Some fixed point theorems and existence of positive solutions of two-point boundary-value problems☆
Introduction
In the study of boundary-value problems (BVPs), the existence of positive solutions has been attracting much attention of mathematicians [1], [2], [3], [4], [5], [6]. Since most results are obtained by the use of the Krasnoselskii’s compression–expansion theorem [7] in cones, the nonlinear item or that appeared in their differential equations is required to be nonnegative or at least bounded below. Among them, V. Anuhadha, D.D. Hai and R. Shivaji researched the two-point BVP under the assumptions
- (A1)
;
- (A2)
with ;
- (A3)
and there exists an such that for each ; and
- (A4)
uniformly on a compact subinterval of .
They proved the following
Theorem A Theorem 2.1 in [2] Suppose (A1)–(A4) hold. Then(1.1)has a positive solution for sufficiently small. R.P. Agarwal, Huei-Lin Hong and Cheh-Chib yeh [4]studied a similar BVP in the formwith the conditions (B1) ; (B2) for and ; (B3) and there exists a positive constant such that for every . LetThey proved
Theorem B Let – hold. Assume that there exist a function and a positive constant such thatandThen BVP(1.2)has at least one positive solution for .
Theorem C Let (B1)–(B3) hold. Assume that there exist a function and a positive constant such that(a) If ,then BVP(1.2)has at least one positive solution for . (b) If and , then BVP(1.2)has at least one positive solution for .
Ge and Ren [6] researched with the assumptions
(C1) and (C2) are the same as (B1) and (B2);
(C3) for , where The existence of positive solutions in [6] is established essentially in a bounded area where is the unique solution of with . Therefore the restriction in (A3) or (B3) is removed.
In this paper we will study the BVP (1.2) under the conditions
(H1) ;
(H2) for and . Different from [4] we give the existence of positive solutions of BVP (1.2) in the cone without the restriction that is bounded below.
To this end we establish some fixed point theorems in the cone at first.
Section snippets
Fixed point theorems
The Krasnoselskii’s compression–expansion theorem is a useful tool in proving the existence of positive solutions to BVPs. The theorem says
Theorem D Let be a Banach space and a cone. Assume that are two open bounded sets in with andis a completely continuous operator such that (i) , and or (ii) , and . Then has a fixed point in . On the other hand, the condition (i) or (ii) inTheorem Dis aKrasnoselskii, [7]
Positive solutions of BVP (1.2)
Let with and . It is well known that the existence of positive solutions to BVP (1.2) is equivalent to the existence of solutions of the abstract equation where and is given in (1.5). Obviously is a completely continuous mapping. Let , then both (2.1), (2.2) hold.
Lemma 3.1 Define by for . Then is also a completely continuous operator. For and [6], Lemma 2.2
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Sponsored by the National Natural Foundation of China (No. 10671023).