# American Institute of Mathematical Sciences

2015, 2015(special): 775-782. doi: 10.3934/proc.2015.0775

## An application of an avery type fixed point theorem to a second order antiperiodic boundary value problem

 1 Department of Mathematics, Nova Southeastern University, 3301 College Avenue, Fort Lauderdale, FL 33314, United States

Received  September 2014 Revised  February 2015 Published  November 2015

In this article, we show the existence of an antisymmetric solution to the second order boundary value problem $x''+f(x(t))=0,\; t\in(0,n)$ satisfying antiperiodic boundary conditions $x(0)+x(n)=0,\; x'(0)+x'(n)=0$ using an Avery et. al. fixed point theorem which itself is an extension of the traditional Leggett-Williams fixed point theorem. The antisymmetric solution satisfies $x(t)=-x(n-t)$ for $t\in[0,n]$ and is nonnegative, nonincreasing, and concave for $t\in[0,n/2]$. To conclude, we present an example.
Citation: Jeffrey W. Lyons. An application of an avery type fixed point theorem to a second order antiperiodic boundary value problem. Conference Publications, 2015, 2015 (special) : 775-782. doi: 10.3934/proc.2015.0775
##### References:
 [1] A. A. Altwaty and P. W. Eloe, The role of concavity in applications of Avery type fixed point theorems to higher order differential equations, J. Math Inequal., 6 (2012), 79-90. Google Scholar [2] A. A. Altwaty and P. W. Eloe, Concavity of solutions of a $2n$-th order problem with symmetry, Opuscula Math., 33 (2013), 603-613. Google Scholar [3] D. R. Anderson and R. I. Avery, Fixed point theorem of cone expansion and compression of functional type, J. Difference Equ. Appl., 8 (2002), 1073-1083. Google Scholar [4] D. R. Anderson, R. I. Avery and J. Henderson, Functional expansion-compression fixed point theorem of Leggett-Williams type,, Electron. J. Differential Equations, 2010 (): 1.   Google Scholar [5] D. R. Anderson, R. I. Avery, J. Henderson and X. Liu, Operator type expansion-compression fixed point theorem,, Electron. J. Differential Equations, 2011 (): 1.   Google Scholar [6] D. R. Anderson, R. I. Avery, J. Henderson and X. Liu, Fixed point theorem utilizing operators and functionals,, Electron. J. Qual. Theory Differ. Equ., 2012 (): 1.   Google Scholar [7] D. R. Anderson, R. I. Avery, J. Henderson, X. Liu and J. W. Lyons, Existence of a positive solution for a right focal discrete boundary value problem, J. Difference Equ. Appl., 17 (2011), 1635-1642. Google Scholar [8] J. Andres and V. Vlček, Green's functions for periodic and anti-periodic BVPs to second-order ODEs, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math., 32 (1993), 7-16. Google Scholar [9] R. I. Avery, D. R. Anderson and J. Henderson, A topological proof and extension of the Leggett-Williams fixed point theorem, Comm. Appl. Nonlinear Anal., 16 (2009), 39-44. Google Scholar [10] R. I. Avery, D. R. Anderson and J. Henderson, Existence of a positive solution to a right focal boundary value problem,, Electron. J. Qual. Theory Differ. Equ., 2010 (): 1.   Google Scholar [11] R. I. Avery, P. W. Eloe and J. Henderson, A Leggett-Willaims type theorem applied to a fourth order problem, Commun. Appl. Anal., 16 (2012), 579-588. Google Scholar [12] C. Bai, On the solvability of anti-periodic boundary value problems with impulse,, Electron. J. Qual. Theory Differ. Equ., 2009 (): 1.   Google Scholar [13] M. Benchohra, N. Hamidi and J. Henderson, Fractional differential equations with anti-periodic boundary conditions, Numer. Funct. Anal. Optim., 34 (2013), 404-414. Google Scholar [14] D. Franco, J. J. Nieto and D. O'Regan, Anti-periodic boundary value problem for nonlinear first order ordinary differential equations, Math. Inequal. Appl., 6 (2003), 477-485. Google Scholar [15] R. W. Leggett and L. R. Williams, Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana Univ. Math. J., 28 (1979), 673-688. Google Scholar [16] X. Liu, J. T. Neugebauer and S. Sutherland, Application of a functional type compression expansion fixed point theorem for a right focal boundary value problem on a time scale, Comm. Appl. Nonlinear Anal., 19 (2012), 25-39. Google Scholar [17] J. W. Lyons and J. T. Neugebauer, Existence of a positive solution for a right focal dynamic boundary value problem, Nonlinear Dyn. Syst. Theory, 14 (2014), 76-83. Google Scholar [18] J. T. Neugebauer and C. L. Seelbach, Positive symmetric solutions of a second order difference equation, Involve, 5 (2012), 497-504. Google Scholar [19] G. F. Roach, Green's Functions, $2^{nd}$ edition, Cambridge University Press, Cambridge-New York, 1982. Google Scholar

show all references

##### References:
 [1] A. A. Altwaty and P. W. Eloe, The role of concavity in applications of Avery type fixed point theorems to higher order differential equations, J. Math Inequal., 6 (2012), 79-90. Google Scholar [2] A. A. Altwaty and P. W. Eloe, Concavity of solutions of a $2n$-th order problem with symmetry, Opuscula Math., 33 (2013), 603-613. Google Scholar [3] D. R. Anderson and R. I. Avery, Fixed point theorem of cone expansion and compression of functional type, J. Difference Equ. Appl., 8 (2002), 1073-1083. Google Scholar [4] D. R. Anderson, R. I. Avery and J. Henderson, Functional expansion-compression fixed point theorem of Leggett-Williams type,, Electron. J. Differential Equations, 2010 (): 1.   Google Scholar [5] D. R. Anderson, R. I. Avery, J. Henderson and X. Liu, Operator type expansion-compression fixed point theorem,, Electron. J. Differential Equations, 2011 (): 1.   Google Scholar [6] D. R. Anderson, R. I. Avery, J. Henderson and X. Liu, Fixed point theorem utilizing operators and functionals,, Electron. J. Qual. Theory Differ. Equ., 2012 (): 1.   Google Scholar [7] D. R. Anderson, R. I. Avery, J. Henderson, X. Liu and J. W. Lyons, Existence of a positive solution for a right focal discrete boundary value problem, J. Difference Equ. Appl., 17 (2011), 1635-1642. Google Scholar [8] J. Andres and V. Vlček, Green's functions for periodic and anti-periodic BVPs to second-order ODEs, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math., 32 (1993), 7-16. Google Scholar [9] R. I. Avery, D. R. Anderson and J. Henderson, A topological proof and extension of the Leggett-Williams fixed point theorem, Comm. Appl. Nonlinear Anal., 16 (2009), 39-44. Google Scholar [10] R. I. Avery, D. R. Anderson and J. Henderson, Existence of a positive solution to a right focal boundary value problem,, Electron. J. Qual. Theory Differ. Equ., 2010 (): 1.   Google Scholar [11] R. I. Avery, P. W. Eloe and J. Henderson, A Leggett-Willaims type theorem applied to a fourth order problem, Commun. Appl. Anal., 16 (2012), 579-588. Google Scholar [12] C. Bai, On the solvability of anti-periodic boundary value problems with impulse,, Electron. J. Qual. Theory Differ. Equ., 2009 (): 1.   Google Scholar [13] M. Benchohra, N. Hamidi and J. Henderson, Fractional differential equations with anti-periodic boundary conditions, Numer. Funct. Anal. Optim., 34 (2013), 404-414. Google Scholar [14] D. Franco, J. J. Nieto and D. O'Regan, Anti-periodic boundary value problem for nonlinear first order ordinary differential equations, Math. Inequal. Appl., 6 (2003), 477-485. Google Scholar [15] R. W. Leggett and L. R. Williams, Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana Univ. Math. J., 28 (1979), 673-688. Google Scholar [16] X. Liu, J. T. Neugebauer and S. Sutherland, Application of a functional type compression expansion fixed point theorem for a right focal boundary value problem on a time scale, Comm. Appl. Nonlinear Anal., 19 (2012), 25-39. Google Scholar [17] J. W. Lyons and J. T. Neugebauer, Existence of a positive solution for a right focal dynamic boundary value problem, Nonlinear Dyn. Syst. Theory, 14 (2014), 76-83. Google Scholar [18] J. T. Neugebauer and C. L. Seelbach, Positive symmetric solutions of a second order difference equation, Involve, 5 (2012), 497-504. Google Scholar [19] G. F. Roach, Green's Functions, $2^{nd}$ edition, Cambridge University Press, Cambridge-New York, 1982. Google Scholar
 [1] Shui-Hung Hou. On an application of fixed point theorem to nonlinear inclusions. Conference Publications, 2011, 2011 (Special) : 692-697. doi: 10.3934/proc.2011.2011.692 [2] Ovide Arino, Eva Sánchez. A saddle point theorem for functional state-dependent delay differential equations. Discrete & Continuous Dynamical Systems, 2005, 12 (4) : 687-722. doi: 10.3934/dcds.2005.12.687 [3] Tiziana Cardinali, Paola Rubbioni. Existence theorems for generalized nonlinear quadratic integral equations via a new fixed point result. Discrete & Continuous Dynamical Systems - S, 2020, 13 (7) : 1947-1955. doi: 10.3934/dcdss.2020152 [4] Nicholas Long. Fixed point shifts of inert involutions. Discrete & Continuous Dynamical Systems, 2009, 25 (4) : 1297-1317. doi: 10.3934/dcds.2009.25.1297 [5] Yakov Krasnov, Alexander Kononovich, Grigory Osharovich. On a structure of the fixed point set of homogeneous maps. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1017-1027. doi: 10.3934/dcdss.2013.6.1017 [6] Jorge Groisman. Expansive and fixed point free homeomorphisms of the plane. Discrete & Continuous Dynamical Systems, 2012, 32 (5) : 1709-1721. doi: 10.3934/dcds.2012.32.1709 [7] Yong Ji, Ercai Chen, Yunping Wang, Cao Zhao. Bowen entropy for fixed-point free flows. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6231-6239. doi: 10.3934/dcds.2019271 [8] Luis Hernández-Corbato, Francisco R. Ruiz del Portal. Fixed point indices of planar continuous maps. Discrete & Continuous Dynamical Systems, 2015, 35 (7) : 2979-2995. doi: 10.3934/dcds.2015.35.2979 [9] Antonio Garcia. Transition tori near an elliptic-fixed point. Discrete & Continuous Dynamical Systems, 2000, 6 (2) : 381-392. doi: 10.3934/dcds.2000.6.381 [10] Cleon S. Barroso. The approximate fixed point property in Hausdorff topological vector spaces and applications. Discrete & Continuous Dynamical Systems, 2009, 25 (2) : 467-479. doi: 10.3934/dcds.2009.25.467 [11] Teck-Cheong Lim. On the largest common fixed point of a commuting family of isotone maps. Conference Publications, 2005, 2005 (Special) : 621-623. doi: 10.3934/proc.2005.2005.621 [12] Mircea Sofonea, Cezar Avramescu, Andaluzia Matei. A fixed point result with applications in the study of viscoplastic frictionless contact problems. Communications on Pure & Applied Analysis, 2008, 7 (3) : 645-658. doi: 10.3934/cpaa.2008.7.645 [13] Parin Chaipunya, Poom Kumam. Fixed point theorems for cyclic operators with application in Fractional integral inclusions with delays. Conference Publications, 2015, 2015 (special) : 248-257. doi: 10.3934/proc.2015.0248 [14] Dou Dou, Meng Fan, Hua Qiu. Topological entropy on subsets for fixed-point free flows. Discrete & Continuous Dynamical Systems, 2017, 37 (12) : 6319-6331. doi: 10.3934/dcds.2017273 [15] Ruhua Wang, Senjian An, Wanquan Liu, Ling Li. Fixed-point algorithms for inverse of residual rectifier neural networks. Mathematical Foundations of Computing, 2021, 4 (1) : 31-44. doi: 10.3934/mfc.2020024 [16] Mark S. Gockenbach, Akhtar A. Khan. Identification of Lamé parameters in linear elasticity: a fixed point approach. Journal of Industrial & Management Optimization, 2005, 1 (4) : 487-497. doi: 10.3934/jimo.2005.1.487 [17] Romain Aimino, Huyi Hu, Matthew Nicol, Andrei Török, Sandro Vaienti. Polynomial loss of memory for maps of the interval with a neutral fixed point. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 793-806. doi: 10.3934/dcds.2015.35.793 [18] Grzegorz Graff, Piotr Nowak-Przygodzki. Fixed point indices of iterations of $C^1$ maps in $R^3$. Discrete & Continuous Dynamical Systems, 2006, 16 (4) : 843-856. doi: 10.3934/dcds.2006.16.843 [19] Hans Koch. A renormalization group fixed point associated with the breakup of golden invariant tori. Discrete & Continuous Dynamical Systems, 2004, 11 (4) : 881-909. doi: 10.3934/dcds.2004.11.881 [20] Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (9) : 4927-4962. doi: 10.3934/dcdsb.2020320

Impact Factor: