# American Institute of Mathematical Sciences

December  2015, 8(6): 1055-1064. doi: 10.3934/dcdss.2015.8.1055

## Reproducing kernel functions for difference equations

 1 Siirt University, Art and Science Faculty, Department of Mathematics, TR-56100 Siirt, Turkey 2 Fırat University, Science Faculty, Department of Mathematics, 23119 Elazıǧ, Turkey, Turkey

Received  May 2015 Revised  September 2015 Published  December 2015

In this work, some new reproducing kernel functions for difference equations are found. Reproducing kernel functions have not been obtained for difference equations until now. We need these functions to solve difference equations with the reproducing kernel method. Therefore, these functions are very considerable functions.
Citation: Ali Akgül, Mustafa Inc, Esra Karatas. Reproducing kernel functions for difference equations. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1055-1064. doi: 10.3934/dcdss.2015.8.1055
##### References:
 [1] S. Abbasbandy, B. Azarnavid and M. S. Alhuthali, A shooting reproducing kernel Hilbert space method for multiple solutions of nonlinear boundary value problems, J. Comput. Appl. Math., 279 (2015), 293-305. doi: 10.1016/j.cam.2014.11.014. [2] M. Adivar, H. C. Koyuncuoǧlu and Y. N. Raffoul, Periodic and asymptotically periodic solutions of systems of nonlinear difference equations with infinite delay, J. Difference Equ. Appl., 19 (2013), 1927-1939. doi: 10.1080/10236198.2013.791688. [3] R. P. Agarwal, Difference Equations and Inequalities, $2^{nd}$ edition, Monographs and Textbooks in Pure and Applied Mathematics, 228, Marcel Dekker Inc., New York, 2000. [4] A. Akgül, A new method for approximate solutions of fractional order boundary value problems, Neural Parallel Sci. Comput., 22 (2014), 223-237. [5] A. Akgül, M. Inc, E. Karatas and D. Baleanu, Numerical solutions of fractional differential equations of Lane-Emden type by an accurate technique, Adv. Difference Equ., 2015 (2015), 12pp. doi: 10.1186/s13662-015-0558-8. [6] S. S. Cheng and W. T. Patula, An existence theorem for a nonlinear difference equation, Nonlinear Anal., 20 (1993), 193-203. doi: 10.1016/0362-546X(93)90157-N. [7] M. Cui and Z. X. Deng, On the best operator of interpolation in $W_2^1$, Math. Numer. Sinica, 8 (1986), 209-216. [8] M. Cui and H. Du, Representation of exact solution for the nonlinear Volterra-Fredholm integral equations, Appl. Math. Comput., 182 (2006), 1795-1802. doi: 10.1016/j.amc.2006.06.016. [9] M. Cui and Y. Lin, Nonlinear Numerical Analysis in the Reproducing Kernel Space, Nova Science Publishers Inc., New York, 2009. [10] S. N. Elaydi, An Introduction to Difference Equations, Second edition, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4757-3110-1. [11] R. A. C. Ferreira and D. F. M. Torres, Fractional h-difference equations arising from the calculus of variations, Appl. Anal. Discrete Math., 5 (2011), 110-121. doi: 10.2298/AADM110131002F. [12] F. Z. Geng, A numerical algorithm for nonlinear multi-point boundary value problems, J. Comput. Appl. Math., 236 (2012), 1789-1794. doi: 10.1016/j.cam.2011.10.010. [13] F. Geng, A new reproducing kernel Hilbert space method for solving nonlinear fourth-order boundary value problems, Appl. Math. Comput., 213 (2009), 163-169. doi: 10.1016/j.amc.2009.02.053. [14] F. Geng and M. Cui, Solving a nonlinear system of second order boundary value problems, J. Math. Anal. Appl., 327 (2007), 1167-1181. doi: 10.1016/j.jmaa.2006.05.011. [15] F. Geng and M. Cui, New method based on the HPM and RKHSM for solving forced Duffing equations with integral boundary conditions, J. Math. Anal. Appl., 233 (2009), 165-172. doi: 10.1016/j.cam.2009.07.007. [16] P. Hartman and A. Wintner, On linear difference equations of second order, Amer. J. Math., 72 (1950), 124-128. doi: 10.2307/2372138. [17] M. Inc and A. Akgül, Approximate solutions for MHD squeezing fluid flow by a novel method, Bound. Value Probl., 2014 (2014), 17pp. doi: 10.1186/1687-2770-2014-18. [18] M. Inc, A. Akgül and A. Kilicman, Numerical solutions of the secondorder one-dimensional telegraph equation based on reproducing kernel Hilbert space method, Abstr. Appl. Anal., (2013), Art. ID 768963, 13pp. [19] S. Javadi, E. Babolian and E. Moradi, New implementation of reproducing kernel Hilbert space method for solving a class of functional integral equations, Commun. Numer. Anal., (2014), Art. ID 00205, 7pp. [20] W. G. Kelley and A. C. Peterson, Difference Equations. An Introduction with Applications, Second edition, Harcourt/Academic Press, San Diego, CA, 2001. [21] R. Ketabchi, R. Mokhtari and E. Babolian, Some error estimates for solving Volterra integral equations by using the reproducing kernel method, J. Comput. Appl. Math., 273 (2015), 245-250. doi: 10.1016/j.cam.2014.06.016. [22] M. Mohammadi and R. Mokhtari, Solving the generalized regularized long wave equation on the basis of a reproducing kernel space, J. Comput. Appl. Math., 235 (2011), 4003-4014. doi: 10.1016/j.cam.2011.02.012. [23] M. N. Phong, A note on a system of two nonlinear difference equations, Electron. J. Math. Anal. Appl., 3 (2015), 170-179. [24] J. Popenda and E. Schmeidel, Some properties of solutions of difference equations, Fasc. Math., 13 (1981), 89-98. [25] R. Salehi and M. Dehghan, A generalized moving least square reproducing kernel method, J. Comput. Appl. Math., 249 (2013), 120-132. doi: 10.1016/j.cam.2013.02.005. [26] K. Seddighi, Reproducing kernel Hilbert spaces, Iranian J. Sci. Tech., 17 (1993), 171-177. [27] S. S. Shishvan, A. Noorzad and A. Ansari, A time integration algorithm for linear transient analysis based on the reproducing kernel method, Comput. Methods Appl. Mech. Engrg., 198 (2009), 3361-3377. doi: 10.1016/j.cma.2009.06.011. [28] Q. Zhang and W. Zhang, On a system of two high-order nonlinear difference equations, Adv. Math. Phys., (2014), Art. ID 729273, 8pp. doi: 10.1155/2014/729273.

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##### References:
 [1] S. Abbasbandy, B. Azarnavid and M. S. Alhuthali, A shooting reproducing kernel Hilbert space method for multiple solutions of nonlinear boundary value problems, J. Comput. Appl. Math., 279 (2015), 293-305. doi: 10.1016/j.cam.2014.11.014. [2] M. Adivar, H. C. Koyuncuoǧlu and Y. N. Raffoul, Periodic and asymptotically periodic solutions of systems of nonlinear difference equations with infinite delay, J. Difference Equ. Appl., 19 (2013), 1927-1939. doi: 10.1080/10236198.2013.791688. [3] R. P. Agarwal, Difference Equations and Inequalities, $2^{nd}$ edition, Monographs and Textbooks in Pure and Applied Mathematics, 228, Marcel Dekker Inc., New York, 2000. [4] A. Akgül, A new method for approximate solutions of fractional order boundary value problems, Neural Parallel Sci. Comput., 22 (2014), 223-237. [5] A. Akgül, M. Inc, E. Karatas and D. Baleanu, Numerical solutions of fractional differential equations of Lane-Emden type by an accurate technique, Adv. Difference Equ., 2015 (2015), 12pp. doi: 10.1186/s13662-015-0558-8. [6] S. S. Cheng and W. T. Patula, An existence theorem for a nonlinear difference equation, Nonlinear Anal., 20 (1993), 193-203. doi: 10.1016/0362-546X(93)90157-N. [7] M. Cui and Z. X. Deng, On the best operator of interpolation in $W_2^1$, Math. Numer. Sinica, 8 (1986), 209-216. [8] M. Cui and H. Du, Representation of exact solution for the nonlinear Volterra-Fredholm integral equations, Appl. Math. Comput., 182 (2006), 1795-1802. doi: 10.1016/j.amc.2006.06.016. [9] M. Cui and Y. Lin, Nonlinear Numerical Analysis in the Reproducing Kernel Space, Nova Science Publishers Inc., New York, 2009. [10] S. N. Elaydi, An Introduction to Difference Equations, Second edition, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4757-3110-1. [11] R. A. C. Ferreira and D. F. M. Torres, Fractional h-difference equations arising from the calculus of variations, Appl. Anal. Discrete Math., 5 (2011), 110-121. doi: 10.2298/AADM110131002F. [12] F. Z. Geng, A numerical algorithm for nonlinear multi-point boundary value problems, J. Comput. Appl. Math., 236 (2012), 1789-1794. doi: 10.1016/j.cam.2011.10.010. [13] F. Geng, A new reproducing kernel Hilbert space method for solving nonlinear fourth-order boundary value problems, Appl. Math. Comput., 213 (2009), 163-169. doi: 10.1016/j.amc.2009.02.053. [14] F. Geng and M. Cui, Solving a nonlinear system of second order boundary value problems, J. Math. Anal. Appl., 327 (2007), 1167-1181. doi: 10.1016/j.jmaa.2006.05.011. [15] F. Geng and M. Cui, New method based on the HPM and RKHSM for solving forced Duffing equations with integral boundary conditions, J. Math. Anal. Appl., 233 (2009), 165-172. doi: 10.1016/j.cam.2009.07.007. [16] P. Hartman and A. Wintner, On linear difference equations of second order, Amer. J. Math., 72 (1950), 124-128. doi: 10.2307/2372138. [17] M. Inc and A. Akgül, Approximate solutions for MHD squeezing fluid flow by a novel method, Bound. Value Probl., 2014 (2014), 17pp. doi: 10.1186/1687-2770-2014-18. [18] M. Inc, A. Akgül and A. Kilicman, Numerical solutions of the secondorder one-dimensional telegraph equation based on reproducing kernel Hilbert space method, Abstr. Appl. Anal., (2013), Art. ID 768963, 13pp. [19] S. Javadi, E. Babolian and E. Moradi, New implementation of reproducing kernel Hilbert space method for solving a class of functional integral equations, Commun. Numer. Anal., (2014), Art. ID 00205, 7pp. [20] W. G. Kelley and A. C. Peterson, Difference Equations. An Introduction with Applications, Second edition, Harcourt/Academic Press, San Diego, CA, 2001. [21] R. Ketabchi, R. Mokhtari and E. Babolian, Some error estimates for solving Volterra integral equations by using the reproducing kernel method, J. Comput. Appl. Math., 273 (2015), 245-250. doi: 10.1016/j.cam.2014.06.016. [22] M. Mohammadi and R. Mokhtari, Solving the generalized regularized long wave equation on the basis of a reproducing kernel space, J. Comput. Appl. Math., 235 (2011), 4003-4014. doi: 10.1016/j.cam.2011.02.012. [23] M. N. Phong, A note on a system of two nonlinear difference equations, Electron. J. Math. Anal. Appl., 3 (2015), 170-179. [24] J. Popenda and E. Schmeidel, Some properties of solutions of difference equations, Fasc. Math., 13 (1981), 89-98. [25] R. Salehi and M. Dehghan, A generalized moving least square reproducing kernel method, J. Comput. Appl. Math., 249 (2013), 120-132. doi: 10.1016/j.cam.2013.02.005. [26] K. Seddighi, Reproducing kernel Hilbert spaces, Iranian J. Sci. Tech., 17 (1993), 171-177. [27] S. S. Shishvan, A. Noorzad and A. Ansari, A time integration algorithm for linear transient analysis based on the reproducing kernel method, Comput. Methods Appl. Mech. Engrg., 198 (2009), 3361-3377. doi: 10.1016/j.cma.2009.06.011. [28] Q. Zhang and W. Zhang, On a system of two high-order nonlinear difference equations, Adv. Math. Phys., (2014), Art. ID 729273, 8pp. doi: 10.1155/2014/729273.
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