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On a quadratic fractional Hammerstein-Volterra integral equation with linear modification of the argument. (English) Zbl 1228.45006

The authors examine a quadratic fractional Hammerstein-Volterra integral equation with a linear modification of the argument:

x(t)=a(t)+f(t,x(t)) Γ(α) 0 t k(t,τ)u(τ,x(τ),x(λτ)) (t-τ) 1-α dτ,t[0,1],0<α,λ<1,

where a:[0,1], f:[0,1]×, k:[0,1]×[0,1] and u:[0,1]×× are functions satisfying suitable assumptions. Using a Darbo-type fixed point theorem and some techniques of the theory of measures of noncompactness, they derive the existence of a nonnegative continuous and nondecreasing solution to the above equation defined on [0,1].

MSC:
45G05Singular nonlinear integral equations
45M20Positive solutions of integral equations
47H08Measures of noncompactness and condensing mappings, K-set contractions, etc.
References:
[1]Argyros, I.: Polynomial operator equations in abstract spaces and applications, (1998)
[2]Banaś, J.; Lecko, M.; El-Sayed, W. G.: Existence theorems of some quadratic integral equations, J. math. Anal. appl. 222, 276-285 (1998) · Zbl 0913.45001 · doi:10.1006/jmaa.1998.5941
[3]Case, K. M.; Zweifel, P. F.: Linear transport theory, (1967) · Zbl 0162.58903
[4]Chandrasekher, S.: Radiative transfer, (1960)
[5]Hu, S.; Khavani, M.; Zhuang, W.: Integral equations arrising in the kinetic theory of gases, Appl. anal. 34, 261-266 (1989) · Zbl 0697.45004 · doi:10.1080/00036818908839899
[6]Kelly, C. T.: Approximation of solutions of some quadratic integral equations in transport theory, J. integral eq. 4, 221-237 (1982) · Zbl 0495.45010
[7]Caballero, J.; Rocha, J.; Sadarangani, K.: On monotonic solutions of an integral equations of Volterra type, J. comput. Appl. math. 174, 119-133 (2005) · Zbl 1063.45003 · doi:10.1016/j.cam.2004.04.003
[8]Banaś, J.; Caballero, J.; Rocha, J.; Sadarangani, K.: Monotonic solutions of a class of quadratic integral equations of Volterra type, Comput. math. Appl. 49, 943-952 (2005) · Zbl 1083.45002 · doi:10.1016/j.camwa.2003.11.001
[9]Oldham, K. B.; Spanier, J.: The fractional calculus: theory and applications of differentiation aand integration to arbitary order, (1974)
[10]Hilfer, R.: Applications of fractional calculus in physics, (2000)
[11]Podlubny, I.: Fractional differential equations, (1999)
[12]Samko, S. G.; Kilbas, A. A.; Marichev, O. I.: Fractional integrals and derivatives: theory and applications, (1993) · Zbl 0818.26003
[13]Deimling, K.: Nonlinear fuctional analysis, (1985) · Zbl 0559.47040
[14]Carr, J.; Dyson, J.: The functional differential equation y’(x)=ay(λx)+by(x), Proc. roy. Soc. Edinburgh sect. A 74, 165-174 (1974/75) · Zbl 0344.34059
[15]Dunkel, G. M.: Function differential equations: examples and problems, Lecture notes in math. 144 (1970) · Zbl 0234.34077
[16]Kulenović, M. R. S.: Oscillation of the Euler differential equation with delay, Czechoslovak math. J. 45, No. 120(1), 1-6 (1995) · Zbl 0832.34069
[17]Melvin, H.: A family of solutions of the IVP for the equation x’(t)=ax(λt), λ1, Aequationes math. 9, 273-280 (1973) · Zbl 0268.34015 · doi:10.1007/BF01832636
[18]Mureşan, V.: On a class of Volterra integral equations with deviating argument, Studia univ. Babeş-bolyai math. 44, No. 1, 47-54 (1999)
[19]Mureşan, V.: Volterra integral equations ith iterations of linear modification of the argument, Novi sad J. Math. 33, No. 2, 1-10 (2003) · Zbl 1091.45002 · doi:emis:journals/NSJOM/33_2/1.html
[20]Caballero, J.; López, B.; Sadarangani, K.: Existence of nondecreasing and continuous solutions of an integral equation with linear modification of the argument, Acta math. Sin. (Engl. Ser.) 2, No. 9, 1719-1728 (2007) · Zbl 1132.45005 · doi:10.1007/s10114-007-0956-2
[21]Benchohra, M.; Darwish, M. A.: On unique solvability of quadratic integral equations with linear modification of the argument, Miskolc math. Notes 10, No. 1, 3-10 (2009) · Zbl 1199.45013
[22]Frigon, M.; Granas, A.: Résultats de type Leray–Schauder pour des contractions sur des espaces de Fréchet, Ann. sci. Math. Québec 22, No. 2, 161-168 (1998) · Zbl 1100.47514
[23]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
[24]Darwish, M. A.: On a quadratic fractional integral equation with linear modification of the argument, Can. appl. Math. Q. 16, No. 1, 45-58 (2008) · Zbl 1179.45004 · doi:http://www.math.ualberta.ca/ami/CAMQ/table_of_content/vol_16/16_1c.htm
[25]Darwish, M. A.: Existence and asymptotic behaviour of solutions of a fractional integral equation, Appl. anal. 88, No. 2, 169-181 (2009) · Zbl 1172.45001 · doi:10.1080/00036810802713800
[26]Darwish, M. A.: On quadratic integral equation of fractional orders, J. math. Anal. appl. 311, 112-119 (2005) · Zbl 1080.45004 · doi:10.1016/j.jmaa.2005.02.012
[27]Banaś, J.; Rzepka, B.: Monotonic solutions of a quadratic integral equation of fractional order, J. math. Anal. appl. 332, 1370-1378 (2007) · Zbl 1123.45001 · doi:10.1016/j.jmaa.2006.11.008
[28]Darwish, M. A.; Ntouyas, S. K.: Monotonic solutions of a perturbed quadratic fractional integral equation, Nonlinear anal. 71, 5513-5521 (2009) · Zbl 1177.45004 · doi:10.1016/j.na.2009.04.041
[29]Argyros, I. K.: Quadratic equations and applications to Chandrasekhar’s and related equations, Bull. austral. Math. soc. 32, 275-292 (1985) · Zbl 0607.47063 · doi:10.1017/S0004972700009953
[30]Leggett, R. W.: A new approach to the H-equation of chandrasekher, SIAM J. Math. 7, 542-550 (1976) · Zbl 0331.45012 · doi:10.1137/0507044
[31]Stuart, C. A.: Existence theorems for a class of nonlinear integral equations, Math. Z. 137, 49-66 (1974) · Zbl 0289.45013 · doi:10.1007/BF01213934
[32]Banaś, J.; Goebel, K.: Measures of noncompactness in Banach spaces, Lecture notes in pure and applied mathematics 60 (1980) · Zbl 0441.47056
[33]Appell, J.; Zabrejko, P. P.: Nonlinear superposition operators, Cambridge tracts in mathematics 95 (1990) · Zbl 0701.47041
[34]Dugundji, J.; Granas, A.: Fixed point theory, Monografie mathematyczne (1982) · Zbl 0483.47038