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J. Numer. Math. 2015; 23 (1):1–11 Francesco A. Costabile and Anna Napoli* A multipoint Birkho type boundary value problem Abstract: A multipoint boundary value problem is considered. The existence and uniqueness of solution is proved. Then, for the numerical solution, a general collocation method is proposed.Numerical experiments conrm theoretical results. Keywords: Boundary value problem, Birkho interpolation. MSC 2010: 65L10, 65D05 DOI: 10.1515/jnma-2015-0001 Received March 14, 2013; accepted January 8, 2014 1 Introduction The present paper is concerned with

Georgian Mathematical Journal Volume 16 (2009), Number 3, 401–411 BOUNDARY VALUE PROBLEMS FOR FRACTIONAL DIFFERENTIAL EQUATIONS RAVI P. AGARWAL, MOUFFAK BENCHOHRA, AND SAMIRA HAMANI Abstract. The sufficient conditions are established for the existence of so- lutions for a class of boundary value problems for fractional differential equa- tions involving the Caputo fractional derivative. 2000 Mathematics Subject Classification: 26A33. Key words and phrases: Boundary value problem, Caputo fractional de- rivative, fractional integral, existence, uniqueness, fixed

1 Introduction The method of quasilinearization, introduced by Bellman [ 4 , 5 ] in the 1960s, offers a numerical method to approximate solutions of nonlinear problems with sequences of solutions of linear problems. Under suitable hypotheses, the sequences of approximate solutions converge monotonically and quadratically. The method has been particularly useful in the study of boundary value problems for ordinary differential equations and we cite a number of its applications here [ 1 , 2 , 9 , 10 , 12 , 19 , 21 , 16 , 17 , 18 ]. Although it appears

GEORGIAN MATHEMATICAL JOURNAL: Vol. 4, No. 5, 1997, 401-412 SINGULAR NONLINEAR (n− 1, 1) CONJUGATE BOUNDARY VALUE PROBLEMS PAUL W. ELOE AND JOHNNY HENDERSON Abstract. Solutions are obtained for the boundary value problem, y(n) + f(x, y) = 0, y(i)(0) = y(1) = 0, 0 ≤ i ≤ n − 2, where f(x, y) is singular at y = 0. An application is made of a fixed point theorem for operators that are decreasing with respect to a cone. § 1. Introduction In this paper, we establish the existence of solutions for the (n − 1, 1) conjugate boundary value problem, y(n) + f(x, y) = 0, 0

References [1] GUSTAFSON, G. B.-RIDENHOUR, J. R.: Lower order branching and conjugate function discontinuity , J. Differential Equations 27 (1978), 167-179. [2] GUSTAFSON, G. B.-RIDENHOUR, J. R.: Uniqueness intervals for multipoint boundary value problems (preprint 2008). [3] HARTMAN, P.: Unrestricted n-parameter families , Rend. Circ. Mat. Palermo (2) 7 (1958), 123-142. [4] HARTMAN, P.: Ordinary Differential Equations, John Wiley and Sons, Inc., New York, 1964. [5] MIKUSI ´NSKI, J.: Sur l’´equation x ( n ) + A ( t ) x = 0, Ann. Polon. Math. 1 (1955

References [1] BITSADZE, A. V.: On the theory of nonlocal boundary value problems, Soviet Math. Dock. 30 (1964), 8-10. [2] BITSADZE, A. V.-SAMARSKII, A. A.: Some elementary generalizations of linear elliptic boundary value problems, Dokl. Akad. Nauk SSSR 185 (1969), 739-740. [3] CHU, J.-ZHOU, Z.: Positive solutions and eigenvalues of nonlocal boundary-value problems, Electron. J. Differential Equations 86 (2005), 1-9. [4] ELOE, P. W.-KHAN, R. A.-ASIF, N.: Positive solutions for a system of singular second order nonlocal boundary value problems, J. Korean Math

Journal of Applied Analysis 17 (2011), 91–103 DOI 10.1515/JAA.2011.005 © de Gruyter 2011 Existence and uniqueness results of some fractional boundary value problem Hanifa Seddiki and Said Mazouzi Abstract. We establish in this paper some existence results of a solution to a boundary value problem of fractional differential equation. We obtain two results, the first one by the Banach fixed point theorem and the second by a nonlinear alternative of Leray–Schauder type. Keywords. Boundary value problem, fractional differential equation, fixed point theorem. 2010

Journal of Applied Analysis 16 (2010), 107–119 DOI 10.1515/JAA.2010.008 © de Gruyter 2010 Positive solutions for discrete boundary value problems with p-Laplacian Dehong Ji and Weigao Ge Abstract. In the paper, we obtain the existence of positive solutions and establish a cor- responding iterative scheme for the following two-point discrete boundary value problem with p-Laplacian: .p.u.k 1///C e.k/f .u.k// D 0; k 2 N.1; T /; u.0/ B0.u.0// D 0; u.T C 1/C B1.u.T // D 0: The main tool is the monotone iterative technique. Keywords. Discrete boundary value problem

GEORGIAN MATHEMATICAL JOURNAL: Vol. 6, No. 6, 1999, 567-590 MULTIPLE SOLUTIONS OF GENERALIZED MULTIPOINT CONJUGATE BOUNDARY VALUE PROBLEMS PATRICIA J. Y. WONG AND RAVI P. AGARWAL Abstract. We consider the boundary value problem y(n)(t) = P (t, y), t ∈ (0, 1) y(j)(ti) = 0, j = 0, . . . , ni − 1, i = 1, . . . , r, where r ≥ 2, ni ≥ 1 for i = 1, . . . , r, ∑r i=1 ni = n and 0 = t1 < t2 < · · · < tr = 1. Criteria are offered for the existence of double and triple ‘positive’ (in some sense) solutions of the boundary value problem. Further investigation on the upper

Analysis 29, 229–258 (2009) / DOI 10.1524/anly.2009.1035 c© Oldenbourg Wissenschaftsverlag, München 2009 A Navier boundary value problem for Willmore surfaces of revolution Klaus Deckelnick, Hans-Christoph Grunau Received: January 19, 2009 Dedicated to Prof. E. Heinz on the occasion of his 85th birthday. Summary: We study a boundary value problem for Willmore surfaces of revolution, where the position and the mean curvature H = 0 are prescribed as boundary data. The latter is a natural datum when considering critical points of the Willmore functional in classes