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

Jeffrey W.  Lyons

doi:10.3934/proc.2015.0775

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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

Jeffrey W. Lyons ^1,

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

Abstract
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Under a Creative Commons license

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.

Keywords: Functional., Antisymmetric, Fixed Point Theorem, Nonlinear, Antiperiodic.

Mathematics Subject Classification: Primary: 34B1.

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.
[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.
[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.
[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, 1-9.
[5]	D. R. Anderson, R. I. Avery, J. Henderson and X. Liu, Operator type expansion-compression fixed point theorem, Electron. J. Differential Equations, 2011, 1-11.
[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, 1-16.
[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.
[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.
[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.
[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, 1-6.
[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.
[12]	C. Bai, On the solvability of anti-periodic boundary value problems with impulse, Electron. J. Qual. Theory Differ. Equ., 2009, 1-15.
[13]	M. Benchohra, N. Hamidi and J. Henderson, Fractional differential equations with anti-periodic boundary conditions, Numer. Funct. Anal. Optim., 34 (2013), 404-414.
[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.
[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.
[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.
[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.
[18]	J. T. Neugebauer and C. L. Seelbach, Positive symmetric solutions of a second order difference equation, Involve, 5 (2012), 497-504.
[19]	G. F. Roach, Green's Functions, $2^{nd}$ edition, Cambridge University Press, Cambridge-New York, 1982.

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.
[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.
[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.
[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, 1-9.
[5]	D. R. Anderson, R. I. Avery, J. Henderson and X. Liu, Operator type expansion-compression fixed point theorem, Electron. J. Differential Equations, 2011, 1-11.
[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, 1-16.
[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.
[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.
[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.
[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, 1-6.
[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.
[12]	C. Bai, On the solvability of anti-periodic boundary value problems with impulse, Electron. J. Qual. Theory Differ. Equ., 2009, 1-15.
[13]	M. Benchohra, N. Hamidi and J. Henderson, Fractional differential equations with anti-periodic boundary conditions, Numer. Funct. Anal. Optim., 34 (2013), 404-414.
[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.
[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.
[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.
[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.
[18]	J. T. Neugebauer and C. L. Seelbach, Positive symmetric solutions of a second order difference equation, Involve, 5 (2012), 497-504.
[19]	G. F. Roach, Green's Functions, $2^{nd}$ edition, Cambridge University Press, Cambridge-New York, 1982.

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