Boundary Value Problems This Provisional PDF corresponds to the article as it appeared upon acceptance Fully formatted PDF and full text (HTML) versions will be made available soon Numerical-analytic technique for investigation of solutions of some nonlinear equations with Dirichlet conditions Boundary Value Problems 2011, 2011:58 doi:10.1186/1687-2770-2011-58 Andrei Ronto (ronto@math.cas.cz) Miklos Ronto (matronto@gold.uni-miskolc.hu) Gabriela Holubova (gabriela@kma.zcu.cz) Petr Necesal (pnecesal@kma.zcu.cz) ISSN Article type 1687-2770 Research Submission date 26 May 2011 Acceptance date 28 December 2011 Publication date 28 December 2011 Article URL http://www.boundaryvalueproblems.com/content/2011/1/58 This peer-reviewed article was published immediately upon acceptance It can be downloaded, printed and distributed freely for any purposes (see copyright notice below) For information about publishing your research in Boundary Value Problems go to http://www.boundaryvalueproblems.com/authors/instructions/ For information about other SpringerOpen publications go to http://www.springeropen.com © 2011 Ronto et al ; licensee Springer This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Numerical-analytic technique for investigation of solutions of some nonlinear equations with Dirichlet conditions Andrei Ront´1 , Miklos Ront´2 , Gabriela Holubov´ and Petr Neˇesal∗3 o o a c Institute of Mathematics, Academy of Sciences of the Czech Republic, Brno, Czech Republic of Analysis, University of Miskolc, Egyetemvaros, Hungary Department of Mathematics, University of West Bohemia, Pilsen, Czech Republic ∗ Corresponding author: pnecesal@kma.zcu.cz Email addresses: AR: ronto@math.cas.cz MR: matronto@gold.uni-miskolc.hu GH: gabriela@kma.zcu.cz Department Abstract The article deals with approximate solutions of a nonlinear ordinary differential equation with homogeneous Dirichlet boundary conditions We provide a scheme of numerical-analytic method based upon successive approximations constructed in analytic form We give sufficient conditions for the solvability of the problem and prove the uniform convergence of the approximations to the parameterized limit function We provide a justification of the polynomial version of the method with several illustrating examples 2000 Mathematics Subject Classification: 34B15; 65L10 Keywords: nonlinear boundary value problem; numerical-analytic method; Chebyshev interpolation polynomials Introduction In studies of solutions of various types of nonlinear boundary value problems for ordinary differential equations side by side with numerical methods, it is often used an appropriate technique based upon some types of successive approximations constructed in analytic form This class of methods includes, in particular, the approach suggested at first in [1, 2] for investigation of periodic solutions Later, appropriate versions of this method were developed for handling more general types of nonlinear boundary value problems for ordinary and functional-differential equations We refer, e.g., to the books [3–5], the articles [6–12], and the series of survey articles [13] for the related references According to the basic idea, the given boundary value problem is replaced by the Cauchy problem for a suitably modified system of integro-differential equations containing some artificially introduced parameters The solution of the perturbed problem is searched in analytic form by successive iterations The perturbation term, which depends on the original differential equation, on the introduced parameters and on the boundary conditions, yields a system of algebraic or transcendental determining equations These equations enable us to determine the values of the introduced parameters for which the original and the perturbed problems coincide Moreover, studying solvability of the approximate determining systems, we can establish existence results for the original boundary value problem In this article, we introduce the Chebyshev polynomial version of the known numerical-analytic method based on successive iterations At the beginning, we follow the ideas presented by Ront´ and Ront´ [14] o o and by Ront´ and Shchobak [15], which contains existence results for a system of two nonlinear differential o equations with separated boundary conditions In order to avoid some technical difficulties, we deal in this article, for simplicity, with nonlinear differential equations with homogeneous Dirichlet boundary conditions On the other hand, our basic recurrence relation has the same general form as it is presented in [15] In Section 2, we state the studied problem and the corresponding setting Sections and contain the construction of the sequence of successive approximations, its convergence analysis, the properties of the limit function, and its correspondence to the solution of the original boundary value problem The existence questions are discussed as well Main results of the article are in Section 5, which contains a justification of the Chebyshev polynomial version of the introduced method with corresponding convergence analysis and error estimates Results in Section allow us to construct the Chebyshev polynomial approximations of the solution of the nonlinear boundary value problem, which essentially simplify the computations of successive approximations in analytic form and simplify also the form of the determining equation In Section 6, we illustrate the applicability of our approach to three Dirichlet boundary value problems: the linear one, the semilinear one, and the quasi-linear one containing the p-Laplace operator Finally, let us note that presented polynomial version of the numerical-analytic method in this article can be extended to more general nonlinear boundary value problems studied in [14] 2 Problem setting and preliminaries We consider the following system of two nonlinear differential equations with Dirichlet boundary conditions   dx1 = f (t, x , x ), t ∈ (0, T ),  1   dt   dx2 (1)  dt = f2 (t, x1 , x2 ),      x1 (0) = x1 (T ) = In the vector form, we have   dx  = f (t, x), t ∈ (0, T ), dt   Ax(0) + C1 x(T ) = 0, (2) where x = col(x1 , x2 ), f (t, x) = col(f1 (t, x), f2 (t, x)) and A= 10 , 00 C1 = 00 10 Let the function f (t, x) be defined and continuous in the domain [0, T ] × D, D = [−a1 , a1 ] × [a2 , b2 ] ⊂ R2 (3) To avoid dealing with singular matrix C1 in (2), which does not enable us to express explicitly x(T ), it is useful to carry out the following parametrization x2 (T ) = λ, (4) λ ∈ Λ ⊆ [a2 , b2 ] (5) where Thus, instead of (2) we use the equivalent problem with two-point parameterized boundary conditions   dx  = f (t, x), t ∈ (0, T ), dt (6)   Ax(0) + Cx(T ) = d(λ), x2 (T ) = λ, where C= 01 , 10 d(λ) = col(λ, 0) The two-point parameterized boundary conditions in (6) allow us to write x(T ) = C −1 d(λ) − C −1 Ax(0), which will be used in the sequel for the construction of the iterative scheme Throughout the text, C([0, T ], R2 ) is the Banach space of vector functions with continuous components and L1 ([0, T ], R2 ) is the usual Banach space of vector functions with Lebesgue integrable components Moreover, the signs | · |, ≤, ≥, max, and operations will be everywhere understood componentwise Let us define the vector δD (f ) := max f (t, u) − f (t, u) (t,u)∈[0,T ]×D , (7) (t,u)∈[0,T ]×D for which the following estimate is true (cf [5, 16]) δD (f ) ≤ max |f (t, u)| (8) (t,u)∈[0,T ]×D For z ∈ R2 of the form z = col(0, z2 ), z2 ∈ [a2 , b2 ] ⊆ [a2 , b2 ] (9) and λ ∈ Λ we define the vector γ : D × Λ → R2 + γ = γ(z, λ) := T δD (f ) + |C −1 d(λ) − (C −1 A + I2 )z|, (10) where I2 is the unit matrix of order In the sequel, we use the following assumptions (A1) The function f : [0, T ] × D → R2 is continuous (A2) The function f satisfies the following Lipschitz condition: there exists a nonnegative constant square matrix K of order such that ∀ t ∈ [0, T ] ∀ u, v ∈ D : |f (t, u) − f (t, v)| ≤ K|u − v| (A3) The subset Dγ := {z = col(0, z2 ) ∈ D : B(z, γ(z, λ)) ⊂ D for all λ ∈ Λ} is non-empty, where B(z, γ(z, λ)) := u ∈ R2 : |u − z| ≤ γ(z, λ) (A4) The greatest eigenvalue r(Q) of the non-negative matrix Q := 3T K 10 satisfies the inequality r(Q) < Remark The history and possible improvements of the constant in [5, 17, 18] 10 in the definition of Q can be found We will use the auxiliary sequence {αm } of continuous functions αm = αm (t), t ∈ [0, T ], defined by α0 (t) := 1, αm+1 (t) := t t 1− T t αm (s) ds + T T αm (s) ds, m = 0, 1, 2, (11) t It is obvious that, in particular, α1 (t) = 1− t t T ds + t T T t T ds = 2t − t , t ∈ [0, T ] According to [16, Lemma 4] or [5, Lemma 2.4], we have the following estimates 3T αm (t), m = 2, 3, , 10 m 10 3T αm+1 (t) ≤ α1 (t), m = 0, 1, 2, 10 αm+1 (t) ≤ (12) (13) Successive approximations and convergence analysis To investigate the solution of the parameterized boundary value problem (6) let us introduce the sequence of functions defined by the recurrence relation t xm+1 (t, z, λ) := z + + t f (s, xm (s, z, λ))ds − T T f (s, xm (s, z, λ))ds (14) t C −1 d(λ) − C −1 A + I2 z , T where d(λ) = col(λ, 0) and x0 (t, z, λ) = z, z ∈ Dγ Let us note that for m = 0, 1, 2, , we have xm (t, z, λ) = col (xm,1 (t, z, λ), xm,2 (t, z, λ)) Moreover, all the functions xm = xm (t, z, λ) are continuously differentiable and satisfy the initial condition xm (0, z, λ) = z as well as the boundary conditions in (6) Let us establish the uniform convergence of the sequence (14) and the relation of the limit function to the solution of some additively modified boundary value problem Theorem Let the assumptions (A1)–(A4) be satisfied Then for all z ∈ Dγ and λ ∈ Λ, the following statements hold The sequence {xm } converges uniformly in t ∈ [0, T ] to the limit function x∗ (t, z, λ) = lim xm (t, z, λ), m→+∞ which satisfies the initial condition x∗ (0, z, λ) = z and the boundary conditions in (6) For all t ∈ [0, T ], the limit function x∗ satisfies the identity t x∗ (t, z, λ) = z + f (s, x∗ (s, z, λ)) ds − t T T f (s, x∗ (s, z, λ)) ds (15) t + C −1 d(λ) − C −1 A + I2 z T Moreover, x∗ is continuously differentiable and it is a unique solution of the Cauchy problem for the additively modified differential equation    dx(t) 1  = f (t, x(t)) + C −1 d(λ) − C −1 A + I2 z − dt T T    x(0) = z T f (s, x(s)) ds, t ∈ (0, T ), (16) The following error estimate holds |x∗ (t, z, λ) − xm (t, z, λ)| ≤ 10 m−1 Q K(I2 − Q)−1 γα1 (t), t ∈ [0, T ] (17) Remark We emphasize that the first component of the vector z is fixed and coincide with the value of x1 (0) in the first boundary condition in (1), while its second component z2 is considered as free parameter Thus, the expression “for all z”, actually means “for all z2 ” Proof (of Theorem 2) First, we show that for all (t, z, λ) ∈ [0, T ] × Dγ × Λ and m ∈ N, all functions xm = xm (t, z, λ) belong to D Indeed, using the estimate in [19, Lemma 2], an arbitrary continuous function u : [0, T ] → R satisfies t  u(s) − T T  α1 (t) u(τ ) dτ  ds ≤ max u(t) − u(t) t∈[0,T ] t∈[0,T ] (18) Thus, we have |x1 (t, z, λ) − x0 (t, z, λ)| = |x1 (t, z, λ) − z|  t f (s, z) − ≤ T  T f (τ, z) dτ  ds + C −1 d(λ) − C −1 A + I2 z ≤ α1 (t)δD (f ) + C −1 d(λ) − C −1 A + I2 z ≤ T δD (f ) + C −1 d(λ) − C −1 A + I2 z = γ (19) Therefore, we conclude that x1 (t, z, λ) ∈ D, whenever (t, z, λ) ∈ [0, T ] × Dγ × Λ By induction, we obtain that for all m ∈ N, we have |xm (t, z, λ) − x0 (t, z, λ)| ≤ γ, i.e., all functions xm are also contained in D For m = 0, 1, 2, , let us define rm+1 (t, z, λ) := xm+1 (t, z, λ) − xm (t, z, λ) Due to the assumption (A2), we have |rm+1 (t, z, λ)| = t 1− T t − T  t [f (s, xm (s, z, λ)) − f (s, xm−1 (s, z, λ))] ds T [f (s, xm (s, z, λ)) − f (s, xm−1 (s, z, λ))] ds t t t ≤ K 1− T t |rm (s, z, λ)| ds + T  T |rm (s, z, λ)| ds (20) t Relation (19) yields |r1 (t, z, λ)| ≤ γ and thus using (20), we obtain  t |r2 (t, z, λ)| ≤ K  − T t t γ ds + T  T γ ds = Kγα1 (t) t By induction, we obtain for m = 1, 2, that |rm+1 (t, z, λ)| ≤ K m γαm (t) Using (12) and (13), we have |rm+1 (t, z, λ)| ≤ 10 3T K 10 m−1 Kγα1 (t) = 10 m−1 Q Kγα1 (t), and thus, for all (t, z, λ) ∈ [0, T ] × Dγ × Λ and j, m ∈ N, we obtain j |rm+i (t, z, λ)| |xm+j (t, z, λ) − xm (t, z, λ)| ≤ i=1 ≤ ≤ 10 j Qm+i−2 Kγα1 (t) = i=1 10 m−1 Q K +∞ Qi γα1 (t) = i=0 10 m−1 Q K j Qi γα1 (t) i=0 10 m−1 −1 Q K (I2 − Q) γα1 (t) (21) Due to the assumption (A4), the sequence {Qm } converges to the zero matrix for m → +∞ Hence, (21) implies that {xm } is a Cauchy sequence in the Banach space C([0, T ], R2 ) and therefore, the limit function x∗ = x∗ (t, z, λ) exists Passing to the limit for j → +∞ in (21), we obtain the final error estimate (17) The limit function x∗ satisfies the initial condition x∗ (0, z, λ) = z as well as the boundary conditions in (6), since these conditions are satisfied by all functions xm = xm (t, z, λ) of the sequence {xm } Passing to the limit in the recurrence relation (14) for xm , we show that the limit function x∗ satisfies the identity (15) If we differentiate this identity, we obtain that x∗ is a unique solution of the Cauchy problem (16) Let us find a relation of the limit function x∗ = x∗ (t, z, λ) of the sequence {xm } and the solution of the parameterized boundary value problem (6) For this purpose, let us define the function ∆ : R2 → R2 1 ∆(z, λ) := C −1 d(λ) − C −1 A + I2 z − T T T f (s, x∗ (s, z, λ)) ds Theorem Let the assumptions (A1)–(A4) be satisfied The limit function x∗ of the sequence {xm } is a solution of the boundary value problem (6) if and only if the value of the vector parameters z ∈ Dγ and λ ∈ Λ are such that ∆(z, λ) = Proof It is sufficient to apply Theorem and notice that the equation in (16) coincides with the original equation in (6) if and only if the relation ∆(z, λ) = holds Remark The function ∆ = ∆(z, λ) is called the determining function and the equation ∆(z, λ) = is called the determining equation, because it determines the values of the unknown parameters z ∈ Dγ and λ ∈ Λ involved in the recurrence relation (14) Properties of the limit function and the existence theorem Let us investigate some properties of the limit function x∗ of the sequence {xm } and the determining function ∆ Lemma Under the assumptions (A1)–(A4), the limit function x∗ satisfies the following Lipschitz condition for all t ∈ [0, T ], all z, y ∈ Dγ and λ ∈ Λ |x∗ (t, z, λ) − x∗ (t, y, λ)| ≤ I2 + where R := sup t∈[0,T ] I2 − t T 10 −1 α1 (t)K (I2 − Q) R |z − y| , C −1 A + I2 Proof Using the assumption (A2), we obtain t |x1 (t, z, λ) − x1 (t, y, λ)| = (z − y) + [f (s, z) − f (s, y)] ds − t T  T [f (s, z) − f (s, y)] ds − t t ≤  1− T t ds + T t C −1 A + I2 (z − y) T  T ds K |z − y| + R |z − y| t = [R + α1 (t)K] |z − y| Similarly, using the above estimate, we have  t |x2 (t, z, λ) − x2 (t, y, λ)| ≤ R + K − T t Kt (R + α1 (s)K) ds + T  T (R + α1 (s)K) ds |z − y| t = R + KRα1 (t) + K α2 (t) |z − y| and by induction, we obtain m−1 K i Rαi (t) + K m αm (t) |z − y| |xm (t, z, λ) − xm (t, y, λ)| ≤ R + i=1 Using the estimates in (13), we get m−2 |xm (t, z, λ) − xm (t, y, λ)| ≤ R + i=0 10 KR 3T K 10 i α1 (t) + 10 K 3T K 10 m−1 α1 (t) |z − y| and passing to the limit for m → +∞, due to the assumption (A4), we obtain the final inequality +∞ |x∗ (t, z, λ) − x∗ (t, y, λ)| ≤ R + i=0 10 KRQi α1 (t) |z − y| , 10 −1 ≤ R + KRα1 (t) (I2 − Q) |z − y| Lemma Under the assumptions (A1)–(A4), the determining function ∆ is well defined and bounded in Dγ × Λ Furthermore, it satisfies the following Lipschitz condition for all z, y ∈ Dγ and λ ∈ Λ |∆(z, λ) − ∆(y, λ)| ≤ 10 −1 R |z − y| C −1 A + I2 + KR + T K (I2 − Q) T 27 Proof It follows from Theorem that the limit function x∗ of the sequence {xm } exists and is continuously differentiable in Dγ × Λ Therefore, ∆ is bounded and the assumption (A2) implies |∆(z, λ) − ∆(y, λ)| ≤ 1 C −1 A + I2 |z − y| + T T T K |x∗ (s, z, λ) − x∗ (s, y, λ)| ds sequence {xq+1 } of vector polynomials xq+1 = col(xq+1 , xq+1 ) of degree (q + 1) m m m,2 m,1 xq+1 (t, z, λ0 ) := z, z = col(0, z2 ), t xq+1 (t, z, λ) m+1 q f := z + (s, xq+1 (s, z, λ)) ds m t − T T f q (s, xq+1 (s, z, λ)) ds + m t + C −1 d(λ) − C −1 A + I2 z , T m = 0, 1, 2, q q where f q = col (f1 , f2 ) is the vector of interpolation Chebyshev polynomial of degree q corresponding to f Let us point out that the coefficients of the interpolation polynomials depend on the parameters z and λ Moreover, all the functions xq+1 = xq+1 (t, z, λ) are continuously differentiable and satisfy the initial condition m m xq+1 (0, z, λ) = z as well as the boundary conditions in (6) m Let us define the domain Dγq := {z ∈ D ⊂ R2 : B(z, γq (z, λ)) ⊂ D for all λ ∈ Λ} ⊂ Dγ , where γq = γq (z, λ) := T (δD (f ) + Lq ) + C −1 d(λ) − (C −1 A + I2 )z (27) Theorem 16 Let the assumptions (A1)–(A4) be satisfied with Dγq instead of Dγ Then for all z ∈ Dγq , λ ∈ Λ, the following statements hold The sequence {xq+1 } converges uniformly in t ∈ [0, T ] to the limit function m x∗ (t, z, λ) = lim lim xq+1 (t, z, λ) = m q→+∞ m→+∞ lim xm (t, z, λ), m→+∞ which satisfies the initial condition x∗ (0, z, λ) = z and the boundary conditions in (6) The following error estimate holds x∗ (t, z, λ) − xq+1 (t, z, λ) ≤ m 10 m−1 10 Q K(I2 − Q)−1 γ(z, λ)α1 (t) + (I2 − Q)−1 α1 (t)Lq 9 (28) Proof We show that for all (t, z, λ) ∈ [0, T ] × Dγq × Λ and m ∈ N, all functions xq+1 = xq+1 (t, z, λ) belong m m 14 to D Similarly as in the proof of Theorem 2, we have xq+1 (t, z, λ) − xq+1 (t, z, λ) = xq+1 (t, z, λ) − z 1   t T f q (s, z) − ≤ f q (τ, z) dτ  ds + C −1 d(λ) − C −1 A + I2 z T 0 t [(f (s, z) − f (s, z)) + f (s, z)] − T T q ≤ [(f q (τ, z) − f (τ, z)) + f (τ, z)] dτ ds + C −1 d(λ) − C −1 A + I2 z ≤ [Lq + δD (f )] α1 (t) + C −1 d(λ) − C −1 A + I2 z ≤ T [Lq + δD (f )] + C −1 d(λ) − C −1 A + I2 z = γq Therefore, we conclude that xq+1 (t, z, λ) ∈ D, whenever (t, z, λ) ∈ [0, T ] × Dγq × Λ By induction, we obtain that for all m ∈ N, we have xq+1 (t, z, λ) − xq+1 (t, z, λ) ≤ γq , m i.e., all functions xq+1 are also contained in D m For j = 1, 2, , m and for all (t, z, λ) ∈ [0, T ] × Dγq × Λ, we estimate t xj (t, z, λ) − xq+1 (t, z, λ) j f (s, xj−1 (s, z, λ)) − f q (s, xq+1 (s, z, λ)) ds j−1 = − ≤ t T T f (s, xj−1 (s, z, λ)) − f q (s, xq+1 (s, z, λ)) ds j−1 t 1− T t + T t f (s, xj−1 (s, z, λ)) − f q (s, xq+1 (s, z, λ)) j−1 ds T f (s, xj−1 (s, z, λ)) − f q (s, xq+1 (s, z, λ)) j−1 ds t Taking into account that f (t, xj−1 (s, z, λ)) − f q (t, xq+1 (t, z, λ)) ≤ f (t, xj−1 (t, z, λ)) − f (t, xq+1 (t, z, λ)) j−1 j−1 + f (t, xq+1 (t, z, λ)) − f q (t, xq+1 (t, z, λ)) j−1 j−1 15 and using the assumption (A2) and the estimate (26), we get  t q+1  1− t xj (t, z, λ) − xj (t, z, λ) ≤ K xj−1 (s, z, λ) − xq+1 (s, z, λ) ds j−1 T  T t + xj−1 (s, z, λ) − xq+1 (s, z, λ) ds j−1 T t t + 1− T t t Lq ds + T T Lq ds t In particular, for j = and j = we have x1 (t, z, λ) − xq+1 (t, z, λ) ≤ α1 (t)Lq , x2 (t, z, λ) − xq+1 (t, z, λ) ≤ α1 (t)Lq + α2 (t)KLq , and by induction xm (t, z, λ) − xq+1 (t, z, λ) ≤ α1 (t) + α2 (t)K + · · · + αm (t)K m−1 Lq m Using (12) and (13), we get xm (t, z, λ) − xq+1 (t, z, λ) ≤ m 10 I2 + 3T K 10 + ··· + 3T K 10 m−1 α1 (t)Lq and due to the assumption (A4), we obtain xm (t, z, λ) − xq+1 (t, z, λ) ≤ m 10 −1 (I2 − Q) α1 (t)Lq By Theorem 2, we can write x∗ (t, z, λ) − xq+1 (t, z, λ) = x∗ (t, z, λ) − xm (t, z, λ) + xm (t, z, λ) − xq+1 (t, z, λ) m m ≤ 10 10 m−1 Q K(I2 − Q)−1 γα1 (t) + (I2 − Q)−1 α1 (t)Lq 9 Recall that the sequence {Qm } converges to the zero matrix for m → +∞ and Lq tends to the zero vector for q → +∞, which implies immediately that the sequence {xq+1 } converges uniformly to x∗ on [0, T ] m Let us define the mth approximate polynomial determining function ∆q (z, λ) m 1 := C −1 d(λ) − C −1 A + I2 z − T T 16 T f q s, xq+1 (s, z, λ) ds m (29) Lemma 17 Let the assumptions (A1)–(A4) be satisfied with Dγq instead of Dγ Then, for all z ∈ Dγq , λ ∈ Λ and m ∈ N, 10T −1 −1 Qm−1 K (I2 − Q) γ + K (I2 − Q) Lq + Lq 27 |∆(z, λ) − ∆q (z, λ)| ≤ m Proof Due to the assumption (A2), (26) and the error estimate (28), we get |∆(z, λ) − ∆q (z, λ)| m = T T f (s, x∗ (s, z, λ)) − f s, xq+1 (s, z, λ) m +f s, xq+1 (s, z, λ) − f q s, xq+1 (s, z, λ) m m K ≤ T ds T x∗ (s, z, λ) − xq+1 (s, z, λ) ds + Lq m 10 −1 −1 K Qm−1 K (I2 − Q) γ + (I2 − Q) Lq ≤ 9T T α1 (s) ds + Lq 10T −1 −1 = Qm−1 K (I2 − Q) γ + K (I2 − Q) Lq + Lq 27 Theorem 18 Let the assumptions (A1)–(A4) be satisfied with Dγq instead of Dγ Moreover, let there exist m ∈ N and nonempty set Ωq ⊂ Dγ × Λ such that the approximate polynomial determining function ∆q m satisfies |∆q (z, λ)| m ∂Ωq 10T −1 −1 Qm−1 K (I2 − Q) γ + K (I2 − Q) Lq + Lq 27 (30) and the Brouwer degree of ∆q over Ωq with respect to satisfies m deg (∆q , Ωq , 0) = m (31) Then there exists a pair (z ∗ , λ∗ ) ∈ Ωq such that ∆(z ∗ , λ∗ ) = and the corresponding limit function x∗ = x∗ (t, z ∗ , λ∗ ) of the sequence {xq+1 } solves the boundary value m problem (6) Proof Using the same steps as in the proof of Theorem 10, we construct the admissible homotopy Pq : [0, 1] × Ωq → R2 Pq (Θ, z, λ) := ∆q (z, λ) + Θ [∆(z, λ) − ∆q (z, λ)] m m 17 and we get deg(∆, Ωq , 0) = deg (∆q , Ωq , 0) m The assumption (31) then guarantees the existence of (z ∗ , λ∗ ) ∈ Ωq such that ∆(z ∗ , λ∗ ) = Applying Theorem 4, we obtain that the limit function x∗ = x∗ (t, z ∗ , λ∗ ) of the sequence {xq+1 } is the m solution of the boundary value problem (6) Examples In this section, we introduce three particular boundary value problems in the form of the system (1) The first problem is a linear one and enables us to build the sequence {xm } directly by the recurrence relation (14) The second problem is nonlinear and it is impossible to integrate in (14) in a closed form Thus, we use the Chebyshev interpolation of the integrand to construct the sequence of successive approximations also in this case In the last example, we use again the polynomial version of presented method in order to approximate a solution of the nonlinear Dirichlet problem containing p-Laplacian Example Let us consider the following linear problem with the Dirichlet boundary conditions  t ∈ (0, 1),  x1 (t) = x2 (t),   1 x (t) = − x1 (t) + 3π + 3π sin(3πt) ,    x (0) = x (1) = 0, 1 (32) which has a unique solution given in a closed form Let us denote this solution by x◦ (t) = col(x◦ (t), x◦ (t)) All the assumptions (A1)-(A4) are satisfied Let us take x0 (t, z, λ) = col(0, z2 ) and construct the successive approximations Xm of the exact solution x◦ for m = 0, 1, 2, in the following way: using the recurrence relation (14), evaluate xm+1 (t, z, λ), solve the (m+1)-th approximate determining equation ∆m+1 (z, λ) = (system of two linear equations) and denote its solution by (zm+1 , λm+1 ), define Xm+1 (t) := xm+1 (t, zm+1 , λm+1 ) Figure contains both components Xm,1 and Xm,2 of approximations Xm for m = 1, 2, 11 and also their differences from the exact solution x◦ Let us point out that the maxima of both components of |X11 (t) − x◦ (t)| for t ∈ 0, are both less then 10−9 18 Example Let us investigate the nonlinear Dirichlet boundary value problem  t ∈ (0, 1),  x1 (t) = sin(x2 (t)),   1 x (t) = − (x1 (t))3 + 3π + 3π sin(3πt) ,    x (0) = x (1) = 0, (33) which has a solution in the form x◦ (t) = col(x◦ (t), x◦ (t)) = 3π sin (3πt) , 3πt + π − 2π In this case, it is not possible to construct the sequence {Xm } of approximations using the iterative scheme as in the previous q+1 Example Thus, we use the following polynomial version of the iterative scheme Choose X0 (t) and for m = 0, 1, 2, , proceed the steps: q+1 q define Fm (t) := f q (t, Xm (t)) and realize the Chebyshev interpolation, define t q+1 Xm+1 (t, z, λ) q Fm (s) ds := z + t − T T q Fm (s) ds + t C −1 d(λ) − C −1 A + I2 z , T q+1 q define Fm+1 (t, z, λ) := f q (t, Xm+1 (t, z, λ)) and realize the Chebyshev interpolation with parameters z and λ q q+1 f1 t, Xm+1 (t, z, λ) = a1,0 (z, λ) + a1,1 (z, λ)t + · · · + a1,q (z, λ)tq , q q+1 f2 t, Xm+1 (t, z, λ) = a2,0 (z, λ) + a2,1 (z, λ)t + · · · + a2,q (z, λ)tq , solve the (m + 1)-th approximate polynomial determining equation 1 C −1 d(λ) − C −1 A + I2 z − T T T q Fm+1 (s, z, λ) ds = 0 and denote its solution by (zm+1 , λm+1 ), q+1 q+1 define Xm+1 (t) := Xm+1 (t, zm+1 , λm+1 ) q+1 In Figure 2, it is possible to compare polynomial approximations Xm for q = 15, m = 1, 2, 11 and the q+1 corresponding differences from the exact solution x◦ Let us note that the maximum of X11,j (t) − x◦ (t) j for t ∈ 0, is less then 10−7 for j = and less then · 10−7 for j = Example Let us consider the following nonlinear problem with the Dirichlet boundary conditions  p t ∈ (0, 1),  x1 (t) = φ p−1 (x2 (t)),   1 p x (t) = − gε (x1 (t)) − (p − 1)πp φp (sinp (πp t)) + gε (sinp (πp t)) , (34)    x1 (0) = x1 (1) = 0, 19 where p > 1, sinp is the generalized sine function (see [21] for the definition), φp (s) := |s|p−1 sgn s, s ∈ R, gε (x) := φp (x + ε) − εp−1 , ε ∈ R, πp := 2π , p sin π p Let us recall that sinp is 2πp -periodic function on R, which coincide with the sin function for p = Moreover, the pair sinp (πp t), (πp cosp (πp t))p−1 is a solution of the following initial value problem  p t ∈ (0, 1),  x1 (t) = φ p−1 (x2 (t)),   p x (t) = −(p − 1)πp φp (x1 (t)) ,    p−1 x1 (0) = 0, x2 (0) = πp For ε = 0, the problem (34) reads as the Dirichlet boundary value problem with p-Laplacian φp (x1 (t)) + φp (x1 (t)) = 2 p − (p − 1)πp φp (sinp (πp t)), t ∈ (0, 1), x1 (0) = x1 (1) = For the problem (34), all the assumptions (A1)–(A4) are satisfied in the linear case p = If p = 2, then there exist bounded domains D = a1 , a2 × b1 , b2 for which the second assumption (A2) concerning the Lipschitz condition of f is not satisfied Thus, we have to take into account the following additional assumptions on D in order to satisfy the assumption (A2): for < p < 2, we have to ensure that −ε < a1 < a2 or a1 < a2 < −ε, for < p, we have to ensure that < b1 < b2 or b1 < b2 < p Let us note that for p > 2, the function φ p−1 , which appears in the first component of f , is not Lipschitz continuous on any interval containing zero On the other hand, in the case of < p < 2, the function gε in the second component of f is not Lipschitz continuous on any interval containing −ε Thus, all the assumptions (A1)–(A4) are satisfied if we take, e.g., p = < 2, ε = 10 and a1 = − 11 The polynomial version of the iterative scheme from the previous Example is applicable in this case q+1 Figure shows Xm for q = 15, m = 1, 2, 11 Their differences from the exact solution (x◦ (t), x◦ (t)) = sinp (πp t), πp cosp (πp t))p−1 of the problem (34) are also available Let us note that the maximum of q+1 X11,j (t) − x◦ (t) for t ∈ 0, is less then · 10−4 for j = and less then · 10−4 for j = j q+1 Last Figure shows Xm for p = > 2, ε = and q = 15, m = 1, 2, 11 In spite of the fact that the q+1 assumption (A2) is not satisfied in this case, we obtain the polynomial approximation X11 , which differs from the exact solution x◦ less than · 10−3 in both components Competing interest The authors declare that they have no competing interests 20 Authors’ contributions All authors contributed to each part of this study equally, and also read and approved the final manuscript Acknowledgements The authors were partially supported by the Ministry of Education, Youth and Sports of the Czech Republic, grant no ME09109 (Program KONTAKT) and by MSM 4977751301 (G Holubov´, P Neˇesal), and by the a c Hungarian Scientific Research Fund OTKA throught grant no 68311 (M Ront´) This research was carried o out as part of the TAMOP-4.2.1.B-10/2/KONV-2010-0001 project with support by the European Union, co-financed by the European Social Fund References Samo˘ ılenko, AM: 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AM, Trofimchuk SI: The theory of the numerical-analytic method: achievements and new directions of development I Ukraă Mat Zh 50:102117 (1998) n 19 Ront, A, Ront´, M: Successive approximation method for some linear boundary value problems for o o differential equations with a special type of argument deviation Miskolc Math Notes 10:69–95 (2009) 20 Gonˇarov, VL: Teoriya interpolirovaniya i pribliˇeniya funkci˘ Gosudarstv Izdat Tehn.-Teor Lit., c z ı Moscow 1954 22 21 del Pino, M, Dr´bek, P, Man´sevich, R: The Fredholm alternative at the first eigenvalue for the onea a dimensional p-Laplacian J Differential Equations 151(2):386–419 (1999) 23 Figure 1: The approximations X1 , X2 and X11 of the exact solution x◦ of (32) Figure 2: The polynomial approximations of the exact solution x◦ of (33) for q = 15 Figure 3: The approximations of the exact solution x◦ of (34) for p = , ε = 10 and q = 15 Figure 4: The approximations of the exact solution x◦ of (34) for p = , ε = and q = 15 24 Figure Figure Figure Figure ... convergence of the sequence (14) and the relation of the limit function to the solution of some additively modified boundary value problem Theorem Let the assumptions (A1)–(A4) be satisfied Then for... λ)) ds Theorem Let the assumptions (A1)–(A4) be satisfied The limit function x∗ of the sequence {xm } is a solution of the boundary value problem (6) if and only if the value of the vector parameters... (0, z, λ) = z as well as the boundary conditions in (6), since these conditions are satisfied by all functions xm = xm (t, z, λ) of the sequence {xm } Passing to the limit in the recurrence relation