ν x 0 on N N+1. Let k x 1 () A 1,k for 0 k N 1, k x 2 () A 2,k for 0 k N 1, nd in generl k x j () A j,k for 0 k N 1 nd 1 j N. Then for ech 1 j N, it holds tht x j is the unique solution to the IVP ν x j(t) 0, t N +1 k x j () A j,k, 0 k N 1. By the linerity of ν, ν [c 1x 1 (t) + + c N x N (t)] c 1 ν x 1(t) c N ν x N(t) 0, so x(t) : c 1 x 1 (t) + + c N x N (t) solves ν x(t) 0. Now suppose x : N N+1 R solves ν x(t) 0. Let k x() M k for 0 k N 1. Then x(t) is the unique solution of ν x(t) 0, t N +1 k x() M k, 0 k N 1. Now it will be shown tht x 1 () x 2 ()... x N () c 1 x 1 () x 2 ()... x N () c N 1 x 1 () N 1 x 2 ()... N 1 x N () c N M 0 M 1. M N 1

30 21 hs unique solution for c 1, c 2,..., c N R. Express the bove eqution s follows: A 1,0 A 2,0... A N,0 c 1 A 1,1 A 2,1... A N,1 c A 1,N 1 A 2,N 1... A N,N 1 c N M 0 M 1. M N 1. Suppose towrds contrdiction tht A 1,0 A 2,0... A N,0 A 1,1 A 2,1... A N, A 1,N 1 A 2,N 1... A N,N 1 Then the columns of the bove mtrix re not linerly independent, so t lest one column is liner combintion of the other columns. Without loss of generlity, sy tht the Nth column is liner combintion of the other N 1 columns. Then α 1 A 1,0 A 1,1. A 1,N 1 + α 2 A 2,0 A 2,1. A 2,N α N 1 A N 1,0 A N 1,1. A N 1,N 1 A N,0 A N,1. A N,N 1 for some α 1, α 2,..., α N 1 R. Note tht X(t) : α 1 x 1 (t)+α 2 x 2 (t)+ +α N 1 x N 1 (t) is solution of ν x(t) 0. Now both X(t) nd x N(t) solve ν y(t) 0, t N +1 k y() α 1 A 1,k + α 2 A 2,k + + α N 1 A N 1,k, 0 k N 1,

31 22 so by uniqueness X(t) x N (t), so x N (t) α 1 x 1 (t) + α 2 x 2 (t) + + α N 1 x N 1 (t). But then x 1, x 2,..., x N re linerly dependent. Thus we hve contrdiction. Theorem 3.5 (Generl Solution of the Nonhomogeneous Eqution). Let ν > 0 nd N ν. Assume x 1, x 2,..., x n re linerly independent solutions of ν x(t) 0 on N N+1 nd y 0 is prticulr solution to ν x(t) h(t) on N N+1 for some h : N +1 R. Then the generl solution of ν x(t) h(t) is given by x(t) c 1 x 1 (t) + + c N x N (t) + y 0 (t), where c 1, c 2,..., c N R re rbitrry constnts nd t N N+1. Proof. By linerity of the opertor ν, ν x(t) c 1 ν x 1(t) + + c N ν x N(t) + ν y 0(t) h(t), so x(t) solves ν x(t) h(t). Conversely, ssume x : N N+1 R solves ν x(t) h(t). Let k x() b k for 0 k N 1. Then x(t) uniquely solves the IVP ν x(t) h(t), t N +1 k x() b k, 0 k N 1. Now there exist constnts c 1, c 2,..., c N R such tht c 1 x 1 (t)+c 2 x 2 (t)+ +c N x N (t) solves ν x h(t) 0, t N +1 k x h () b k k y 0 (), 0 k N 1.

32 23 Note tht x h (t) + y 0 (t) solves ν x(t) h(t). Also k x h () + k y 0 () b k k y 0 () + k y 0 () b k for 0 k N 1. Hence x(t) nd x h (t) + y 0 (t) both solve the sme IVP. Hence by uniqueness, x(t) c 1 x 1 (t) + c 2 x 2 (t) + + c N x N (t) + y 0 (t). Therefore ny solution to ν x(t) 0 cn be written in this form. Now consider the non-homogeneous boundry vlue problem (3.2) nd the corresponding homogeneous boundry vlue problem (3.3). Note tht some rguments in the proof below re nlogous to those given in [3]. Theorem 3.6. The nonhomogeneous boundry vlue problem (3.2) hs unique solution. Proof. By Theorem 3.4, generl solution to ν x(t) 0 is given by x(t) c 0x 0 (t)+ + c N 1 x N 1 (t), where c 0,..., c N 1 R nd x 0, x 1,..., x N 1 re N linerly independent solutions to ν x(t) 0 on N N+1. Thus, by Theorem 3.1, the generl solution to ν x(t) 0 is given by x(t) c 0 + c 1 H 1 (t, ) + c 2 H 2 (t, ) + + c N 1 H N 1 (t, ), for t N b N+1, where c 0, c 1,..., c N 1 R. Without loss of generlity, sy x 0 (t) 1, x 1 (t) H 1 (t, ),..., x N 1 (t) H N 1 (t, ) for t N N+1. Sy x(t) solves (3.3). It will be shown tht x(t) is the trivil solution. It holds tht

33 24 x(t) is the trivil solution if nd only if c 0 c 1... c N 1 0. This holds if nd only if the system of equtions c 0 k x 0 ( 1) + + c N 1 k x N 1 ( 1) 0, 0 k N 2 c 0 N 1 x 0 (b) + + c N 1 N 1 x N 1 (b) 0 hs only the trivil solution. Define x 0 ( 1) x 1 ( 1)... x N 1 ( 1) x 0 ( 1) x 1 ( 1)... x N 1 ( 1) M : N 2 x 0 ( 1) N 2 x 1 ( 1)... N 2 x N 1 ( 1) N 1 x 0 (b) N 1 x 1 (b)... N 1 x N 1 (b) If det(m) 0, then the system of equtions hs only the trivil solution. For 0 i N 1 nd 0 j N 2, it holds tht j x i ( 1) j H i (t, ) t 1 H i j ( 1, ). Now, for i j, H i j ( 1, ) H 0 ( 1, ) 1. Also N 1 x N 1 (b) H 0 (b, ) 1. Thus M hs 1 s on the digonl. For i j + 1, H i j ( 1, ) H (j+1) j ( 1, ) ( 1)1 Γ(2) 1. This implies tht M hs -1 s on the superdigonl. Next, if 0 i < j nd 0 j N 2, then H i j ( 1, ) 0, since i j < 0. If j + 1 i N 1 for 0 j N 2, then H i j ( 1, ) ( 1)i j Γ(i j+1) 0. Finlly, if

34 25 0 i N 2, then N 1 x i (b) H i N+1 (b, ) H i N+1 (b, ) 0. Thus M......, so det(m) 1. Then the bove system hs unique solution, so homogeneous BVP (3.3) hs only the trivil solution. Next, by Theorem 3.5, the generl solution to ν x(t) h(t) is given by y(t) c 0 x 0 (t) + + c N 1 x N 1 (t) + y 0 (t) c 0 + c 1 H 1 (t, ) + + c N 1 H N 1 (t, ) + y 0 (t), where c 0,..., c N 1 R nd y 0 : N N+1 R is prticulr solution to ν y 0(t) h(t). Then y(t) solves (3.2) if nd only if c 0 k x 0 ( 1) + + c N 1 k x N 1 ( 1) + k y 0 ( 1) A k, 0 k N 2 c 0 N 1 x 0 (b) + + c N 1 N 1 x N 1 (b) + N 1 y 0 (b) B

35 26 hs unique solution if nd only if c 0 k x 0 ( 1) + + c N 1 k x N 1 ( 1) A k k y 0 ( 1), 0 k N 2 c 0 N 1 x 0 (b) + + c N 1 N 1 x N 1 (b) B N 1 y 0 (b) hs unique solution. Thus y(t) uniquely stisfies the boundry conditions in (3.2) if nd only if det(m) 0. Therefore, since det(m) 1, the BVP (3.2) hs unique solution. 3.3 The Green s Function In this section, the Green s function for the BVP (3.3) will be defined. Then in Theorem 3.11, three bounds will be estblished for the Green s function in the cse where 1 < ν 2 tht will be key in considering the nonliner cse in Section Definition nd Bsic Properties of the Green s Function Note here tht the proofs through Theorem 3.9 use rguments nlogous to the derivtion of different Green s function in [3]. Definition 3.7 (Green s Function). Let x(t, s) be the Cuchy function for ν x(t) 0. The Green s function, G(t, s) : N b N+1 Nb +1 R, for the BVP (3.3) is given by u(t, s), G(t, s) : v(t, s), t ρ(s) ρ(s) t (3.4)

36 27 where u(t, s) is defined to be the unique solution of the BVP ν u(t, s) 0, t Nb +1 k u( 1, s) 0, 0 k N 2 N 1 u(b, s) N 1 x(b, s) for ech fixed s N b +1, nd v(t, s) : u(t, s) + x(t, s). Theorem 3.8 below shows tht the Green s function, mong other things, is useful in clculting solutions to the BVP (3.3). Theorem 3.8. Let G(t, s) be the Green s function for the BVP (3.3), nd ssume h : N b +1 R. Then the unique solution to ν t Nb +1 k y( 1) 0, 0 k N 2 (3.5) N 1 y(b) 0, is given by y(t) G(t, s)h(s) s, t N b N+1. Proof. Let y(t) : G(t, s)h(s) s for t Nb N+1. Then, for t Nb N+1, y(t) t+1 G(t, s)h(s) s v(t, s)h(s) s + t+1 u(t, s)h(s) s

37 28 t+1 [u(t, s) + x(t, s)]h(s) s + u(t, s)h(s) s + u(t, s)h(s) s + u(t, s)h(s) s + u(t, s)h(s) s + t+1 t t t t+1 x(t, s)h(s) s u(t, s)h(s) s x(t, s)h(s) s + x(t, t + 1)h(t + 1) x(t, s)h(s) s + H ν 1 (t, ρ(t + 1))h(t + 1) x(t, s)h(s) s. Define z(t) : t x(t, s)h(s) s. By the vrition of constnts formul in Theorem 3.3, z(t) solves the IVP ν z(t) h(t), t Nb +1 k z() 0, 0 k N 1 for t N b N+1. Thus ν y(t) ν [ h(t) ] u(t, s)h(s) s + z(t) ν u(t, s)h(s) s + ν z(t) for t N b N+1. It will now be shown tht the boundry conditions hold. First, sy 0 k N 2. Then k y( 1) 0 + k u( 1, s)h(s) s + k z( 1) 1 k x( 1, s)h(s) s

38 29 0. According to the Leibniz Formul in [31, Theorem 3.41 on p. 175], if f : N N +1 ( ) t R, then f(t, τ) τ t tf(t, τ) τ + f(ρ(t), t). Hence, N 1 y(b) N 1 u(b, s)h(s) s + N 1 z(b) t N 1 u(b, s)h(s) s + [ N 1 ] x(t, s)h(s) s t ] N 1 u(b, s)h(s) s + [ N 2 x(t, s)h(s) s + x(ρ(b), b)h(b) t ] N 1 u(b, s)h(s) s + [ N 2 x(t, s)h(s) s tb tb tb + H ν 1 (ρ(b), ρ(b))h(b). 0. t N 1 u(b, s)h(s) s + [ N 2 N 1 u(b, s)h(s) s + N 1 x(b, s)h(s) s + ] x(t, s)h(s) s tb N 1 x(b, s)h(s) s N 1 x(b, s)h(s) s Hence the boundry conditions hold. Theorem 3.9. The Green s function G(t, s) : N b N+1 Nb +1 R for the homoge-

39 30 neous BVP (3.3) is given by (3.4) where u(t, s) : H N 1 (t, 1)H ν N (b, ρ(s)) nd v(t, s) : H N 1 (t, 1)H ν N (b, ρ(s)) + H ν 1 (t, ρ(s)). Proof. By Theorem 3.8, it suffices to show tht, for ech fixed s N b +1 when t ρ(s), u(t, s) is the solution to the BVP ν u(t, s) 0, t Nb +1 k u( 1, s) 0, 0 k N 2 N 1 u(b, s) N 1 x(b, s). Let t ρ(s). First, ν u(t, s) (N ν) N u(t, s) (N ν) H ν N (b, ρ(s)) H N 1 (N 1) (t, 1) (N ν) H ν N (b, ρ(s)) 1 (N ν) 0 0. Next, it will be shown tht the boundry conditions hold. Let 0 k N 3. Then k u( 1, s) H ν N (b, ρ(s)) H N k 1 (t, 1) t 1 H ν N (b, ρ(s)) Γ(t + N k) Γ(t + 1)Γ(N k) t 1

40 31 H ν N (b, ρ(s)) 0. Γ(N k 1) Γ(0)Γ(N k) Also N 2 u( 1, s) H ν N (b, ρ(s)) H N 1 N+2 (t, 1) t 1 H ν N (b, ρ(s)) t + 1 Γ(N k) t 1 0. Next, note tht N 1 x(t, s) N 1 H ν 1 (t, ρ(s)) H ν N (t, ρ(s)). Then, N 1 u(b, s) H ν N (b, ρ(s)) H N 1 N+1 (t, 1) tb H ν N (b, ρ(s)) N 1 x(b, s). Consider the cse where ν N N. Then u(t, s) H N 1 (t, 1) (t + 1)N 1, Γ(N) nd v(t, s) H N 1 (t, 1) + H N 1 (t, ρ(s)) (t + 1)N 1 + Γ(N) (t s + 1)N 1. Γ(N)

41 32 Note tht this ligns well with the Green s function for the continuous cse with non-frctionl derivtives. In the continuous cse, ccording to [41, Exmple 6.30 on p. 296], the Green s function for the BVP x (n) 0, x (i) () 0, 0 i n 2 x (n 1) (b) 0. is given by G(t, s) : (t )n 1 (n 1)!, t s b (t )n 1 (n 1)! + (t s)n 1 (n 1)!, s t b. In Exmple 3.10, Theorems 3.8 nd 3.9 will be used to clculte the solution to specil cse of the BVP (3.3). Exmple Sy ν 15, 0, b 6, nd h(t) 1. Tht is, consider the 8 following BVP x(t) 1, t N6 1 x( 1) 0, (3.6) x(6) 0. By Theorem 3.9, the Green s function for the BVP (3.6) is given by u(t, s), G(t, s) v(t, s), t ρ(s) ρ(s) t

42 33 where u(t, s) (t + 1)H 1 (6, ρ(s)) 8 nd v(t, s) u(t, s) + x(t, s), where x(t, s) H 7 8 (t, ρ(s)). By Theorem 3.8 nd by its proof, the solution for t N 6 1 to the BVP (3.6) is given by x(t) (t + 1) [ (t + 1) G(t, s)h(s) s u(t, s)h(s) s H 7 8 (t + 1)H 7 8 t 0 x(t, s)h(s) s H 1 (6, ρ(s))h(s) s + 8 [ (6, s) ] s6 s0 (6, 0) H 15 8 H 15 8 (t, s) t 0 ] s6 s0 (t, 6) + H 15 (t, 0). 8 H 7 (t, ρ(s))h(s) s 8 Note tht by rewriting the bove solution s ) x(t) Γ ( Γ(6)Γ ( ) Γ ( t Γ(t 6)Γ ( ) + Γ ( t Γ(t)Γ ( ), for t N 6 1, we my provide grphicl solution to the BVP (3.6), s shown in Figure 3.1. ) )

43 34 Figure 3.1: A solution to the homogenous BVP (3.6) Bounds on the Green s Function We will now estblish bounds on G(t, s) with the gol of eventully pproching the nonliner cse using this Green s function. Consider G(t, s) for the BVP (3.3). If t ρ(s), it holds tht the Cuchy function H ν 1 (t, ρ(s)) 0 so u(t, s) v(t, s). Additionlly, for t N, define : R N such tht t t +, for t nd such tht t for t <. Then, to be used in Theorem 3.11, define the constnt r : b b ( + 1). 4 Note tht, in order to estblish the bound in prt (iii) of Theorem 3.11, the proof requires restriction on the domin of solutions to the BVP (3.3). In prticulr, it

44 35 requires tht 1 b Note tht 1 1, which implies tht ν 3. 2 ν 2 ν 2 Hence we get the restriction mentioned in prt (iii) of Theorem 3.11 nd in Section 3.4. Theorem Let 1 < ν < 2 nd 1 b 1 2 ν. Let G(t, s) : Nb 1 N b +1 R be the Green s function for the homogeneous BVP (3.3). Then the following hold: (i) G(t, s) 0 for (t, s) N b 1 N b +1, nd (ii) min t N b 1 G(t, s) G(ρ(s), s) for ech fixed s N b +1. If we further ssume 1 b 1 2 ν 1 nd thus tht 3 2 ν 2, then (iii) G(t, s) k G(ρ(s), s), where k is constnt stisfying 0 < k 1 for t N b r nd s N b +1. Proof. (i) Note tht N 2. Let (t, s) N b 1 N b +1 nd t ρ(s). Then G(t, s) u(t, s) H 1 (t, 1)H ν 2 (b, ρ(s)) (t + 1)(b s + 1)ν 2 Γ(ν 1) (t + 1)Γ(b s + ν 1) Γ(b s + 1)Γ(ν 1) 0 Next let (t, s) N b 1 N b +1 nd ρ(s) t. Then G(t, s) v(t, s) H 1 (t, 1)H ν 2 (b, ρ(s)) + H ν 1 (t, ρ(s)).

45 36 Now it will be shown tht v(t, s) is incresing with respect to t where t N b 1 nd ρ(s) t for ech fixed s N b +1. Note tht for ν 2, t H ν 2 (t, ρ(s)) t H 0 (t, ρ(s)) t 1 0. For 1 < ν < 2, t H ν 2 (t, ρ(s)) H ν 3 (t, ρ(s)) (t s + 1)ν 3 Γ(ν 2) Γ(t s + ν 2) Γ(t s + 1)Γ(ν 2) 0, provided t s, since Γ(ν 2) 0. Hence t v(t, s) H ν 2 (b, ρ(s)) + H ν 2 (t, ρ(s)) 0 for t s. If t s b, then t v(b, b) H ν 2 (b, ρ(b)) + H ν 2 (b, ρ(b)) 0. Also, in generl if t s, t v(t, t) H ν 2 (b, ρ(t)) + H ν 2 (t, ρ(t)) (b t + 1)ν 2 + 1ν 2 Γ(ν 1) Γ(ν 1)

46 37 Γ(b t + ν 1) Γ(b t + 1)Γ(ν 1) since 0 < ν 1 1. Hence t v(t, s) 0 for t s for every fixed s N b 1. Thus v(t, s) is incresing with respect to t where t N b 1 nd ρ(s) t for ech fixed s N b 1. Next let (t, s) N b 1 N b +1 nd t ρ(s). Then v(b, s) H 1 (b, 1)H ν 2 (b, ρ(s)) + H ν 1 (b, ρ(s)) (b + 1)(b s + 1)ν 2 + Γ(ν 1) (b + 1)Γ(b s + ν 1) + Γ(b s + 1)Γ(ν 1) (b + 1)Γ(b s + ν 1)(ν 1) Γ(b s + 1)Γ(ν) Γ(b s + ν 1) Γ(b s + 1)Γ(ν) Γ(b s + ν 1) Γ(b s + 1)Γ(ν) Γ(b s + ν 1) Γ(b s + 1)Γ(ν) Γ(b s + ν 1) [(b )(2 ν) 1] Γ(b s + 1)Γ(ν) Γ(b s + ν 1) Γ(b s + 1)Γ(ν) 0. (b s + 1)ν 1 Γ(ν) Γ(b s + ν) Γ(b s + 1)Γ(ν) + (b s + ν 1)Γ(b s + ν 1) Γ(b s + 1)Γ(ν) [ (b + 1)(ν 1) + (b s) + ν 1] [ (b + 1)(ν 1) + (b ) + ν 2] [(b )(1 ν) (ν 1) + (b ) + ν 2] [ 1 (2 ν) 1 2 ν ] Therefore v(t, s) 0.

47 38 (ii) Let t ρ(s) nd (t, s) N b 1 N b +1. Then (b s + 1)ν 2 t u(t, s) Γ(ν 1) Γ(b s + ν 1) Γ(b s + 1)Γ(ν 1) 0. Note tht by the proof of (i) it lso holds tht t v(t, s) 0 for ρ(s) t where (t, s) N b 1 N b +1. Hence min t N b 1 G(t, s) G(ρ(s), s) for ech fixed s N b +1. (iii.) Let for t N b r, s N b +1, 1 b 1 1. Define 2 ν It will first be shown tht Let t ρ(s). Then { 3(b ) k : min, 4(b ) b G(t,s) G(ρ(s),s) k. [ (b )(ν 2) + 1 ν 1 ]}. G(t, s) G(ρ(s), s) H 1 (t, 1)H ν 2 (b, ρ(s)) + H ν 1 (t, ρ(s)) H 1 (ρ(s), 1)H ν 2 (b, ρ(s)) + H ν 1 (ρ(s), ρ(s)) (t + 1)H ν 2(b, ρ(s)) H ν 1 (t, ρ(s)) 1 s (s )H ν 2 (b, ρ(s)) [ t + 1 H ν 1(t, ρ(s)) H ν 2 (b, ρ(s)) ]. Note [ ] Hν 1 (t, ρ(s)) s H ν 2 (b, ρ(s)) H ν 2(b, ρ(s))[ H ν 2 (t, s 2)] H ν 1 (t, ρ(s))[ H ν 3 (b, s 2)]. H ν 2 (b, ρ(s))h ν 2 (b, s 2)

48 39 Then, considering the numertor of the bove expression, H ν 2 (b, ρ(s))[ H ν 2 (t, s 2)] H ν 1 (t, ρ(s))[ H ν 3 (b, s 2)] (3.7) (b s + 1)ν 2 (t s + 2) ν 2 + (t s + 1)ν 1 (b s + 2) ν 3 Γ(ν 1)Γ(ν 1) Γ(ν)Γ(ν 2) Γ(b s + ν 1)Γ(t s + ν) (Γ(ν 1)) 2 Γ(b s + 1)Γ(t s + 2) + Γ(t s + ν)γ(b s + ν 1) Γ(ν)Γ(ν 2)Γ(t s + 1)Γ(b s + 2) [ ] Γ(b s + ν 1)Γ(t s + ν) 1 Γ(b s + 1)Γ(t s + 1)Γ(ν 1)Γ(ν 2) (t s + 1)(ν 2) + 1 (b s + 1)(ν 1) 0, since, provided ν 2, 1 < ν 2 < 0 nd Γ(ν 2) 0. It cn be shown tht if ν 2, the expression in (3.7) is nonpositive. Additionlly t [(t + 1)H ν 2 (b, ρ(s)) H ν 1 (t, ρ(s))] H ν 2 (b, ρ(s)) H ν 2 (t, ρ(s)) 0 by the proof of prt (i). Hence, [ 1 t + 1 H ] ν 1(t, ρ(s)) 1 [ t + 1 H ] ν 1(t, ρ( + 1)) s H ν 2 (b, ρ(s)) b H ν 2 (b, ρ( + 1)) [ ] 1 (t ) ν 1 t + 1 b (ν 1)(b ) ν 2 [ ] 1 (b ) ν 1 b + 1 b (ν 1)(b ) ν 2 1 [ b + 1 b ] Γ(b + ν 1)Γ(b ) (ν 1)Γ(b + ν 2)Γ(b ) 1 [ b + 1 b + ν 2 ] b ν 1

49 1 [ ] (b + 1)(ν 1) (b + ν 2) b ν 1 1 [ ] (b )(ν 2) + 1 b ν 1 k. 40 Let t ρ(s). Becuse t N b r, t b b ( + 1) 4 b ( + 1) b 4 + b ( + 1) b + 4 b ( + 1) b. 4 Then G(t, s) G(ρ(s), s) H 1(t, 1)H ν 2 (b, ρ(s)) H 1 (ρ(s), 1)H ν 2 (b, ρ(s)) t + 1 s b b (+1) b 3(b ) + 5 4(b ) k. Next it will be shown tht 0 < k 1. First, for 3 2 ν 2, 1 b [ ] (b )(ν 2) + 1 ν 1 [( 1 1 b 2 ν 1) (ν 2) + 1 ν 1 ]

50 41 1 b 1 b 0. [ ν 1 2 ν [ 2 ν ν 1 ] (ν 2) + 1 ν 1 ] For ν 2, note 1 b [ ] (b )(ν 2) + 1 ν 1 1 b > 0. Also, 1 b [ ] (b )(ν 2) + 1 ν 1 1 [ 1 b [ 1 b 1. ν 1 ] 1 ( 1 2) ] Note tht if 1 b 4, then N b r contins only the element b. Thus the cses where 1 b 4 re contined within the cse where ρ(s) t. If b 5, then, for the t ρ(s) cse, 3(b )+5 4(b ) 1, nd certinly 3(b )+5 4(b ) 0. Note tht Theorem 3.11 lso holds for ν 2. However, the bound 1 b 1 2 ν is not necessry for prts (i) nd (ii) to hold in this cse. Additionlly prt (iii) holds without the bound 1 b 1 2 ν 1. Next we give some exmples to illustrte our results.

51 42 Exmple Consider the boundry vlue problem x(t) h(t), t N8 1 x( 1) 0 x(8) 0 Tht is, ν 15, b 8, nd 0. Note tht ν Hence, Theorem 3.11, prt (i) nd (ii) pplies to this exmple. Set s 5. Then, for 1 t 4, G(t, 5) u(t, 5) H 1 (t, 1) H 1 (b, ρ(s)) 8 ) (t + 1)Γ ( b s Γ(b s + 1)Γ ( ) 7 8 ) (t + 1)Γ ( Γ(4)Γ ( ) 7 8 (t + 1) ( ) ( ( ) 8) (t + 1) Also, for 4 t 8, G(t, 5) v(t, 5) (t + 1) + H 7 (t, ρ(s)) 8

52 (t 4) (t + 1) + Γ ( ) 15 8 ( ) Γ t (t + 1) + Γ(t 4)Γ ( ) Note tht in Figure 3.2, G(t, 5) 0 for 1 t 8 nd min t {1,2,,8} G(t, 5) G(4, 5). Figure 3.2: A grph of G(t, 5) for 1 t 8. In Exmple 3.13, we increse the domin in Exmple 3.12 by 1 so tht the bound in Theorem 3.11 no longer holds. Tht is, we find in this instnce tht the Green s function lredy no longer stisfies G(t, s) 0 for ll (t, s) N 1 N +1. Exmple Now consider the boundry vlue problem x(t) h(t), t N9 1 x( 1) 0 x(9) 0

53 44 Tht is ν 15 8, b 9, nd 0. Set s 1. Then if t N9 1, Then, G(t, 1) v(t, 1) (t + 1)Γ ( b s Γ(b s + 1)Γ ( ) + Γ ( t ρ(s) Γ(t ρ(s))γ ( ) v(8, 1) (9)Γ ( Γ(9)Γ ( ) + Γ ( Γ(8)Γ ( ) ) ) ) ) Note tht in this exmple, incresing the vlue of b by 1 pst the region to which Theorem 3.11, prt (i) pplies cuses G(t, s) 0 to no longer be true. 3.4 The Nonliner Cse Consider the following nonliner boundry vlue problem for 3 2 ν 2 ν x(t) h(t, x(t 1)), t Nb +1 x( 1) 0, (3.8) x(b) 0, where h : N b +1 R + R + 1, b Z, 1 b 1, nd where the solutions 2 ν x re defined on N b 1. Write r : b b (+1). 4 In this section, we will use Theorem 3.11 nd Theorem 3.15 (Guo-Krsnosel skiĭ Theorem) to find solutions to the BVP (3.8). In the pproch to this problem, rguments nlogous to those used in [27, Section 3], where Erbe nd Peterson found

54 45 positive solutions to boundry vlue problem involving whole-order derivtives in time scles context, re used below. Define the Bnch spce E : {x : N b 1 R : x( 1) 0 nd x(b) 0} with norm x : mx{ x(t), t N b 1}. Tht is, ll elements of E stisfy the boundry conditions in BVP (3.8). Define the cone K : {x E : both x(t) 0 for t N b 1 nd x(t) k x for t N b r}, where 0 < k 1 is constnt s defined in Theorem 3.11 prt (iii). The generl definition of cone is given in Definition Define the opertor A by Ax(t) G(t, s)h(s, x(s 1)) s s+1 b G(t, s)h(s, x(s 1)) s for t N b 1 nd x K. By Theorem 3.11 prt (i), G(t, s) 0 for (t, s) N b 1 N b +1, nd h : N b +1 R + R +. Hence Ax(t) 0 for t N b 1. Also by Theorem 3.11 prt (ii), min Ax(t) t N b r k k min G(t, s)h(s, x(s 1)) s t N b r G(ρ(s), s)h(s, x(s 1)) s mx t N b 1 k mx t N b 1 G(t, s)h(s, x(s 1)) s G(t, s)h(s, x(s 1)) s

55 46 k Ax. Thus it holds tht Ax K, so A : K K. In the theorems tht follow, we will be finding fixed points of the opertor A, since fixed points give solutions to the BVP (3.8). The generl definition of cone from [38] is given below. Definition Let E be Bnch spce. A nonempty closed convex set C E is clled cone if it stisfies the following two conditions: (i.) x C, λ 0 implies λx C; (ii.) x C, x C implies x 0. where 0 denotes the identity element of E. Next we will be mking use of theorem from Krsnosel skiĭ [42] nd Deimling [26] stted below. The lrger context of the theorem in cone theory nd its pplictions to nonliner problems is more fully elborted in the text by Guo nd Lkshmiknthm [35]. Theorem 3.15 (Guo-Krsnosel skiĭ). Let E be Bnch spce nd let P E be cone. Assume Ω 1, Ω 2 re open subsets of E with Ω 1 Ω 2 nd 0 Ω 1, nd ssume tht (i.) Ax x, x P Ω 1, nd Ax x, x P Ω 2, or (ii.) Ax x, x P Ω 1, nd Ax x, x P Ω 2. Then A hs fixed point in P (Ω 2 \ Ω 1 ).

56 Existence of Positive Solutions In Theorems 3.16 nd Theorem 3.17, we will be mking use of Theorem 3.15 to find fixed points of A nd thus to show the existence of positive solutions. Define 1 m :, G(ρ(s), s) s 1 w : k G(t r 0, s) s, nd where t 0 N b r is fixed. Note tht w m, since k G(t 0, s) s G(t 0, s) s G(ρ(s), s) s r r r G(ρ(s), s) s. In the following, let B r (0) E denote n open bll of rdius r bout the origin in E. For ll theorems in this section ssume tht h : N b 1 R + R + is continuous. Theorem Consider the following hypotheses: (i.) h(t, x) mp 1 for ll t N b 1 nd 0 x p 1 ; (ii.) if r < b, h(t, x) wx for ll t N b r+1 nd kp 2 x p 2, or if r b, h(t, x) wx for t b nd kp 2 x p 2 ; (iii.) kp 2 p 1 whenever p 1 < p 2 nd w m; nd (iv.) wp 2 < mp 1 whenever p 2 < p 1. If there exist p 1, p 2 (0, ) such tht either (i), (ii), nd (iii) hold nd p 1 < p 2 OR (i), (ii), nd (iv) hold nd p 2 < p 1, then the boundry vlue problem (3.8) hs positive solution.

57 48 Proof. Suppose first tht p 1 < p 2. Then p 1 kp 2, so h(t, x(t 1)) mkes sense. Define Ω 1 : B p1 (0) E nd Ω 2 : B p2 (0) E. Sy x is such tht x p 1, x 0, nd x kp 1 on t N b r. Then x K Ω 1. Hence, Ax(t) G(t, s)h(s, x(s 1)) s G(ρ(s), s)h(s, x(s 1)) s mp 1 G(ρ(s), s) s p 1. Then Ax p 1 x. Next sy x is such tht x p 2, x 0, nd x kp 2 on t N b r. Then x K Ω 2. Hence, Ax(t 0 ) w r G(t 0, s)h(s, x(s 1)) s G(t 0, s)h(s, x(s 1)) s r wk x x G(t 0, s)x(s 1) s r G(t 0, s) s so since Ax(t 0 ) Ax, it holds tht Ax x. Therefore the boundry vlue problem (3.8) hs solution in K (Ω 2 \ Ω 1 ). Tht is, the solution x(t) is such tht p 1 x p 2, x 0 on t N b 1, nd x kp 1. Now sy p 2 < p 1. Then wp 2 < mp 1, so wx wp 2 < mp 1 for x p 2, so h(t, x(t 1)) mkes sense. Define Ω 1 : B p2 (0) E nd Ω 2 : B p1 (0) E. If x is such tht x p 2, x 0, nd x(t) kp 2 on t N b r, then x K Ω 1, nd by the bove,

58 49 Ax x. If x is such tht x p 1, x 0, nd x(t) kp 1 on t N b r, then x K Ω 2, nd by the bove, Ax x. Then by the second prt of the Guo- Krsnosel skiĭ Theorem (Theorem 3.15), the boundry vlue problem hs solution x(t) such tht p 2 x p 1, x 0 on t N b 1, nd x kp 2. Next, define λ λ(t) : lim x 0 + h(t, x) x nd h(t, x) γ γ(t) : lim. x x for t N b 1. Theorem Assume λ nd γ exist pointwise on the extended rel numbers nd furthermore tht either (i) λ 0 nd γ or (ii) λ nd γ 0. Then the nonliner boundry vlue problem (3.8) hs solution. Proof. (i) Assume λ 0 nd γ. Becuse λ 0, we define δ > 0 such tht 0 < x δ implies, for ll t N b 1, h(t, x) x m.

59 50 Thus for 0 < x δ, h(t, x) mx, for ll t N b 1. Define Ω 1 : B δ (0) E nd sy x Ω 1 K. Then for t N b 1, Ax(t) m mδ mδ δ. G(t, s)h(s, x(s 1)) s G(t, s)x(s 1) s G(t, s) s G(ρ(s), s) s Since x δ, Ax x. Next, becuse γ, we define δ > 0 such tht x δ implies, for ll t N b 1, h(t, x) x w. Thus for x δ, h(t, x) wx, for ll t N b 1. Then define r : mx{δ, 2δ} Define Ω 2 : B r (0) E nd sy x Ω 2 K. Then Ax(t 0 ) w r G(t 0, s)h(s, x(s 1)) s G(t 0, s)h(s, x(s 1)) s r wk x G(t 0, s)x(s 1) s r G(t 0, s) s

60 51 x. Since Ax(t 0 ) Ax, Ax x. Therefore the nonliner boundry vlue problem (3.8) hs solution in K (Ω 2 \ Ω 1 ). (ii) Assume λ nd γ 0. Becuse λ, define ζ > 0 such tht 0 < x ζ implies, for ll t N b 1, h(t, x) x w. Thus for 0 < x ζ, h(t, x) wx. Define Ω 1 : B ζ (0) E nd sy x Ω 1 K. Then Ax(t 0 ) w r G(t 0, s)h(s, x(s 1)) s G(t 0, s)h(s, x(s 1)) s r kw x x, G(t 0, s)x(s 1) s r G(t 0, s) s so Ax x. Becuse γ 0, define ξ > 0 such tht x ξ implies, for ll t N b 1, h(t, x) x m. Then for x ξ, h(t, x) mx for ll t N b 1. Define α : mx{ξ, 2ζ}, nd define

61 52 Ω 2 : B α (0) E. Then if x Ω 2 K, Ax(t) m G(t, s)h(s, x(s 1)) s m x m x x. G(t, s)x(s 1) s G(t, s) s G(ρ(s), s) s Thus Ax x. Therefore the nonliner boundry vlue problem (3.8) hs solution in K (Ω 2 \ Ω 1 ) Conditions for the Existence of Multiple Positive Solutions Next note tht Theorems 3.16 nd Theorem 3.17, while showing the existence of positive solutions, did not gurntee uniqueness of solutions. Theorem 3.18 below shows the possibility of not only two solutions but even countbly mny solutions when prticulr conditions on the function h re met. Recll tht, s in the previous two theorems, the constnt k is s defined in Theorem 3.11 nd is 0 < k 1. Additionlly recll tht 1 m :, G(ρ(s), s) s 1 w : k G(t r 0, s) s nd where t 0 N b r is fixed, nd recll tht w m.

62 53 Theorem Suppose we hve sequence of positive rel vlues such tht 0 < p 1 < kp 2 p 2 < p 3 < kp 4 p 4 < p 5 < kp 6 p 6 <. Consider the following hypotheses: (i.) h(t, x) mp 1 for ll t N b 1 nd 0 x p 1, (ii.) h(t, x) mp i for i odd for ll t N b 1 nd p i 1 + ε x p i, where 0 < ε < p i p i 1, (iii.) h(t, x) wx for i even for ll t N b r+1 or t b if r b nd kp i x p i. Then there re countbly mny positive solutions to the BVP (3.8). Proof. For ech n N, define Ω n : B pn (0) E. Then by Theorem 3.16, there exists solution in K (Ω 2 \ Ω 1 ). Assume x p i when i is odd. Then h(t, x) mp i. Also ssume x 0 nd x kp i on t N b r. Then x K Ω i. Hence Ax(t) G(t, s)h(s, x(s 1)) s G(ρ(s), s)h(s, x(s 1)) s mp i G(ρ(s), s) s p i x. Hence Ax x. Next ssume x p i+1. Then h(t, x) wx. Also ssume tht x 0 nd x kp i+1 on t N b r. Then x K Ω i+1. Ax(t) G(t, s)h(s, x(s 1)) s

63 54 w r G(t 0, s)h(s, x(s 1)) s r k x w x. G(t 0, s)x(s 1) s r G(t 0, s) s Hence Ax x. From the bove one cn conclude tht there exists solution in K (Ω i+1 \ Ω i ) for ll i N. Hence, there exist countbly mny positive solutions to the BVP (3.8). Note tht while the theorem bove specifies conditions on h(t, x) under which countbly mny solutions my be gurnteed, the proof of the bove theorem my be used to gurntee other finite numbers of solutions. The exmple below will illustrte how to use the bove proof to gurntee two solutions to the boundry vlue problem (3.8). Exmple Consider the boundry vlue problem x(t) h(t, x(t 1)), t N6 1 x( 1) 0, (3.9) x(6) 0, where solutions x re defined on N 6 1 nd h : N 6 1 R + R +. In this exmple, we will find conditions on h tht gurntee two solutions to the BVP (3.9). First, we must find the vlue of constnts k, m, nd w. By the definition of k from Theorem