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Fixed points and stability of an integral equation: nonuniqueness. (English) Zbl 1066.45002

Summary: We consider a paper of J. Banaś and 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.

MSC:

45G10 Other nonlinear integral equations
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47N20 Applications of operator theory to differential and integral equations
45M05 Asymptotics of solutions to integral equations
45M10 Stability theory for integral equations

Citations:

Zbl 1015.47034
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References:

[1] Banaś, J.; Rzepka, R., An application of a measure of noncompactness in the study of asymptotic stability, Appl. math. lett., 16, 1, 1-6, (2003) · Zbl 1015.47034
[2] Smart, D.R., Fixed point theorems, (1980), Cambridge University Press New York · Zbl 0427.47036
[3] Burton, T.A., A fixed-point theorem of Krasnoselskii, Appl. math. lett., 11, 1, 85-88, (1998) · Zbl 1127.47318
[4] Burton, T.A.; Furumochi, T., A note on stability by Schauder’s theorem, Funkcialaj ekvacioj, 44, 73-82, (2001) · Zbl 1158.34329
[5] Burton, T.A., Differential inequalities for integral and delay equations, (), 43
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