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.

[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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