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Positivity 12 (2008), 725732c[circlecopyrt] 2008 Birkhauser Verlag Basel/Switzerland1385-1292/040725-8, published online May 27, 2008DOI 10.1007/s11117-008-2209-8 Positivity
Abstract. Using the mixed monotone method we establish existence and uniqueness results for a singular integral equation. The theorem obtained is very general and complements previous known results.
Mathematics Subject Classication (2000). 34B15; 34B16.
Keywords. Mixed monotone operator, integral equation, singular, existence, uniqueness.
1. Introduction
In recent years integral equation have been studied extensively in the literature (see [19] and references therein). Most of the results concern single or multiple positive solutions. However very few uniqueness results are available.
Recently some results have appeared on the uniqueness of solutions for singular higher-order boundary value problems (see [10,11] and [12]). In this paper we present a uniqueness result for the singular integral equation
x(t)=
1 0
K(t, s)f(s, x(s))ds, 0 t 1, > 0. (1.1) where f(t, x) C((0, 1)(0, +), (0, +)),and f may be singular at t = 0 and/or t = 1, and also may be singular at x =0.
2. Preliminaries
Let P be a normal cone of a Banach space E,and e P with e 1, e = . Dene
The work was supported by the National Natural Science Foundation of China (No.10571021 and No.10701020) and Key Laboratory for Applied Statistics of MOE(KLAS) and Subject Foundation of Harbin University (No. HXK200714).
Existence and uniqueness of solutions for singular integral equation
Zhongwei Cao, Daqing Jiang, Chengjun Yuan and Donal ORegan
726 Z. Cao et al. Positivity
Qe = {x P |x = , there exist constants m, M > 0 such that me x Me}.The following denition and results can be found in [10,12].
Denition 2.1. Assume A : Qe Qe Qe. Now A is said to be mixed monotone if A(x, y) is nondecreasing in x and nonincreasing in y, i.e., if x1 x2(x1,x2 Qe) implies A(x1,y) A(x2,y) for any y Qe, and y1 y2(y1,y2 Qe) implies A(x, y1) A(x, y2) for any x Qe.Wesay x Qe is axedpoint of A if A(x,x)= x.
Theorem 2.1. Suppose that A: Qe Qe Qe is a mixed monotone operator and a constant , 0 < 1 such that
A(tx, 1t y) tA(x, y), x, y Qe, 0 <t< 1. (2.1)
Then Ahas auniquexedpoint x Qe. Moreover, for...