Abstract
In this chapter, we present a review of results concerning measures of noncompactness in the space of real functions defined, continuous and bounded on the real half-axis and furnished with the supremum norm. We will also investigate measures of noncompactness in a more general space of functions defined and continuous on the real half-axis and tempered by a given function. Moreover, we show the applicability of those measures of noncompactness in the theory of nonlinear functional integral equations.
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References
Agarwal, R.P., O’Regan, D., Wong, P.J.Y.: Positive Solutions of Differential. Difference and Integral Equations. Kluwer Academic Publishers, Dordrecht (1999)
Akhmerov, R.R., Kamenski, M.I., Potapov, A.S., Rodkina, A.E., Sadovskii, B.N.: Measures of Noncompactness and Condensing Operators. Birkhäuser, Basel (1992)
Appell, J., Banaś, J., Merentes, N.: Measures of noncompactness in the study of asymptotically stable and ultimately nondecreasing solutions of integral equations. Zeitsch. Anal. Anwend. 29, 251–273 (2010)
Appell, J., Zabrejko, P.P.: Nonlinear Superposition Operators. Cambridge University Press, Cambridge (1990)
Arino, O., Pituk, M.: Convergence in asymptotically autonomous functional differential equations. J. Math. Anal. Appl. 237, 376–392 (1999)
Ayerbe Toledano, J.M., Domínguez Benavides, T., López Acedo, G.: Measures of Noncompactness in Metric Fixed Point Theory. Birkhäuser, Basel (1997)
Azbelev, N.V., Maksimov, V.P., Rakhmatullina, L.F.: Introduction to Theory of Functional-Differential Equations. Nauka, Moscow (1990)
Banaś, J.: Measures of noncompactness in the space of continuous tempered functions. Demonstratio Math. 14, 127–133 (1981)
Banaś, J.: Applications of Measures of Noncompactness to Various Problems. Zeszyty Naukowe Politech. Rzeszow, Rzeszów (1987)
Banaś, J.: Existence results for Volterra-Stieltjes quadratic integral equations on an unbounded interval. Math. Scand. 98, 143–160 (2006)
Banaś, J., Cabrera, I.J.: On solutions of a neutral differential equation with deviating argument. Math. Comput. Model. 44, 1080–1088 (2006)
Banaś, J., Cabrera, I.J.: On existence and asymptotic behaviour of solutions of a functional integral equation. Nonlin. Anal. 66, 2246–2254 (2007)
Banaś, J., Chlebowicz A.: On an elementary inequality and its application in the theory of integral equations. J. Math. Ineq. (to appear)
Banaś, J., Dhage, B.C.: Global asymptotic stability of solutions of a functional integral equation. Nonlin. Anal. 69, 1945–1952 (2008)
Banaś, J., Dudek, S.: The technique of measures of noncompactness in Banach algebras and its applications to integral equations. Abstract Appl. Anal. 2013(537897), 15 (2013)
Banaś, J., Goebel, K.: Measures of Noncompactness in Banach Spaces. Lecture Notes in Pure and Applied Mathematics, vol. 60. Marcel Dekker, New York (1980)
Banaś, J., Mursaleen, M.: Sequence Spaces and Measures of Noncompactness with Applications to Differential and Integral Equations. Springer, New Delhi (2014)
Banaś, J., Olszowy, L.: On solutions of a quadratic Urysohn integral equation on an unbounded interval. Dyn. Syst. Appl. 17, 255–270 (2008)
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)
Banaś, J., O’Regan, D., Sadarangani, K.: On solutions of a quadratic Hammerstein integral equation on an unbounded interval. Dyn. Syst. Appl. 18, 251–264 (2009)
Banaś, J., Rzepka, B.: An application of a measure of noncompactness in the study of asymptotic stability. Appl. Math. Lett. 16, 1–6 (2003)
Banaś, J., Rzepka, B.: On existence and asymptotic stability of solutions of a nonlinear integral equation. J. Math. Anal. Appl. 284, 165–173 (2003)
Banaś, J., Zaja̧c, T.: Solvability of a functional integral equation of fractional order in the class of functions having limits at infinity. Nonlin. Anal. 71, 5491–5500 (2009)
Benchohra, M., Graef, J.R., Hamani, S.: Existence results for boundary value problems with nonlinear fractional differential equations. Appl. Anal. 87, 851–863 (2008)
Burton, T.A.: Volterra Integral and Differential Equations. Academic Press, New York (1983)
Burton, T.A., Zhang, B.: Fixed points and stability of an integral equation: nonuniqueness. Appl. Math. Lett. 17, 839–846 (2004)
Cahlon, B., Eskin, M.: Existence theorems for an integral equation of the Chandrasekhar \(H\)-equation with perturbation. J. Math. Anal. Appl. 83, 159–171 (1981)
Chandrasekhar, S.: Radiative Transfer. Dover, New York (1960)
Cichoń, M., El-Sayed, A.M.A., Salem, H.A.H.: Existence theorem for nonlinear functional integral equations of fractional orders. Comment. Math. 41, 59–67 (2001)
Corduneanu, C.: Integral Equations and Applications. Cambridge University Press, Cambridge (1991)
Czerwik, S.: The existence of global solutions of a functional-differential equation. Colloq. Math. 36, 121–125 (1976)
Darwish, M.A.: On quadratic integral equation of fractional orders. J. Math. Anal. Appl. 311, 112–119 (2005)
Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1985)
Dhage, B.C., Ntouyas, S.K.: Existence results for nonlinear functional integral equations via a fixed point theorem of Krasnoselskii-Schaefer type. Nonlin. Stud. 9, 307–317 (2002)
Dronka, J.: Note on the Hausdorff measure of noncompactness in \(L^p\)-spaces. Bull. Pol. Acad. Sci. Math. 41, 39–41 (1993)
Dunford, N., Schwartz, J.T.: Linear Operators I. International Publications, Leyden (1963)
El-Sayed, A.M.A.: Nonlinear functional differential equations of arbitrary order. Nonlin. Anal. 33, 181–186 (1998)
El-Sayed, W.G.: Solvability of a neutral differential equation with deviated argument. J. Math. Anal. Appl. 327, 342–350 (2007)
Fichtenholz, G.M.: Differential and Integral Calculus, vol. II. PWN, Warsaw (2007)
Garg, M., Rao, A., Kalla, S.L.: Fractional generalization of temperature field problem in oil strata. Mat. Bilten 30, 71–84 (2006)
Gaul, L., Klein, P., Kemple, S.: Damping description involving fractional operators. Mech. Syst. Signal Process. 5, 81–88 (1991)
Goldens̆tein, L.S., Gohberg, I.T., Markus, A.S.: Investigations of some properties of bounded linear operators with their \(q\)-norms. Učen. Zap. Kishinevsk. Univ. 29, 29–36 (1957)
Goldens̆tein, L.S., Markus, A.S.: On a measure of noncompactness of bounded sets and linear operators. In: Studies in Algebra and Mathematical Analysis, Kishinev, 45–54 (1965)
Graef, J.R., Grammatikopoulos, M.K., Spikes, P.W.: Classification of solutions of functional differential equations of arbitrary order. Bull. Inst. Math. Acad. Sinica 9, 517–532 (1981)
Grimm, L.J.: Existence and uniqueness for nonlinear neutral-differential equations. Bull. Amer. Math. Soc. 77, 374–376 (1971)
Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)
Hu, S., Khavanin, M., Zhuang, W.: Integral equations arising in the kinetic theory of gases. Appl. Anal. 34, 261–266 (1989)
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)
Jaradat, O.K., Al-Omari, A., Momani, S.: Existence of the mild solution for fractional semilinear initial value problems. Nonlin. Anal. 69, 3153–3159 (2008)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier Science B.V, Amsterdam (2006)
Kilbas, A.A., Trujillo, J.J.: Differential equations of fractional order: methods, results, problems I. Appl. Anal. 78, 153–192 (2001)
Lakshmikantham, V., Devi, J.V.: Theory of fractional differential equations in a Banach space. Eur. J. Pure Appl. Math. 1, 38–45 (2008)
Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)
O’Regan, D., Meehan, M.: Existence Theory for Nonlinear Integral and Integrodifferential Equations. Kluwer Academic, Dordrecht (1998)
Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)
Rzepka, B.: On attractivity and asymptotic stability of solutions of a quadratic Volterra integral equation of fractional order. Topol. Meth. Nonlin. Anal. 32, 89–102 (2008)
Rzepka, B.: On local attractivity and asymptotic stability of solutions of nonlinear Volterra-Stieltjes integral equations in two variables. Zeitsch. Anal. Anwend. (to appear)
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Amsterdam (1993)
Saxena, R.K., Kalla, S.L.: On a fractional generalization of the free electron laser equation. Appl. Math. Comput. 143, 89–97 (2003)
Saxena, R.K., Mathai, A.M., Haubold, H.J.: On generalized kinetic equations. Physica A 344, 657–664 (2004)
Srivastava, H.M., Saxena, R.K.: Operators of fractional integration and their applications. Appl. Math. Comput. 118, 1–52 (2001)
Zaja̧c, T.: Solvability of fractional integral equations on an unbounded interval through the theory of Volterra-Stieltjes integral equations. Zeitsch. Anal. Anwend. 33, 65–85 (2014)
Zaja̧c, T.: On monotonic and nonnegative solutions of a nonlinear Volterra-Stieltjes integral equation. J. Funct. Spaces 2014(601824), 5 (2014)
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Banaś, J., Merentes, N., Rzepka, B. (2017). Measures of Noncompactness in the Space of Continuous and Bounded Functions Defined on the Real Half-Axis. In: Banaś, J., Jleli, M., Mursaleen, M., Samet, B., Vetro, C. (eds) Advances in Nonlinear Analysis via the Concept of Measure of Noncompactness. Springer, Singapore. https://doi.org/10.1007/978-981-10-3722-1_1
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