Three Point Boundary Value Problems on Time Scales

Journal of Difference Equations and Applications, Vol. 10, No. 9, 10 August 2004, pp. 843–849 Three Point Boundary Value Problems on Time Scales † A.C. PETERSONa,*, Y.N. RAFFOULb, and C.C. TISDELLc,‡ a Department of Mathematics, University of Nebraska – Lincoln, Lincoln, NE 68588-0323, USA; bDepartment of Mathematics, University of Dayton, Dayton, OH 45469-2316, USA; c School of Mathematics, The University of New South Wales, Sydney, NSW 2052, Australia (Received 8 August 2003; Revised 5 January 2004; In final form 22 March 2004) This work formulates existence theorems for solutions to three-point boundary value problems on time scales. The ideas are based on a relationship between the three point boundary conditions, lower and upper solutions and topological degree theory. Keywords: Time scale; Degree theory; Three-point boundary value problem; Second-order dynamic equation AMS Subject Classification: 39A12 INTRODUCTION This paper considers the existence of solutions to the second-order dynamic equation y DD ðtÞ ¼ f ðt; y s ðtÞÞ; t [ ½a; b
; ð1Þ subject to the three point boundary conditions gðð yðaÞ; yðs 2 ðbÞÞ; yðeÞ; ð y D ðaÞ; y D ðsðbÞÞÞÞ ¼ ð0; 0Þ; a , e , s 2 ðbÞ; e [ T; ð2Þ where f : ½a; b
£ R ! R; g : R2 £ R £ R2 ! R2 are continuous, and t is from a so-called “time scale” T. It is assumed that the reader is familiar with the time scale calculus and associated definitions such as delta derivative, jump operators and right-dense continuity. If not, then we refer the reader to Ref. [2]. In Ref. [5], the authors introduced the idea of compatibility of boundary conditions for two point boundary value problems (BVPs) on time scales. In this paper, we extend these ideas to three point BVPs on time scales. The new compatibility conditions are then applied to give some results for the existence of solutions to three point BVPs on time scales. The BVPs treated in this paper include a very wide range of boundary conditions, including nonlinear BVPs. A solution y to Eq. (1) is a function y : ½a; s 2 ðbÞ
: ! R satisfying Eq. (1) with y [ C 2rd : *Corresponding author. E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected] Journal of Difference Equations and Applications ISSN 1023-6198 print/ISSN 1563-5120 online q 2004 Taylor & Francis Ltd http://www.tandf.co.uk/journals DOI: 10.1080/10236190410001702481 844 A.C. PETERSON et al. COMPATIBILITY OF BOUNDARY CONDITIONS We will need the two following results, the proofs are found in Ref. [5]. Lemma 1 Let a; b [ C 2rd ð½a; s 2 ðbÞ
Þ satisfy aðtÞ # bðtÞ; for t [ ½a; s 2 ðbÞ
; a DD ðtÞ . f ðt; uÞ; for t [ ½a; b
; u , a s ðtÞ; ð3Þ b DD ðtÞ , f ðt; uÞ; for t [ ½a; b
; u . b s ðtÞ: ð4Þ If y is a solution to Eq. (1) with aðaÞ # yðaÞ # bðaÞ and aðs 2 ðbÞÞ # yðs 2 ðbÞÞ # bðs 2 ðbÞÞ; then aðtÞ # yðtÞ # bðtÞ for t [ ½a; s 2 ðbÞ
: Similarly, the following result holds. Lemma 2 Let a; b [ C 2rd ð½a; s 2 ðbÞ
Þ satisfy aðtÞ , bðtÞ; t [ ½a; s 2 ðbÞ
; a DD ðtÞ . f ðt; uÞ; b DD ðtÞ , f ðt; uÞ; for for t [ ½a; b
; t [ ½a; b
; u # a s ðtÞ; s u $ b ðtÞ: ð5Þ ð6Þ If y is a solution to Eq. (1) with aðaÞ , yðaÞ , bðaÞ and aðs 2 ðbÞÞ , yðs 2 ðbÞÞ , bðs 2 ðbÞÞ; then aðtÞ , yðtÞ , bðtÞ for t [ ½a; s 2 ðbÞ
: The functions a and b satisfying the inequalities in Lemmas 1 and 2 are usually referred to as lower and upper solutions, respectively. For more on upper and lower solutions on time scales see Refs. [1, 2 Chapter 6, 3 Chapter 6]. The notion of compatible boundary conditions for BVPs on time-scales will now be naturally extended from the theory in Ref. [5]. In the remainder of this paper assume D ¼ ðaðaÞ; bðaÞÞ £ ðaðs 2 ðbÞÞ; bðs 2 ðbÞÞÞ – Y:  2 Þ is strongly inwardly Definition 1 We say that the vector field C ¼ ðc 0 ; c 1 Þ [ CðDR pointing on D̄ if c 0 ðC; DÞ . a D ðaÞ for C ¼ aðaÞ; and aðs 2 ðbÞÞ # D # bðs 2 ðbÞÞ; c 0 ðC; DÞ , b D ðaÞ for C ¼ bðaÞ; and aðs 2 ðbÞÞ # D # bðs 2 ðbÞÞ; c 1 ðC; DÞ , a D ðsðbÞÞ for D ¼ aðs 2 ðbÞÞ; and aðaÞ # C # bðaÞ; c 1 ðC; DÞ . b D ðsðbÞÞ for D ¼ bðs 2 ðbÞÞ; and aðaÞ # C # bðaÞ: If we replace the strict inequalities by weak inequalities, then we say C is inwardly pointing.  £ R £ R2 ; R2 Þ: We say that g is strongly compatible with a Definition 2 Let g [ CðD and b if for all strongly inwardly pointing vector fields C on D̄ and all continuous functions f : ½aðaÞ; bðaÞ
! ½aðeÞ; bðeÞ
; GðC; DÞ – ð0; 0Þ; for all ðC; DÞ [ ›D; dðG; D; ð0; 0ÞÞ – 0; ð7Þ ð8Þ  and dðG; D; ð0; 0ÞÞ is the where GðC; DÞ ¼ gððC; DÞ; fðCÞ; CðC; DÞÞ for all ðC; DÞ [ D degree of G at (0,0) relative to D. If Eqs. (7) and (8) hold for all inwardly pointing vector fields C and all f then we call g very strongly compatible with a and b. THREE POINT BVPS ON TIME SCALES 845 EXISTENCE OF SOLUTIONS In this section, some existence results are presented for the BVP (1) and (2). The proofs rely on the a priori bounds on solutions of “Compatibility of Boundary Conditions Section” and on the following “Homotopy Principle”, the proof of which can be found in Ref. [4]. Assume E is a bounded, open subset of the normed space X and p [ X: See Ref. [4] for the definition of K1 in Theorem 1. Theorem 1 Let H [ K 1 ðE £ ½0; 1
; XÞ such that Hðz; lÞ – p for all z [ ›E and all l [ ½0; 1
: Then dðHðz; lÞ; D; pÞ is independent of l [ ½0; 1
: The Homotopy principle above will be applied to the modified BVP y DD ¼ mðt; y s Þ; t [ ½a; b
; ð9Þ where 8 ð12jJðu2 b s ðtÞÞjÞf ðt; b s ðtÞÞþJðu2 b s ðtÞÞðjf ðt; b s ðtÞÞjþ1Þ; u $ b s ðtÞ; > > < s s mðt;uÞ ¼ f ðt;uÞ; a ðtÞ # u # b ðtÞ; > > : ð12jJðu2 a s ðtÞÞjÞf ðt; a s ðtÞÞþJðuÞ2 a s ðtÞÞðjf ðt; a s ðtÞÞjþ1Þ; u # a s ðtÞ; and J is given by 8 1; v $ 1; > > < JðvÞ ¼ v; jvj , 1; > > : 21; v # 1; subject to the boundary conditions (2) via the following lemma. Lemma 3 Let V £ D , Cð½a; s 2 ðbÞ
Þ £ R2 with V £ D open and bounded. Let H [  £D  £ ½0; 1
; Cð½a; s 2 ðbÞ
Þ £ R2 Þ be such that Hð y; C; D; 1Þ ¼ 0 is equivalent to the K 1 ðV BVP (9) and (2). If all solutions ð y; C; DÞ to Hð y; C; D; lÞ ¼ 0; satisfy ð y; C; DÞ Ó ›ðV £ DÞ for all l [ ½0; 1
and if dðHð y; C; D; 0Þ; V £ D; 0Þ – 0; then the BVP (9) and (2) has at least one solution. Proof Note that the conditions of Theorem 1 are satisfied and therefore dðHð·; 1Þ; V £ D; 0Þ ¼ dðHð·; 0Þ; V £ D; 0Þ – 0: Hence Hð y; C; D; 1Þ ¼ 0 has a solution ð y; C; DÞ [ V £ D: Since Hð y; C; D; 1Þ ¼ 0 is equivalent to the BVP (9) and (2) the BVP has at least one solution. A Theorem 2 Assume a; b [ C 2rd ð½a; s 2 ðbÞ
Þ with aðtÞ # bðtÞ for t [ ½a; s 2 ðbÞ
and f : ½a; b
£ R ! R is continuous. Further assume a DD ðtÞ . f ðt; uÞ; for t [ ½a; b
; u , a s ðtÞ; ð10Þ b DD ðtÞ , f ðt; uÞ; for t [ ½a; b
; u . b s ðtÞ: ð11Þ If g [ CðD £ R £ R2 ; R2 Þ is strongly compatible with a and b then the BVP (1) and (2) has at least one solution y [ Cð½a; s 2 ðbÞ
Þ with y DD [ Crd ð½a; b
Þ satisfying aðtÞ # yðtÞ # bðtÞ for t [ ½a; s 2 ðbÞ
: 846 A.C. PETERSON et al. Proof (i) Modification Consider the following modified equation of (9) with respect to a and b y ¼ mðt; y s Þ; t [ ½a; b
; ð0; 0Þ ¼ gðð yðaÞ; yðs 2 ðbÞÞ; yðeÞ; ð y D ðaÞ; y D ðsðbÞÞÞÞ: ð12Þ ð13Þ The approach now is to show that the BVP (12) and (13) has a solution y satisfying aðtÞ # yðtÞ # bðtÞ for t [ ½a; s 2 ðbÞ
: As f and m agree in this region then y will also be the required solution to the BVP (1) and (2). Notice that m; a and b satisfy a ðtÞ . mðt; uÞ; for t [ ½a; b
; u , a s ðtÞ; b ðtÞ , mðt; uÞ; for t [ ½a; b
; u . b s ðtÞ:  then aðtÞ # Therefore by Lemma 1, if y is a solution to Eq. (12) and ð yðaÞ; yðs 2 ðbÞÞÞ [ D 2 yðtÞ # bðtÞ for t [ ½a; s ðbÞ
: Hence y is the required solution to Eq. (1). (ii) Existence Consider the equation Hð y; C; D; lÞ ¼ ðH 1 ð y; C; D; lÞ; H 2 ð y; C; D; lÞÞ ¼ ð0; 0; 0Þ; where 8 y 2 3lwðC; DÞ 2 ð1 2 3lÞða þ bÞ=2; for 0 # l # 1=3; > > < for 1=3 # l # 2=3; ; H 1 ð y; C; D; lÞ ¼ y 2 ð3l 2 1ÞTð yÞ 2 wðC; DÞ; > > : y 2 Ty 2 wðC; DÞ; for 2=3 # l # 1; where a ¼ wðC; DÞðtÞ ¼ min t[½a;s 2 ðbÞ
b ¼ aðtÞ 2 1; C s 2 ðbÞ 2 Da þ ðD 2 CÞt ; s 2 ðbÞ 2 a ðTyÞðtÞ ¼ ð sðbÞ max t[½a;s 2 ðbÞ
bðtÞ þ 1; for C; D [ R Gðt; sÞmðs; y s ðsÞÞDs; and a # t # s 2 ðbÞ; t [ ½a; s 2 ðbÞ
; a where Gðt; sÞ ¼ and 8 ðt2aÞðs 2 ðbÞ2sðsÞÞ > ; <2 s 2 ðbÞ2a 2 ðsðsÞ2aÞðs ðbÞ2tÞ > ; :2 s 2 ðbÞ2a for t # s; for sðsÞ # t: H 2 ð y; C; D; lÞ ¼ 8 gððC; DÞ; fðCÞ; CðC; DÞÞ; for 0 # l # 2=3; > > > > < gðC; DÞ; ð3l 2 2ÞyðeÞ þ 3ð1 2 lÞfðCÞ; ð3l 2 2Þð y D ðaÞ; y D ðsðbÞÞ þ 3ð1 2 lÞCðC; DÞÞ; > > > > : for 2=3 # l # 1; THREE POINT BVPS ON TIME SCALES 847 Clearly H is completely continuous and Hð y; C; D; 1Þ ¼ 0 is equivalent to the modified BVP (12) and (13). Let V ¼ {y [ Cð½a; s 2 ðbÞ
Þ : a , yðtÞ , b on ½a; s 2 ðbÞ
} and G ¼ V £ D: Therefore, to apply Lemma 3 we need to show that solutions ð y; C; DÞ to Hð y; C; D; lÞ ¼ 0 satisfy ð y; C; DÞ Ó ›G for all l [ ½0; 1
: We investigate the cases l [ ½2=3; 1
and ð1=3; 2=3Þ; the case l [ ½0; 1=3
is trivial. Case (i) l [ ½2=3; 1
: By assumption there is no solution with l ¼ 1; so we assume there is a solution ð y; C; DÞ with l [ ½2=3; 1Þ: Note that Hð y; C; D; lÞ ¼ 0 is equivalent to the BVP y DD ¼ mðt; y s Þ; t [ ½a; b
; ð14Þ yðaÞ ¼ C; yðs 2 ðbÞÞ ¼ D; ð0;0Þ ¼ gððC;DÞ;ð3l 2 2ÞyðeÞ þ 3ð1 2 lÞfðCÞ;ð3l 2 2Þð y D ðaÞ; y D ðsðbÞÞ þ 3ð1 2 lÞCðC;DÞÞÞ:  By Lemma 1 we have Now suppose y is a solution of Eq. (14) and ð yðaÞ; yðs 2 ðbÞÞÞ [ D: 2 aðtÞ # yðtÞ # bðtÞ for t [ ½a; s ðbÞ
: Hence y Ó ›V: Assume that ðC; DÞ [ ›D: If C ¼ yðaÞ ¼ aðaÞ; then y D ðaÞ $ a D ðaÞ: Thus ð3l 2 2Þy D ðaÞ þ 3ð1 2 lÞc 0 ð yðaÞ; yðs 2 ðbÞÞÞ . a D ðaÞ; since C is strongly inwardly pointing. Similarly, the other cases ðC; DÞ ¼ ð yðaÞ; yðs 2 ðbÞÞÞ [ ›D lead to ð3l 2 2Þy D ðaÞ þ 3ð1 2 lÞc 0 ð yðaÞ; yðs 2 ðbÞÞÞ , b D ðaÞ; ð3l 2 2Þy D ðsðbÞÞ þ 3ð1 2 lÞc 1 ð yðsðbÞÞ; yðs 2 ðbÞÞÞ , a D ðsðbÞÞ; ð3l 2 2Þy D ðsðbÞÞ þ 3ð1 2 lÞc 1 ð yðs 2 ðbÞ; yðs 2 ðbÞÞÞ . b D ðsðbÞÞ: It follows that ð3l 2 2Þð y D ðaÞ; y D ðsðbÞÞÞ þ 3ð1 2 lÞCðC; DÞ; is a strongly inwardly pointing vector field for all l [ ½2=3; 1Þ: Since g is strongly compatible, H 2 ð y; yðaÞ; yðs 2 ðbÞÞ; lÞ – 0; a contradiction. Thus ðC; DÞ Ó ›D: Case (ii) l [ ð1=3; 2=3Þ: Note that Hð y; C; D; lÞ ¼ 0 is equivalent to the BVP y DD ¼ ð3l 2 1ÞÞmðt; y s Þ; yðaÞ ¼ C; yðs 2 ðbÞÞ ¼ D; t [ ½a; b
; gððC; DÞ; fðCÞ; CðC; DÞÞ ¼ ð0; 0Þ: Let aðtÞ ¼ a and bðtÞ ¼ b; for t [ ½a; s 2 ðbÞ
: From the boundary conditions we see that aðaÞ , yðaÞ , bðaÞ and aðs 2 ðbÞÞ , yðs 2 ðbÞÞ , bðs 2 ðbÞÞ: Notice that for u # a s ðtÞ ¼ a 848 A.C. PETERSON et al. and u $ b s ðtÞ ¼ b we have, respectively ð3l 2 1Þmðt; uÞ ¼ 2ð3l 2 1Þðj f ðt; uÞj þ 1Þ , 0 ¼ a DD ðtÞ; ð3l 2 1Þmðt; uÞ ¼ ð3l 2 1Þðj f ðt; uÞj þ 1Þ . 0 ¼ b DD ðtÞ: Hence Lemma 2 is applicable and aðtÞ ¼ a , yðtÞ , bðtÞ ¼ b on ½a; s 2 ðbÞ
: Therefore y Ó ›V: Since C is strongly inwardly pointing and g is strongly compatible, by the compatibility conditions there are no solutions ð y; C; DÞ with ðC; DÞ [ ›D: Thus there are no solutions of Hð y; C; D; lÞ ¼ 0 with ð y; C; DÞ [ ›G for l [ ½0; 1
and H satisfies the conditions of Lemma 3. Therefore, dðHð·; 1Þ; V £ D; 0Þ ¼ dðHð·; 0Þ; V £ D; 0Þ; ¼ dð y 2 ða þ bÞ=2; V; 0Þ £ dðG; D; ð0; 0Þ; ¼ dðG; D; ð0; 0ÞÞ – 0: Thus there is a solution ð y; C; DÞ [ G of Hð y; C; D; 1Þ ¼ 0; and hence a solution y [ Cð½a; s 2 ðbÞ
Þ of problem (1) and (2). Since y is continuous and s is right-dense continuous, the composition y s is right-dense continuous [2]. Since f is continuous we have y DD ¼ f ðt; y s Þ [ C rd ð½a; b
Þ: This concludes the proof. A Remark There are many variants of Theorem 2 concerning the inequalities involving a, b and f. For example, inequalities (10) and (11) may be replaced with a DD ðtÞ . f ðt; a s ðtÞÞ; for t [ ½a; b
; b DD ðtÞ , f ðt; b s ðtÞÞ; for t [ ½a; b
; and the conclusion of Theorem 2 still holds with at least one of the solutions satisfying aðtÞ , yðtÞ , bðtÞ for t [ ½a; s 2 ðbÞ
: Remark Existence theorems for m . 3 point BVPs follow by extending the notion of compatibility through the introduction of appropriate functions fi for i ¼ 1; . . .; m 2 2 such that each fi : ½aðaÞ; bðaÞ
! ½aðei Þ; bðei Þ
and the compatibility conditions hold. Acknowledgements The first author greatly appreciates the support of NSF Grant 0072505 and the third author gratefully acknowledges the financial support of UNSW. References [1] E. Akin, Boundary value problems for a differential equation on a measure chain, PanAmerican Math. J. 10 (2000), 17–30. THREE POINT BVPS ON TIME SCALES 849 [2] Martin Bohner and Allan Peterson, Dynamic Equations on Time Scales: An Introduction With Applications, Birkhäuser, Boston, 2001. [3] Martin Bohner and Allan Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, 2003. [4] N. G. Lloyd, Degree Theory, Cambridge University Press, London, 1978. [5] C.C. Tisdell and H.B. Thompson, On the existence of solutions to boundary value problems on time scales, Discrete Continuous Dynamic Systems, in press.