Positive solutions of some three-point boundary value problems via fixed point index for weakly inward -proper maps
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We use the theory of fixed point index for weakly inward A-proper maps to establish the existence of positive solutions of some second-order three-point boundary value problems in which the highest-order derivative occurs nonlinearly. 1. Introduction In the present paper, we discuss the existence of positive solutions of the nonlinear threepoint boundary value problem (BVP) −u (t) = f (t,u,u ,u ),
t ∈ (0,1),
(1.1)
with the nonlocal boundary conditions (BCs) u(0) = 0,
αu(η) = u(1),
0 < η < 1, αη < 1,
(1.2)
in which the second derivative may occur nonlinearly. Positive solutions for the case f (t,u,u ,u ) = g(t)h(u) have been studied by Ma [15] and Webb [20, 21], when f (t,u,u ,u ) = h(t,u) by He and Ge [5] and also by Lan [11]. The case f (t,u,u ,u ) = g(t)h(u,u ) has been studied by Feng [4]. The results in [4, 15] are obtained by means of Krasnosel’ski˘ı’s theorem [8], the ones in [5] use Leggett and Williams’ theorem [14] and the results in [11, 20, 21] are achieved via the classical fixed point index for compact maps, see for example [1]. Lafferriere and Petryshyn [9] and Cremins [2] studied existence of positive solutions of the so-called Picard boundary value problem −u (t) = f (t,u,u ,u ),
(1.3)
u(0) = u(1) = 0,
(1.4)
with BCs
Copyright © 2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005:2 (2005) 177–184 DOI: 10.1155/FPTA.2005.177
178
Positive solutions of some BVPs
by means of fixed point index theory for A-proper maps. A key restriction in [2, 9] is that f must take positive values. Lan and Webb [13] improved the results of [2, 9] by allowing f to possibly take some negative values. Here we will exploit Lan and Webb’s theory [12] of fixed point index for weakly inward A-proper maps, to prove new results on the existence of positive solutions of the BVP (1.1)-(1.2). We mention that with very little change, this technique may be applied to a variety of BCs, (e.g., other three-point BCs [6, 7], or m-point BCs [16]), but for brevity, we refrain from discussing other cases. 2. Preliminaries Let X denote an infinite-dimensional Banach space endowed with a fixed projection scheme Γ = {Xn ,Pn }, where {Xn } is a sequence of finite-dimensional subspaces of X and Pn : X → Xn is a linear projection with Pn x → x for every x ∈ X. We recall below the concept of A-proper mapping, introduced by Petryshyn, and we refer to his book [18] for further information on projection schemes, properties, and applications of A-proper maps. Definition 2.1. Given a map T : D ⊂ X → X, T is said to be A-proper at a point y ∈ X relative to Γ if Tn := Pn T : D ∩ Xn −→ Xn
(2.1)
is continuous for each n ∈ N and if {xn j |xn j ∈ Xn j } is a bounded sequence such that Pn T xn − y −→ 0 j
as j −→ ∞,
(2.2)
there exists a subsequence {xn j(k) } of {xn j } and x ∈ X such that xn j(k) → x and T(x) = y. T is A-proper on a set K if it is A-proper at all points of K. A-proper alone means A-proper on X. In a similar way, for a fixed γ ≥ 0, T is said to be Pγ -compact at a point
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