The upper and lower solution method for some fourth-order boundary value problems

• Published in: Nonlinear analysis Vol. 67; no. 6; pp. 1704 - 1709
• Main Author: Bai, Zhanbing
• Format: Journal Article
• Language: English
• Published: Oxford Elsevier Ltd 15.09.2007; Elsevier
• Subjects: Exact sciences and technology; Fourth-order boundary value problem; Integral equations; Integral–differential equation; Lower and upper solutions; Mathematical analysis; Mathematics; Nagumo-type condition; Numerical analysis; Numerical analysis. Scientific computation; Ordinary differential equations; Partial differential equations, boundary value problems; Sciences and techniques of general use; Fourth-order boundary value problem; Lower and upper solutions; 34B15; Integral–differential equation; Nagumo-type condition; Boundary value problem; Differential equation; Nonlinear analysis; Integral equation; Fourth-order boundary value problem; Lower and upper solutions; Nagumo-type condition; Integral-differential equation; Mathematical model; Multidisciplinary
• ISSN: 0362-546X; 1873-5215
• DOI: 10.1016/j.na.2006.08.009

In this paper, we are concerned with the fourth-order two-point boundary value problem u ( i v ) ( t ) = f ( t , u ( t ) , u ′ ( t ) , u ″ ( t ) , u ‴ ( t ) ) , 0 < t < 1 , u ( 0 ) = u ′ ( 1 ) = u ″ ( 0 ) = u ‴ ( 1 ) = 0 . By placing certain restrictions on the nonlinear term f , we obtain the existence results for the fourth-order two-point boundary value problem via the lower and upper solution method. In particular, a new truncating technique and an appropriate Nagumo-type condition are introduced and employed.