# 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 & 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. doi: 10.1016/j.cam.2014.11.014. Google Scholar [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. doi: 10.1080/10236198.2013.791688. Google Scholar [3] R. P. Agarwal, Difference Equations and Inequalities,, $2^{nd}$ edition, (2000). Google Scholar [4] A. Akgül, A new method for approximate solutions of fractional order boundary value problems,, Neural Parallel Sci. Comput., 22 (2014), 223. Google Scholar [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). doi: 10.1186/s13662-015-0558-8. Google Scholar [6] S. S. Cheng and W. T. Patula, An existence theorem for a nonlinear difference equation,, Nonlinear Anal., 20 (1993), 193. doi: 10.1016/0362-546X(93)90157-N. Google Scholar [7] M. Cui and Z. X. Deng, On the best operator of interpolation in $W_2^1$,, Math. Numer. Sinica, 8 (1986), 209. Google Scholar [8] M. Cui and H. Du, Representation of exact solution for the nonlinear Volterra-Fredholm integral equations,, Appl. Math. Comput., 182 (2006), 1795. doi: 10.1016/j.amc.2006.06.016. Google Scholar [9] M. Cui and Y. Lin, Nonlinear Numerical Analysis in the Reproducing Kernel Space,, Nova Science Publishers Inc., (2009). Google Scholar [10] S. N. Elaydi, An Introduction to Difference Equations,, Second edition, (1999). doi: 10.1007/978-1-4757-3110-1. Google Scholar [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. doi: 10.2298/AADM110131002F. Google Scholar [12] F. Z. Geng, A numerical algorithm for nonlinear multi-point boundary value problems,, J. Comput. Appl. Math., 236 (2012), 1789. doi: 10.1016/j.cam.2011.10.010. Google Scholar [13] F. Geng, A new reproducing kernel Hilbert space method for solving nonlinear fourth-order boundary value problems,, Appl. Math. Comput., 213 (2009), 163. doi: 10.1016/j.amc.2009.02.053. Google Scholar [14] F. Geng and M. Cui, Solving a nonlinear system of second order boundary value problems,, J. Math. Anal. Appl., 327 (2007), 1167. doi: 10.1016/j.jmaa.2006.05.011. Google Scholar [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. doi: 10.1016/j.cam.2009.07.007. Google Scholar [16] P. Hartman and A. Wintner, On linear difference equations of second order,, Amer. J. Math., 72 (1950), 124. doi: 10.2307/2372138. Google Scholar [17] M. Inc and A. Akgül, Approximate solutions for MHD squeezing fluid flow by a novel method,, Bound. Value Probl., 2014 (2014). doi: 10.1186/1687-2770-2014-18. Google Scholar [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). Google Scholar [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). Google Scholar [20] W. G. Kelley and A. C. Peterson, Difference Equations. An Introduction with Applications,, Second edition, (2001). Google Scholar [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. doi: 10.1016/j.cam.2014.06.016. Google Scholar [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. doi: 10.1016/j.cam.2011.02.012. Google Scholar [23] M. N. Phong, A note on a system of two nonlinear difference equations,, Electron. J. Math. Anal. Appl., 3 (2015), 170. Google Scholar [24] J. Popenda and E. Schmeidel, Some properties of solutions of difference equations,, Fasc. Math., 13 (1981), 89. Google Scholar [25] R. Salehi and M. Dehghan, A generalized moving least square reproducing kernel method,, J. Comput. Appl. Math., 249 (2013), 120. doi: 10.1016/j.cam.2013.02.005. Google Scholar [26] K. Seddighi, Reproducing kernel Hilbert spaces,, Iranian J. Sci. Tech., 17 (1993), 171. Google Scholar [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. doi: 10.1016/j.cma.2009.06.011. Google Scholar [28] Q. Zhang and W. Zhang, On a system of two high-order nonlinear difference equations,, Adv. Math. Phys., (2014). doi: 10.1155/2014/729273. Google Scholar

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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. doi: 10.1016/j.cam.2014.11.014. Google Scholar [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. doi: 10.1080/10236198.2013.791688. Google Scholar [3] R. P. Agarwal, Difference Equations and Inequalities,, $2^{nd}$ edition, (2000). Google Scholar [4] A. Akgül, A new method for approximate solutions of fractional order boundary value problems,, Neural Parallel Sci. Comput., 22 (2014), 223. Google Scholar [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). doi: 10.1186/s13662-015-0558-8. Google Scholar [6] S. S. Cheng and W. T. Patula, An existence theorem for a nonlinear difference equation,, Nonlinear Anal., 20 (1993), 193. doi: 10.1016/0362-546X(93)90157-N. Google Scholar [7] M. Cui and Z. X. Deng, On the best operator of interpolation in $W_2^1$,, Math. Numer. Sinica, 8 (1986), 209. Google Scholar [8] M. Cui and H. Du, Representation of exact solution for the nonlinear Volterra-Fredholm integral equations,, Appl. Math. Comput., 182 (2006), 1795. doi: 10.1016/j.amc.2006.06.016. Google Scholar [9] M. Cui and Y. Lin, Nonlinear Numerical Analysis in the Reproducing Kernel Space,, Nova Science Publishers Inc., (2009). Google Scholar [10] S. N. Elaydi, An Introduction to Difference Equations,, Second edition, (1999). doi: 10.1007/978-1-4757-3110-1. Google Scholar [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. doi: 10.2298/AADM110131002F. Google Scholar [12] F. Z. Geng, A numerical algorithm for nonlinear multi-point boundary value problems,, J. Comput. Appl. Math., 236 (2012), 1789. doi: 10.1016/j.cam.2011.10.010. Google Scholar [13] F. Geng, A new reproducing kernel Hilbert space method for solving nonlinear fourth-order boundary value problems,, Appl. Math. Comput., 213 (2009), 163. doi: 10.1016/j.amc.2009.02.053. Google Scholar [14] F. Geng and M. Cui, Solving a nonlinear system of second order boundary value problems,, J. Math. Anal. Appl., 327 (2007), 1167. doi: 10.1016/j.jmaa.2006.05.011. Google Scholar [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. doi: 10.1016/j.cam.2009.07.007. Google Scholar [16] P. Hartman and A. Wintner, On linear difference equations of second order,, Amer. J. Math., 72 (1950), 124. doi: 10.2307/2372138. Google Scholar [17] M. Inc and A. Akgül, Approximate solutions for MHD squeezing fluid flow by a novel method,, Bound. Value Probl., 2014 (2014). doi: 10.1186/1687-2770-2014-18. Google Scholar [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). Google Scholar [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). Google Scholar [20] W. G. Kelley and A. C. Peterson, Difference Equations. An Introduction with Applications,, Second edition, (2001). Google Scholar [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. doi: 10.1016/j.cam.2014.06.016. Google Scholar [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. doi: 10.1016/j.cam.2011.02.012. Google Scholar [23] M. N. Phong, A note on a system of two nonlinear difference equations,, Electron. J. Math. Anal. Appl., 3 (2015), 170. Google Scholar [24] J. Popenda and E. Schmeidel, Some properties of solutions of difference equations,, Fasc. Math., 13 (1981), 89. Google Scholar [25] R. Salehi and M. Dehghan, A generalized moving least square reproducing kernel method,, J. Comput. Appl. Math., 249 (2013), 120. doi: 10.1016/j.cam.2013.02.005. Google Scholar [26] K. Seddighi, Reproducing kernel Hilbert spaces,, Iranian J. Sci. Tech., 17 (1993), 171. Google Scholar [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. doi: 10.1016/j.cma.2009.06.011. Google Scholar [28] Q. Zhang and W. Zhang, On a system of two high-order nonlinear difference equations,, Adv. Math. Phys., (2014). doi: 10.1155/2014/729273. Google Scholar
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