Generalized monotone iterative method for integral boundary value problems with causal operators

Wang, Wenli¹ ; Tian, Jingfeng²

¹ Department of Information Engineering, China University of Geosciences Great Wall College, Baoding, Hebei 071000, People's Republic of China
² College of Science and Technology, North China Electric Power University, Baoding, Hebei 071051, People's Republic of China

Journal of nonlinear sciences and its applications, Tome 8 (2015) no. 5, p. 600-609.

Résumé

This paper investigates the existence of solutions for a class of integral boundary value problems with causal operators. The arguments are based upon the developed monotone iterative method. As applications, two examples are worked out to demonstrate the main results.

Détail

DOI : 10.22436/jnsa.008.05.12

Classification : 39A10, 34A34
Keywords: Generalized monotone iterative method, integral boundary value problems, causal operators, upper and lower solutions.

Bibliographie

[1] Ahmad, B.; Sivasundaram, S. Existence of solutions for impulsive integral boundary value problems of fractional order , Nonlinear Anal. Hybrid Syst., Volume 4 (2010), pp. 134-141

[2] Amini, S.; Sloan, I. H. Collocation methods for second kind integral equations with non-compact operators, J. Integral Equations Appl., Volume 2 (1989), pp. 1-30

[3] Benchohra, M.; S. Hamani The method of upper and lower solutions and impulsive fractional differential inclusions, Nonlinear Anal. Hybrid Syst., Volume 3 (2009), pp. 433-440

[4] Bhaskar, T. G.; McRae, F. A. Monotone iterative techniques for nonlinear problems involving the difference of two monotone functions, Appl. Math. Comput., Volume 133 (2002), pp. 187-192

[5] Cabada, A.; Z. Hamdi Nonlinear fractional differential equations with integral boundary value conditions, Appl. Math. Comput., Volume 228 (2014), pp. 251-257

[6] Han, J.; Liu, Y.; J. Zhao Integral boundary value problems for first order nonlinear impulsive functional integrodifferential differential equations, Appl. Math. Comput., Volume 218 (2012), pp. 5002-5009

[7] Jankowski, T. Differential equations with integral boundary conditons, J. Comput. Appl. Math., Volume 147 (2002), pp. 1-8

[8] Lakshmikantham, V.; Leela, S.; Drici, Z.; McRae, F. A. Theory of Causal Differential Equations, Atlantis Press, Paris, 2010

[9] Liu, Z.; Han, J.; Fang, L. Integral boundary value problems for first order integro-differential equations with impulsive integral conditions, Comput. Math. Appl., Volume 61 (2011), pp. 3035-3043

[10] Nanware, J. A.; Dhaigude, D. B. Existence and uniqueness of solutions of differential equations of fractional order with integral boundary conditions, J. Nonlinear Sci. Appl., Volume 7 (2014), pp. 246-254

[11] Song, G.; Zhao, Y.; Sun, X. Integral boundary value problems for first order impulsive integro-differential equations of mixed type, J. Comput. Appl. Math., Volume 235 (2011), pp. 2928-2935

[12] Wang, T.; Xie, F. Existence and uniqueness of fractional differential equations with integral boundary conditions, J. Nonlinear Sci. Appl., Volume 1 (2008), pp. 206-212

[13] Wang, P.; Tian, S.; Y. Wu Monotone iterative method for first-order functional difference equations with nonlinear boundary value conditions, Appl. Math. Comput., Volume 203 (2008), pp. 266-2721

[14] Wang, P.; Wu, M. Oscillation of certain second order nonlinear damped difference equations with continuous variable, Appl. Math. Lett., Volume 20 (2007), pp. 637-644

[15] Wang, P.; Wu, M.; Wu, Y. Practical stability in terms of two measures for discrete hybrid systems, Nonlinear Anal. Hybrid Syst., Volume 2 (2008), pp. 58-64

[16] Wang, P.; Zhang, J. Monotone iterative technique for initial-value problems of nonlinear singular discrete systems, J. Comput. Appl. Math., Volume 221 (2008), pp. 158-164

[17] Wang, P.; Wang, W. Anti-periodic boundary value problem for first order impulsive delay difference equations, Adv. Difference Equ., Volume 2015 (2015), pp. 1-13

[18] West, I. H.; Vatsala, A. S. Generalized monotone iterative method for initial value problems, Appl. Math. Lett., Volume 17 (2004), pp. 1231-1237

[19] Yang, W. Positive solutions for nonlinear semipositone fractional q-difference system with coupled integral boundary conditions, Appl. Math. Comput., Volume 244 (2014), pp. 702-725

[20] Zhang, X.; Ge, W. Positive solutions for a class of boundary-value problems with integral boundary conditions, Comput. Math. Appl., Volume 58 (2009), pp. 203-215

[21] Zhang, X.; Feng, M.; W. Ge Existence of solutions of boundary value problems with integral boundary conditions for second-order impulsive integro-differential equations in Banach spaces, J. Comput. Appl. Math., Volume 233 (2010), pp. 1915-1926

[22] Zhao, Y.; Song, G.; Sun, X. Integral boundary value problems with causal operators, Comput. Math. Appl., Volume 59 (2010), pp. 2768-2775