The Existence and Application of Unbounded Connected Components

Let X be a Banach space and C n a family of connected subsets of R × X. We prove the existence of unbounded components in superior limit of {C n }, denoted by limC n , which have prescribed shapes. As applications, we investigate the global behavior of the set of positive periodic solutions to nonlinear first-order differential equations with delay, which can be used for modeling physiological processes.


Introduction and the Main Results
The connectivity result on the fixed set of a 1-parameter family of maps, which goes back to Leray and Schauder [1] and was proved in its full generality by Browder [2], is a useful tool in the study of global continua of solutions on nonlinear differential equations.Costa and Gonc ¸alves [3] stated and proved a suitable version for the study of nonlinear boundary value problems at resonance.Massabò and Pejsachowicz [4] generalized the main results of [1,2] to the -parameter family of compact vector fields.The above results were established when the parameter(s) changes in a bounded set.Sun and Song [5] proved the existence of unbounded connected component of 1-parameter family of compact vector fields, where the parameter varies on whole real line.All of these results play important roles in the study of nonlinear functional analysis and nonlinear differential equations.
For clearly reading, we firstly recall Kuratowski's definitions and notations in [6].
Let M be a metric space.Let {  |  = 1, 2, . ..} be a family of subsets of M. Then the superior limit D of {  } is defined by A component of a set M means a maximal connected subset of M. Definition 1.Let  be a Banach space with the norm ‖ ⋅ ‖.Let  be a component of solutions in R × . meets (, 0) and infinity means that there existed a sequence {(  ,   )} ⊂ [ \ {(, 0)}] such that (  ,   ) → (, 0) as  → ∞.
For ,  ∈ (0, ∞), let us denote Let {C  } be a family of connected subsets of R × .The purpose of this paper is to study the existence of unbounded components in lim C  which have prescribed shapes.
More precisely, we will prove the following theorems.

Proofs of the Main Results
To prove Theorems 2 and 3, we need the following preliminary result, which is proved by Ma and An [7] Then there exists an unbounded component C in lim   and (b) Assume, on the contrary, that the conclusion is not true.Then there exists ( * ,  * ) ∈ C with  * ≤  * and ‖ * ‖ = .Hence, there exists Thus, there exists  0 ∈ N, such that, for  ≥  0 , However, this contradicts (3).Proof of Theorem 3. (a) Since   → ∞, we may assume that So, it follows from conditions (ii) and (iii) that C  meets {} ×   and infinity in Set Then Π ̸ = 0 since From Lemma 4, it follows that Π is closed in [0, ∞) × , and, furthermore, By Lemma 4, lim If (, V) = +∞ for some (, V) ∈ Π, then Theorem 3 holds.
(b) By a fully analogous argument as in the proof of Theorem 2(b) (with minor modifications), one can immediately obtain the desired results.

Application to Functional Differential Equations
In recent years, there has been considerable interest in the existence of -periodic solutions of the equation where , ℎ ∈ (R, [0, ∞)) are -periodic functions and  is a continuous -periodic function.Equation (24) has been proposed as a model for a variety of physiological processes and conditions including production of blood cells, respiration, and cardiac arrhythmias.See, for example, [8][9][10][11][12][13][14][15][16][17][18][19][20] and the references therein.Recently, Wang [18] used the fixed point index [20,21] to study the existence, multiplicity, and nonexistence of positive solutions of (24) under the following assumptions.( His results provide no any information about the global behavior of the set of positive solutions of (24).
In this section, we will use Theorems 2 and 3 to establish several results on the global behavior of the set of positive solutions of (24), and, accordingly, we get some existence and multiplicity results of positive solutions of (24).
We will work essentially in the Banach space By a positive solution of (24), we mean a pair (, ), where  > 0 and  is a solution of (24) with  > 0 on [0, ].
Let Σ ⊂ R + ×  be the closure of the set of positive solutions of (24).
We extend the function  to a continuous function f defined on R in such a way that f > 0 for all  < 0. For  > 0, we then look at arbitrary solutions  of the eigenvalue problem It was shown in [18] that (26) is equivalent to where By the positivity of Green's function   (⋅, ⋅), ℎ(⋅), and (⋅), such solutions are positive.Therefore, the closure of the set of nontrivial solutions (, ) of (24) in R + ×  is exactly Σ.
Next, we consider the spectrum of the linear eigenvalue problem and for  > 0, let Define an operator   :  →  by Lemma 8 (see [18]).
>  for all  ∈ N,   → +∞ and C  meets (  , 0) Then there exists a component C in lim C  such that (A2)   → 0 + and C  meets (  , 0) and infinity;