## Dr. Leigh C. Becker

## Publications, Presentations and Teaching Awards

# Teaching Awards

#

May 2012 ─ The Dr. Marguerite Cooper Distinguished Professor Award for Excellence In Teaching.

May 2002 ─ The Dr. Marguerite Cooper Distinguished Professor Award for Excellence In Teaching.

Research Papers

Leigh C. Becker

Solutions of Fractional Relaxation Equations via Resolvents and a Variation of Parameters Formula.

(work in progress).

Leigh C. Becker and Ioannis K. Purnaras

Fractional Relaxation Equations and a Cauchy Formula for Repeated Integration of the Resolvent.

Advances in the Theory of Nonlinear Analysis and its Applications, Vol. 2 (2018) No. 1, 11-32.

Leigh C. Becker

Properties of the resolvent of a linear Abel integral equation: implications for a complementary fractional equation.

Electron. J. Qual. Theory Differ. Equ., No. 64, 2016, pp. 1-38.

L. C. Becker, T. A. Burton, and I. K. Purnaras

Integral and fractional equations, positive solutions, and Schaefer`s fixed point theorem,

Opuscula Mathematica, 36, No. 4 (2016), pp. 431-458.

L. C. Becker, T. A. Burton, and I. K. Purnaras

Existence of Solutions of Nonlinear Fractional Differential Equations of Riemann-Liouville Type,

Journal of Fractional Calculus and Applications, Vol. 7(2), July 2016, pp. 20-39.

L. C. Becker, T. A. Burton, and I. K. Purnaras

Fractional differential equations, transformations, and fixed points,

Dynamics of Continuous, Discrete and Impulsive Systems, Series A: Mathematical Analysis,

Vol. 22, No. 5 (2015), pp. 333-361.

L. C. Becker, T. A. Burton, and I. K. Purnaras

An Inversion of a Fractional Differential Equation and Fixed Points,

Nonlinear Dynamics and Systems Theory, Vol. 15, No. 3, 2015, pp. 242-271.

L. C. Becker, T. A. Burton, and I. K. Purnaras

Complementary equations: a fractional differential equation and a Volterra integral equation,

Electron. J. Qual. Theory Differ. Equ., No. 12, 2015, pp. 1-24.

Leigh C. Becker

Resolvents and solutions of singular Volterra integral equations with separable kernels

Applied Mathematics and Computation, 219, Issue 24 (15 August 2013), pp. 11265-11277.

Leigh C. Becker

Resolvents for weakly singular kernels and fractional differential equations

Nonlinear Analysis: Theory, Methods & Applications, Vol. 75, Issue 13 (September 2012), pp. 4839-4861.

Leigh C. Becker, T. A. Burton, I. K. Purnaras

Singular integral equations, Liapunov functionals, and resolvents

Nonlinear Analysis: Theory, Methods & Applications, 75, Issue 7 (May 2012), pp. 3277-33291.

Digital Object Identifier Information: 10.1016/j.na.2011.10.050

Leigh C. Becker

Resolvents and solutions of weakly singular linear Volterra integral equations

Nonlinear Analysis: Theory, Methods & Applications, 74, Issue 5 (March 2011), pp. 1892-1912.

Digital Object Identifier Information: 10.1016/j.na.2010.10.060

Leigh C. Becker

Uniformly Continuous L1 Solutions of Volterra Equations and Global Asymptotic Stability

CUBO, A Mathematical Journal, Vol 11, No. 3 (August 2009), pp. 1-24.

This paper appears in the special issue: Qualitative Properties of Functional Equations.

The series of papers in this issue begins with a preface and overview by T.A. Burton (guest editor).

Leigh C. Becker

Function bounds for solutions of Volterra equations and exponential asymptotic stability

Nonlinear Analysis: Theory, Methods & Applications 67, Issue 2 (July 2007), pp. 382-397.

Digital Object Identifier Information: 10.1016/j.na.2006.05.016

Leigh C. Becker

Principal matrix solutions and variation of parameters for a Volterra integro-differential equation and its adjoint

Electronic Journal of Qualitative Theory of Differential Equations, No. 14 (2006), pp. 1-22.

Note: A version of the paper can also be found on pp. 36 - 54 in the monograph:

Liapunov Theory for Integral Equations with Singular Kernels and Fractional Differential Equations

by T. A. Burton, amazon.com (2012). To view this version on the web, click on the above link and search for Becker; then select page 36.

Leigh C. Becker and T. A. Burton

Stability, fixed points and inverses of delays

Proceedings of the Royal Society of Edinburgh, 136A, No. 2 (May 2006), pp. 245-275.

Abstract (PDF format) Abstract (DVI format) Abstract (IngentaConnect [www.ingentaconnect.com])

Leigh C. Becker and T. A. Burton

Jensen's Inequality and Liapunov's Direct Method

CUBO, A Mathematical Journal, Vol 6, No. 3 (October 2004), pp. 67-90.

Abstract (PDF format) Abstract (DVI format) Abstract (PostScript format) Abstract (MS Word)

Download preprint (PDF format)

Photos of authors:

Leigh Becker & T. A. Burton at the AMS Meeting in Portland, Oregon June 20-22, 2002

Leigh Becker at the Oregon coast June 23, 2002

L. C. Becker, T. A. Burton, and S. Zhang

Functional differential equations and Jensen's inequality

Journal of Mathematical Analysis and Applications, 138, Issue 1 (1989), Academic Press, Orlando, pp. 137-156.

L. C. Becker and T. A. Burton

Asymptotic stability criteria for delay-differential equations

Proceedings of the Royal Society of Edinburgh 110A (1988), pp. 31-44.

L. C. Becker, T. A. Burton, and T. Krisztin

Floquet Theory for a Volterra equation

Journal of the London Mathematical Society (2) 37 (1988), pp. 141-147.

L. C. Becker, T. A. Burton, and S. Zhang

Functional differential equations and Jensen's Inequality

Dynamics of Infinite Dimensional Systems (Hale, J. K. and Chow, S. N., eds.)

NATO ASI Series, Vol. F37 (1987), Springer-Verlag, New York, pp. 31-38.

Other Publications

Leigh C. Becker

Review of the monograph:

Liapunov theory for integral equations with singular kernels and fractional differential equations

by Theodore Allen Burton in Mathematical Reviews (MR3088486), American Mathematical Society (2014).

(Amazon.com link to monograph)

Leigh C. Becker

Constant Delay Differential Equations and the Method of Steps

Maple Application Center (June 8, 2009).

(To view the HTML version or to download the Maple worksheet, click on the above hyperlink.)

Leigh C. Becker

Scalar Volterra Integro-Differential Equations

Maple Application Center (August 2007).

(To view the HTML version or to download the Maple worksheet, click on the above hyperlink.)

Leigh C. Becker and Micah Wheeler

Numerical and Graphical Solutions of Volterra Integral Equations of the Second Kind

Maple Application Center (June 2005).

(To view the HTML version or to download the Maple worksheet, click on the above hyperlink.)

Leigh C. Becker

Setting the Stage with APR Problems

Mathematics and Computer Education (1984, Fall), 18(3), pp. 189 - 195.

Ph.D. Dissertation

Leigh C. Becker

Stability Considerations for Volterra Integro-differential Equations

Ph.D. dissertation, Southern Illinois University at Carbondale, Advisor: T. A. Burton, Dissertation Research Award 1979.

Download dissertation (pdf format)

Note: Revised and more concise versions of the part of the dissertation dealing with the principal matrix solution (aka the resolvent)

can also be found in

(1) the paper ''Principal matrix solutions and variation of parameters for a Volterra integro-differential equation and its adjoint'',

Electronic Journal of Qualitative Theory of Differential Equations, No. 14 (2006), pp. 1 - 22

and

(2) the monograph Liapunov Theory for Integral Equations with Singular Kernels and Fractional Differential Equations by T. A. Burton, amazon.com (2012).

(To view this version on the web, click on the above link and search for Becker; then select page 36.)

Acknowledgements of solutions of problems

Problem 82-19: A binomial coefficient summation, SIAM Review 25 (1983), p. 576.

Problem 184: TYCMJ (now the College Mathematics Journal) 13, No. 3 (1982), p. 211.

Problem 194: TYCMJ (now the College Mathematics Journal) 13, No. 5 (1982), p. 341.

Unpublished Textbook with ancillaries

Leigh C. Becker

Ordinary Differential Equations: Concepts, Methods, and Models

XanEdu Press, 2015 - 2016 edition, 518 pages (work in progress)

Leigh C. Becker

Complete Solutions of Selected Problems for Ordinary Differential Equations: Concepts, Methods, and Models

Web site: http:///facstaff.cbu.edu/~lbecker/231Ans.htm

Leigh C. Becker

Maple worksheets for Ordinary Differential Equations: Concepts, Methods, and Models

Web site: http://facstaff.cbu.edu/~lbecker/231maple.ht

Miscellanea

L. C. Becker

ODE Experiments Using Maple, CBU Press, 1998.

L. C. Becker

The function y = (1 + 1/x)x and the number e, (Lab 4 in Laboratory Manual for Calculus II), Christian Brothers University (1991), 29-39.

Reviewed research papers & monographs for the following:

Applied Mathematics and Computation

Applied Mathematics Letters

Differential Equations & Applications

Electronic Journal of Differential Equations

Electronic Journal of Qualitative Theory of Differential Equations

Filomat

International Journal of Dynamical Systems and Differential Equations

Journal of Integral Equations and Applications

Journal of Mathematical Analysis and Applications

Journal of Taibah University for Science

Mathematical Reviews

Nonlinear Analysis

Positivity

Presentations

November 15, 2016 ─ Rhodes College, Introduction to fractional derivatives.

October 17, 2015 ─ American Mathematical Society Southeastern Sectional Meeting hosted by the University of Memphis.

Special session speaker on analysis of differential and integral equations: Existence of solutions of nonlinear fractional differential

equations of Riemann-Liouville type.

October 11, 2014 ─ The 34th Southeastern Atlantic Regional Conference on Differential Equations hosted by the University of Memphis.

Title of talk: A resolvent and fractional differential equation.

November 7, 2009 ─ Differential Equations Weekend Conference hosted by the University of Memphis.

Title of talk: Uniformly continuous and asymptotically stable solutions of Volterra integro-differential equations.

July 4, 2008 ─ WCNA 2008 in Orlando, Florida (Fifth World Congress of Nonlinear Analysts, July 2 - July 9). Presented a 45-minute

invited lecture entitled Uniformly continuous L1 solutions of Volterra equations and global asymptotic stability in the session

''Qualitative Properties and Application of Functional Equations'' and chaired part of this session.

Photos:

Speakers at the ''Qualitative Properties & Application of Functional Equations'' session (July 4 & 5, 2008):

Muhammad Islam, Min He, Alphonso Casal, Colleen Kirk, T. A. Burton, Tetsuo Furumochi, Leigh C. Becker, Bo Zhang, Youssef Raffoul, Liancheng Wang, & Henri Schurz (not shown).

Lynne Marie Becker at the Hyatt Grand Cypress Resort Pool in Orlando (July 7, 2008).

March 16, 2007 ─ Asymptotic Stability for Scalar Linear Volterra Integro-Differential Equations, Mathematical Association of America Southeastern Section 86th Annual Meeting, Georgia Southern University, Statesboro, GA.

January 5-8, 2005 ─ Joint Mathematics Meetings in Atlanta, Georgia. Co-chaired the American Mathematical Society Session on Ordinary Differential Equations (Jan. 7). Presented a paper (coauthor T.A. Burton, Northwest Research Institute) entitled Fixed Points and Stability of a Volterra Equation with Variable Delay.

July 1, 2004 ─ WCNA 2004 in Orlando, Florida (Fourth World Congress of Nonlinear Analysts, June 30 - July 7). Presented a 45-minute invited lecture

entitled Stability, Fixed Points, and Inverses of Delays in the session ''Application of Fixed Point Theory to Functional Equations'' and chaired part of this session.

March 26, 2004 ─ Fixed Points and Stability of an Equation with Variable Delay (with T. A. Burton), Mathematical Association of America Southeastern Section meeting,

Austin Peay State University, Clarksville, TN.

June 21, 2002 ─ Jensen's Inequality and Liapunov's Direct Method, ''Special Session on ''Qualitative Properties and Applications of Functional Equations'', American Mathematical Society Meeting in Portland, Oregon.

November 6, 2001 ─ Mathematical Scientists & What They Do, Mu Alpha Theta Induction Ceremony 2001 at Ridgeway High School.

October 21, 2000 ─ At the request of Institutional Advancement, talked at Sacred Heart Catholic Church in Memphis about CBU.

September 25, 1997 ─ Maple in CBU Mathematics Courses, CBU President's Circle.

August 22, 1996 ─ presented a demonstration of a Maple lesson entitled Integral Curves and Contours of Exact Equations, CBU President's Workshop.

February 9, 1996 ─ talk to the CBU student chapter of the MAA entitled She loves me, she loves me not (linear system of ordinary differential equations).

August 30, 1990 ─ Simulating and Stimulating with Baseball Cards, Mathematics & Computer In-Service, Ridgeway High School, Memphis.

November 10-11, 1989 ─ Asymptotic Stability Criteria for Delay-Differential Equations, 18th Midwest Differential Equations Conference, Southern Illinois University at Carbondale.

August 1988 ─ conducted two days of a four-day workshop for Memphis high school mathematics teachers on elementary differential equations in the BC calculus curriculum. Participants were shown how to use the HP-28S calculator when solving and graphing ordinary differential equations.

December 2, 1987 ─ Floquet Theory for Volterra Equations I, Differential Equations Seminar, Memphis State University.

November 25, 1987 ─ Floquet Theory for Volterra Equations I, Differential Equations Seminar, Memphis State University.

October 23-24, 1987 ─ co-author of Floquet Theory for Volterra Equations, Midwest-Southeast Differential Equations Conference, Vanderbilt University, Nashville, TN.

1983 ─ consulting work on a mathematical model of a shutter for a certain type of camera for DeZign Corporation.

May 13, 1982 ─ A Variation of Parameters Formula for Volterra Integral Equations, Mathematical Sciences Colloquium, Memphis State University.

April, 1982 ─ Setting the Stage with APR Problems, regional Mathematical Association of America meeting at Emory University in Atlanta, Georgia.

January 25, 1979 ─ Some Results for Volterra Integrodifferential Equations, American Mathematical Society Session on Ordinary Differential Equations, Biloxi, Mississippi.