Smoothness of solutions with respect to multi-strip integral boundary conditions for nth order ordinary differential equations

Transcription

1 396 Nonlinear Analysis: Modelling and Control, 2014, Vol. 19, No. 3, ttp://dx.doi.org/ /na Smootness of solutions wit respect to multi-strip integral boundary conditions for nt order ordinary differential equations Jonny Henderson Department of Matematics, Baylor University Waco, Texas, USA Received: 4 December 2013 / Revised: 17 Marc 2014 / Publised online: 30 June 2014 Abstract. Under certain conditions, solutions of te boundary value problem, y n) = fx, y, y,..., y n 1) ), a < x < b, y i 1) x 1) = y i, i = 1,..., n 1, yx m 2) γi yx) dx = y n, a < x 1 < ξ 1 < η 1 < ξ 2 < η 2 < < ξ m < η m < x 2 < b, are differentiated wit respect to te boundary conditions. Keywords: nt order ordinary differential equation, multi-strip integral boundary conditions, smoot dependence, boundary data. 1 Introduction In tis paper, we will be concerned wit differentiating solutions of boundary value problems wit respect to boundary data for te nt order ordinary differential equation, y n) = f x, y, y,..., y n 1)), a < x < b, 1) satisfying te Diriclet and multi-strip integral boundary conditions, y i 1) x 1 ) = y i, i = 1,..., n 1, yx 2 ) yx) dx = y n, 2) were a < x 1 < ξ 1 < η 1 < ξ 2 < η 2 < < ξ m < η m < x 2 < b, R, i = 1,..., m, and y 1,..., y n R, and were we assume: i) fx, s 1, s 2,..., s n ) : a, b) R n R is continuous, ii) f/ s i x, s 1, s 2,..., s n ) : a, b) R n R is continuous, i = 1, 2,..., n, and iii) Solutions of initial value problems for 1) extend to a, b). c Vilnius University, 2014

2 Smootness of solutions wit respect to multi-strip integral BC 397 Condition iii) is not necessary for te results of tis paper, yet, by assuming iii), we avoid continually making statements in terms of solutions maximal intervals of existence. Under uniqueness assumptions on solutions of 1), 2), we will establis analogues of a result tat Hartman 1 attributes to Peano concerning differentiation of solutions of 1) wit respect to initial conditions. For our differentiation wit respect to 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 s i x, yx), y x),..., y n 1) x) ) z i 1). 3) Interest in nonlocal boundary value problems for differential equations involving integral boundary conditions as been ongoing for several years. To see only few of tese papers, we refer te reader to te papers 2 9. And very recently, Amad and Ntouyas 10 initiated researc regarding multipoint nonlocal integral boundary conditions suc as seen in 2). In describing suc boundary conditions, tey coined te term multistrip integral boundary conditions. Suc boundary conditions can be interpreted in te sense tat a controller at te rigt end of te interval under consideration is influenced by a discrete distribution of finite many sensors or strips) of arbitrary lengt expressed in terms of integral boundary conditions. Subsequent to tat paper, Amad and Ntouyas ave put fort a couple of additional papers devoted to solutions of boundary value problems involving multi-strip integral boundary conditions for bot fractional differential equations and fractional differential inclusions; see 11, 12. It can also be pointed out tat, under suitable measures, te boundary conditions can be considered in te form of Stieltjes integrals; readers can find of interest te papers, and In te same way, tere ave been many papers devoted to smootness of solutions of boundary value problems in regard to smootness of te differential equation s nonlinearity, as well as in regard to te smootness of te boundary conditions. For a view of ow tis work as evolved, involving not only boundary value problems for ordinary differential equations, but also discrete versions, we suggest te manifold results in te classical papers and 8, 9, 29 34, as well as te more current papers 35, 36 and 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 Peano). Assume tat wit respect of 1), conditions i) iii) are satisfied. Let x 0 a, b) and yx) := yx, x 0, c 1,..., c n ) denote te solution of 1) satisfying te initial conditions y i 1) x 0 ) = c i, i = 1,..., n. Ten, a) For j = 1,..., n, α j := y/ c j exists on a, b), is te solution of te variational equation 3) along yx), and satisfies te initial conditions, α i 1) j x 0 ) = δ ij, i = 1,..., n. Nonlinear Anal. Model. Control, 2014, Vol. 19, No. 3,

3 398 J. Henderson b) 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 ), i = 1,..., n. c) y/ x 0 x) = n j=1 yj) x 0 ) y/ c j x). In addition, our analogue of Teorem 1 depends on uniqueness of solutions of 1), 2), a condition we list as an assumption: iv) Given a < x 1 < ξ 1 < η 1 < ξ 2 < η 2 < < ξ m < η m < x 2 < b, if y i 1) x 1 ) = z i 1) x 1 ), i = 1,..., n 1, and yx 2 ) m γ i yx) dx = zx 2 ) m γ i zx) dx, 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 < ξ 1 < η 1 < ξ 2 < η 2 < < ξ m < η m < x 2 < b and a solution yx) of 1), if u i 1) x 1 )=0,,..., n 1, and ux 2 ) m γ i ux) dx=0, were ux) is a solution of 3) along yx), ten ux) 0. Remark 1. We observe tat, if v) is assumed, ten for α n x) in Teorem 1, α n x 2 ) 2 An analogue of Peano s teorem for 1), 2) α n x) dx 0. 4) In tis section, we state and prove our analogue of Teorem 1 for boundary value problem 1), 2). Continuous dependence of solutions on boundary conditions plays a fundamental role for suc a differentiation result. Proof of continuous dependence usually makes application of te Brouwer teorem on invariance of domain. Te spirit of suc arguments can be found in 36 or 38; we state te continuity result ere, but we omit te details. Teorem 2. 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 < ξ 1 < η 1 < ξ 2 < η 2 < < ξ m < η m < x 2 < d < b be given. Ten, tere exists a δ > 0 suc tat, for x j t j < δ, j = 1, 2, ρ i < δ and η i σ i < δ, i = 1,..., m, u j 1) x 1 ) y j < δ, j = 1,..., n 1, ux 2 ) m γ i ux) dx y n < δ, and γ k ζ k < δ, k = 1,..., m, tere exists a unique solution u δ x) of 1) suc tat u j 1) δ t 1 ) = y j, j = 1,..., n 1, u δ t 2 ) m ζ σi i ρ i u δ x) dx = y n, and {u j) δ x)} converges uniformly to uj) x), as δ 0, on c, d, for j = 0, 1,..., n 1. We now present te result of tis paper.

4 Smootness of solutions wit respect to multi-strip integral BC 399 Teorem 3. Assume conditions i) v) are satisfied. Let ux) be a solution of 1) on a, b). Let a < x 1 < ξ 1 < η 1 < ξ 2 < η 2 < < ξ m < η m < x 2 < b be given, so tat ux) = ux, x 1, x 2, ξ 1, η 1,..., ξ m, η m, u 1,..., u n, γ 1,..., γ m ), were u j 1) x 1 ) = u j, j = 1,..., n 1, and ux 2 ) m ux) dx = u n. Ten, a) For j = 1,..., n 1, r j := u/ u j exists on a, b), is te solution of te variational equation 3) along ux), and satisfies te boundary conditions, r i 1) j x 1 ) = δ ij, i = 1,..., n 1, r j x 2 ) ξ 1 r j x) dx = 0. b) r n := u/ u n exists on a, b), is te solution of 3) along ux), and satisfies te boundary conditions, r i 1) n x 1 ) = 0, i = 1,..., n 1, r n x 2 ) ξ 1 r n x) dx = 1. c) z 1 := u/ x 1 and z 2 := u/ x 2 exist on a, b), are solutions of 3) along ux), and satisfy te respective boundary conditions, z i 1) 1 x 1 ) = u i) x 1 ), i = 1,..., n 1, z 1 x 2 ) z i 1) 2 x 1 ) = 0, i = 1,..., n 1, z 2 x 2 ) z 1 x) dx = 0, z 2 x) dx = u x 2 ). d) For eac j = 1,..., m, w j := u/ exists on a, b), is te solution of 3) along ux), and satisfies te boundary conditions, w i 1) j x 1 ) = 0, i = 1,..., n 1, w j x 2 ) w j x) dx = γ j u ). e) For eac j = 1,..., m, q j := u/ η j exists on a, b), is te solution of 3) along ux), and satisfies te boundary conditions, q i 1) j x 1 ) = 0, i = 1,..., n 1, q j x 2 ) q j x) dx = γ j uη j ). f) For eac j = 1,..., m, p j := u/ γ j exists on a, b), is te solution of 3) along ux), and satisfies te boundary conditions, p i 1) j x 1 ) = 0, i = 1,..., n 1, p j x 2 ) p j x) dx = ux) dx. Nonlinear Anal. Model. Control, 2014, Vol. 19, No. 3,

5 400 J. Henderson g) Te partial derivatives satisfy, n 1 u x) = u j) x 1 ) u x), x 1 u j j=1 u x) = γ j u ) u x), u n u γ j x) = u ux) dx x). u n u x) = u x 2 ) u x), x 2 u n u x) = γ j uη j ) u x), η j u n Proof. Wit ux) = ux, x 1, x 2, ξ 1, η 1,..., ξ m, η m, u 1,..., u n, γ 1,..., γ m ), as given in te statement of te teorem, many of te results will be establised by considering ux) as a solution of an initial value problem for 1). In particular, from te boundary value notation, u i 1) x 1 ) = u i, i = 1,..., n 1, and we will let β n = u n 1) x 1 ). 5) Ten, using te notation of Teorem 1 for solutions of initial value problems for 1) and viewing ux) as a solution of an initial value problem, we will frequently intercange notation by using, ux) = ux, x 1, x 2, ξ 1, η 1,..., ξ m, η m, u 1,..., u n, γ 1,..., γ m ) = yx, x 1, u 1,..., u n 1, β n ). For part a), we fix j = 1,..., n 1, and for notational sortand purposes, we denote ux) by ux,, u j ). Let δ > 0 be as in Teorem 2. Let 0 < < δ be given and define Ten, for every 0, r j x) = 1 ux,, uj + ) ux,, u j ). r j 1) j x 1 ) = 1 u j + u j = 1, and r i 1) j x 1 ) = 1 u i u i = 0, i {1,..., n 1} \ {j}, r j x 2 ) r j x) dx = 1 ux2,, u j + ) ux 2,, u j ) 1 = 1 u n u n = 0. ux,, uj + ) ux,, u j ) dx

6 Smootness of solutions wit respect to multi-strip integral BC 401 Wit β n as defined in 5), let = ) = u n 1) x 1,, u 1 + ) β n. By Teorem 2, = ) 0, as 0. Using te notation of Teorem 1 for solutions of initial value problems for 1) and viewing te solutions u as solutions of initial value problems, we ave r j x) = 1 yx, x1, u 1, u 2,..., u j +,..., u n 1, β n + ) yx, x 1, u 1, u 2,..., u j,..., u n 1, β n ). Ten, by utilizing a telescoping sum, we ave r j x) = 1 { yx, x1, u 1, u 2,..., u j +,..., u n 1, β n + ) yx, x 1, u 1, u 2,..., u j,..., u n 1, β n + ) } + { yx, x 1, u 1, u 2,..., u j,..., u n 1, β n + ) yx, x 1, u 1, u 2,..., u j,..., u n 1, β n ) }. By Teorem 1 and te Mean Value teorem, we obtain r j x) = 1 α j x, yx, x1, u 1,..., u j +,..., u n 1, β n + ) ) u j + u j ) + 1 α n x, yx, x1, u 1,..., u n 1, β n + ) ) β n + β n ), were α j x, y )) is te solution of te variational equation 3) along y ) and satisfies α i 1) j x 1 ) = δ ji, i = 1,..., n, and α n x, y )) is te solution of te variational equation 3) along y ) and satisfies α i 1) n x 1 ) = δ ni, i = 1,..., n. Furtermore, u j + is between u j and u j +, and β n + is between β n and β n +. Now simplifying, r j x) = α j x, yx, x1, u 1,..., u j +,..., u n 1, β n + ) ) + α n x, yx, x1, u 1,..., u n 1, β n + ) ). Tus, to sow lim r j x) exists, it suffices to sow lim / exists. Now, by Remark 1, α n x2, y ) ) ) α n x, y ) dx 0. Nonlinear Anal. Model. Control, 2014, Vol. 19, No. 3,

7 402 J. Henderson However, we derived above tat r j x 2 ) m obtain + γ i r j x) dx = 0, from wic we = α j x 2, yx, x 1,, u j +,, β n + )) α n x 2, yx, x 1, u 1,, β n + )) m γ i α n x, yx, x 1, u 1,, β n + )) dx m γ i α j x, yx, x 1,, u j +,, β n + )) dx α n x 2, yx, x 1, u 1,, β n + )) m α n x, yx, x 1, u 1,, β n + )) dx. As a consequence of continuous dependence, we can let 0, so tat lim = α jx 2, yx, x 1, u 1,, β n )) + m α n x 2, yx, x 1, u 1,, β n )) m γ i = α jx 2, ux)) + m α n x 2, ux)) m γ i γ i α j x, ux)) dx := A. α n x, ux)) dx Let r j x) = lim r j x), and note by construction of r j x), Furtermore, α j x, yx, x 1, u 1,, β n )) dx α n x, yx, x 1, u 1,, β n )) dx r j x) = u u j x, x 1, x 2, ξ 1, η 1,..., ξ m, η m, u 1,..., u n, γ 1,..., γ m ). r j x) = lim r j x) = α j x, yx, x1, u 1,, β n ) ) + Aα 2 x, yx, x1, u 1,, β n ) ) = α 1 x, ux) ) + Aα2 x, ux) ), wic is a solution of te variational equation 3) along ux). In addition because of te boundary conditions satisfied by r j x), we also ave, r i 1) j x 1 ) = δ ji, i = 1,..., n 1, and r j x 2 ) Tis completes te argument for u/ u j. r j x) dx = 0. For part b), tere are similarities wit te previous argument, yet tere are significant enoug differences for us to include te details concerning te caracterization of u/ u n. For tis consideration, we denote ux) by ux,, u n ). Again, let δ > 0 be as in Teorem 2. Let 0 < < δ be given and define Tis time, for 0, r n x) = 1 ux,, un + ) ux,, u n ). r i 1) n x 1 ) = 1 u i u i = 0, i = 1,..., n 1,

8 Smootness of solutions wit respect to multi-strip integral BC 403 and r n x 2 ) r n x) dx = 1 ux2,, u n + ) ux 2,, u n ) 1 Again wit β n defined in 5), let = 1 u n + u n = 1. ux,, un + ) ux,, u n ) dx = ) = u n 1) x 1,, u n + ) β n. As before, = ) 0, as 0. Employing te notation of Teorem 1 for solutions of initial value problems for 1) and viewing te solutions u as solutions of initial value problems, we ave r n x) = 1 yx, x1, u 1,..., u n 1, β n + ) yx, x 1, u 1,..., u n 1, β n ). By Teorem 1 and te Mean Value teorem, we obtain r n x) = 1 α n x, yx, x1, u 1,..., u n 1, β n + ) ) β n + β n ) = α n x, yx, x1, u 1,..., u n 1, β n + ) ), were β n + is between β n and β n +, and α n x, y )) is te solution of te variational equation 3) along y ) and satisfies, α i 1) n x 1 ) = δ ni, i = 1,..., n. Tus, to sow lim r n x) exists, it suffices to sow lim / exists. By Remark 1, α n x2, y ) ) ) α n x, y ) dx 0, and we also ave above tat r n x 2 ) m γ i r n x) dx = 1, from wic we obtain = 1 α n x 2, yx, x 1, u 1,, β n + )) m γ i α n x, yx, x 1, u 1,, β n + )) dx. Nonlinear Anal. Model. Control, 2014, Vol. 19, No. 3,

9 404 J. Henderson By continuous dependence, we can let 0, so tat lim = 1 α n x 2, yx, x 1, u 1,, β n )) m γ i α n x, yx, x 1, u 1,, β 2 )) dx 1 = α n x 2, ux)) m α n x, ux)) dx = B. Let r n x) = lim r n x), and ten by construction of r n x), Moreover, r n x) = u u n x, x 1, x 2, ξ 1, η 1,..., ξ m, η m, u 1,..., u n, γ 1,..., γ m ). r n x) = lim r n x) = Bα n x, yx, x1, u 1,, u n 1, β n ) ) = Bα n x, ux) ), wic is a solution of te variational equation 3) along ux). Because of te boundary conditions satisfied by r n x), we also ave, r i 1) n x 1 ) = 0, i = 1,..., n 1, and r n x 2 ) And tis completes te argument for u/ u n. r n x) dx = 1. For part c) of te teorem, we will produce te details for u/ x 2, wit te arguments for u/ x 1 being somewat along te same lines. For tis case, we denote ux) by ux, x 2, ). Wit δ > 0 as in Teorem 2, let 0 < < δ be given, and define z 2 x) = 1 ux, x2 +, ) ux, x 2, ). First, we consider boundary conditions. We ave z i 1) 2 x 1 ) = 1 u i u i = 0, i = 1,..., n 1. Next, by employing te Mean Value teorem, z 2 x 2 ) = 1 1 z 2 x) dx ux 2, x 2 +, ) ux 2, x 2, ) ux, x 2 +, ) dx ux, x 2, ) dx

10 Smootness of solutions wit respect to multi-strip integral BC 405 = 1 1 ux 2 +, x 2 +, ) ux 2, x 2, ) ux, x 2, ) dx ux, x 2 +, ) dx 1 ux2 +, x 2 +, ) ux 2, x 2 +, ) = 1 u n u n 1 u ν, x 2 +, ) = u ν, x 2 +, ), were ν is between x 2 and x 2 +. In passing to te limit, we ave lim { z 2 x 2 ) z 2 x) dx } = u x 2, x 2, ) = u x 2 ). Next, we deal wit te existence of lim z 2 x). Wit β n as defined in 5), tis time we let = ) = u n 1) x 1, x 2 +, ) β n, and by Teorem 2, = ) 0, as 0. As in parts a) and b), we use te notation of Teorem 1 for solutions of initial value problems for 1) and viewing te solutions u as solutions of initial value problems, we ave z 2 x) = 1 ux, x2 +, ) ux, x 2, ) = 1 yx, x1, u 1,..., u n 1, β n + ) yx, x 1, u 1,..., u n 1, β n ) = 1 α n x, yx, x1, u 1,..., u n 1, β n + ) ) = α n x, yx, x1, u 1,..., u n 1, β n + ) ), were α n x, y )) is te solution of 3) along y ) and satisfies α i 1) n x 1 ) = δ ni, i = 1,..., n, and β n + lies between β n and β n +. As before, to sow lim z 2 x) exists, it suffices to sow lim / exists. Now, recalling from above tat z 2 x 2 ) m γ i z 2 x) dx = u ν, x 2 +, ), it follows from Remark 1 tat, = u ν, x 2 +, ) α n x 2, yx, x 1, u 1,, β n + )) m γ i α n x, yx, x 1, u 1,, β n + )) dx. Nonlinear Anal. Model. Control, 2014, Vol. 19, No. 3,

11 406 J. Henderson And passing to te limit due to continuous dependence, we ave, lim From above, = u x 2 ) α n x 2, ux)) m α n x, ux)) dx z 2 x) = α n x, yx, x1, u 1,..., β n + ) ), := C. from wic we can evaluate te limit as 0, and if we let z 2 x) = lim z 2 x), we ave z 2 x) = u/ x 2. Tat is, we obtain z 2 x) = u x 2 x, x 1, x 2, ξ 1, η 1,..., ξ m, η m, u 1,..., u n, γ 1,..., γ m ) = lim z 2 x) = Cα n x, ux) ), wic is a solution of 3) along ux). In addition, from above computations, z 2 x) satisfies te boundary conditions, z 2 x 2 ) z i 1) 2 x 1 ) = lim z i 1) 2 x 1 ) = 0, i = 1,..., n 1, z 2 x) dx = lim Tis completes te proof of part c). For part d), fix j {1,..., m}, and define z 2 x 2 ) J := {1,..., m} \ {j}. z 2 x) dx = u x 2 ). In dealing wit caracterization of u/, we denote ux) by ux,, ). Let δ > 0 as in Teorem 2 and let 0 < < δ be given. Define w j x) = 1 ux,, ξj + ) ux,, ). We first look at boundary conditions satisfied by w j x). To begin wit, w i 1) j x 1 ) = 1 u i 1) x 1,, + ) u i 1) x 1,, ) = 1 u i u i = 0, i = 1,..., n 1.

12 Smootness of solutions wit respect to multi-strip integral BC 407 Next, by employing te Mean Value teorem for integrals, w j x 2 ) = 1 = 1 γ j γ j + γ j ξ j+ w j x) dx ux,, + ) dx + ux,, + ) dx + } { { ux 2,, + ) i J ux 2,, ) ux,, + ) dx + u n u n = 1 γ juc,, + ) = γ j uc,, + ), ux,, + ) dx ux,, ) dx for some c inclusively between and +. By Teorem 2, we can compute te limit, lim w j x 2 ) w j x) dx = γ j u ). Now, we deal wit te existence of lim w j x). Let β n be as in 5), and set = u n 1) x 1,, + ) β n. By Teorem 2, 0, as 0, and upon employing initial value solutions notation, w j x) = 1 yx, x1, u 1,..., u n 1, β n + ) yx, x 1, u 1,..., u n 1, β n ) = α n x, yx, x1, u 1,..., u n 1, β n + ) ), were β n + lies between β n and β n +, and α n x, y )) is as in te cases above. From Remark 1, we can solve for = w j x 2 ) m γ i α n x 2, y )) m from wic, using te above limit, we can calculate, lim = γ j u ) α n x 2, ux)) m w j x) dx α n x, y )) dx, α n x, ux)) dx := D. } Nonlinear Anal. Model. Control, 2014, Vol. 19, No. 3,

13 408 J. Henderson As as consequence, lim w j x) exists, and we define w j x) := lim w j x). In particular, w j x) = u x, x 1, x 2, ξ 1, η 1,..., ξ m, η m, u 1,..., u n, γ 1,..., γ m ) = lim w j x) = Dα n x, ux) ), wic is a solution of 3) along ux). In addition, w j x) satisfies te boundary conditions, w j x 2 ) w i 1) j x 1 ) = lim w j x) dx = lim Te proof of part d) is complete. j x 1 ) = 0, i = 1,..., n 1, η i w j x 2 ) w j x) dx = γ j u ). w i 1) For part e), te arguments are completely analogous to tose just given for part d). For part f), fix j {1,..., m}, and let J be as defined in te proof of part d). In caracterizing u/ γ j, we will designate ux) by ux,, γ j ). Wit δ > 0 and 0 < < δ given as usual, define p j x) = 1 ux,, γj + ) ux,, γ j ). As in te previous cases, we first consider boundary conditions satisfied by p j x). To begin wit, and p j x 2 ) = 1 = 1 i J p i 1) j x 1 ) = 1 u i 1) x 1,, γ j + ) u i 1) x 1,, γ j ) = 1 ui u i = 0, i = 1,..., n 1, p j x) dx ux,, γ j + ) dx + ux 2,, γ j + ) γ j + ) ux,, γ j + ) dx { ux,, γ j + ) dx + u n u n ux 2,, γ j ) = ux,, γ j + ) dx, ux,, γ j + ) dx ux,, γ j ) dx }

14 Smootness of solutions wit respect to multi-strip integral BC 409 from wic we can take te limit, lim p j x 2 ) p j x) dx = ux) dx. Finally, in considering te existence of lim p j x), let β n be as in 5), and again, define = u n 1) x 1,, γ j + ) β n, for wic by continuous dependence 0, as 0. Using initial value solutions notation, we ave p j x) = 1 yx, x1, u 1,..., u n 1, β n + ) yx, x 1, u 1,..., u n 1, β n ) = α 2 x, yx, x1, u 1,..., u n 1, β n + ) ), wit β n + between β n and β n +, and α n x, y )) as usual. In view of Remark 1, we can solve for = p j x 2 ) m γ i p j x) dx α n x 2, y )) η m. i α n x, y )) dx Using te last above limit, we can calculate, lim = ηj ux) dx α n x 2, ux)) m It follows tat p j x) := lim p j x) exists, and α n x, ux)) dx := E. p j x) = u γ j x, x 1, x 2, ξ 1, η 1,..., ξ m, η m, u 1,..., u n, γ 1,..., γ m ) = Eα n x, ux) ), is a solution of 3) along ux). And p j x) satisfies te boundary conditions, p i 1) j x 1 ) = 0, i = 1,..., n 1, and p j x 2 ) Tis completes te proof of part f). p j x) dx = ux) dx. Part g) of te teorem is immediate by verifying tat eac side of te respective equations are solutions of 3) along ux) and satisfy te same boundary conditions, and ten assumption v) establises te equalities. Te proof is complete. Nonlinear Anal. Model. Control, 2014, Vol. 19, No. 3,

15 410 J. Henderson References 1. P. Hartman, Ordinary Differential Equations, Wiley, New York, B. Amad, S.K. Ntouyas, A study of iger-order nonlinear ordinary differential equations wit four-point nonlocal integral boundary conditions, J. Appl. Mat. Comput., 39:97 108, 2012). 3. M. Bencora, F. Beroun, J.J. Nieto, Existence results for impulsive boundary value problem wit integral boundary conditions, Dyn. Syst. Appl., 19: , M. Bencora, S. Hamani, J.J. Nieto, Te metod of upper and lower solutions for second order differential inclusions wit integral boundary conditions, Rocky Mt. J. Mat., 40:13 26, T. Jankowski, Differential equations wit integral boundary conditions, J. Comput. Appl. Mat., 147:1 8, S. Liu, M. Jia, Y. Tian, Existence of positive solutions for boundary-value problems wit integral boundary conditions and sign canging nonlinearities, Electron. J. Differ. Equ., ):1 12, S. Pečiulytė, O. Štikonienė, A. Štikonas, Sturm Liouville problem for stationary differential operator wit nonlocal integral boundary condition, Mat. Model. Anal., 10: , M. Sapagovas, R. Čiupaila, Ž. Jokšienė, T. Meškauskas, Computational experiment for stability analysis of difference scemes wit nonlocal conditions, Informatica, 24: , M. Sapagovas, K. Jakubėlienė, Alternating direction metod for two-dimensional parabolic equation wit nonlocal integral condition, Nonlinear Anal. Model. Control, 17:91 98, B. Amad, S.K. Ntouyas, Nonlinear fractional differential equations and inclusions of arbitrary order and multi-strip boundary condtions, Electron. J. Differ. Equ., ):1 22, B. Amad, S.K. Ntouyas, A study of nonlinear fractional differential equations of arbitrary order wit Riemann Liouville type multi-strip boundary conditions, Mat. Probl. Eng., 2013, Article ID , 9 pp., B. Amad, S.K. Ntouyas, Existence results for iger order fractional differential inclusions wit multi-strip fractional integral boundary conditions, Electron. J. Qual. Teory Differ. Equ., ):1 19, G. Infante, J.R.L. Webb, Nonlinear nonlocal boundary value problems and perturbed Hammerstein integral equations, Proc. Edinb. Mat. Soc., 49: , G.L. Karakostas, P.C. Tsamatos, Existence of multiple positive solutions for a nonlocal boundary value problem, Topol. Metods Nonlinear Anal., 19: , G.L. Karakostas, P.C. Tsamatos, Multiple positive solutions of some Fredolm integral equations arisen from nonlocal boundary value problem, Electron. J. Differ. Equ., ):1 17, J.R.L. Webb, G. Infante, Positive solutions of nonlocal boundary value problems: A unified approac, J. Lond. Mat. Soc., 74: ,

16 Smootness of solutions wit respect to multi-strip integral BC J.R.L. Webb, Existence of positive solutions for a termostat model, Nonlinear Anal., Real World Appl., 13: , W.M. Wyburn, Differential equations wit general boundary conditions, Bull. Am. Mat. Soc., 48: , A. Datta, Differences wit respect to boundary points for rigt focal boundary conditions, J. Differ. Equ. Appl., 4: , J. Eme, Differentiation of solutions of boundary value problems wit respect to nonlinear boundary conditions, J. Differ. Equations, 101: , J. Eme, P.W. Eloe, J. Henderson, Differentiability wit respect to boundary conditions and deviating argument for functional-differential systems, Differ. Equ. Dyn. Syst., 1:59 71, J. Eme, J. Henderson, Differentiation of solutions of boundary value problems wit respect to boundary conditions, Appl. Anal., 46: , J. Eme, J. Henderson, Functional boundary value problems and smootness of solutions, Nonlinear Anal., Teory Metods Appl., 26: , J. Eme, B. Lawrence, Linearized problems and continuous dependence for finite difference equations, Panam. Mat. J., 10:13 24, J. Henderson, Rigt focal point boundary value problems for ordinary differential equation and variational equations, J. Mat. Anal. Appl., 98: , J. Henderson, Disconjugacy, disfocality and differentiation wit respect to boundary conditions, J. Mat. Anal. Appl., 121:1 9, J. Henderson, B. Lawrence, Smoot dependence on boundary matrices, J. Differ. Equ. Appl., 2: , B. Lawrence, A variety of differentiability results for a multi-point boundary value problem, J. Comput. Appl. Mat., 141: , A.C. Peterson, Comparison teorems and existence teorems for ordinary differential equations, J. Mat. Anal. Appl., 55: , A.C. Peterson, Existence-uniqueness for ordinary differential equations, J. Mat. Anal. Appl., 64: , A.C. Peterson, Existence-uniqueness for focal point boundary value problems, SIAM J. Mat. Anal., 12: , A.C. Peterson, Existence and uniqueness teorems for nonlinear difference equations, J. Mat. Anal. Appl., 125: , J. Spencer, Relations between boundary value functions for a nonlinear differential equation and its variational equation, Can. Mat. Bull., 18: , D. Sukup, On te existence of solutions to multipoint boundary value problems, Rocky Mt. J. Mat., 6: , J. Erke, J. Henderson, C. Kunkel, Q. Seng, Boundary data smootness for solutions of nonlocal boundary value problems for second order differential equations, J. Mat. Anal. Appl., 333: , Nonlinear Anal. Model. Control, 2014, Vol. 19, No. 3,

17 412 J. Henderson 36. J. Henderson, B. Karna, C.C. Tisdell, Uniqueness implies existence for multipoint boundary value problems second order equations, Proc. Am. Mat. Soc., 133: , J. Henderson, J. Lyons, Caracterization of partial derivatives wit respect to boundary conditions for nonlocal boundary value problems for n-t order differential equations, Int. J. Pure Appl. Mat., 56: , J. Henderson, C.C. Tisdell, Boundary data smootness for solutions of tree point boundary value problems for second order ordinary differential equations, Z. Anal. Anwend., 44: , B. Hopkins, E. Kim, J. Lyons, B. Speer, Boundary data smootness for solutions of nonlocal boundary value problems for second order difference equations, Commun. Appl. Nonlinear Anal., 16:1 12,