POSITIVE SOLUTIONS IN AN ANNULUS FOR NONLINEAR DIFFERENTIAL EQUATIONS ON A MEASURE CHAIN

Authors

  • CHUAN-JEN CHYAN Department of Mathematics, Tarnkang University, Taipei, Taiwan, 251
  • JOHNNY HENDERSON Department of Mathematics, Auburn University, Auburn, Alabama 36849-5310 USA.
  • HUI-CHUN LO Department of Mathematics, Tamkang University, Taipei, Taiwan, 251.

DOI:

https://doi.org/10.5556/j.tkjm.30.1999.4230

Keywords:

Differential equation, measure chain, boundary value problem

Abstract

We study the existence of positive solutions of the second order differential equation in an annulus on a measure chain, $u^{\Delta\Delta}(t) + f(u(\sigma(t))) = 0$, $t \in [0, 1]$, satisfying the boundary conditions, $\alpha y(0)-\beta y^\Delta(0)=0$ and $\gamma y(\sigma(1))+\delta y^\Delta ((1))=0$, where $f$ is a positive function and $f(x)$ is sublinear (respectively supcrlinear) at $x = 0$ and is superlinear (respectively sublinear) at $x = \infty$· The methods involve applications of a fixed point theorem for operators on a cone in a Banach space.

References

R. P. Agarwal and M. Bohner, BASIC calculus on time scales and some of its applications, preprint.

R. P. Agarwal, M. Bohner and P. Wong, Sturm-Liouville eigenvalue problems on time scales, Appl. Math. Comput., in press.

F. Atici, Two positive solutions of a boundary value problem for difference equations, J. Difference Eqns. Appl. 1(1995), 263-270.

B. Aulback and S. Hilger, Linear dynamic proces邳 s with inhomogeneous time scale, Nonlin­ ear Dynamics and Quantum Dynamical Systems, Math. Res. 59, Akademie Verlag, Berlin, 1990.

C. J. Chyan and J. Henderson, Eigenvalue problems for nonlinear differential equations on a measure chain, J. Math. Anal. Appl., in press.

C. J. Chyan, J. M. Davis, J. Henderson and W. K. C. Yin, Eigenvalue comparisons for non­ linear differential equations on a measure chain, Electronic Journal of Differential Equations 1998(1998), 1-7.

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, New York, 1985.

P. W. Eloe and J. Henderson, Positive solutions and nonlinear $(k, n- k)$ conjugate eigenvalue problems, Differential Equations Dynam. Systems 6(1998), 309-317.

L. H. Erbe and S. Hilger, Sturmian theory on measure chains, Differential Equations Dynam Systems 1(1993), 223-246.

L. H. Erbe and A. Peterson, Green's functions and comparison theorems for differential equations on measure chains, Dynam. Contin. Discrete Impuls. Systems, in press.

L. H. Erbe and A. Peterson, Positive solutions for a nonlinear differential equation on a measure chain, Math. Comput. Modelling, in press.

L. H. Erbe and A. Peterson, Eigenvalue conditons and positive solutions, preprint.

L. H. Erbe, S. Hu and H. Wang, Multiple positive solutions of some boundary value problems, J. Math. Anal. Appl. 184(1994), 640-648.

L. H. Erbe and M Tang, Existence and multiplicity of positive solutions to nonlinear bound­ary value problems, Differential Equations Dynam. Systems 4(1996), 313-320.

S. Hilger, Analysis on measure chains - a unified approach to continuous and discrete cal­culus, Resultate Math. 18(1990), 18-56.

J. Henderson and H. Wang, Positive solutions for nonlinear eigenvalue problems, J. Math Anal. Appl. 208(1997), 252-259.

E. R. Kaufmann, Multiple positive solutions for higher order boundary value problems, Rocky Mountain J. Math. 28(1998), 1017-1028.

J. Henderson and E. R. Kaufmann, Multiple positive solutions for focal boundary value problems, Comm. Appl. Anal. 1(1997), 53-60.

B. Kaymakcalan, V. Lakshmikantham and S. Sivasundaram, Dynamical Systems on Mea­sure Chains, Kluwer Academic Publishers, Boston, 1996.

M. A. Krasnosel'skii, Positive Solutions of Operator Equations, Noordhoof Groningen, 1964.

S. Lauer, Positive Solutions of Boundary Value Problems for Nonlinear Difference Equa­tions, Ph. D. dissertation, Auburn University, 1997.

H. Wang, On the existence of positive solutions for semilinear elliptic equations in the annulus, J. Diff. Eqns. 109(1994), 1-7.

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Published

1999-09-01

How to Cite

CHYAN, C.-J., HENDERSON, J., & LO, H.-C. (1999). POSITIVE SOLUTIONS IN AN ANNULUS FOR NONLINEAR DIFFERENTIAL EQUATIONS ON A MEASURE CHAIN. Tamkang Journal of Mathematics, 30(3), 231–240. https://doi.org/10.5556/j.tkjm.30.1999.4230

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Papers