J. Numer. Math. 2015; 23 (1):1–11
Francesco A. Costabile and Anna Napoli*
A multipoint Birkho type boundaryvalueproblem
Abstract: A multipoint boundaryvalueproblem 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: Boundaryvalueproblem, Birkho interpolation.
MSC 2010: 65L10, 65D05
Received March 14, 2013; accepted January 8, 2014
The present paper is concerned with
Georgian Mathematical Journal
Volume 16 (2009), Number 3, 401–411
BOUNDARYVALUEPROBLEMS FOR FRACTIONAL
RAVI P. AGARWAL, MOUFFAK BENCHOHRA, AND SAMIRA HAMANI
Abstract. The sufficient conditions are established for the existence of so-
lutions for a class of boundaryvalueproblems for fractional differential equa-
tions involving the Caputo fractional derivative.
2000 Mathematics Subject Classification: 26A33.
Key words and phrases: Boundaryvalueproblem, 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 boundaryvalueproblems 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
PAUL W. ELOE AND JOHNNY HENDERSON
Abstract. Solutions are obtained for the boundaryvalueproblem,
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)
y(n) + f(x, y) = 0, 0
References  GUSTAFSON, G. B.-RIDENHOUR, J. R.: Lower order branching and conjugate function discontinuity , J. Differential Equations 27 (1978), 167-179.  GUSTAFSON, G. B.-RIDENHOUR, J. R.: Uniqueness intervals for multipoint boundaryvalueproblems (preprint 2008).  HARTMAN, P.: Unrestricted n-parameter families , Rend. Circ. Mat. Palermo (2) 7 (1958), 123-142.  HARTMAN, P.: Ordinary Differential Equations, John Wiley and Sons, Inc., New York, 1964.  MIKUSI ´NSKI, J.: Sur l’´equation x ( n ) + A ( t ) x = 0, Ann. Polon. Math. 1 (1955
References  BITSADZE, A. V.: On the theory of nonlocal boundaryvalueproblems, Soviet Math. Dock. 30 (1964), 8-10.  BITSADZE, A. V.-SAMARSKII, A. A.: Some elementary generalizations of linear elliptic boundaryvalueproblems, Dokl. Akad. Nauk SSSR 185 (1969), 739-740.  CHU, J.-ZHOU, Z.: Positive solutions and eigenvalues of nonlocal boundary-valueproblems, Electron. J. Differential Equations 86 (2005), 1-9.  ELOE, P. W.-KHAN, R. A.-ASIF, N.: Positive solutions for a system of singular second order nonlocal boundaryvalueproblems, J. Korean Math
GEORGIAN MATHEMATICAL JOURNAL: Vol. 6, No. 6, 1999, 567-590
MULTIPLE SOLUTIONS OF GENERALIZED MULTIPOINT
PATRICIA J. Y. WONG AND RAVI P. AGARWAL
Abstract. We consider the boundaryvalueproblem
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,
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 boundaryvalueproblem. Further investigation on the upper