Multiple Anti-Periodic Solutions to a Discrete Fourth Order Nonlinear Equation

Differential Equations and Dynamical Systems - Tập 27 - Trang 601-610 - 2016
John R. Graef1, Lingju Kong1, Xueyan Liu1
1Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, USA

Tóm tắt

This paper is concerned with the existence of multiple anti-periodic solutions to a nonlinear fourth order difference equation. The analysis is based on variational methods and critical point theory. Clark’s critical point theorem is used to prove the main results. An example illustrates the applicability of the results.

Tài liệu tham khảo

Abdurahman, A., Anton, F., Bordes, J.: Half-string oscillator approach to string field theory (Ghost sector: I). Nuclear Phys. B 397, 260–282 (1993) Agarwal, R.P.: Difference equations and inequalities, theory, methods, and applications, 2nd edn. Marcel Dekker, New York (2000) Aftabizadeh, A.R., Aizicovici, S., Pavel, N.H.: Anti-periodic boundary value problems for higher order differential equations in Hilbert spaces. Nonlinear Anal. 18, 253–267 (1992) Aftabizadeh, A.R., Aizicovici, S., Pavel, N.H.: On a class of second-order anti-periodic boundary value problems. J. Math. Anal. Appl. 171, 301–320 (1992) Ahn, C., Rim, C.: Boundary flows in general coset theories. J. Phys. A Math. Gen. 32, 2509–2525 (1999) Cabada, A., Dimitrov, N.: Multiplicity results for nonlinear periodic fourth order difference equations with parameter dependence and singularities. J. Math. Anal. Appl. 371, 518–533 (2010) Cabada, A., Ferreiro, J.B.: Existence of positive solutions for \(n\)th-order periodic difference equations. J. Differ. Equ. Appl. 17, 935–954 (2011) Cai, X., Yu, J.: Existence of periodic solutions for fourth-order difference equations. J. Math. Anal. Appl. 320, 649–661 (2006) Chen, H.L.: Antiperiodic wavelets. J. Comput. Math. 14, 32–39 (1996) Clark, D.C.: A variant of the Liusternik-Schnirelman theory. Indiana Univ. Math. J. 22, 65–74 (1972/73) Delvos, F.J., Knoche, L.: Lacunary interpolation by antiperiodic trigonometric polynomials. BIT 39, 439–450 (1999) Deng, X.Q.: Nonexistence and existence results for a class of fourth-order difference mixed boundary value problems. J. Appl. Math. Comput. 45, 1–14 (2014) Djiakov, P., Mityagin, B.: Simple and double eigenvalues of the Hill operator with a two term potential. J. Approx. Theory 135, 70–104 (2005) Du, J.Y., Han, H.L., Jin, G.X.: On trigonometric and paratrigonometric Hermite interpolation. J. Approx. Theory 131, 74–99 (2004) Franco, D., Nieto, J.J., O’Regan, D.: Anti-periodic boundary value problem for nonlinear first order ordinary differential equations. J. Math. Inequal. Appl. 6, 477–485 (2003) Graef, J.R., Kong, L., Kong, Q.: On a generalized discrete beam equation via variational methods. Commun. Appl. Anal. 16, 293–308 (2012) Graef, J., Kong, L.J., Liu, X.Y.: Existence of solutions to a discrete fourth order periodic boundary value problem, to appear Graef, J.R., Kong, L., Tian, Y., Wang, M.: On a discrete fourth order periodic boundary value problem. Indian J. Math. 55, 163–184 (2013) Graef, J.R., Kong, L., Wang, M.: Solutions of a nonlinear fourth order periodic boundary value problem for difference equations. Dyn. Contin. Discrete Impuls. Syst. Ser. A 20, 53–63 (2013) Graef, J.R., Kong, L., Wang, M.: Existence of multiple solutions to a discrete fourth order periodic boundary value problems, Discrete Contin. Dyn. Syst. suppl. 291–299 (2013) Graef, J.R., Kong, L., Wang, M.: Multiple solutions to a periodic boundary value problem for a nonlinear discrete fourth order equation. Adv. Dyn. Syst. Appl. 8, 203–215 (2013) Graef, J.R., Kong, L., Wang, M.: Solutions of a nonlinear fourth order periodic boundary value problems for difference equations. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 20, 53–63 (2013) Graef, J.R., Kong, L., Wang, M., Yang, B.: Uniqueness and parameter dependence of positive solutions of a discrete fourth order problem. J. Differ. Equ. Appl. 19, 1133–1146 (2013) Graef, J.R., Kong, L., Yang, B.: Positive solutions for boundary value problems of discrete and continuous beam equations. J. Appl. Math. Comput. 41, 197–208 (2013) Haraux, A.: Anti-periodic solutions of some nonlinear evolution equations. Manuscr. Math. 63, 479–505 (1989) Kelly, W.G., Peterson, A.C.: Difference equations, an introduction with applications, 2nd edn. Academic Press, New York (2001) Kleinert, H., Chervyakov, A.: Functional determinants from Wronski Green function. J. Math. Phys. 40, 6044–6051 (1999) Liu, X., Zhang, Y.B., Shi, H.P., Deng, X.Q.: Existence of periodic solutions of fourth-order nonlinear difference equations. RACSAM 108, 811–825 (2014) Okochi, H.: On the existence of periodic solutions to nonlinear abstract parabolic equations. J. Math. Soc. Japan 40, 541–553 (1988) Pinsky, S., Tritman, U.: Antiperiodic boundary conditions to supersymmetric discrete light cone quantization. Phys. Rev. D 62(8), 087701 (2000) Rabinowitz, P.H.: Minimax methods in critical point theory with applications to differential equations. CBMS Regional Conference Series in Mathematics, 65. American Mathematical Society, Providence, RI (1986) Tian, Y., Henderson, J.: Anti-periodic solutions of higher order nonlinear difference equations: a variational approach. J. Differ. Equ. Appl. 19, 1380–1392 (2013)