Oscillating global continua of positive solutions of second order Neumann problem with a set-valued term
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Oscillating global continua of positive solutions of second order Neumann problem with a setvalued term Dongming Yan Correspondence: [email protected] Department of Mathematics, Sichuan University, Chengdu 610064, China
Abstract In this note, we study the oscillating global continua of the differential inclusion of the form
−u + qu ∈ λF(·, u), u (0) = 0, u (1) = 0,
where F is a “set-valued representation” of a function with jump discontinuities along the line segment [0, 1] × {0}, and l Î [0, ∞) is a parameter. The proof of our main result relies on an approximation procedure. Mathematics Subject Classification 2000: 34B16; 34B18. Keywords: climate model, differential inclusion, eigenvalue, positive solutions
1 Introduction In recent years, nonsmooth analysis has come to play an important role in functional analysis [1], dynamical systems [2], control theory [3], optimization [4], mechanical systems [5], differential equation [6,7] etc. Since many mathematical and physical problems may be reduced to ODES or PDES with discontinuous nonlinearities, the existence of multiple solutions for differential inclusion problems has been widely investigated [8-19]. In this article, we are concerned with the following differential inclusion problem which raises from a Budyko-North type energy balance climate models:
−u (x) + q(x)u(x) ∈ λF(x, u(x)), u (0) = 0, u (1) = 0
a.e.x ∈ (0, 1),
(1:1)
(see [20-25] and the references therein). In particular, the set-valued right hand side arise from a jump discontinuity of the albedo at the ice-edge in these models. By filling such a gap, one arrives at the set-valued problem (1.1). As in [25], we are here interested in a considerably simplify version as compared to the situation from climate modeling, e.g. a one-dimensional regular Sturm-Liouville differential operator substitutes for a two-dimensional Laplace-Beltrami operator or a singular Legendre-type operator, and the jump discontinuity is transformed to u = 0 in a way, which resembles only locally the climatological problem. © 2012 Yan; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Yan Boundary Value Problems 2012, 2012:47 http://www.boundaryvalueproblems.com/content/2012/1/47
We are concerned with the set-valued problem (1.1) under the following assumptions (H1) qÎC([0, 1],(0,+∞)); (H2) f+Î C ([0, 1] × [0,+∞), (0,+∞)), f + (x, s) = b(x) ∈ C([0, 1], (0, ∞)) . s→+∞ x∈[0,1] s Let the set-valued function F in (1.1) is given by + {f (x, y)}, x ∈ [0, 1], y > 0, F(x, y) = [0, f + (x, 0)], x ∈ [0, 1]. inf f + (x, 0) > 0,
lim
Notice that if f+(x, 0) ≡ 0, x Î [0, 1], then the differential inclusion problem (1.1) reduces to the BVP of differential equation −u (x) + q(x)u(x) = λf + (x, u(x)), x ∈ (0, 1), (1:2) u (0) = 0, u (1) = 0. In the last
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