Existence results and iterative method for a fully fourth-order nonlinear integral boundary value problem
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Existence results and iterative method for a fully fourth-order nonlinear integral boundary value problem Quang A Dang1
· Quang Long Dang2
Received: 24 April 2019 / Accepted: 1 November 2019 / © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract We consider the boundary value problem u (t) = f (t, u(t), u (t), u (t), u (t)), 0 < t < 1, 1 u (0) = u (0) = u (1) = 0, u(0) = g(s)u(s)ds, 0
where f : [0, 1] × R4 → R+ , a : [0, 1] → R+ are continuous functions. For f = f (u(t)), very recently in Benaicha and Haddouchi (An. Univ. Vest Timis. Ser. Mat.-Inform. 1(54): 73–86, 2016) the existence of positive solutions was studied by employing the fixed point theory on cones. In this paper, by the method of reducing the boundary value problem to an operator equation for the right-hand sides we establish the existence, uniqueness, and positivity of solution and propose an iterative method on both continuous and discrete levels for finding the solution. We also give error analysis of the discrete approximate solution. Some examples demonstrate the validity of the obtained theoretical results and the efficiency of the iterative method. Keywords Fourth-order nonlinear equation · Integral boundary condition · Existence, uniqueness, and positivity of solution · Iterative method · Total error
Dang Quang A
[email protected] Dang Quang Long [email protected] 1
Centre for Informatics and Computing, VAST, 18 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam
2
Institute of Information Technology, VAST, 18 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam
Numerical Algorithms
1 Introduction Boundary value problems for nonlinear differential equations with integral boundary conditions appear in many applied fields, such as flexibility mechanics, chemical engineering, and heat conduction. These problems have been studied extensively, among which are problems for fourth-order differential equations (see e.g. [3, 4, 16– 29]). Below we mention some of these works. In 2009, Zhang and Ge [29] considered the problem u (t) = w(t)f (t, u(t), u (t)), 0 < t < 1, 1 g(s)u(s)ds, u(1) = 0, u(0) = 0
u (0) =
1
h(s)u (s)ds, u (1) = 0,
0
where w may be singular at t = 0 and/or t = 1, f : [0, 1] × R+ × R− → R+ is continuous, and g, h ∈ L1 [0, 1] are nonnegative. Using the fixed point theorem of cone expansion and compression of norm type, the authors established the existence and nonexistence of positive solutions. In 2013, Li et al. [17] studied the fully nonlinear fourth-order boundary value problem u (t) = f (t, u(t), u (t), u (t), u (t)), t ∈ [0, 1], 1 u(0) = u (1) = u (1) = 0, u (0) = h(s, u(s), u (s), u (s))ds, 0
where f : [0, 1]×R4 → R, h : [0, 1]×R3 → R are continuous functions. Based on a fixed point theorem for a sum of two operators, one is completely continuous and the other is a nonlinear contraction, the authors established the existence of solutions and monotone positive solutions for the problem. Later, in 2015, Lv et al. [18] considered a simplified form
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