# Construction of measures of noncompactness of $C^k(\Omega)$ and $C^k_0$ and their application to functional integral-differential equations

Document Type : Research Paper

Authors

1 Department of Mathematics, Sari Branch, Islamic Azad University, Sari, Iran.

Abstract

‎In this paper‎, ‎first‎, ‎we investigate the construction of compact sets of $C^k$ and $C_0^k$‎ ‎by proving $C^k‎, ‎C_0^k-version$‎" ‎of Arzel\`{a}-Ascoli theorem‎, ‎and then introduce new measures of noncompactness on these spaces‎. ‎Finally‎, ‎as an application‎, ‎we study the existence of entire solutions for a class of the functional integral-differential equations by using Darbo's fixed point theorem associated with these new measures of noncompactness‎. ‎Further‎, ‎some examples are presented to show the efficiency of our results.

Keywords

Main Subjects

#### References

R. P. Agarwal, S. Arshad, D. O'Regan and V. Lupulescu, A Schauder fixed point theorem in semilinear spaces and applications, Fixed Point Theory Appl. 2013 (2013), no. 306, 13 pages.
R. P. Agarwal, M. Benchohra and D. Seba, On the application of measure of noncom pactness to the existence of solutions for fractional differential equations, Results Math. 55 (2009), no. 3-4, 221--230.
R. P. Agarwal, M. Meehan and D. O'Regan, Fixed Point Theory and Applications, Cambridge Univ. Press, Cambridge, 2004.
R. P. Agarwal and D. O'Regan, Fredholm and Volterra integral equations with integrable singularities, Hokkaido Math. J. 33 (2004), no. 2, 443--456.
A. Aghajani, J. Banas and Y. Jalilian, Existence of solutions for a class of nonlinear Volterra singular integralm equations, Comput. Math. Appl. 62 (2011), no. 3, 1215--1227.
A. Aghajani and Y. Jalilian, Existence and global attractivity of solutions of a nonlinear functional integral equation, Commun. Nonlinear Sci. Numer. Simul. 15 (2010), no. 11, 3306--3312.
J. Bana, Measures of noncompactness in the study of solutions of nonlinear differential and integral equations, Cent. Eur. J. Math. 10 (2012), no. 6, 2003--2011.
J. Bana and K. Goebel, Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics 60, Marcel Dekker, New York, 1980.
J. Bana and R. Rzepka, An application of a measure of noncompactness in the study of asymptotic stability, Appl. Math. Lett. 16 (2003), no. 1, 1--6.
F. Bloom, Asymptotic bounds for solutions to a system of damped integrodifferential equations of electromagnetic theory, J. Math. Anal. Appl. 73 (1980), no. 2, 524--542.
G. Darbo, Punti uniti in trasformazioni a codominio non compatto, Rend. Semin. Mat. Univ. Padova 24 (1955) 84--92.
M. A. Darwish, On monotonic solutions of a quadratic integral equation with supremum, Dynam. Systems Appl. 17 (2008), no. 3-4, 539--549.
M. A. Darwish, J. Henderson and D. O'Regan, Existence and asymptotic stability of solutions of a perturbed fracttional functional-integral equation with linear modification of the arrgument, Bull. Korean Math. Soc. 48 (2011), no. 3, 539--553.
B. C. Dhage and S. S. Bellale, Local asymptotic stability for nonlinear Quadratic functional integral equations, Electron. J. Qual. Theory Differ. Equ. 2008 (2008), no. 10, 13 pages.
C. Gonalez and A. Jimenez-Melado, Existence of monotonic asymptotically constant solutions for second order differential equations, Glasg. Math. J. 49 (2007), no. 3, 515--523.
D. Guo, Existence of solutions for nth-order integro-differential equations in Banach spaces, Comput. Math. Appl. 41 (2001) 597--606.
H. Hanche-Olsen and H. Holden: The Kolmogorov-Riesz compactness theorem, Expo. Math. 28 (2010), no. 4, 385--394.
R. P. Kanwal, Linear Integral Differential Equations: Theory and Technique, Academic Press, New York, 1971.
L. Liu, F. Guo, C. Wu and Y. Wu, Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces, J. Math. Anal. Appl. 309 (2005), no. 2, 638--649.
Z. Liu and S. M. Kang, Applications of Schauder's fixed point theorem with respect to iterated functional equations, Appl. Math. Lett. 14 (2001), no. 8, 955--962.
M. Mursaleen and S. A. Mohiuddine, Applications of noncompactness to the infinite system of differential equations in lp spaces, Nonlinear Anal. 75 (2012), no. 4, 2111--2115.
L. Olszowy, Solvability of infinite systems of singular integral equations in Frechet space of coninuous functions, Comput. Math. Appl. 59 (2010), no. 8, 2794--2801.

### History

• Receive Date: 17 December 2014
• Revise Date: 08 October 2015
• Accept Date: 13 February 2017