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

##### MSC:
 45G10 Nonsingular nonlinear integral equations 47H09 Mappings defined by “shrinking” properties 47N20 Applications of operator theory to differential and integral equations 45M05 Asymptotic theory of integral equations 45M10 Stability theory of integral equations
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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, No. 1, 1-6 (2003) · Zbl 1015.47034 [2] Smart, D. R.: Fixed point theorems. (1980) · Zbl 0427.47036 [3] Burton, T. A.: A fixed-point theorem of Krasnoselskii. Appl. math. Lett. 11, No. 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. Comparison methods and stability theory, lecture notes in mathematics, 43 (1994)