inv lve a journal of mathematics 2008 Vol. 1, No. 2

inv lve a journal of matematics Boundary data smootness for solutions of nonlocal boundary value problems for n-t order differential equations Jonny Henderson, Britney Hopkins, Eugenie Kim and Jeffrey Lyons matematical sciences publisers 2008 Vol. 1, No. 2

INVOLVE 1:22008) Boundary data smootness for solutions of nonlocal boundary value problems for n-t order differential equations Jonny Henderson, Britney Hopkins, Eugenie Kim and Jeffrey Lyons Communicated by Kennet S. Berenaut) Under certain conditions, solutions of te boundary value problem y n) = f x,y, y,..., y n 1) ), y i 1) x 1 ) = y i for 1 i n 1, and yx 2 ) m i=1 r i yη i ) = y n, are differentiated wit respect to boundary conditions, were a < x 1 < η 1 < < η m < x 2 < b, and r 1,..., r m, y 1,..., y n. 1. Introduction In tis paper, we will be concerned wit differentiating solutions of certain nonlocal boundary value problems wit respect to boundary data for te n-t order ordinary differential equation satisfying y n) = f x, y, y,..., y n 1) ), a < x < b, 1) y i 1) x 1 ) = y i, 1 i n 1, yx 2 ) r k yη k ) = y n, 2) were a < x 1 < η 1 < < η m < x 2 < b, and y 1,..., y n, r 1,..., r m, and were we assume i) f x, u 1,..., u n ) : a, b) n is continuous, ii) f/ u i x, u 1,..., u n ) : a, b) n are continuous, i = 1, 2,..., n, and iii) solutions of initial value problems for 1) extend to a, b). MSC2000: primary 34B15, 34B10; secondary 34B08. Keywords: nonlinear boundary value problem, ordinary differential equation, nonlocal boundary condition, boundary data smootness. 167

168 JOHNNY HENDERSON, BRITNEY HOPKINS, EUGENIE KIM AND JEFFREY LYONS We remark tat condition iii) is not necessary for te spirit of tis work s results, owever, by assuming iii), we avoid continually making statements in terms of solutions maximal intervals of existence. Under uniqueness assumptions on solutions of 1) and 2), we will establis analogues of a result tat Hartman [1964] attributes to Peano concerning differentiation of solutions of 1) wit respect to initial conditions. For our differentiation wit respect to te boundary conditions results, given a solution yx) of 1), we will give muc attention to te variational equation for 1) along yx), wic is defined by z n) = n f u k x, yx), y x),..., y n 1) x))z k 1). 3) Tere as long been interest in multipoint nonlocal boundary value problems for ordinary differential equations, wit muc attention given to positive solutions. To see only a few of tese papers, we refer te reader to [Bai and Fang 2003; Gupta and Trofimcuk 1998; Ma 1997; 2002; Yang 2002]. Likewise, many papers ave been devoted to smootness of solutions of boundary value problems wit respect to boundary data. For a view of ow tis work as evolved, involving not only boundary value problems for ordinary differential equations, but also discrete versions, functional differential equations versions and dynamic equations on time scales versions, we suggest results from among te many papers [Datta 1998; Eme 1993; Eme et al. 1993; Eme and Henderson 1996; Eme and Lawrence 2000; Hartman 1964; Henderson 1984; 1987; Henderson et al. 2005; Henderson and Lawrence 1996; Lawrence 2002; Peterson 1976; 1978; 1981; 1987; Spencer 1975]. In fact, smootness results ave been given some consideration for 1) and 2) wen n = 2 and for specific and general values of m [Erke et al. 2007; Henderson and Tisdell 2004]. Te teorem for wic we seek an analogue and attributed to Peano by Hartman can be stated in te context of 1) as follows Teorem 1.1. [Peano] Assume tat, wit respect to 1), conditions i) iii) are satisfied. Let x 0 a, b) and yx) yx, x 0, c 1, c 2,..., c n ) denote te solution of 1) satisfying te initial conditions y i 1) x 0 ) = c i, 1 i n. Ten, i) For eac 1 i n, y/ c i exists on a, b) and α i y/ c i is a solution of te variational equation 3) along yx) and satisfies te initial condition, α j 1) i x 0 ) = δ i j, 1 i, j n. ii) y/ x 0 exists on a, b), and β y/ x 0 is te solution of te variational equation 3) along yx) satisfying te initial conditions, β i 1) x 0 ) = y i) x 0 ), 1 i n.

SMOOTHNESS OF SOLUTIONS FOR NONLOCAL BVPS 169 iii) y/ x 0 x) = n yk) x 0 ) y/ c k x). In addition, our analogue of Teorem 1.1 depends on uniqueness of solutions of 1) and 2), a condition we list as an assumption. iv) Given a < x 1 < η 1 < < η m < x 2 < b, if y i 1) x 1 ) = z i 1) x 1 ) for eac 1 i n 1, and yx 2 ) m r k yη k ) = zx 2 ) m r kzη k ), were yx) and zx) are solutions of 1), ten yx) zx). We will also make extensive use of a similar uniqueness condition on 3) along solutions yx) of 1). v) Given a < x 1 < η 1 < < η m < x 2 < b, and a solution yx) of 1), if u i 1) x 1 ) = 0, 1 i n 1, and ux 2 ) m r kuη k ) = 0, were ux) is a solution of 3) along yx), ten ux) 0. 2. An analogue of Peano s Teorem for Equations 1) and 2) In tis section, we derive our analogue of Teorem 1.1 for boundary value problem 1), 2). For suc a differentiation result, we need continuous dependence of solutions on boundary conditions. Te arguments for tis continuous dependence follow muc along te lines of tose in [Henderson and Tisdell 2004], wen 1) is of second order. For tat reason, we omit te details of te proof. Teorem 2.1. Assume i) iv) are satisfied wit respect to 1). Let ux) be a solution of 1) on a, b), and let a < c < x 1 < η 1 < < η m < x 2 < d < b be given. Ten, tere exists a δ > 0 suc tat, for x i t i < δ, i = 1, 2, η i τ i < δ and r i ρ i <δ, 1 i m, ux 2 ) u i 1) x 1 ) y i < δ, 1 i n 1 r k uη k ) y n < δ, tere exists a unique solution u δ x) of 1) suc tat u i 1) δ t 1 ) = y i, 1 i n 1, u δ t 2 ) ρ k u δ τ k ) = y n and {u j 1) δ x)} converges uniformly to u j 1) x), as δ 0, on [c, d], for 1 j n. We now present te result of te paper.

170 JOHNNY HENDERSON, BRITNEY HOPKINS, EUGENIE KIM AND JEFFREY LYONS Teorem 2.2. Assume conditions i) v) are satisfied. Let ux) be a solution 1) on a, b). Let a < x 1 < η 1 < < η m < x 2 < b be given, so tat ux) = ux, x 1, x 2, u 1,..., u n, η 1,..., η m, r 1,..., r m ), were u i 1) x 1 ) = u i, 1 i n 1, and ux 2 ) m r kuη k ) = u n. Ten, i) For eac 1 i n, u/ u i exists on a, b). Moreover, for eac 1 j n 1, y j u/ u j solves Equation 3) along ux) and satisfies te boundary conditions, y i 1) j x 1 ) = δ i j, 1 i n 1, y j x 2 ) r k y j η k ) = 0, and y n u/ u n solves 3) along ux) and satisfies te boundary conditions, y n i 1) x 1 ) = 0, 1 i n 1, y n x 2 ) r k y n η k ) = 1. ii) u/ x 1 and u/ x 2 exist on a, b), and z i u/ x i, i = 1, 2, are solutions of 3) along ux) and satisfy te respective boundary conditions, z i 1) 1 x 1 ) = u i) x 1 ), 1 i n 1, z 1 x 2 ) r k z 1 η k ) = 0, z i 1) 2 x 1 ) = 0, 1 i n 1, z 2 x 2 ) r k z 2 η k ) = u x 2 ). iii) For 1 j m, u/ η j exists on a, b), and w j u/ η j, j = 1,..., m, is a solution of 3) along ux) and satisfies w i 1) j x 1 ) = 0, 1 i n 1, w j x 2 ) r k w j η k ) = r j u η j ). iv) For 1 j m, u/ r j exists on a, b), and v j u/ r j, j = 1,..., m, is a solution of 3) along ux) and satisfies, v i 1) j x 1 ) = 0, 1 i n 1, v j x 2 ) r k v j η k ) = uη j ). Proof. For part i), let 1 j n 1, and consider u/ u j, since te argument for u/ u n is similar. In tis case we designate, for brevity, ux, x 1, x 2, u 1,..., u n, η 1,..., η m, r 1,..., r m ) by ux, u j ). Let δ > 0 be as in Teorem 2.1. Let 0 < < δ be given and define y j x) = 1 [ ux, u j + ) ux, u j ) ].

SMOOTHNESS OF SOLUTIONS FOR NONLOCAL BVPS 171 Note tat u j 1) x 1, u j + ) = u j +, and u j 1) x 1, u j ) = u j, so tat, for every = 0, y j 1) j x 1 ) = 1 [u j + u j ] = 1. Also, for every = 0, 1 i n 1, i = j, and y j x 2 ) y i 1) j x 1 ) = 1 [ u i 1) x 1, u j + ) u i 1) x 1, u j ) ] = 1 [u i u i ] = 0, r k y j η k ) = 1 [ ux2, u j + ) ux 2, u j ) ] r k [ uηk, u j + ) uη k, u j ) ] = 1 [u n u n ] = 0. Let β = u n 1) x 1, u j ), and ɛ = ɛ) = u n 1) x 1, u j + ) β. By Teorem 2.1, ɛ = ɛ) 0, as 0. Using te notation of Teorem 1.1 for solutions of initial value problems for Equation 1) and viewing te solutions u as solutions of initial value problems and denoting yx, x 1, u 1,..., u j,..., u n 1, β) by yx, x 1, u j, β), we ave y j x) = 1 [ yx, x1, u j +, β + ɛ) yx, x 1, u j, β) ]. Ten, by utilizing a telescoping sum, we ave y j x) = 1 [ {yx, x1, u j +, β + ɛ) yx, x 1, u j, β + ɛ)} By Teorem 1.1 and te Mean Value Teorem, we obtain + {yx, x 1, u j, β + ɛ) yx, x 1, u j, β)} ]. y j x) = 1 α j x, yx, x1, u j +, β + ɛ) ) u j + u j ) + 1 α n x, yx, x1, u j, β + ɛ) ) β + ɛ β), were α k x, y )), k { j, n}, is te solution of te variational Equation 3) along y ) and satisfies, in eac case, α i 1) j x 1 ) = δ i j α i 1) n x 1 ) = δ in, 1 i n, respectively. Furtermore, u j + is between u j and u j +, and β + ɛ is between β and β + ɛ. Now simplifying, y j x) = α j x, yx, x1, u j +, β + ɛ) ) + ɛ α n x, yx, x1, u j, β + ɛ) ).

172 JOHNNY HENDERSON, BRITNEY HOPKINS, EUGENIE KIM AND JEFFREY LYONS Tus, to sow lim y j x) exists, it suffices to sow lim ɛ/ exists. 0 0 Now α n x,y )) is a nontrivial solution of Equation 3) along y ), and α i 1) n x 1,y )) = 0, 1 i n 1. So, by assumption v), α n x 2, y )) m r kα n η k, y )) = 0. However, we observed tat y j x 2 ) m r k y j η k ) = 0, from wic we obtain m ɛ = r kα j ηk, yx, x 1, u j +, β + ɛ) ) α j x2, yx, x 1, u j +, β + ɛ) ) α n x2, yx, x 1, u j, β + ɛ) ) m r kα n ηk, yx, x 1, u j, β + ɛ) ). As a consequence of continuous dependence, we can let 0, so tat ɛ lim 0 = α j x2, yx, x 1, u j, β 2 ) ) m r kα j ηk, yx, x 1, u j, β) ) α n x2, yx, x 1, u j, β) ) m r kα n ηk, yx, x 1, u j, β) ) = α jx 2, ux)) m r kα j η k, ux)) α n x 2, ux)) m r =: D. kα n η k, ux)) Let y j x) = lim 0 y j x), and note by construction of y j x) tat Furtermore, y j x) = u u j x). y j x) = lim 0 y j x) = α j x, yx, x 1, u j, β)) + Dα n x, ux)), wic is a solution of te variational Equation 3) along ux). In addition because of te boundary conditions satisfied by y j x), we also ave y i 1) j x 1 ) = δ i j, 1 i n 1, y j x 2 ) r k y j η k ) = 0. Tis completes te argument for u/ u j. In part ii) of te teorem, we will produce te details for u/ x 1, wit te arguments for u/ x 2 being similar. Tis time, we designate ux, x 1, x 2, u 1,..., u n, η 1,..., η m, r 1,..., r m ) by ux, x 1 ). So, let δ > 0 be as in Teorem 2.1, let 0 < < δ be given, and define z 1 x) = 1 [ux, x 1 + ) ux, x 1 )].

SMOOTHNESS OF SOLUTIONS FOR NONLOCAL BVPS 173 Note tat, for 1 i n 1, z i 1) 1 x 1 ) = 1 [ui 1) x 1, x 1 + ) u i 1) x 1, x 1 )] = 1 [ui 1) x 1, x 1 + ) u i 1) x 1 +, x 1 + )] = 1 [ui) c x1,, x 1 + ) ] = u i) c x1,, x 1 + ), were c x1, lies between x 1 and x 1 +. In addition, we note tat, for every = 0, z 1 x 2 ) r k z 1 η k ) = 1 [ux 2, x 1 + ) r k uη k, x 1 + ) Next, let Let us note at tis point tat {ux 2, x 1 ) β = u n 1) x 1, x 1 ), r k uη k, x 1 )}] = 1 [u n u n ] = 0. ɛ j = ɛ j ) = u j 1) x 1, x 1 + ) u j, ɛ β = ɛ β ) = u n 1) x 1, x 1 + ) β. ɛ j j 1) = z 1 x 1 ) = u j) c x1,, x 1 + ). By Teorem 2.1, bot ɛ j 0 and ɛ β 0, as 0. As in part i), we employ te notation of Teorem 1.1 for solutions of initial value problems for 1). Viewing te solutions u as solutions of initial value problems, and denoting by yx, x 1, u j, β), we ave yx, x 1, u 1,..., u j,..., u n 1, β) z 1 x) = 1 [yx, x 1, u j + ɛ j, β + ɛ β ) yx, x 1, u j, β)] By te Mean Value Teorem, = 1 [yx, x 1, u j + ɛ j, β + ɛ β ) yx, x 1, u j, β + ɛ β ) +yx, x 1, u j, β + ɛ β ) yx, x 1, u j, β)]. z 1 x) = 1 [ ɛ j α j x, yx, x1, u j + ɛ j, β + ɛ β ) ) + ɛ β α n x, yx, x1, u j, β + ɛ β ) )],

174 JOHNNY HENDERSON, BRITNEY HOPKINS, EUGENIE KIM AND JEFFREY LYONS were u j + ɛ j lies between u j and u j + ɛ j, β + ɛ β lies between β and β + ɛ β, and α j x, y )) and α n x, y )) are te solutions of Equation 3) along y ) and satisfy, respectively, α i 1) j x 1 ) = δ i j, 1 i n, α i 1) n x 1 ) = δ in, 1 i n. As before, to sow tat lim 0 z 1 x) exists, it suffices to sow tat exist. Now, from above, ɛ j lim 0 and lim 0 ɛ β ɛ j lim 0 = lim 0 j 1) z 1 x 1 ) = lim u j) c x1,, x 1 + ) = u j) x 1 ). 0 Since α n x,y )) is a nontrivial solution of 3) along y ) and it follows from assumption v) tat Since we ave ɛ β = ɛ j were ) α i 1) n x 1,y )) = 0, 1 i n 1, α n x 2,y )) z 1 x 2 ) r k α n η k,y )) =0. r k z 1 η k ) = 0, A α n x2, yx, x 1, u j, β + ɛ β ) ) m r kα n ηk, yx, x 1, u j, β + ɛ β ) ), A = α j x2, yx, x 1, u j + ɛ j, β + ɛ β ) ) And so, r k α j ηk, yx, x 1, u j + ɛ j, β + ɛ β ) ). ɛ β lim 0 = u j) x 1 ) [ α j x2, yx, x 1, u j, β) ) m i=1 r iα j ηi, yx, x 1, u j, β) )] α n x2, yx, x 1, u j, β) ) m i=1 r iα n ηi, yx, x 1, u j, β) ) = u j) x 1 ) [ α j x2, ux) ) m i=1 r iα j ηi, ux) )] α n x2, ux) ) m i=1 r ) =: E. iα n ηi, ux)

SMOOTHNESS OF SOLUTIONS FOR NONLOCAL BVPS 175 From te above expression, z 1 x) = ɛ j α j x, yx, x1, u j + ɛ j, β + ɛ β ) ) + ɛ β α n x, yx, x1, u j, β + ɛ β ) ), and we can evaluate te limit as 0. If we let z 1 x) = lim 0 z 1 x), ten z 1 x) = u/ x 1, and z 1 x) = lim 0 z 1 x) = u j) x 1 )α j x, yx, x1, u j, β) ) + Eα n x, yx, x1, u j, β) ) = u j) x 1 )α j x, ux) ) + Eαn x, ux) ), wic is a solution of Equation 3) along ux). In addition, from above observations, z 1 x) satisfies te boundary conditions, and z i 1) 1 x 1 ) = lim 0 z i 1) 1 x 1 ) = u i) x 1 ), 1 i n 1, z 1 x 2 ) r k z 1 η k )) = lim 0 z 1 x 2 ) r k z 1 η k )) = 0. Tis completes te proof for u/ x 1. Te proofs of iii) and iv) are in very muc te same spirit. For iii), we fix 1 j m, and tis time we designate ux,x 1,x 2,u 1,...,u n,η 1,...,η m,r 1,...,r m ) by ux, η j ). Let δ > 0 be as in Teorem 2.1 and 0 < < δ be given. Define Note tat for every = 0, Next, let β = u n 1) x 1, η j ), and w j x) = 1 [ux, η j + ) ux, η j )]. w i 1) j x 1 ) = 0, 1 i n 1. ɛ = ɛ) = u n 1) x 1, η j + ) β. By Teorem 2.1, ɛ 0, as 0. Again, we use te notation of Teorem 1.1 for solutions of initial value problems for 1); viewing te solutions u as solutions of initial value problems and denoting yx, x 1, u 1,..., u n 1, β)

176 JOHNNY HENDERSON, BRITNEY HOPKINS, EUGENIE KIM AND JEFFREY LYONS by yx, x 1, β), we ave By te Mean Value Teorem, w j x) = 1 [yx, x 1, β + ɛ) yx, x 1, β)]. w j x) = ɛ α nx, yx, x 1, β + ɛ)), were α n x, y )) is te solution of Equation 3) along y ) and satisfies α i 1) n x 1 ) = δ in, 1 i n 1, and β + ɛ lies between β and β + ɛ. Once again, to sow lim 0 w j x) exists, it suffices to sow lim 0 ɛ/ exists. Since α n x,y )) is a nontrivial solution of 3) along y ) and it follows from assumption v) tat Hence, α i 1) n x 1,y )) = 0, 1 i n 1, α n x 2,y )) r k α n η k,y )) = 0. ɛ = w j x 2 m r kw j η i ) α n x2, yx, x 1, β 2 + ɛ) ) m r kα n ηk, yx, x 1, β + ɛ) ). We look in more detail at te numerator of tis quotient. Consider w j x 2 ) r k w j η k ) = 1 [ ux2, η j + ) r k uη k, η j + ) [ ux 2, η j ) r k uη k, η j ) ]] = 1 [ ux2, η j + ) k {1,...,m}\{ j} r k uη k, η j + ) r j uη j +, η j + ) + r j uη j +, η j + ) r j uη j, η j + ) ] u n = u n u n + r juη j +, η j + ) r j uη j, η j + ) = r j [ uη j +, η j + ) uη j, η j + ) ]

= r j η j + η j SMOOTHNESS OF SOLUTIONS FOR NONLOCAL BVPS 177 u s, η j + )ds = r j u c j,, η j + )η j + η j ) = r j u c j,, η j + ), were c j, is between η j and η j +. So, as 0 we obtain r j u c, η j + ) r j u η j, η j ) = r j u η j ). Wen we return to te quotient defining ɛ/, we compute te limit, ɛ lim 0 = r j u η j ) α n x2, yx, x 1, u 1, β) ) m r kα n ηk, yx, x 1, u 1, β) ) From = r j u η j ) α n x2, ux) ) m r kα n ηk, ux) ) =: E j. w j x) = ɛ α n x, yx, x1, u 1, β + ɛ) ), if we let w j x) = lim 0 w j x), ten w j x) = u/ η j, and w j x) = lim 0 w j x) = E j α n x, yx, x 1, u 1, β)) = E j α n x, ux)), wic is a solution of Equation 3) along ux). In addition, from above observations, w j x) satisfies te boundary conditions, w i 1) j x 1 ) = lim w i 1) j x 1 ) = 0, 1 i n 1, 0 w j x 2 ) r k w j η k ) = r j u η j ). Tis concludes te proof of iii). It remains to verify part iv). Fix 1 j m as before and consider u/ r j. Again, let δ > 0 be as in Teorem 2.1 and 0 < < δ. Define were, for brevity, we designate by ux, r j ). Note tat v j x) = 1 [ux, r j + ) ux, r j )], ux, x 1, x 2, u 1,..., u n, η 1,..., η m, r 1,..., r m ) v i 1) j x 1 ) = 1 u i u i ) = 0,

178 JOHNNY HENDERSON, BRITNEY HOPKINS, EUGENIE KIM AND JEFFREY LYONS for every = 0 and 1 i n 1. Also, we see tat v j x 2 ) r k v j η k ) = 1 [ ux2, r j + ) ux 2, r j ) r k uηk, r j + ) uη k, r j ) )] = 1 [ ux2, r j + ) ux 2, r j ) r k uη k, r j + ) + r k uη k, r j ) ] = 1 ux 2, r j + ) 1 = 1 [ ux2, r j + ) k {1,...,m}\{ j} r k uη k, r j + ) u n r k uη k, r j + ) r j uη j, r j + ) uη j, r j + ) + uη j, r j + ) ] u n = 1 [ ux2, r j + ) r k uη k, r j + ) k {1,...,m}\{ j} r j + )uη j, r j + ) ] + uη j, r j + ) u n = u n + uη j, r j + ) u n = uη j, r j + ). And so by Teorem 2.1, lim v jx 2 ) 0 r k v j η k ) = uη j, r j ). Now recall tat u n 2) x 1, r j ) = u n 1, and define β = u n 1) x 1, r j ), and ɛ = ɛ) = u n 1) x 1, r j + ) β. As usual, ɛ 0 as 0. Once again, using te notation for solutions of initial value problems for 1) and denoting yx, x 1, u 1,..., u n 1, β) by yx, x 1, β), we ave By te Mean Value Teorem, v j x) = 1 [ yx, x1, β + ɛ) yx, x 1, β) ]. v j x) = 1 α n x, yx, x1, β + ɛ) ) β + ɛ β) = ɛ α n x, yx, x1, β + ɛ) ),

SMOOTHNESS OF SOLUTIONS FOR NONLOCAL BVPS 179 were α n x, y )) is te solution of Equation 3) along y ) and satisfies α i 1) n x 1, y )) = 0, 1 i n 1, α n 1) n x 1, y )) = 1, and β+ɛ lies between β and β+ɛ. As in previous cases, it follows from assumption v) tat α n x 2, y )) r k α n η k, y )) = 0. Hence, ɛ = v j x 2 m r kv j η k ) α n x2, yx, x 1, β + ɛ) ) m r kα n ηk, yx, x 1, β + ɛ) ), and so from above, From ɛ lim 0 = r j uη j ) α n x2, yx, x 1, β) ) m r kα n ηk, yx, x 1, β) ) r j uη j ) = α n x2, ux) ) m r kα n ηk, ux) ) =: E j. v j x) = ɛ α n x, yx, x1, β + ɛ) ), if we set v j x) = lim 0 v j x), we obtain v j x) = u/ r j. In particular, v j x) = lim 0 v j x) = E j α n x, yx, x 1, β)) = E j α n x, ux)), wic is a solution of 3) along ux). In addition, v j x) satisfies te boundary conditions, v j x 1 ) = lim v i 1) j x 1 ) = 0, 1 i n 1, 0 v j x 2 ) r k v j η k ) = uη j ). Tis completes case iv), wic in turn completes te proof of te teorem. We conclude te paper wit a corollary to Teorem 2.2, wose verification is a consequence of te n-dimensionality of te solution space for te variational Equation 3). In addition, tis corollary establises an analogue of part iii) of Teorem 1.1.

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