Nonsmooth and nonlocal differential equations in lattice-ordered Banach spaces
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We derive existence results for initial and boundary value problems in lattice-ordered Banach spaces. The considered problems can be singular, functional, discontinuous, and nonlocal. Concrete examples are also solved. 1. Introduction In this paper, we apply fixed point results for mappings in partially ordered function spaces to derive existence results for initial and boundary value problems in an ordered Banach space E. Throughout this paper, we assume that E satisfies one of the following hypotheses. (A) E is a Banach lattice whose every norm-bounded and increasing sequence is strongly convergent. (B) E is a reflexive lattice-ordered Banach space whose lattice operation E x → x+ = sup{0,x} is continuous and x+ ≤ x for all x ∈ E. We note that condition (A) is equivalent with E being a weakly complete Banach lattice, see, for example, [11]. The problems that will be considered in this paper include many kinds of special types, such as, for example, the following: (1) the differential equations may be singular; (2) both the differential equations and the initial or boundary conditions may depend functionally on the unknown function; (3) both the differential equations and the initial or boundary conditions may contain discontinuous nonlinearities; (5) problems on unbounded intervals; (6) finite and infinite systems of initial and boundary value problems; (7) problems of random type. The plan of the paper is as follows. In Section 2, we provide the basic abstract fixed point result which will be used in later sections. In Section 3, we deal with first-order initial value problems, and in Sections 4 and 5, second-order initial and boundary value problems are considered. Concrete examples are solved to demonstrate the applicability of the obtained results. Copyright © 2005 Hindawi Publishing Corporation Boundary Value Problems 2005:2 (2005) 165–179 DOI: 10.1155/BVP.2005.165
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Nonsmooth and nonlocal differential equations
2. Preliminaries We will start with the following auxiliary result. Lemma 2.1. Let J = (a,b) ⊂ R be some interval. Given a function w : J → R+ , denote
P = u ∈ C(J,E) | u(t) ≤ w(t) for each t ∈ J ,
(2.1)
and assume that C(J,E) is ordered pointwise. Then the following results hold. (a) The zero function 0 is an order center of P in the sense that sup{0,v} and inf {0,v} belong to P for each v ∈ P. (b) If U is an equicontinuous subset of P, then U is relatively well-order complete in P in the sense that all well-ordered and inversely well-ordered chains of U have supremums and infimums in P. Proof. (a) In both cases (A) and (B), the mapping x → x+ is continuous in E and x+ ≤ x for each x ∈ E. Thus, for each v ∈ C(J,E), the mapping v + = sup{0,v } = t → sup{0, v(t)} belongs to C(J,E), and v+ (t) ≤ v(t) for all t ∈ J. These properties ensure that v+ = sup{0,v}, v− = sup{0, −v}, and inf {0,v} = −v− belong to P for each v ∈ P. (b) Assume next that U is an equicontinuous subset of P. If E is reflexive, then bounded and monotone sequences converge weakly in E. Consequently,
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