## On local attractivity and asymptotic stability of solutions of a quadratic Volterra integral equation.(English)Zbl 1175.45002

The authors study the integral equation
$x(t) = p(t) +f(t, x(t))\int_0^t v(t, s, x(s))ds.$
They write the following: “We will study the solvability of this equation in the space $$BC (\mathbb{R}_{+})$$ consisting of all real functions defined, continuous and bounded on the interval $$\mathbb{R}_{+} = [0; \infty).$$ More precisely, we will look for assumptions concerning the functions involved in this equation which guarantee that this equation has solution belonging to $$BC (\mathbb{R}_{+})$$ and being locally attractive or asymptotic stable on $$\mathbb{R}_{+}$$.”
As example the authors consider the equation
$x(t) =t\exp(-2t) +\frac{1}{2\pi}\arctan(\sqrt{t} +tx(t)) \int_{0}^{t} \left( \frac{2x(s)^{2/3}+x(s)}{(s+1)(t^2+1)} +\frac{1}{10(t^2+1)} \right)\,ds .$

### MSC:

 45G10 Other nonlinear integral equations 47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) 45M05 Asymptotics of solutions to integral equations 45M10 Stability theory for integral equations 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.
Full Text:

### References:

 [1] Agarwal, R.P.; O’Regan, D.; Wong, P.I.Y., Positive solutions of differential, difference and integral equations, (1999), Kluwer Academic Publishers Dordrecht [2] Agarwal, R.P.; O’Regan, D., Infinite interval problems for differential, difference and integral equations, (2001), Kluwer Academic Publishers Dordrecht · Zbl 1003.39017 [3] Banaś, J.; Goebel, K., Measures of noncompactness in Banach spaces, Lecture notes in pure and applied mathematics, vol. 60, (1980), Marcel Dekker New York · Zbl 0441.47056 [4] Banaś, J.; Martinon, A., Monotonic solutions of a quadratic integral equation of Volterra type, Comput. math. appl., 47, 271-279, (2004) · Zbl 1059.45002 [5] Banaś, J.; O’Regan, D., On existence and local attractivity of solutions of a quadratic Volterra integral equation of fractional order, J. math. anal. appl., 345, 573-582, (2008) · Zbl 1147.45003 [6] Banaś, J.; Rzepka, B., An application of a measure of noncompactness in the study of asymptotic stability, Appl. math. lett., 16, 1-6, (2003) · Zbl 1015.47034 [7] Banaś, J.; Rzepka, B., On existence and asymptotic stability of solutions of a nonlinear integral equation, J. math. anal. appl., 284, 165-173, (2003) · Zbl 1029.45003 [8] Burton, T.A.; Zhang, B., Fixed points and stability of an integral equation: nonuniqueness, Appl. math. lett., 17, 839-846, (2004) · Zbl 1066.45002 [9] Caballero, J.; Mingarelli, A.B.; Sadarangani, K., Existence of solutions of an integral equation of Chandrasekhar type in the theory of radiative transfer, Electronic J. diff. eq., 57, 1-11, (2006) · Zbl 1113.45004 [10] Corduneanu, C., Integral equations and applications, (1991), Cambridge University Press Cambridge · Zbl 0714.45002 [11] Chandrasekhar, S., Radiative transfer, (1960), Dover New York · Zbl 0037.43201 [12] Deimling, K., Nonlinear functional analysis, (1985), Springer Verlag Berlin · Zbl 0559.47040 [13] Hu, S.; Khavanin, M.; Zhuang, W., Integral equations arising in the kinetic theory of gases, Appl. anal., 34, 261-266, (1989) · Zbl 0697.45004 [14] Hu, X.; Yan, J., The global attractivity and asymptotic stability of solution of a nonlinear integral equation, J. math. anal. appl., 321, 147-156, (2006) · Zbl 1108.45006 [15] Kelly, C.T., Approximation of solutions of some quadratic integral equations in transport theory, J. integral eq., 4, 221-237, (1982) · Zbl 0495.45010 [16] O’Regan, D.; Meehan, M., Existence theory for nonlinear integral and integrodifferential equations, (1998), Kluwer Academic Publishers Dordrecht · Zbl 0932.45010 [17] Väth, M., Volterra and integral equations of vector functions, Pure and applied math, (2000), Marcel Dekker New York [18] Zabrejko, P.P.; Koshelev, A.I.; Krasnosel’skii, M.A.; Mikhlin, M.A.; Rakovschik, S.G.; Stetsenko, V.J., Integral equations, (1975), Nordhoff Leyden
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.