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Hindawi Publishing Corporation Boundary Value Problems Volume 2011, Article ID 929061, 17 pages doi:10.1155/2011/929061 Research Article Approximation of Solutions for Second-Order m-Point Nonlocal Boundary Value Problems via the Method of Generalized Quasilinearization Ahmed Alsaedi Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O Box 80203, Jeddah 21589, Saudi Arabia Correspondence should be addressed to Ahmed Alsaedi, aalsaedi@hotmail.com Received 11 May 2010; Revised 29 July 2010; Accepted October 2010 Academic Editor: Gennaro Infante Copyright q 2011 Ahmed Alsaedi This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited We discuss the existence and uniqueness of the solutions of a second-order m-point nonlocal boundary value problem by applying a generalized quasilinearization technique A monotone sequence of solutions converging uniformly and quadratically to a unique solution of the problem is presented Introduction The monotone iterative technique coupled with the method of upper and lower solutions 1–7 manifests itself as an effective and flexible mechanism that offers theoretical as well as constructive existence results in a closed set, generated by the lower and upper solutions In general, the convergence of the sequence of approximate solutions given by the monotone iterative technique is at most linear 8, To obtain a sequence of approximate solutions converging quadratically, we use the method of quasilinearization 10 This method has been developed for a variety of problems 11–20 In view of its diverse applications, this approach is quite an elegant and easier for application algorithms The subject of multipoint nonlocal boundary conditions, initiated by Bicadze and Samarski˘ 21 , has been addressed by many authors, for instance, 22–32 The multipoint ı boundary conditions appear in certain problems of thermodynamics, elasticity and wave propagation, see 23 and the references therein The multipoint boundary conditions may be understood in the sense that the controllers at the endpoints dissipate or add energy according to censors located at intermediate positions 2 Boundary Value Problems In this paper, we develop the method of generalized quasilinearization to obtain a sequence of approximate solutions converging monotonically and quadratically to a unique solution of the following second-order m−point nonlocal boundary value problem −x t f t, x t , x t , t ∈ 0, , m−2 px − qx 1.1 m−2 τi x ηi , px qx σi x ηi , i ηi ∈ 0, , 1.2 i where f : 0, × R × R → R is continuous and τi , σi i 1, 2, , m − are nonnegative real constants such that m−2 τi < 1, m−2 σi < 1, and p, q > with p > i i Here we remark that 26 studies 1.1 with the boundary conditions of the form m−2 δx − γx 0, ηi ∈ 0, αi x ηi , x 1.3 i A perturbed integral equation equivalent to the problem 1.1 and 1.3 considered in 26 is m−2 k t, s f s, x s , x s ds x t t2 , αi x ηi 1.4 i where k t, s δ ⎧ ⎨ γ δt − s , ≤ t ≤ s, γ ⎩ δ γs − t , s ≤ t ≤ 1.5 It can readily be verified that the solution given by 1.4 does not satisfy 1.1 On the other hand, by Green’s function method, a unique solution of the problem 1.1 and 1.3 is m−2 k t, s f s, x s , x s ds x t αi x ηi i γ δt , δ γ 1.6 where k t, s is given by 1.5 Thus, 1.6 represents the correct form of the solution for the problem 1.1 and 1.3 Boundary Value Problems Preliminaries For x ∈ C1 0, , we define x x x , where x max{|x t | : t ∈ 0, } It can easily be verified that the homogeneous problem associated with 1.1 - 1.2 has only the trivial solution Therefore, by Green’s function method, the solution of 1.1 - 1.2 can be written as m−2 G t, s f s, x s , x s ds x t τi x ηi i m−2 σi x ηi 2q i p q p p 2q p 2.1 q p 2q p t −t 2q p , where G t, s is the Green’s function and is given by G t, s ⎧ ⎨ q pt q p − s , ≤ t ≤ s, 2q ⎩ q ps q p − t , s ≤ t ≤ 1 p p 2.2 Note that G t, s > on 0, × 0, We say that α ∈ C2 0, is a lower solution of the boundary value problem 1.1 and 1.2 if −α t ≤ f t, α t , α t , pα − qα ≤ t ∈ 0, , m−2 τi α ηi , qα ≤ pα i m−2 2.3 σi α ηi , i and β ∈ C2 0, is an upper solution of 1.1 and 1.2 if −β t ≥ f t, β t , β t , pβ − qβ ≥ m−2 τi β ηi , pβ i t ∈ 0, , qβ ≥ m−2 2.4 σi β ηi i Definition 2.1 A continuous function h : 0, ∞ → 0, ∞ is called a Nagumo function if ∞ λ sds h s ∞, 2.5 for λ ≥ We say that f ∈ C 0, × R × R satisfies a Nagumo condition on 0, relative to α, β if for every t ∈ 0, and x ∈ mint∈ 0,1 α t , maxt∈ 0,1 β t , there exists a Nagumo function h such that |f t, x, x | ≤ h |x | We need the following result 33 to establish the main result 4 Boundary Value Problems Theorem 2.2 Let f : 0, × R2 → R be a continuous function satisfying the Nagumo condition on E { t, x, y ∈ 0, × R2 : α ≤ x ≤ β} where α, β : 0, → R are continuous functions such that α t ≤ β t for all t ∈ 0, Then there exists a constant M > (depending only on α, β, the Nagumo function h) such that every solution x of 1.1 - 1.2 with α t ≤ x t ≤ β t , t ∈ 0, satisfies |x | ≤ M If α, β ∈ C2 0, are assumed to be lower and upper solutions of 1.1 - 1.2 , respectively, in the statement of Theorem 2.2, then there exists a solution, x t of 1.1 and 1.2 such that α t ≤ x t ≤ β t , t ∈ 0, Theorem 2.3 Assume that α, β ∈ C2 0, are, respectively, lower and upper solutions of 1.1 - 1.2 If f t, x, y ∈ C 0, × R × R is decreasing in x for each t, y ∈ 0, × R, then α ≤ β on 0, α t − β t so that u ∈ C2 0, Proof Let us define u t conditions pu − qu ≤ m−2 τi u ηi , pu and satisfies the boundary qu ≤ i m−2 2.6 σi u ηi i For the sake of contradiction, let u have a positive maximum at some t0 ∈ 0, If t0 ∈ 0, , and u t0 ≤ On the other hand, in view of the decreasing property of then u t0 f t, x, y in x, we have u t0 α t0 − β t0 ≥ −f t0 , α t0 , α t0 f t , β t0 , β t0 > 0, which is a contradiction If we suppose that u has a positive maximum at t0 follows from the first of boundary conditions 2.6 that pu − qu ≤ 2.7 0, then it m−2 τi u ηi ≤ u , 2.8 i which implies that p − u ≤ qu Now as p > 1, q > 0, u > 0, u ≤ 0, therefore we obtain a contradiction We have a similar contradiction at t0 Thus, we conclude that α t ≤ β t , t ∈ 0, Main Results Theorem 3.1 Assume that A1 the functions α, β ∈ C2 0, are, respectively, lower and upper solutions of 1.1 - 1.2 such that α ≤ β on 0, ; A2 the function f ∈ C2 0, × R × R satisfies a Nagumo condition relative to α, β and fx ≤ on 0, × mint∈ 0,1 α t , maxt∈ 0,1 β t × −M, M , where M is a positive constant depending on α, β, and the Nagumo function h Further, there exists a function φ ∈ C2 0, × R2 such that Ψ f φ ≥ with Ψ φ ≥ on 0, × mint∈ 0,1 α t , maxt∈ 0,1 β t × −M, M , where Ψ x−y ∂2 ∂x2 x−y x −y ∂2 ∂x∂x x −y ∂2 ∂x 3.1 Boundary Value Problems Then, there exists a monotone sequence {αn } of approximate solutions converging uniformly to a unique solution of the problems 1.1 - 1.2 Proof For y ∈ R, we define ω y modified m-point BVP −x t max{−M, min{y, M}} and consider the following t ∈ 0, , , f t, x t , ω x t m−2 px − qx 3.2 m−2 τi x ηi , σi x ηi qx px i i We note that α, β are, respectively, lower and upper solutions of 3.2 and for every t, x ∈ 0, × mint∈ 0,1 α t , maxt∈ 0,1 β t , we have f ≤h ω x where h · h x 3.3 , h ω · As ∞ sds M h s sds h s ∞ M sds h M ∞, 3.4 so h is a Nagumo function Furthermore, there exists a constant N depending on α, β, and Nagumo function h such that M sds h s ≥ N sds > max β t : t ∈ 0, hs − min{α t : t ∈ 0, } , 3.5 where M > max{N, α , β } Thus, any solution x of 3.2 with α t ≤ x t ≤ β t , t ∈ 0, satisfies |x | ≤ M on 0, and hence it is a solution of 1.1 - 1.2 Let us define a function F : 0, × R2 → R by F t, x, x f t, x, x φ t, x, x − ω x 3.6 In view of the assumption A2 , it follows that F ∈ C2 0, × R2 and satisfies Ψ F ≥ on 0, × mint∈ 0,1 α t , maxt∈ 0,1 β t × −M, M Therefore, by Taylor’s theorem, we obtain f t, x, ω x ≥ f t, y, ω y Fx t, y, ω y ≥ f t, y, ω y Fx t, y, ω y Fx t, y, ω y ω x −ω y Fx t, y, ω y ω x −ω y x−y − φ t, x, − φ t, y, − φx t, β, x−y 3.7 Boundary Value Problems We set H t, x, x ; y, y f t, y, ω y Fx t, y, ω y − φx t, β, Fx t, y, ω y ω x −ω y x−y 3.8 , and observe that f t, x, ω x ≥ H t, x, x ; y, y , f t, x, ω x H t, x, x ; x, x 3.9 By the mean value theorem, we can find α ≤ c1 ≤ y and α ≤ c2 ≤ y c1 , c2 depend on y, y , resp , such that f t, y, ω y − f t, α t , α t fx t, c1 , c2 y − α t fx t, c1 , c2 ω y − α t 3.10 Letting H1 t, x, x ; y, y fx t, c1 , c2 x − α t f t, α t , α t fx t, c1 , c2 ω x − α t , 3.11 we note that f t, y, ω y f t, α t , α t H1 t, y, y ; y, y , 3.12 H1 t, α t , α t ; y, y Let us define H as H ⎧ ⎨H t, x, x ; y, y , for x ≥ y, ⎩H t, x, x ; y, y , for x ≤ y 3.13 Clearly H is continuous and bounded on 0, × mint∈ 0,1 α t , maxt∈ 0,1 β t ×R and satisfies a Nagumo condition relative to α, β For every α t ≤ y ≤ β t and y ∈ R, we consider the m-point BVP −x px − qx H t, x, x ; y, y , t ∈ 0, , m−2 τi x ηi , i m−2 px σi x ηi qx i 3.14 Boundary Value Problems Using 3.9 , 3.12 and 3.13 , we have H t, α t , α t ; y, y H1 t, α t , α t ; y, y pα − qα ≤ m−2 τi α ηi , ≥ −α t , f t, α t , α t qα ≤ pα i m−2 σi α ηi , i 3.15 H t, β t , β t ; y, y H t, β t , β t ; y, y pβ − qβ ≥ m−2 τi β ηi , ≤ −β t , ≤ f t, β t , β t pβ qβ ≥ i m−2 σi β ηi i Thus, α, β are lower and upper solutions of 3.14 , respectively Since H satisfies a Nagumo condition, there exists a constant M1 > max{ α , β } depending on α, β and a Nagumo function such that any solution x of 3.14 with α t ≤ x t ≤ β t satisfies |x | < M1 on 0, Now, we choose α0 α and consider the problem −x t ∈ 0, , H t, x, x ; α0 , α0 , m−2 px − qx 3.16 m−2 τi x ηi , σi x ηi qx px i i Using A1 , 3.9 , 3.12 and 3.13 , we obtain H t, α0 , α0 ; α0 , α0 pα0 − qα0 ≤ f t, α0 , α0 ≥ −α0 t , m−2 τi α0 ηi , qα0 ≤ pα0 i m−2 σi α0 ηi , i 3.17 H t, β t , β t ; α0 , α0 pβ − qβ ≥ H t, β t , β t ; α0 , α0 ≤ f t, β t , β t m−2 τi β ηi , pβ qβ ≥ i ≤ −β t , m−2 σi β ηi , i which imply that α0 and β are lower and upper solutions of 3.16 Hence by Theorems 2.2 and 2.3, there exists a unique solution α1 of 3.16 such that α0 ≤ α1 ≤ β t , α1 ≤ M1 , t ∈ 0, 3.18 Boundary Value Problems Note that the uniqueness of the solution follows by Theorem 2.3 Using 3.9 and 3.13 together with the fact that α1 is solution of 3.16 , we find that α1 is a lower solution of 3.2 , that is, −α1 H t, α1 , α1 ; α0 , α0 ≤ f t, α1 , ω α1 t ∈ 0, , , m−2 pα1 − qα1 3.19 m−2 τi α1 ηi , pα1 σi α1 ηi qα1 i i In a similar manner, it can be shown by using A1 , 3.12 , 3.13 , and 3.19 that α1 and β are lower and upper solutions of the following m-point BVP −x H t, x, x ; α1 , α1 , t ∈ 0, , m−2 px − qx 3.20 m−2 τi x ηi , px σi x ηi qx i i Again, by Theorems 2.2 and 2.3, there exists a unique solution α2 of 3.20 such that α1 t ≤ α2 t ≤ β t , ≤ M1 , α2 t t ∈ 0, 3.21 Continuing this process successively, we obtain a bounded monotone sequence {αn } of solutions satisfying α1 t ≤ α2 t ≤ α3 t ≤ · · · ≤ αn t ≤ β t , t ∈ 0, , 3.22 where αn is a solution of the problem −x H t, x, x ; αn−1 , αn−1 , t ∈ 0, , m−2 px − qx 3.23 m−2 τi x ηi , px σi x ηi , qx i i and is given by x t m−2 G t, s H s, αn , αn ; αn−1 , αn−1 ds τi x ηi i m−2 σi x ηi i t 2q p q p 2q p −t 2q p q p p 2q p 3.24 Since H is bounded on 0, × mint∈ 0,1 α t , maxt∈ 0,1 β t × R × mint∈ 0,1 α t , j 0, are uniformly maxt∈ 0,1 β t × R, therefore it follows that the sequences {αn } j bounded and equicontinuous on 0, Hence, by Ascoli-Arzela theorem, there exist the j subsequences and a function x ∈ C1 0, such that αn → x j uniformly on 0, as Boundary Value Problems n → ∞ Taking the limit n → ∞, we find that H t, αn , αn ; αn−1 , αn−1 → f t, x, ω x consequently yields m−2 G t, s f s, x s , ω x s x t −t 2q p τi x ηi ds i m−2 σi x ηi 2q i p q p p 2q p 3.25 q p 2q p t which This proves that x is a solution of 3.2 Theorem 3.2 Assume that A1 and A2 hold Further, one assumes that my2 ≤ for |y| ≥ A3 the function F ∈ C2 0, × R × R satisfies y ∂/∂x F t, x, y M, where m max{|Fx x t, x, y | : t, x, y ∈ 0, × mint∈ 0,1 α t , maxt∈ 0,1 β t × −M, M }, and F f φ Then, the convergence of the sequence {αn } of approximate solutions (obtained in Theorem 3.1) is quadratic Proof Let us set en pen − qen x t − αn t 1 t ≥ so that en ηi , satisfies the boundary conditions m−2 m−2 τi en pen 1 qen σi en i 1 ηi 3.26 i In view of the assumption A3 , for every t, x ∈ 0, × mint∈ 0,1 α t , maxt∈ 0,1 β t , it follows that 2mM ≤ 0, Fx t, x, M Fx t, x, −M − 2mM ≥ 3.27 Now, by Taylor’s theorem, we have −en t F t, x, x − φ t, x, − f t, αn , ω αn Fx t, α, ω αn − φx t, β, αn Fx t, αn , ω αn − αn x − αn 1 x − αn Fxx t, z1 , z2 x − ω αn αn − αn − ω αn Fx t, αn , ω αn ω αn Fx t, αn , ω αn x − ω αn x − αn x − ω αn 1 Fxx t, z1 , z2 Fx x t, z1 , z2 − φ t, x, − φ t, αn , − φx t, β, αn ≤ Fx t, αn , ω αn x − ω αn M2 − αn |x − αn | x − ω αn ρ1 x − αn , 3.28 10 Boundary Value Problems max{|Fxx |, |Fxx |, |Fx x |} on where αn ≤ z1 ≤ x, ω αn ≤ z2 ≤ x , αn ≤ ξ ≤ β, M2 ρ max{φxx t, x, : t, x, ∈ 0, × mint∈ 0,1 α t , maxt∈ 0,1 β t × −M, M and ρ1 0, × mint∈ 0,1 α t , maxt∈ 0,1 β t } with ρ > satisfying β − αn ≤ ρ x − αn on 0, Also, in view of 3.13 , we have −en f t, x, x − H t, αn , αn ; αn , αn t ≥ f t, x, x − f t, αn , ω αn fx t, c3 , c4 en ≥ −γen fx t, c3 , c4 x − ω αn fx t, c3 , c4 x − ω αn 1 3.29 , max{|fx t, x, y | : t, x, y ∈ 0, × where αn ≤ c3 ≤ x, ω αn ≤ c4 ≤ x and γ mint∈ 0,1 α t , maxt∈ 0,1 β t × −M, M } Now we show that ω αn t αn t By the mean value theorem, for every y1 ∈ −M, M and ω αn t ≤ c5 ≤ y1 , we obtain Fx t, αn t , y1 Let αn Fx t, αn t , ω αn > M for some t ∈ 0, Then ω αn Fx t, αn t , y1 Fx x t, αn t , c5 y1 − ω αn t t t 3.30 M and 3.30 becomes Fx x t, αn t , c5 y1 − M Fx t, αn t , M ≤ Fx t, αn t , M − m y1 − M In particular, taking y1 3.31 −M and using 3.27 , we have Fx t, αn t , −M ≤ Fx t, αn t , M 2mM ≤ 0, 3.32 which contradicts that Fx t, αn t , −M ≥ 2mM > Similarly, letting αn < −M for some t ∈ 0, , we get a contradiction Thus, it follows that |αn t | ≤ M for every t ∈ 0, , which αn t and consequently, 3.28 and 3.29 take the form implies that ω αn t −en where M3 ρ1 t ≤ Fx t, αn , ω αn t en t M3 en , 3.33 M2 /2 and −en t ≥ −γen t fx t, c3 , c4 en Now, by a comparison principle, we can obtain en of the problem −r t pr − qr Fx t, αn , ω αn t i t 3.34 t ≤ r t on 0, , where r t is a solution r t M3 en , m−2 τi en 3.35 m−2 ηi , pr σi en qr i 1 ηi Boundary Value Problems 11 Since Fx is continuous and bounded on 0, × mint∈ 0,1 α t , maxt∈ 0,1 β t × R, there exist ζ2 , ζ1 > independent of n such that −ζ1 ≤ Fx ≤ ζ2 on 0, × mint∈ 0,1 α t , maxt∈ 0,1 β t × −M, M Since ζ2 − Fx t, αn , ω αn ≥ on 0, , so we can rewrite 3.35 as r t ζ2 − Fx t, αn , ω αn ζ2 r t pr − qr r t − M3 en m−2 3.36 m−2 τi en ηi , pr qr σi en i 1 ηi , i whose solution is given by r t ζ2 − Fx t, αn , ω αn Gζ2 t, s m−2 τi en i ηi −t 2q p r s − M3 en ds m−2 q p p 2q p σi en t ηi 2q i p q p 2q p 3.37 where Gζ2 t, s ⎧ ⎪ ⎪ − p ζ2 q e−ζ2 ⎪ ⎪ ⎨ p ζ2 p −1 qζ2 /p − e−ζ2 ⎪ ⎪ −ζ ⎪ ⎪ e ⎩ Introducing the integrating factor μ t takes the form ζ2 q −ζ2 e p t e Fx s,αn s ,ω α n s ds such that e−ζ1 t < μ ≤ eζ2 t , 3.34 −1 m−2 τi en q i 1 3.39 m−2 i τi en Integrating 3.39 from to t and using r ≥ −1/q ηi − M e n ≤ t ≤ s, ζ2 q − e−ζ2 s , s ≤ t ≤ 1, p 3.38 p −M3 en μ t r t μt r tμ t ≥ ζ2 q − e−ζ2 t , p 1−s p − t−s p 1−s ηi , we obtain t μ s ds, 3.40 which can alternatively be written as r t ≥ −1 m−2 τi en qeζ1 t i −1 m−2 ≥ τi en q i 1 ηi − M3 en ζ2 eζ1 t eζ2 − 3.41 M3 − en ζ2 eζ2 − −ρ1 en − ρ2 e n , 12 Boundary Value Problems m−2 i τi , 1/q where ρ1 3.41 yields M3 /ζ2 eζ2 − Using the fact that Gζ2 t, s ≤ together with ρ2 Gζ2 t, s ζ2 − Fx r t ≤ Gζ2 t, s ζ2 − F x ρ1 e n ρ2 e n 3.42 ≤ Gζ2 t, s ζ2 ρ1 e n ζ1 ρ2 e n , which, on substituting in 3.37 , yields en 1 ≤r t ≤ Gζ2 t, s ρ1 e n ζ1 ζ2 ρ2 e n m−2 τi en −t 2q p ηi i ≤ ds σi en ζ1 ζ2 ρ1 e n 2q i Gζ2 t, s ds ρ2 ζ2 p ζ1 M3 en m−2 m−2 τi p q p 2q p σi i i m−2 m−2 τi B i ds ηi p q p 2q p σi i en q p 2q p t ηi Gζ2 t, s ≤ M3 e n m−2 q p p 2q p en A en , 3.43 where A ρ2 ζ2 ζ1 Gζ2 t, s ds, M3 max B ζ2 ≤ , we obtain A 1−B− 3.44 Taking the maximum over 0, and then solving 3.43 for en en Gζ2 t, s ds ζ1 ρ1 max m−2 i τi m−2 i σi p q/p 2q en p 3.45 Also, it follows from 3.33 that en μ t ≥ −M3 en μ t ≥ −M3 eζ2 t en , 1 t ∈ 0, m−2 Integrating 3.46 from to t and using ≥ −1/q i τi en m−2 τi en ηi , we obtain condition pen − qen i en t μt ≥ −1 m−2 τi en q i 1 ηi − M3 eζ2 t − ζ2 ηi en , 3.46 from the boundary 3.47 Boundary Value Problems 13 which, in view of the fact e−ζ1 t < μ ≤ eζ2 t and 3.45 , yields ⎛ ⎡ en −1 q ⎢ t ≥ eζ1 t ⎣ − ⎞ m−2 ⎜ τi ⎝ i A m−2 i τi 1−B− M3 eζ2 t − ζ2 en m−2 i σi p q /p 2q p ⎟ ⎠ 3.48 ≥ −δ1 en , where ⎧ ⎪ ⎨ ⎢ max eζ1 t ⎣ ⎪ ⎩ δ1 ⎛ ⎡ ⎞ m−2 ⎜ τi ⎝ qi 1−B− M3 e − ζ2 ζ2 t A m−2 i τi , t ∈ 0, m−2 i σi p q /p 2q p ⎟ ⎠ 3.49 As en ∈ C1 0, , there exists t ∈ 0, such that en t en ≤ 1 − en m−2 σi en pi ⎡ ⎢ ≤⎣ 1 ≤ en ηi − q e p n 1 1 ≤ m−2 σi en pi qδ en p ⎤ m−2 A m−2 i p 1−B− m−2 i τi σi p q /p 2q p σi i 3.50 qδ ⎥ ⎦ en p Integrating 3.46 from t to t t ≤ t and using 3.50 , we have ⎡ en ⎢ t ≤ eζ1 t ⎣ eζ2 t A m−2 i p 1−B− qδ p σi m−2 i m−2 i τi M3 eζ2 t − eζ2 t ζ2 σi p q /p 2q p 3.51 ⎤ ⎥ ⎦ en Using 3.45 in 3.34 , we obtain en t μ1 t ≤ γAμ1 t 1−B− m−2 i σi m−2 i τi p q /p 2q p en , 3.52 14 Boundary Value Problems t e fx s,c3 ,c4 ds Since fx is bounded on 0, × mint∈ 0,1 α t , maxt∈ 0,1 β t × where μ1 t −M, M , we can choose ζ3 , ζ4 > such that −ζ3 ≤ fx t,c3 ,c4 ≤ ζ4 on 0, × mint∈ 0,1 α t , maxt∈ 0,1 β t × −M, M and e−ζ3 t < μ1 t ≤ eζ4 t so that 3.52 takes the form en ≤ t μ1 t γAeζ4 t m−2 i 1−B− m−2 i τi σi p q /p 2q en p 3.53 Integrating 3.53 from t to t t ≥ t , and using 3.51 , we find that ⎡ en t ≤ ⎢ ⎣en μ1 t ⎤ γA eζ4 t − eζ4 t t μ1 t m−2 i L2 − B − m−2 i τi σi p q /p 2q p ⎥ en ⎦ ⎡ ⎢ ≤ eζ3 t ⎣ Aeζ4 t m−2 i p 1−B− m−2 i m−2 i τi σi qδeζ4 t p σi p q /p 2q p ⎤ γA eζ4 t − eζ4 t ζ4 − B − m−2 i m−2 i τi σi p q /p 2q p ⎥ ⎦ en 3.54 Letting ⎧ ⎪ ⎨ δ2 ⎧ ⎪ ⎨ ⎡ ⎢ max max eζ1 t ⎣ ⎪ ⎪ ⎩ ⎩ p 1−B− qδ p ⎧ ⎪ ⎨ eζ2 t A m−2 i m−2 i σi m−2 i τi σi M3 eζ2 t − eζ2 t ζ2 p ⎤ ⎥ ⎦, t ∈ 0, t q /p 2q p ⎫ ⎪ ⎬ , ⎪ ⎭ ⎡ ⎢ max eζ3 t ⎣ ⎪ ⎩ p 1−B− Aeζ4 t m−2 i m−2 i m−2 i τi σi qδeζ4 t p σi p q /p 2q p ⎤ γA eζ4 t − eζ4 t ζ4 − B − m−2 i σi m−2 i τi p q /p 2q p ⎥ ⎦, t ∈ t, , 3.55 Boundary Value Problems 15 it follows from 3.51 and 3.54 that en t ≤ δ2 e n 3.56 Hence, from 3.48 and 3.56 , it follows that en where δ3 ≤ δ3 e n , 1 3.57 max{δ1 , δ2 } From 3.45 and 3.57 with Q A m−2 i 1−B− m−2 i τi σi p q /p 2q δ3 , p 3.58 we obtain en 1 en ≤ Q en 3.59 This proves the quadratic convergence in C1 norm Example 3.3 Consider the boundary value problem −x − x t x te − x−1 − 720 35 16 x , t ∈ 0, , 3.60 11 x − x 20 x 4 x , 5 x 11 x 20 x Let α t and β t t be, respectively, lower and upper solutions of 3.60 Clearly α t and β t are not the solutions of 3.60 and α t < β t , t ∈ 0, Also, the assumptions of Theorem 3.1 are satisfied Thus, the conclusion of Theorem 3.1 applies to 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Theorem 3.1 applies to the. .. 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