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Existence and asymptotic stability of solutions of a perturbed fractional functional-integral equation with linear modification of the argument. (English) Zbl 1220.45011

A certain perturbed quadratic fractional functional integral equation is studied and the existence of solutions of it is proved in the space of real functions, those defined as continuous and bounded on an unbounded interval. In the process the Riemann-Liouville fractional integral is used. Interesting to note that (those mentioned in Section 1, p. 540 of the paper) under some specific substitutions (or functional relations) the equation (mentioned above) reduces to a quadratic functional-integral equation of fractional order, to a quadratic Urysohn-Volterra integral fractional equation of fractional order, to a perturbed quadratic integral equation of fractional order, and to a quadratic integral equation of fractional order with linear modification of the argument.

Incidentally all these produce, as particular cases, results obtained (references are mentioned by authors). This justifies the character to be of more general nature of the equation [see Eq. (1), p. 539] considered by the authors. Further, in this paper, some asymptotic characterization of solutions of the equation (mentioned above as Eq. (1) of this paper) are obtained. The proof of which depends on a suitable combination of the technique of measures of non-compactness and the Schauder fixed point principle (this has a mention on p. 542 of this paper).

##### MSC:
 45M10 Stability theory of integral equations 45G05 Singular nonlinear integral equations 26A33 Fractional derivatives and integrals (real functions) 45M05 Asymptotic theory of integral equations