Solvability of Functional Integral-Differential Equations in the Sobolev space w^{k,infinity}(R^n)
Subject Areas : StatisticsMasoome Hosseini Farahi 1 , Mahmoud Hassani 2 , Reza Allahyari 3
1 - Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran.
2 - Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran.
3 - Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran.
Keywords: شرایط کاراتئودوری, اندازه های نافشردگی, قضیه نقطه ثابت داربو, معادلات انتگرال- دیفرانسیل, فضاهای سوبولوف,
Abstract :
In 1930, Kuratowski introduced the concept of measure of noncompactness. Later, Banas and Goebel generalized this concept axiomatically, which is more convenient in applications. The principal application of measures of noncompactness in fixed point theory is contained in the Darbo'sfixed point theorem. This is a tool to investigate the existence and behaviour of solutions of manyclasses of integral equations such as Volterra, Fredholm and Uryson types.The technique of measure of noncompactness is applicable in several branches of nonlinear analysis. In particular, it is a very useful tool for several types of integral and integral-differential equations. In addition, the measure of noncompactness is also used in functional equations, fractional partial differential equations, ordinary and partial differential equations, operator theory and optimal control theory. The purpose of this article is to introduce a new measure of noncompactness in the Sobolev space W^(k,∞) (R^n). The results are obtained to solve integral-differential equations. Finally, by providing an example to show the efficiency of our results.
[1] Bloom, F., 1980. Asymptotic bounds for solutions to a system of damped integro-differential equations of electromagnetic theory. J. Math. Anal.
Appl., 73(2), 524–542.
[2] Guo, D., 2001. Existence of Solutions for nth-Order Integro-Differential Equations in Banach Spaces. Computers and Mathematics with Applications, 4, 597–606.
[3] Forbes, L. K., Crozier, S.D., Doddrell,
M., 1997. Calculating current densities and fields produced by shielded magnetic resonance imaging probes. SIAM J. Appl. Math., 57(2), 401–425.
[4] Holmaker, K., 1993. Global asymptotic stability for a stationary solution of a system of integro-differential equations describing the formation of liver zones. SIAM J. Math. Anal., 24(1), 116–128.
[5] Behiry, S. H., Hashish, H., 2002. Wavelet methods for the numerical solution of Fredholm integro-differential equations. Int. J. Appl. Math., 11(1),
27–35.
[6] Bica, A. M., Caus, V. A., Muresan, S., 2006. Application of a trapezoid tnequality to neutral Fredholm integro-differential equations in Banach spaces. J. Inequal Pure and Appl. Math.,7, Art. 173.
[7] Hosseini, S. M., Shahmorad, S., 2003. Tau numerical solution of Fredholm integro-differential equations with arbitrary polynomial base. Appl. Math
Modelling, 27(2), 145–154.
[8] Kuratowski, K., 1930. Sur les espaces, Fund. Math., 15, 301–309.
[9] Bana's, J., & Goebel, K. (1980). Measures of Noncompactness in Banach Spaces, Lect. Notes Pure Appl. Math., 60, Dekker, New York.
[10] Arab, R., Allahyari, R., Shole Haghighi, A., 2014. Existence of solutions of infinite systems of integral equations in two variables via measure of noncompactness. Applied Mathematics and Computation, 246, 283–291.
[11] Bana's, J., 2012. Measures of noncompactness in the study of solutions of nonlinear differential and integral equations. Cent. Eur. J. Math., 10(6), 2003–2011.
[12] Bana's, J., O'Regan, D., Sadarangani, K., 2009. On solutions of a quadratic hammerstein integral equation on an unbonded interval. Dynam. Systems Appl., 18, 251–264.
[13] Olszowy, L., 2014. A Family of Measures of Noncompactness in the Space and its Application to Some Nonlinear Volterra Integral Equation. Mediterr. J. Math., 11, 687–701.
[14] Agarwal, R. P., Benchohra, M., Seba, D., 2009. On the application of measure of noncompactness to the existence of solutions for fractional differential equations. esults Math., 55, 221–230.
[15] Aghajani, J. A., Shole Haghighi, A., 2014. Existence of solutions for a class of functional integral equations of Volterra type in two variables via measure of noncompactness. Iran. J. Sci. Technol. Trans. A: Sci., 38 , A1: 1–8.
[16] Aghajani, A., Mursaleen, M., Shole Haghighi, A., 2015. Fixed point theorems for Meir-Keeler condensing operators via measure of noncompactness. Acta Mathematica Scientia, 35B (3), 552–566.
[17] Allahyari, R., Arab, R., Shole Haghighi, A., 2016. Construction of a Measure of Noncompactness on and its Application to Volterra Integral Equations. Mediterr. J. Math., 13(3), 1197–1210.
[18] Arab, R., 2016. The existence of fixed points via the measure of non-compactness and its application to functional integral equations. Mediterr. J. Math., 13(2), 759–773.
[19] Darbo, G., 1955. Punti uniti in transformazioni a condomino non compatto. Rend. Sem. Mat.Uni. Padova, 24, 84–92.
[20] Agarwal, R., Meehan, M., O'Regan, D., (2004). Fixed point theory and applications, Cambridge University Press.
[21] Aghajani, A., Bana's, J., Sabzali, N., 2013. Some generalizations of Darbo fixed point theorem and applications Bull. Belg. Math. Soc. Simon Stevin, 20, 2: 345–358.
[22] Aghajani, A., Bana's, J., Jalilian, Y., 2011. Existence of solutions for a class of nonlinear Volterra singular integralm equations. Comput. Math. Appl., 62,
1215 –1227.
[23] Aghajani, A., Sabzali, N., 2014. Existence of coupled fixed points via measure of noncompactness and applications. J. Nonlinear Convex A,
941 –952.
[24] Aghajani, A., Allahyari, R., Mursaleen, M., 2014. A generalization of Darbo's theorem with application to the solvability of systems of integral equations. Journal of Computational and Applied Mathematics, 260, 68–77.
[25] Arab, R., 2015. Some fixed point theorems in generalized Darbo fixed point theorem and the existence of solutions for system of integral equations. J. Korean Math. Soc., 52(1) , 125–139.
[26] Ayad, A., 1999. Spline approximation for first order Fredholm delay integro-differential equations. Int. J. Comput. Math.,70(3), 467–476.
[27] Olszowy, L., 2010. Solvability of infinite systems of singular integral equations in Frechet space of coninuous functions. Comp. Math. Appl., 59, 2794–2801.
[28] Olszowy, L., 2012. Fixed point theorems in the Frechet space and functional integral equations on an unbounded interval, Appl. Math Comput. 218, 9066–9074.
[29] Brezis, H., (2011). Functional analysis, Sobolev spaces and partial differential equations, Springer Science, Businnes Media, LLC.
[30] Runst, T., Sickel, W., (1996). Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, deGruyter, Berlin.
[31] Hanche-Olsen, H., Holden, H., 2010. The Kolmogorov-Riesz compactness theorem. Expo. Math., 28, 385-394.