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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)) 0 t v(t,s,x(s))ds·

They write the following: “We will study the solvability of this equation in the space BC( + ) consisting of all real functions defined, continuous and bounded on the interval + =[0;)· More precisely, we will look for assumptions concerning the functions involved in this equation which guarantee that this equation has solution belonging to BC( + ) and being locally attractive or asymptotic stable on + .”

As example the authors consider the equation

x(t)=texp(-2t)+1 2πarctan(t+tx(t)) 0 t 2x(s) 2/3 +x(s) (s+1)(t 2 +1)+1 10(t 2 +1)ds·

MSC:
45G10Nonsingular nonlinear integral equations
47H30Particular nonlinear operators
45M05Asymptotic theory of integral equations
45M10Stability theory of integral equations
47H09Mappings defined by “shrinking” properties
References:
[1]Agarwal, R. P.; O’regan, D.; Wong, P. I. Y.: Positive solutions of differential, difference and integral equations, (1999)
[2]Agarwal, R. P.; O’regan, D.: Infinite interval problems for differential, difference and integral equations, (2001)
[3]Banaś, J.; Goebel, K.: Measures of noncompactness in Banach spaces, Lecture notes in pure and applied mathematics 60 (1980) · 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 · doi:10.1016/S0898-1221(04)90024-7
[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 · doi:10.1016/j.jmaa.2008.04.050
[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 · doi:10.1016/S0893-9659(02)00136-2
[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 · doi:10.1016/S0022-247X(03)00300-7
[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 · doi:10.1016/j.aml.2004.06.015
[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 · doi:emis:journals/EJDE/Volumes/2006/57/abstr.html
[10]Corduneanu, C.: Integral equations and applications, (1991)
[11]Chandrasekhar, S.: Radiative transfer, (1960)
[12]Deimling, K.: Nonlinear functional analysis, (1985) · 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 · doi:10.1080/00036818908839899
[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 · doi:10.1016/j.jmaa.2005.08.010
[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)
[17]Väth, M.: Volterra and integral equations of vector functions, Pure and applied math (2000)
[18]Zabrejko, P. P.; Koshelev, A. I.; Krasnosel’skii, M. A.; Mikhlin, M. A.; Rakovschik, S. G.; Stetsenko, V. J.: Integral equations, (1975)