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Zbl 1066.45002
Burton, T.A.; Zhang, Bo
Fixed points and stability of an integral equation: nonuniqueness.
(English)
[J] Appl. Math. Lett. 17, No. 7, 839-846 (2004). ISSN 0893-9659

Summary: We consider a paper of {\it J. Banaś} and {\it B. Rzepka} [ibid. 16, No. 1, 1--6 (2003; Zbl 1015.47034)] which deals with existence and asymptotic stability of an integral equation by means of fixed-point theory and measures of noncompactness. By choosing a different fixed-point theorem, we show that the measures of noncompactness can be avoided and the existence and stability can be proved under weaker conditions. Moreover, we show that this is actually a problem about a bound on the behavior of a nonunique solution. In fact, without nonuniqueness, the theorems of stability are vacuous.
[Ulrich Kosel (Freiberg)]
MSC 2000:
*45G10 Nonsingular nonlinear integral equations
47H09 Mappings defined by "shrinking" properties
47N20 Appl. of operator theory to differential and integral equations
45M05 Asymptotic theory of integral equations
45M10 Stability theory of integral equations

Keywords: Fixed points; asymptotic stability; Integral equations; Nonuniqueness; measures of noncompactness

Citations: Zbl 1015.47034

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