Monotone iterative technique and positive solutions of lidstone boundary value problems. (English) Zbl 1049.34028

Consider the boundary value problem \[ (-1)^n w^{(2n)} (t) = f(t,w(t)), \quad 0 \leq t \leq 1,\quad w^{(2i)} (0) = w^{(2i)} (1) = 0, \quad 0 \leq i \leq n-1, \tag{\(*\)} \] under the conditions (i) \(f:[0,1] \times [0,+\infty) \rightarrow [0,+\infty)\) is continuous. (ii) \(f(t,.)\) is nondecreasing for any \(t \in [0,1].\) (iii) \(f(t,w)=f(1-t,w)\) for all \(t\in [0,1], w\in [0,\infty).\) A solution of \((*)\) is said to be symmetric, if \(w(t)=w(1-t)\). By using the monotone iterative technique, the author proves the existence of \(N\) symmetric positive solutions of \((*)\).


34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34A45 Theoretical approximation of solutions to ordinary differential equations
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[1] Agarwal, R.P., Boundary value problems for higher order differential equations, (1986), World Scientific Singapore · Zbl 0598.65062
[2] Agarwal, R.P.; Wong, P.J.Y., On lidstone polynomials and boundary value problems, Comput. math. appl., 17, 1397-1421, (1989) · Zbl 0682.65049
[3] Twizell, E.H.; Boutayeb, A.; Djidjli, K., Numerical methods for eighth-, tenth-, and twelveth-order eigenvalue problems arising in thermal instability, Adv. comput. math., 2, 407-436, (1994) · Zbl 0847.76057
[4] Agarwal, R.P., Focal boundary value problems for differential and difference equations, (1998), Kluwer Academic Publishers Dordrecht · Zbl 0914.34001
[5] Wong, P.J.Y.; Agarwal, R.P., Eigenvalues of lidstone boundary value problems, Appl. math. comput., 104, 15-31, (1999) · Zbl 0933.65089
[6] Henderson, J.; Thompson, H.B., Multiple symmetric positive solutions for a second order boundary value problem, Proc. am. math. soc., 128, 2373-2379, (2000) · Zbl 0949.34016
[7] Yao, Q.; Bai, Z., Existence of positive solutions of BVP for u(4)(t)−λh(t)f(u(t))=0, Chinese annal. math., 20, A, 575-578, (1999), (in Chinese) · Zbl 0948.34502
[8] Ladde, G.S.; Lakshmikantham, V.; Vatsala, A.S., Monotone iterative techniques for nonlinear differential equations, (1985), Pitman Advanced Publishing Program Boston, MA · Zbl 0658.35003
[9] Guo, D.; Lakshmikantham, V., Nonlinear problems in abstract cones, (1988), Academic Press New York · Zbl 0661.47045
[10] Zeidler, E., Nonlinear functional analysis and its applications II/B: nonlinear monotone operators, (1992), Springer-Verlag New York
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