Leggett–Williams norm-type fixed point theorems for multivalued mappings
Introduction
In 1980, Leggett and Williams proved a generalization of Krasnoselskii’s theorem on cone expansion and compression for completely continuous single valued operators [7]. Some refinements of their result can be found for example in [1], [5], [15], [16]. In this paper we establish several multivalued norm-type versions of the Leggett–Williams theorem. We also present applications of our results to the second order boundary value problem with reflection of argumentand the Fredholm integral inclusion
Section snippets
Fixed point theorems
We begin this section by reviewing some facts on cone theory in Banach spaces. A nonempty subset P, P ≠ {θ}, of a real Banach space E is called a cone if P is closed, convex and
- (i)
λx ∈ P for all x ∈ P and λ ⩾ 0,
- (ii)
x, −x ∈ P implies x = θ.
It is well-known that every cone induces a partial ordering in E as follows: for x, y ∈ E we say that x ⪯ y if and only if y − x ∈ P. We will write x ⪯̸ y if y − x ∉ P. A cone P is called solid if int P ≠ ∅. We say that P is a normal cone if there exists γ > 0 such thatThe
Boundary value problem
In this section we will study the existence of positive solutions for the following boundary value problem with reflection of the argumentwhere t ∈ [−1, 1], a, c ⩾ 0, b, d > 0 and ad + bc + ac > 0. Similar problems have been studied for example in [6], [13], [14]. The Green’s function for (8) is given bywhere ρ = 2ac + bc + ad. It is easy to show that (9) has the following properties:
Integral inclusion
In this section we will use Theorem 1 to establish results which guarantee the existence of nonnegative solution to the following Fredholm integral inclusionwith and f : [0, 1] × [0, ∞) → K([0, ∞)), where K([0, ∞)) denotes the family of nonempty, compact, convex subsets of [0, ∞). Assume that:
3. f : [0, 1] × [0, ∞) → K([0, ∞)), t → f(t, x) is measurable for every x ∈ [0, ∞) and x → f(t, x) is upper semicontinuous for a.e. t ∈ [0, 1],
4. for each a > 0 there exists ga ∈ L1[0, 1] such
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