Philip Korman


BOOKS


Lectures on Linear Algebra and its Applications. In the series De Gruyter Textbook, Walter de Gruyter, Oct 24, 2023 - Mathematics - 258 pages. ISBN: 9783111085401.


Chapters 1-5 provide a textbook for the course ``Linear Algebra" for sophomores in science and engineering. Chapter 6 has in depth presentation of systems of differential equations. Chapter 7 has applications to Mathematical Analysis and Differential Geometry. The book has many interesting exercises, and detailed hints are given. This book is well suited for self-study, or for a refresher course, as well as a regular textbook. An extensive solution manual, written by the author, is available. The publisher lets university libraries buy E-books, which makes them free to students. Please consider adapting this book.
Lectures on Differential Equations. AMS/MAA Textbooks 54, MAA Press, Providence, RI, 2019. xii+399 pp. ISBN: 978-1-4704-5173-8


Chapters 1-4 provide a textbook for the course ``Differential Equations" for sophomores in science and engineering. Chapters 7 and 8 cover the course ``Fourier Series and PDE". Chapters 5 and 6 are suitable for a graduate ODE course. An extensive solution manual, written by the author, is available. The publisher lets university libraries buy E-books, which makes them free to students. Please consider adapting this book.
Solutions Manual for "Lectures on Differential Equations" 195 pp.


available in pdf
Global solution curves for semilinear elliptic equations. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012. xii+241 pp. ISBN: 978-981-4374-34-7; 981-4374-34-2


Please ask your library to order a hard copy, or write to the author

PAPERS

Necessary and sufficient condition for existence for a case of eigenvalues of multiplicity two.
Electron. J. Differential Equations 2024, Paper No. 09.

available in pdf
Periodic and unbounded solutions of periodic systems
Bull. Malays. Math. Sci. Soc. 47 (2024), no. 4, Paper No. 107.

available in pdf
Infinitely many solutions for a class of resonant problems
NoDEA Nonlinear Differential Equations Appl. 30 (2023), no. 4, Paper No. 48, 17 pp.


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Generalized Pohozhaev's identity for radial solutions of $p$-Laplace equations
Georgian Math. J. 30 (2023), no. 2, 263–267.


available in pdf
Iterative solution of elliptic equations
with D. Schmidt
Electron. J. Qual. Theory Differ. Equ. 2022, No. 34, 1-9.

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Infinitely many solutions and asymptotics for resonant oscillatory problems
with D. Schmidt
Special issue in honor of Alan C. Lazer,
Electron. J. Diff. Equ., Special Issue 01, 301-313 (2021).

available in pdf
Non-singular solutions of $p$-Laplace problems, allowing multiple changes of sign in the nonlinearity
Electron. Res. Arch. 30 (2022), no. 4, 1414-1418.

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Continuation of global solution curves using global parameters
with D. Schmidt
J. Comput. Appl. Math. 440 (2024), Paper No. 115621, 18 pp.

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Calculating Global Solution Curves for Boundary Value Problems
with D. Schmidt
From the Wolfram Notebook Archive (2022)

available: https://notebookarchive.org/2022-07-9pt96oa
Solving partial differential equations on balls in R^n
with D. Schmidt
From the Wolfram Notebook Archive (2021)

available: https://notebookarchive.org/2021-08-cgkai6z
On the interaction of species capable of explosive growth
Applied Mathematics and Nonlinear Sciences, Sciendo (2021), 1-10.

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Global solution curves in harmonic parameters, and multiplicity of solutions
Journal of Differential Equations 296 (2021), 186-212.

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A remark on Pinney's equation
Communications on Applied Nonlinear Analysis 26 (2019), no 4, 98 - 100.

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Global solution curves for first order periodic problems, with applications
with D. Schmidt
Nonlinear Anal. Real World Appl. 56 (2020), 103161, 18 pp.
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Infinitely many solutions and Morse index for non-autonomous elliptic problems
Commun. Pure Appl. Anal. 19 (2020), no. 1, 31-46.

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A computer assisted study of uniqueness of nodal ground state solutions
with Y. Li and D. Schmidt
J. Comput. Appl. Math. 356 (2019), 402–406.
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Bounded solutions for a class of Hamiltonian systems
with G. Peng
Electronic Journal of Qualitative Theory of Differential Equations (2018), No. 81, 1-7.
available in pdf
A simplified proof of a conjecture for the perturbed Gelfand equation from combustion theory
with Y. Li and T. Ouyang
J. Differential Equations 263 (2017), no. 5, 2874–2885.
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On separated solutions of logistic population equation with harvesting

J. Math. Analysis and Applic. 455 (2017), no. 2, 2024-2029.
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Exact multiplicity of solutions for some semilinear Dirichlet problems

Applicable Anal. 98 (2019), no. 8, 1483–1495.
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Explicit solutions and multiplicity results for some equations with the $p$-Laplacian

Quart. Appl. Math. 75 (2017), no. 4, 635-647.
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Global solution curves for several classes of singular periodic problems

Nonlinear Anal. Real World Appl. 34 (2017), 1–16.
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A periodic model for the dynamics of cell volume

Annales Polonici Mathematici 116, no. 3 (2016), 243-249.
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Infinitely many solutions for three classes of self-similar equations with p-Laplace operator: Gelfand, Joseph–Lundgren and MEMS problems

Proc. Roy. Soc. Edinburgh Sect. A 148 (2018), no. 2, 341-356.
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Nonlinear elliptic equations and systems with linear part at resonance

Electron. J. Diff. Equ., Vol. 2016 (2016), No. 67, pp. 1-17.
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Non-singular solutions of two-point problems, with multiple changes of sign in the nonlinearity


Proc. Amer. Math. Soc. 144 (2016), no. 6, 2539–2546.
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Families of solution curves for some non-autonomous problems

Acta Applicandae Mathematicae (2016) 143(1), 165-178.
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Regularization of radial solutions of $p$-Laplace equations, and computations using infinite series

Electron. J. Qual. Theory Differ. Equ. (2015), No. 40.
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A global solution curve for a class of free boundary value problems arising in plasma physics

Appl. Math. Optim. 71 (2015), no. 1, 25-38.
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Harmonic oscillators at resonance, perturbed by a non-linear friction force
with Y. Li
Acta Math. Sci. Ser. B Engl. Ed. 34 (2014), no. 4, 1025-1028.
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Global solution curves for self-similar equations

J. Differential Equations 257 (2014), no. 7, 2543-2564.
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Exact multiplicity of positive solutions for concave-convex and convex-concave nonlinearities
with Y. Li
J. Differential Equations 257 (2014), no. 10, 3730–3737.
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A global solution curve for a class of periodic problems, including the relativistic pendulum

Appl. Anal. 93 (2014), no. 1, 124-136.
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Curves of equiharmonic solutions, and problems at resonance

Discrete Contin. Dyn. Syst. 34 (2014), no. 7, 2847-2860.
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Non-existence of solutions for non-autonomous elliptic systems of Hamiltonian type

Comm. Appl. Nonlinear Anal. 21 (2014), no. 3, 13-20.
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Exact multiplicity and numerical computation of solutions for two classes of non-autonomous problems with concave-convex nonlinearities

Nonlinear Anal. 93 (2013), 226–235.
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Direction of bifurcation for some non-autonomous problems

Electron. J. Differential Equations 2012, No. 214, 13 pp.
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Nonlinear perturbations of linear elliptic systems at resonance

Proc. Amer. Math. Soc. 140 (2012), no. 7, 2447-2451.
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A computer assisted study of uniqueness of ground state solutions
with Y. Li
J. Comput. Appl. Math. 236 (2012), no. 11, 2838-2843.
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Existence and uniqueness of solutions for a class of p-Laplace equations on a ball

Adv. Nonlinear Stud. 11 (2011), no. 4, 875-888.
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Curves of equiharmonic solutions and solvability of elliptic systems

Differential Integral Equations 24 (2011), no. 11-12, 1147-1162.
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Remarks on time map for quasilinear equations
with T. Adamowicz
J. Math. Anal. Appl. 376 (2011), no. 2, 686-695.
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Global solution curves for a class of quasilinear boundary-value problems
with Y. Li
Proc. Roy. Soc. Edinburgh Sect. A 140 (2010), no. 6, 1197-1215.
available in pdf
Global solution curves for a class of elliptic systems

Appl. Anal. 89 (2010), no. 6, 849-860.
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Pohozaev's identity and non-existence of solutions for elliptic systems

Comm. Appl. Nonlinear Anal. 17 (2010), no. 4, 81-88.
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Computing the location and the direction of bifurcation for sign changing solutions
with Y. Li
Differ. Equ. Appl. 2 (2010), no. 1, 1-13.
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Curves of equiharmonic solutions, and ranges of nonlinear equations

Adv. Differential Equations 14 (2009), no. 9-10, 963-984.
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Global solution curves for boundary value problems, with linear part at resonance

Nonlinear Anal. 71 (2009), no. 7-8, 2456-2467.
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An oscillatory bifurcation from infinity, and from zero

NoDEA Nonlinear Differential Equations Appl. 15 (2008), no. 3, 335-345.
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A global approach to ground state solutions

Electron. J. Differential Equations 2008, No. 122, 13 pp.
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An exact multiplicity result for a class of symmetric problems

Port. Math. 65 (2008), no. 2, 257-260.
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Uniqueness and exact multiplicity of solutions for a class of Dirichlet problems

J. Differential Equations 244 (2008), no. 10, 2602-2613.
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A global solution curve for a class of periodic problems, including the pendulum equation

Z. Angew. Math. Phys. 58 (2007), no. 5, 749–766.
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Stability and instability of solutions of semilinear problems

Appl. Anal. 86 (2007), no. 2, 135-147.
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On the direction of pitchfork bifurcation
with X. Hou and Y. Li
Proceedings of the Sixth Mississippi State–UBA Conference on Differential Equations and Computational Simulations, 153–158, Electron. J. Differ. Equ. Conf., 15, Southwest Texas State Univ., San Marcos, TX, 2007.
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Computation of radial solutions of semilinear equations

Electron. J. Qual. Theory Differ. Equ. 2007, No. 13, 14 pp.
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Global solution branches and exact multiplicity of solutions for two point boundary value problems

Handbook of differential equations: ordinary differential equations. Vol. III, 547–606, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2006.
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Positivity for the linearized problem for semilinear equations
with T. Ouyang
Topol. Methods Nonlinear Anal. 28 (2006), no. 1, 53–60.
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On the oscillations of the solution curve for a class of semilinear equations
with A. Galstyan and Y. Li
J. Math. Anal. Appl. 321 (2006), no. 2, 576-588.
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Uniqueness and exact multiplicity of solutions for non-autonomous Dirichlet problems

Adv. Nonlinear Stud. 6 (2006), no. 3, 461-481.
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Existence and uniqueness of solutions for a class of non-autonomous Dirichlet problems

Funkcial. Ekvac. 48 (2005), no. 1, 99-111.
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On a principle of predatory exclusion

Appl. Anal. 84 (2005), no. 7, 707-712.
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Application of generalized averages to uniqueness of solutions, and to a non-local problem

Comm. Appl. Nonlinear Anal. 12 (2005), no. 4, 69–78.
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Computing the location and the direction of bifurcation
with Y. Li and T. Ouyang
Mathematical Research Letters 12, 933–944 (2005).
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On Lane-Emden type systems
with J. Shi
Discrete Contin. Dyn. Syst. 2005, suppl., 510-517.
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Further remarks on the non-degeneracy condition

Results Math. 45 (2004), no. 3-4, 293-298.
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Uniqueness and exact multiplicity of solutions for a class of fourth-order semilinear problems

Proc. Roy. Soc. Edinburgh Sect. A 134 (2004), no. 1, 179-190.
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Curves of positive solutions for supercritical problems

Appl. Anal. 82 (2003), no. 1, 45-54.
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Perturbation of global solution curves for semilinear problems
with Y. Li and T. Ouyang
Adv. Nonlinear Stud. 3 (2003), no. 2, 289–299.
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An accurate computation of the global solution curve for the Gelfand problem through a two point approximation

Appl. Math. Comput. 139 (2003), no. 2-3, 363–369.
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Similarity of solution branches for two-point semilinear problems

Proceedings of the Fifth Mississippi State Conference on Differential Equations and Computational Simulations (Mississippi State, MS, 2001), 187–191 (electronic), Electron. J. Differ. Equ. Conf., 10, Southwest Texas State Univ., San Marcos, TX, 2003.
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A remark on the non-degeneracy condition

Results Math. 41 (2002), no. 3-4, 334-336.
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Global solution curves for semilinear systems

Math. Methods Appl. Sci. 25 (2002), no. 1, 3-20.
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Curves of sign-changing solutions for semilinear equations

Nonlinear Anal. 51 (2002), no. 5, Ser. A: Theory Methods, 801–820.
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On uniqueness of positive solutions for a class of semilinear equations

Discrete Contin. Dyn. Syst. 8 (2002), no. 4, 865-871.
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Stability and Morse indices of solutions for two classes of semilinear problems

Comm. Appl. Nonlinear Anal. 9 (2002), no. 2, 1–17.
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Exact multiplicity of solutions for a class of semilinear Neumann problems

Comm. Appl. Nonlinear Anal. 9 (2002), no. 4, 1–17.
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The global solution set for a class of semilinear problems

Differential equations and control theory (Athens, OH, 2000), 177–192, Lecture Notes in Pure and Appl. Math., 225, Dekker, New York, 2002.
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New exact multiplicity results with an application to a population model
with J. Shi
Proc. Roy. Soc. Edinburgh Sect. A 131 (2001), no. 5, 1167-1182.
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On the multiplicity of solutions of semilinear equations

Math. Nachr. 229 (2001), 119-127.
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Monotone approximations of unstable solutions

J. Comput. Appl. Math. 136 (2001), no. 1-2, 309-315.
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Instability and exact multiplicity of solutions of semilinear equations
with J. Shi
Proceedings of the Conference on Nonlinear Differential Equations (Coral Gables, FL, 1999), 311–322 (electronic), Electron. J. Differ. Equ. Conf., 5, Southwest Texas State Univ., San Marcos, TX, 2000.
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Infinitely many solutions at a resonance
with Y. Li
Proceedings of the Conference on Nonlinear Differential Equations (Coral Gables, FL, 1999), 105–111 (electronic), Electron. J. Differ. Equ. Conf., 5, Southwest Texas State Univ., San Marcos, TX, 2000.
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Bounded solution to a differential equation (Problem Proposal-1577)

Mathematical Magazine (?) (2000).
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A pedagogically correct way to square it out

Amer. J. Math. Management Sci. 20 (2000), no. 3-4, 227–228.
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On the exactness of an S-shaped bifurcation curve
with Y. Li
Proc. Amer. Math. Soc. 127 (1999), no. 4, 1011-1020.
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Generalized averages for solutions of two-point Dirichlet problems
with Y. Li
J. Math. Anal. Appl. 239 (1999), no. 2, 478-484.
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Multiplicity and Morse indices of sign-changing solutions for semilinear equations
with T. Ouyang
Canad. Appl. Math. Quart. 7 (1999), no. 3, 239–250.
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Multiplicity of positive solutions for semilinear equations on circular domains

Comm. Appl. Nonlinear Anal. 6 (1999), no. 3, 17–35.
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Exact multiplicity of positive solutions for a class of semilinear equations on a ball

Electron. J. Qual. Theory Differ. Equ. 1999, No. 8, 15 pp. (electronic).
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Uniqueness and exact multiplicity results for two classes of semilinear problems

Nonlinear Anal. 31 (1998), no. 7, 849–865.
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Remarks on Nagumo's condition

Portugal. Math. 55 (1998), no. 1, 1-9.
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A global solution curve for a class of semilinear equations

Proceedings of the Third Mississippi State Conference on Difference Equations and Computational Simulations (Mississippi State, MS, 1997), 119–127 (electronic), Electron. J. Differ. Equ. Conf., 1, Southwest Texas State Univ., San Marcos, TX, 1998.
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Steady states and long time behavior of some convective reaction-diffusion equations

Funkcial. Ekvac. 40 (1997), no. 2, 165-183.
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Solution curves for semilinear equations on a ball

Proc. Amer. Math. Soc. 125 (1997), no. 7, 1997-2005.
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On homoclinic and heteroclinic orbits for Hamiltonian systems
with A. Lazer and Y. Li
Differential Integral Equations 10 (1997), no. 2, 357-368.
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An exact multiplicity result for a class of semilinear equations
with Y. Li and T. Ouyang
Comm. Partial Differential Equations 22 (1997), no. 3-4, 661–684.
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Average Temperature in a Reaction-Diffusion Process (Problem Proposal 97-8)

SIAM Review (1997).
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Exact multiplicity results for boundary value problems with nonlinearities generalising cubic
with Y. Li and T. Ouyang
Proc. Roy. Soc. Edinburgh Sect. A 126 (1996), no. 3, 599-616.
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Solution curves for two classes of boundary-value problems
with T. Ouyang
Nonlinear Anal. 27 (1996), no. 9, 1031–1047.
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Multiplicity results for two classes of boundary-value problems
with T. Ouyang
SIAM J. Math. Anal. 26 (1995), no. 1, 180-189.
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Exact multiplicity results for a class of boundary-value problems with cubic nonlinearities
with T. Ouyang
J. Math. Anal. Appl. 194 (1995), no. 1, 328–341.
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Exact multiplicity results for two classes of periodic equations
with T. Ouyang
J. Math. Anal. Appl. 194 (1995), no. 3, 763–779.
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Symmetry of positive solutions for elliptic problems in one dimension

Appl. Anal. 58 (1995), no. 3-4, 351-365.
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On computation of solution curves for semilinear elliptic problems
with T. Ouyang
Numer. Funct. Anal. Optim. 16 (1995), no. 1-2, 219–231.
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Review of "An Introduction to Partial Differential Equations" by M. Renardy and R.C. Rogers

SIAM Review 37 (1995), no. 5, 469-470.
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Periodic solutions of a differential equation (Problem Proposal 95-9)

SIAM Review (1995).
available in pdf
Homoclinic orbits for a class of symmetric Hamiltonian systems
with A. Lazer
Electron. J. Differential Equations 1994, No. 01, approx. 10 pp. (electronic only).
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On the dynamics of two classes of periodic ecological models

Oscillations in nonlinear systems: applications and numerical aspects. J. Comput. Appl. Math. 52 (1994), no. 1-3, 267–275.
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Exact multiplicity results for two classes of boundary value problems
with T. Ouyang
Differential Integral Equations 6 (1993), no. 6, 1507-1517.
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An algorithm for computing unstable solutions of semilinear boundary value problems

Computing 51 (1993), no. 3-4, 327-334.
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On periodic solutions for singular perturbation problems

Proc. Roy. Soc. Edinburgh Sect. A 120 (1992), no. 1-2, 143-152.
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Some new results on the periodic competition model

J. Math. Anal. Appl. 171 (1992), no. 1, 131-138.
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On computation of solutions of fully nonlinear elliptic problems
with G. Choudury
J. Comput. Appl. Math. 41 (1992), no. 3, 301-311.
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Dynamics of the Lotka-Volterra systems with diffusion

Appl. Anal. 44 (1992), no. 3-4, 191-207.
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A computer-assisted study of periodic parabolic systems

Appl. Math. Comput. 50 (1992), no. 2-3, 203-221.
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Approximating optimal controls for elliptic obstacle problem by monotone iteration schemes
with V. Barbu
Numer. Funct. Anal. Optim. 12 (1991), no. 5-6, 429-442.
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Computation of solutions of nonlinear boundary value problems
with G. Choudury
Comput. Math. Appl. 22 (1991), no. 8, 49-55.
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An upper bound (Problem Proposal 91-8)

SIAM Review (1991).
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Monotone iterations for nonlinear obstacle problem
with A. Leung and S. Stojanovic
J. Austral. Math. Soc. Ser. B 31 (1990), no. 3, 259-276.
available in pdf
On nonlinear singular perturbation problems

Nonlinear Anal. 15 (1990), no. 5, 467–478.
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On existence of periodic solutions for a class of quasilinear noncoercive problems

Funkcial. Ekvac. 33 (1990), no. 1, 127–138.
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On existence of solutions for two classes of nonlinear problems

Comm. Partial Differential Equations 14 (1989), no. 4, 519–539.
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On existence of solutions for a class of fully nonlinear noncoercive problems

J. Math. Anal. Appl. 137 (1989), no. 2, 477–484.
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A maximum principle for fourth order ordinary differential equations

Appl. Anal. 33 (1989), no. 3-4, 267-273.
available in pdf
On computation of solutions of elliptic systems

Numer. Funct. Anal. Optim. 10 (1989), no. 9-10, 977-990.
available in pdf
On existence of solutions for several classes of free boundary problems

SIAM J. Math. Anal. 19 (1988), no. 4, 814–823.
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On blow up of solutions of nonlinear evolution equations

Proc. Amer. Math. Soc. 103 (1988), no. 1, 189-197.
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Computation of displacements for nonlinear elastic beam models using monotone iterations

Internat. J. Math. Math. Sci. 11 (1988), no. 1, 121–128.
available in pdf
On the existence and uniqueness of positive steady states in the Volterra-Lotka ecological models with diffusion
with A. Leung
Appl. Anal. 26 (1987), no. 2, 145-160.
available in pdf
Existence of solutions for a class of semilinear noncoercive problems

Nonlinear Anal. 10 (1986), no. 12, 1471–1476.
available in pdf
Existence and approximation of solutions of inverse-positive problems

Nonlinear Anal. 10 (1986), no. 9, 863–872.
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A general monotone scheme for elliptic systems with applications to ecological models
with A. Leung
Proc. Roy. Soc. Edinburgh Sect. A 102 (1986), no. 3-4, 315-325.
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On application of the monotone iteration scheme to noncoercive elliptic and hyperbolic problems

Nonlinear Anal. 8 (1984), no. 2, 97–105.
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Existence of periodic solutions for a class of nonlinear problems

Nonlinear Anal. 7 (1983), no. 8, 873–879.
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Existence of solutions for a class of nonlinear noncoercive problems

Comm. Partial Differential Equations 8 (1983), no. 8, 819–846.
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