CAMERON UNIVERSITY DEPARTMENT OF MATHEMATICS LAWTON, OKLAHOMA 73505-6377 U.S.A. NAME: IOANNIS K. ARGYROS DATE-PLACE OF BIRTH: 20 February 1956, Athens Greece CITIZENSHIP: U.S.A. ADDRESS: Cameron University Department of Mathematics Lawton, OK 73505 U.S.A. E-MAIL: ioannisa@cua.cameron.edu TELEPHONE: (405) 581-2908 or (405) 581-2481 (Office) (405) 355-4504 (Home) 1.STUDIES (a) 1983-1984 : Ph.D. in Mathematics. University of Georgia, Athens, GA. (b) 1982-1983 : M.Sc. in Mathematics. University of Georgia, Athens, GA. (c) 1974-1979 : B.Sc. in Mathematics. University of Athens, Greece. 2. ACADEMIC AND TEACHING EXPERIENCE (a) 1994-Present : Full Professor, Cameron University, U.S.A. (b) 1993-1994 : Tenured Professor, Cameron University, U.S.A. (c) 1990-1993 : Associate Professor, Cameron University, U.S.A. (d) 1986-1990 : Assistant Professor, New Mexico State University, U.S.A. (e) 1984-1986 : Assistant Professor, University of Iowa, U.S.A. (f) 1982-1984 : Teaching-Research Assistant University of Georgia, U.S.A. (g) 1979-1982 : Serving the Greek Army as Technical Consultant, GREECE. COURSES TAUGHT GRADUATE (a) Advanced Applied Analysis (b) Numerical solution of Functional Equations (c) The Finite Difference & the Finite-Element Method for O.D.E. & P.D.E. (d) Advanced Numerical Analysis (e) Thesis in Mathematics (f) Special topics in Functional Analysis and Numerical Functional Analysis. (g) Optimization UNDERGRADUATE COURSES (a) Calculus courses (b) Differential equations (c) Numerical Analysis (d) Linear Algebra (e) Real Analysis (f) History of Mathematics (g) Geometry (h) Statistics (i) Abstract Algebra (j) Independent Study in Mathematics (k) Matrix Algebra 3. TEACHING EFFECTIVENESS I believe that I have had some success in using computer softwares for some of the applied math courses taught in the department. Since my research area is applied mathematics it was not difficult for me to use existing software as well as produce my own. It has been desirable for students to use computer softwares as a facilitating tool in many courses. I have been attending seminars and converences as well as constantly reviewing the developments in my field in order to have a broad knowledge of Mathematical subjects. I am trying to be aware of its increasing relevance in our technological age, and be able to stimulate my students to understand and possibly use some of these concepts in their future careers. I am also concerned with the communication of these ideas to students. Throughout the course I try to make the concepts as understandable as possible by giving examples that help them relate these ideas to topics in that course. I have also provided opportunities to my students in which they can express their views to the class to sharpen their skills in discovering and communicating the concepts. I have used my teaching effectiveness throughout my teaching career. I have also produced a textbook (1993) (see (M) for details) to be used by students in Mathematics, Economics, Physics, Engineering and the applied sciences. A second textbook will be published in (1996) and a third in (1998). The books of course bear the Cameron University name. I have also reviewed a Numerical Analysis textbook entitled "Introduction to Numerical Analysis", by Kendall Atkinson, University of Iowa, published by Wiley and Jons (1992). The author in his preface recognizes and praises my talents in teaching and expresses his gratitude for my contribution in the improvement of his book. His textbook is considered to be the best book in Numerical Analysis in this country. I have helped several students be accepted in graduate programs at the top universities in this country. I have also helped them find jobs and still keep in contact with them and their careers after they leave the University. The following Ph.D. students have obtained their Ph.D. degree under my supervision: (a) Losta Mansor, Ph.D. dissertation title: Numerical Methods for solving singular perturbation problems appearing in elasticity and astrophysics, (1989). (b) Joan Peeples, Ph.D. dissertation title: Point to set mappings and oligopoly theory, (1989). Writer and Grader for the Funct-Num. Analysis, comprehensive examination (a) January 87, 88, 89 (b) August 87, 88 I prepared the Ph.D. Comprehensive Examination Guide in Fun. Analysis. (a) Chair, Master's Examination Committee, (b) Mitra Ashan, Spring, 1987. (c) Christopher Stuart, Spring, 1988. (d) Anis Shahrour, Fall, 1988. Member, Master's Examination Committee, (a) Juji Hiratsuka, Spring, 1987. (Dean's Representative, Art Department). (b) Alice Lynn Bertini, Spring, 1988. (c) Daniel Patrick Eshner, Summer, 1989. (Dean's Representative, Computer Sci.). Chair, Doctoral Oral Examination Committee, (a) Losta, Mansor, Fall 1987 and Summer 1989 (Final Defense). (b) Joan Peeples, Spring 1989 and Summer 1989 (Final Defense). Member, Doctoral Oral Examination Committee, (a) Aomar Ibenbrahim, Spring 1987. (b) Maragoudakis Christos, Spring 1988. (Dean's Representative for both, Electrical Engineering Department). 4. SCIENTIFIC ACTIVITY (A) Fields of interest: (a) Applied Analysis. (b) Numerical Functional Analysis. (c) Numerical Analysis, Numerical methods. (d) Optimization. (e) Parallel computing. (f) Differential-Integral Equations. (g) Fixed point theory. (h) Statistics. Most of my research falls into three categories, Applied Analysis, Numerical Analysis (Mainly Numerical Methods, Prediction Theory, Error Analysis, Numerical Functional Analysis), Optimization and Statistics. (B) APPLIED ANALYSIS The so-called polynomial operators are a natural generalization of linear operators. Equations in such operators are the linear space analog of ordinary polynomials in one or several variables over the fields of real or complex numbers. Such equations encompass a broad spectrum of applied problems including all linear equations. Often the polynomial nature of many non-linear problems goes unrecognized by researchers. This is most likely due to the fact that unlike polynomials in a single variable, polynomial operators have received little attention. Whether this situation is due to an inherent intractability of these operators or to simple oversight remains to be seen. Hopefully, one should be able to exploit their semi-linear character to wrest more extensive results for these equations than one can obtain in the general non-linear setting. Examples of equations involving polynomial operators can be found in the literature. My contribution in this area can be found in papers #3, 4, 6-12, 16, 22, 23, 25, 35, 84. Many of the equations of elasticity theory are of this type #3,4. The problem discussed there pertains to the buckling of a thin shallow spherical shell clamped at the edge and under uniform external pressure. Some equations in heat transfer, kinetic theory of gases and neutron transport, including the famous Chandrasekhar (Nobel in Physics, 1983) integral equation are of quadratic type. Numerical methods for finding small or large solutions of the above equations and their variations as well as results on the number of solutions of the above equations can be found in papers #1-4, 21, 24, 37, 55, 85, 99. Some pursuit and bending of beams problems can be formulated as polynomial equations. My investigations on such equations can be found in paper #6. Paper #11, contains results on the study of feedback systems containing an arbitrary finite number of time-varying amplifiers and the study of electromechanical networks containing an arbitrary number of time-varying nonlinear dissipative elements. Scientists that have worked in this area, agree that much work both of theoretical and computational nature remains to be done on polynomials in a normed linear space. A summary of some of the remaining problems can be found in my third book entitled The Solution of Polynomial Operator Equations in Abstract Spaces and Applications". It must certainly be mentioned that the existence theory is far from complete and what little is there it is confined to local small solutions in neighborhoods which are often of very small radius #1-5, 7-9, 13, 14, 17, 26, 30, 33. In my papers #5, 6, 8, 10, 23, 30, 33, 34, 37, 42, 69, 72, I have provided numerical methods for approximating distinct solutions of polynomial equations under various hypotheses. As far as I know the above mentioned authors are the only researchers that have worked on global existence theorems not related with contractions. My contribution in this area is contained in papers #7, 14, 23, 34, 35, 44, 69. Moreover for those of a qualitative rather than computational frame of mind, it has been suggested that polynomial operators should carry a Galois theory. Such a theory, should it exist, may be very limited, but nonetheless, interesting. The pessimistic note is prompted by the fact that a complete general spectral theory does not exist for polynomial operators. In an attempt to produce such a theory at least the way an analyst understands it, I wrote the relevant papers #18, 23, 34, 35, 45. Finally papers #15, 41, 45, 47, 83 deal with the related area of fixed point theory. (C) NUMERICAL ANALYSIS, NUMERICAL METHODS, PREDICTION THEORY, OPTIMIZATION The most important iterative procedures for solving nonlinear equations in a Banach space are undoubtedly the so called Newton-Like methods. Indeed, L.V. Kantorovich has given sufficient conditions for the quadratic convergence of Newton's iteration to a locally unique solution of the abstract nonlinear equation in Banach space. His conditions are in some sense the best possible. For the scalar case these conditions coincide with those given earlier by A. I. Ostrowski. Simple sharp apriori estimates were given independently (by different methods) by W. B. Gragg and R. A. Tapia. The method of nondiscrete mathematical induction was used later by V. Ptak, F. Potra, X. Chen, T. Yamamoto, P. Zabrejko, D. Ngyen, I. Moret et. al.; this method yields not only sharp apriori estimates but also convergence proofs through the induction theorem. This method, in which the rate of convergence is now a function and not a number, is closely related with the closed graph theorem. My contribution in this area can be found in the papers #19, 20, 31, 40, 44, 47, 50, 51, 57, 60, 61, 63, 65, 68, 70, 81, 82, 83,89, 90, 92, 95, 96, 97, 101, 125, 129, 145. One of the basic assumptions for the use of Newton's method is the condition that the Frechet derivative of the nonlinear operator involved be Frechet-differentiable. There are however interesting differential equations and singular integral equations (see for example the work of Etzio Venturino) where the nonlinear operator is only Holder continuous. It turns out that the error analysis of Newton-Like methods changes dramatically and the results obtained by the above authors do not hold in this setting. My contribution in this area can be found in the papers #19, 20, 31, 32, 38, 63, 68, 71, 100. Papers #65, 73, 104, 106 deal with the solutions of nonlinear operator equations containing a nondifferentiable term. Papers #61, 80, 89, 101, 104, 113 deal with the approximation of implicit functions. Papers #60, 66, 79, 81, 95, 104 deal with projection methods for the approximate solution of nonlinear equations. Papers #64, 125, 143 deal with iterative procedures for the solution of nonlinear equations in generalized Banach spaces. Papers #88, 114, 128, 130, 152 deal with inexact iterative procedures. Papers #54, 56, 67, 98, 124 deal with the solution of nonlinear operator equations and their discretizations in relation with the mesh-independence principle. Papers #82, 105, 116 deal with the solution of linear and nonlinear perturbed two-point boundary value problems with left, right and interior boundary layers. I have applied the above numerical methods, in particular Newton's and its variations to concrete integral equations arising in radiative transfer. See for example papers #21, 37. Since the numerical solution of integral equations is closely related to compact operators, I tried in the papers #28, 39, 53, 74 to find some results relating numerical methods and compactness. Work on this subject has already been conducted (see e.g. the work of P. Anselone & K. Atkinson), but the results so obtained are too general or too particular to be used for my purposes. Papers #91, 121, 131, 132, 133, 138, 140, 149, 153-218 deal with the convergence and error analysis of multipoint iterative methods in Banach spaces. Paper #103 deals with the introduction of an optimization algorithm based on the gradient projection technique and the Karmarkar's projective scaling method for linear programming. Papers 214-218 involve perturbed nondifferentiable operator equations on generalized Banach spaces with a convergence structure and Newton methods. (D) STATISTICS Paper #123 deals with t-estimates of parameters of general nonlinear models in finite dimensional spaces. The method is highly insensitive to outliers. It can also be applied to solve a system of nonlinear equations. (E) ECONOMY Papers #62, 74, 76, 93, 94, 107, 134, 139 deal with the convergence of iteration schemes generated by the recursive application of a point-to-set mapping. Our results have been applied to solve dynamic economic as well as input-output systems. (F) SEMINARS At the University of Iowa I gave eight seminars per academic year in Numerical and Functional Analysis. I continue doing so at New Mexico State and Cameron University. During my talks I either explain my current work or I go through some papers of areas mentioned in (A)-(E). (G) PAPERS PRESENTED (1) University of Berkeley. International Summer Institute on Nonlinear Functional Analysis and Applications. (1983). Title : "On a contraction theorem and applications". (2) Los Alamos Laboratories (organizers). Conference on Invariant Imbedding, transport theory, and Integral Equations. Eldorado Hotel, Santa Fe, N.M. (1988). Title: "On a class of nonlinear equations arising in neutron transport." (3) Annual meeting of the American Mathematical Society #863. San Francisco, California June 16-19, 1991. Title: "On the convergence of Algorithmic models" (Chairman of the Numerical Analysis Session (#516), 7:00 p.m. - 9:55 p.m. Thursday Jan. 17, 1991). (4) Mathematical Association of America, Oklahoma-Arkansas Section, Spring 1991. Title: "Improved bounds for the zeros of polynomials". (5) Annual meeting of the American Mathematical Society #871. Baltimore, Maryland Jan. 8-11, 1992. Title: "On the midpoint iterative method for solving nonlinear operator equations in Banach spaces". (6) CAM 92, Edmond, OK, March 27, 1992. University of Central Oklahoma, Title: "On the secant method under weak assumptions" (7) CAM 93, Edmond, OK, February 5, 1993. University of Central Oklahoma, Title: "On a two-point Newton method in Banach spaces of order four and applications". (8) As in (7), Title: "Sufficient convergence conditions for itertions schemes modeled by point-to-set mappings". (9) As in (7), Title: "On a two-point Newton method in Banach spaces and the Ptak error estimates". (10) CAM 94, Edmond, OK, February 4, 1994. University of Central Oklahoma, Title: "On the monotone convergence of fast iterative methods in partially ordered topological spaces". (11) CAM 94, Edmond, OK, February 4, 1994. University of Central Oklahoma, Title: "On a multistep Newton method in Banach spaces and the Ptak error estimates". (12) 56th Annual Meeting of the Oklahoma-Arkansas session of the Mathematical Association of America, March 24, 1994. Title: "On an inequality from applied analysis". (Analysis section). It was held at the University of Searcy, Searcy, Arkansas. (13) CAM 95, Edmond, OK, February 10, 1995, University of Central Oklahoma. "A mesh independence principle for nonlinear equations in Banach spaces and their discretizations. (14) 57th Annuale Meeting of the Oklahoma-Arkansas session of the Mathematical Association of America, March 1995. Title: "On the discretization of Newton-like methods", Southwestern Oklahoma State University, Weatherford, Oklahoma. (H) SELECTED LECTURES PRESENTED University of Georgia, 1982-1984 University of Iowa, 1984-1986 State University of Iowa, 1985 Northern University of Virginia, 1986, 1988 New Mexico State University, 1986-1990 University of Ohio, 1986 University of Iowa, 1986, 1988 University of New York, 1986-1988 University of Texas at El-Paso, 1987-1990 University of Arizona, I.E.D., 1989, 1990 Portland State University, 1990 Cameron University, 1990 University of Central Oklahoma, 1992, 1993 University of Cyprus, Nicosia Cyprus, 1993 (I) OTHER MEETINGS ATTENDED (1) American Mathematical Society/Mathematical Association of America annual meetings, Denver, Colorado, 1983, & Phoenix, Arizona, 1989. (2) SIAM Mathematical meetings, Des Moines, Iowa, 1985. (3) International Conference on Theory and Applications of Differential Equations, Ohio University, Athens, Ohio, 1988. (4) Annual research conferences of the Bureau of the Cencus, Arlington , Virginia, March 21- 24, 1993 and 1995. (J) COMPUTING EXPERIENCE IN (a) Fortran (b) Pascal (c) Cobol (d) Basic (e) Parallel computing. (K) GRANTS RECEIVED (a) New Mexico State University Grant, (1986), #1-3-43841, RC #87-01. (b) New Mexico State University International Grant, (1987), #1-3-44770. (c) U.S.A. Army, (1988-1990), #DAEA, 26-87-R-0013 (F.M.)ARMY jointly the Mechanical Engineering Department (N.M.S.U.). (d) Cameron University, research support July, (1992) - June, (1994). (L) RESEARCH ARTICLES The scientific papers listed below have been published in the following continents and at the top refereed journals in the following countries repeatedly: America: U.S.A., Canada, Chile. Europe: U.K., Sweeden, Belgium, Holland, Spain, Germany, Austria, Hungary, Chechoslovakia, Romania, Poland, Yugoslavia, Italy. Asia: Peoples Republic of China, Republic of China, India, Pakistan, Japan, Saudi Arabia. Australia: Australia. A 7% of the scientific papers listed below have been published jointly with professors Mohammad Tabatabai (Cameron, U.S.A.) Dong Chen (University of Arkansas, U.S.A.), Ferenc Szidarovszky (University of Arizona, U.S.A.) and Losta Mansor (Saudi Arabia). Finally 68% of the papers are published with the name of Cameron University. 1. A contribution to the theory of nonlinear operator equations in Banach space, Master of Science Dissertation, (1983). 2. Quadratic equations in Banach space, perturbation techniques and applications to Chandrasekhar's and related equations, Doctor of Philosophy Dissertation, (1984). 3. Quadratic equations and applications to Chandrasekhar's and related equations, Bull. Austral. Math. Soc. Vol. 32, 2, (1985), 275-292; Not. Amer. Math. Soc. 85T-46-142; Z.F.M.6074063 (1987); Math. Rev.87d: 47077, Gerard Lebourg (Paris). 4. On a contraction theorem and applications, Proc. Amer. Math. Soc., Symposium on Nonlinear Functional Analysis and Applications, 45, 1, (1986) 51-53; Math. Rev. 87h: 65108, Sh.Singh, Z.F.M.6224077 (1988). 5. Iterations converging to distinct solutions of some nonlinear equations in Banach space, Internat. J. Math. & Math. Sci. Vol. 9 No. 3 (1986), 583-587; Z.F.M.61447044 (1986); Math. Rev. 87j47097, P.P. Zabrejko (Minsk). 6. On the cardinality of solutions of multilinear differential equations and applications, Internat. J. Math. & Math. Sci. Vol. 9 No. 4 (1986), 757-766. Math. Rev. 88e34017, Achmadjon Soleev (Samarkand); Z.F.M.66334008 (89), A. Soleev. 7. Uniqueness-Existence of solutions of polynomial equations in linear space, P.U.J.M., Vol. XIX, (1986), 39-57; Z.F.M.62547050 (1988); Math. Rev. 88g47116 B.G. Pachpatte (6-Mara). 8. On a theorem for finding "large" solutions of multilinear equations in Banach space, P.U.J.M., Vol. XIX, (1986), 29-37; Z.F.M.62547051 (1988); Math. Rev. 88g47115, B.G. Pachpatte (6-Mara). 9. On the approximation of some nonlinear equations, Aequationes Mathematicae 32, (1987), 87-95; Z.F.M.61447043 (1986); Math. Rev. 88g47124, P.P. Zabrejko (Minsk). 10. An Improved condition for solving multilinear equations, P.U.J.M., Vol. XX, (1987), 43-46; Math. Rev. 89c47075; Z.F.M.64747015, (1989). 11. On a class of nonlinear equations, Tamkang J. Math. Vol. 18 No 2, (1987), 19-25; Math. Rev. 89f47091, Ramendra Krishna Bose (1-SUNYF); Z.F.M.65347042, (1989), J. Appel. 12. On polynomial equations in Banach space, perturbations techniques and applications, Internat. J. Math. & Math. Sci. Vol. 10 No. 1, (1987), 69-78; Math. Rev. 88c47123, Heinrich Steinlein (Munich); Z.F.M.61747038, (1987). 13. A note on quadratic equations in Banach space, P.U.J.M., Vol. XX, (1987), 47-50; Math. Rev. 89c47076; Z.F.M.64747016, (1989). 14. Quadratic finite rank operator equations in Banach space, Tamkang J. Math. Vol. 18 No 4. (1987), 8-19; Z.F.M.66247011(89); Math. Rev. 89k47100, Nicole Brillouet- Belluot, (Nantes). 15. On some Theorems of Mishra Ciric and Iseki, Mat. Vesnik, Vol. 39, (1987), 377- 380; Math. Rev. 89c54083; Z.F.M.64854035, (1989). 16. An iterative solution of the polynomial equation in Banach space, Bull. Inst. Math. Acad. Sin. Vol. 15, No. 4. (1987), 403-410, (Not. in Math. Rev. 89) 47H17, 46G99, 58C15. 17. A survey on the ideals of the space of bounded linear operators on a separable Hilbert space, Rev. Acad. Ci. Exactas Fis. Quim. Nat. Zaragoza, II. Ser. 42, (1987), 24-43; Math. Rev. 89g47059. 18. On the solution by series of some nonlinear equations, Rev. Acad. Ci. Exactas Fis. Quim. Nat. Zaragoza, II. Ser. 42, (1987), 18-23. Z.F.M.64947048, (1989). Math. Rev. 90f65085, V.V. Vasin (Sverdlosk). 19. Newton-Like methods under mild differentiability conditions with error analysis, Bull. Austral. Math. Soc. Vol. 37, 1, (1988), 131-147; Z.F.M.62965061, (1988), S. Reich; Math. Rev. 89b65142, A.V. Dzhishkariani (Tbilisi). 20. On Newton's method and nondiscrete mathematical induction, Bull. Austral. Math. Soc. Vol. 38, (1988), 131-140; Math. Rev. 90a65136, A.M. Galperin (Ben-Gurion Intern. Airp.). 21. On a class of nonlinear integral equations arising in neutron transport, Aequationes Mathematicae. Vol. 35, (1988), 99-111; Math. Rev. 89M47058 H.E. Gollwitzer (1-DREX). 22. On multilinear equations, Pr. Rev. Mat. Vol. 14, July, (1988), 95-105. 23. New ways for finding solutions of polynomial equations in Banach space, Tamkang J. Math. Vol. 19, 1, (1988), 37-42. Math Rev. 90f47093, V. V. Vasin, (Sverdlosk). 24. On a new iteration for solving the Chandrasekhar's H-equations, Pr. Rev. Mat. No 15, (1988), 21-31. 25. On a new iteration for solving polynomial equations in Banach space, Funct. et Approx. Comment. Math. Vol. XIX (1988). 26. Conditions for faster convergence of contraction sequences to the fixed points of some equations in Banach space, Tamkang J. Math. Vol. 19, 3, (1988), 19-22. Math. Rev. 90j47074, Roman Manka (Mogilno). 27. On some nonlinear equations, Pr. Rev. Mat. No 15, (1988), 75-82. 28. On the approximation of solutions of compact operator equations, PR. Rev. Mat. Vol. 14, July, (1988), 29-46. 29. Approximating the fixed points of some nonlinear equations, Mathem. Slovaca, 38, #4 (1988), 409-417; Z.F.M.667 (1989), S.L. Singh. Math. Rev. 90g47109 (O.P.Kapoor (6-11TK)). 30. Some sufficient conditions for finding a second solution of the quadratic equation in Banach space, Mathem. Slovaca, 4, (1988). Math. Rev. 90g47108. (O.P. Kapoor (6-11TK)). 31. Concerning the approximate solutions of operator equations in Hilbert space under mild differentiability conditions, Tamkang J. Math. Vol 19, 4, (1988), 81-87. 32. On the Secant method and fixed points of nonlinear equations, Monatshefte fur Mathematik, 106, (1988), 85-94; Z.F.M.65265043, (1989). Math. Rev. 90b6511, A.M.Galperin, Ben-Gurion Intern. Airport. 33. An iterative procedure for finding "large" solutions of the quadratic equation in Banach space, P.U.J.M. Vol.XXI, (1988), 13-21. 34. Vietta-Like relations in Banach space, Rev.Acad.Ci.Exactas Fis.Quim.Nat.Zaragoza, I, Ser.43, (1988), 103-107. Math. Rev. 47f47095, V.V. Vasin, (Sverdlovsk). 35. A global theorem for the solutions of polynomial equations, Rev. Acad. Ci. Exactas Fis. Quim. Nat. Zaragoza, I, Ser. 43, (1988), 93-101. Math. Rev. 90f47094, V.V. Vasin, (Sverdlosk). 36. Concerning the convergence of Newton's method, P.U.J.M. Vol. XXI, (1988), 1-11. 37. On the number of solutions of some integral equations arising in radiative transfer, Internat. J. Math. & Math. Sci. Vol. 12, No. 2, (1989), 297-304. Math. Rev. 90h86004 S. Rajasekar (Ticuchirapalli). 38. On the approximate solutions of operator equations in Hilbert space under mild differentiability conditions, J. Pure & Appl. Sc. Vol. 8, No. 1, (1989), 51-56. 39. On the fixed points of some compact operator equations, Tamkang J. Math. vol. 20, No. 3, (1989), 203-209. Math Rev. 91a47088 Jing Xian Sum (PRC-Shan). 40. Error bounds for a certain class of Newton-Like methods, Tamkang J. Math. Vol. 20, No. 4, (1989). 41. On a generalization of fixed and common fixed point theorems of operators in complete metric spaces, Rev. Mat. Cubo, 5, (1989), 17-25. 42. Approximating distinct solutions of quadratic equations in Banach space, Rev. Mat. Cubo, 5, (1989), 1-16. 43. Concerning the convergence of iterates to fixed points of nonlinear equations in Banach space, Bull. Malays. Math. Soc. Vol. 12, 2, (1989), 15-24. 44. A series solution of the quadratic equation in Banach space, Chinese J. Math. Vol. 27, No. 4, (1989). Math. Rev. 90k47131. 45. On a fixed point in a 2-Banach space, Rev. Acad. Ciencias, Zaragoza 44, (1989), 19-21. Math. Rev. 91a47077. 46. Some matrices in oligopoly theory, New Mexico J. Sci. 29, 1, (1989), 22. 47. On a theorem of Fisher and Khan, Rev. Acad. Ciencias, Zaragoza, 44, (1989), 13- 17. 48. On quadratic equations, Mathematica-Rev. Anal. Numer. Theor. Approximation 18, 1, (1989), 19-26. 49. Concerning the approximate solutions of nonlinear functional equations under mild differentiability conditions. Bull. Malays. Math. Soc. Vol. 12, 1, (1989), 55-65. 50. On the convergence of certain iterations to the fixed points of nonlinear equations, Annales sectio computatorica. Ann. Univ. Sci. Budapest. Sect. Computing 9, (1989), 21-31. 51. On the secant method and nondiscrete mathematical induction. Mathematica-Revue D'analyse Numerique et de theorie de l'approximation tome 18, No. 1, (1989), 27-36. 52. On Newton's method for solving nonlinear equations and multilinear projections, Functiones et approximatio Comment. Math., XIX, (1990), 41-52. 53. Nonlinear operator equations and pointwise convergence, Functiones et approximatio Comment. Math., XIX, (1990), 29-39. 54. Iterations converging faster than Newton's method to the solutions of nonlinear equations in Banach space, Functiones et approximatio Comment. Math., XIX, (1990), 23-28. 55. On some quadratic integral equations, Functiones et Approximmatio, XIX, (1990), 159-166. 56. A mesh independence principle for nonlinear equations using Newton's method and nonlinear projections, Rev. Acad. Ciencias. Zaragoza 45, (1990), 19-35. 57. Error bounds for the modified secant method, BIT, 30, (1990), 92-100. 58. Improved error bounds for a certain class of Newton-Like methods, J. Approximation Theory and its Applications, (6:1), (1990), 80-98. 59. On the solution of some equations satisfying certain differential equations, P.U.J.M., Vol XXIII, (1990), 47-59. 60. On some projection methods for approximating the fixed points of nonlinear equations in Banach space, Tamkang J. Math., Vol. 21, 4, (1990), 351-357. 61. On some projection methods for the approximation of implicit functions, Appl. Math. Lett. Vol. 3, No. 2, (1990), 5-7. Math. Rev. 91b65066. 62. On the monotone convergence of some iterative procedures in partially ordered Banach spaces, Tamkang J. Math. Vol. 21, No. 3, (1990), 269-277. 63. The Newton-Kantorovich method under mild differentiability conditions and the Ptak error estimates, Monatshefte fur Mathematik, Vol. 109, No. 3, (1990). 64. The secant method in generalized Banach spaces, Appl. Math. & Comput., 39, (1990), 111-121. 65. On the solution of equations with nondifferentiable operators and Ptak error estimates, BIT, 30, (1990), 752-754. 66. On some projection methods for enclosing the root of a nonlinear operator equation, P.U.J.M., Vol XXIII, (1990), 35-46. 67. A mesh independence principle for operator equations and their discretizations under mild differentiability conditions, Computing, 45, (1990), 265-268. 68. On Newton's method under mild differentiability conditions, Arabian J. Math. Vol. 15, 1, (1990), 233-239. 69. Remarks on quadratic equations in Banach space, Intern. J. Math. & Math. Sc., Vol. 13, No. 3, (1990), 611-616. 70. On the improvement of the speed of convergence of some iterations converging to solutions of quadratic equations, Acta Math. Hungarica, Vol. 57/3-4, (1990), 71. A note on Newton's method, Rev. Acad. Ciencias Zaragoza, 45, (1990), 37-45. 72. On the solution of compact linear and quadratic operator equations in Hilbert space, Rev. Acad. Ciencias Zaragoza, 45, (1990), 47-52. 73. On some generalized projection methods for solving nonlinear operator equations with a nondifferentiable term, Bull. Malays. Math. J., Vol 13, No. 2, (1990), 85-91. 74. Comparison theorems for algorithmic models, Applied Math. and Comput., Vol. 40, No. 2, Nov. (1990), 179-187. 75. On an iterative algorithm for solving nonlinear equations, Beitrage zur Numerischen Math. (Renamed Z.A.A.), Vol. 10, No. 1, (1991), 83-92. 76. On time dependent multistep dynamic processes with set valued iteration functions on partially ordered topological spaces, Bull. Austal. Math. So., Vol. 43, (1991), 51-61. 77. Error bounds for the secant method, Math. Slovaca, Vol. 41, 1, (1991), 69-82. 78. On the approximate solutions of nonlinear functional equations under mild differentiability conditions, Acta Math. Hungarica, Vol. 58 (1-2), (1991), 3-7. 79. On the convergence of some projection methods with perturbation, J. Comput. and Appl. Math. 36, (1991), 255-258. 80. On an application of a modification of the Zincenko method to the approximation of implicit functions, Z.A.A. 10, 3(1991), 391-396. 81. On some projection methods for solving nonlinear operator equations with a nondifferentiable term, Rev. Academia de Ciencias, Zaragoza, 46, (1991), 17-24. 82. Integral equations for two-point boundary value problems. Rev. Academia de Ciencias, Zaragoza 46, (1991), 25-35. 83. A fixed point theorem for orbitally continuous functions, Pr. Rev. Mat. Vol. 10, No. 7, (1991), 53-57. 84. Bounds for the zeros of polynomials, Rev. Academia de ciencias. Zaragoza, 47, (1992), 61-66.. 85. On a class of quadratic equations with perturbation, Functiones et Approximmatio, XX, (1992), 51-63. 86. On a new iteration for finding "almost" all solutions of the quadratic equation in Banach space, Studia Scientiarum Mathematicarum Hungarica, 27 (3-4), (1992), 361-368. 87. A Newton-like method for solving nonlinear equations in Banach space, Studia Scientiarum Mathematicarum Hungarica, 27 (3-4), (1992), 369-378. 88. On the convergence of nonstationary Newton-methods, Func. et. Approx., Vol. XXI, (1992), 7-16. 89. On an application of the Zincenko method to the approximation of implicit functions. Publicationes Mathematicae, Vol. 40/1-2; (1992), 43-49. 90. Improved error bounds for the modified secant method, Intern. J. Computer Math. Vol. 43, No. 1+2, (1992), 99-109. 91. On the midpoint method for solving nonlinear operator equations in Banach spaces, Appl. Math. Letters, Vol. 5, No. 4, (1992), 7-9. 92. On an application of a Newton-like method to the approximation of implicit functions, Math. Slovaca, 42, No. 3, (1992), 339-347. 93. On the monotone convergence of general Newton-like methods, Bull. Austral. Math. Soc., 45, (1992), 489-502. 94. Convergence of general iteration schemes, J. Math. Anal. and Applic., 168, No. 1, (1992), 42-62. 95. Some generalized projection methods for solving operator equations. Journ. Comp. Appl. Math., 39, No. 1 (1992), 1-6. 96. Sharp error bounds for a class of Newton-like methods under weak smoothness assumptions., Bull. Austral. Math. Soc. 45, (1992), 415-422. 97. Approximating Newton-like procedures, Appl. Math. Lett., Vol. 5, No. 1, (1992), 27-29. 98. On a mesh independence principle for operator equations and the secant method, Acta Math. Hungarica, 60, 1-2, (1992), 7-19. 99. On the solution of quadratic integral equations, P.U.J.M., Vol. XXV , (1992), 131- 143. 100. The secant method under weak assumptions, Proceedings CAM 92, Edmond, OK, June (1992), 7-18. 101. On the convergence of generalized Newton-methods and implicit functions, Journ. Comp. Appl. Math. 43, (1992), 335-342. 102. On a Stirling-like method, P.U.J.M., Vol. XXV (1992), 83-94. 103. An algorithm for solving nonlinear programming problems using Karmarkar's technique, Proceedings CAM 92, Edmond, OK, June (1992), 287-296. 104. On the approximate construction of implicit functions and Ptak error estimates, P.U.J.M., Vol. XXV, (1992), 95-98. 105. On the numerical solution of linear perturbed two-point boundary value problems with left, right and interior boundary layers. Arabian J. Science and Engineering 17 : 4B, October (1992), 611-624. 106. On the convergence of Newton-like methods. Tamkang J. Math, Vol. 23, No. 3 (1992), 165-170. 107. On the monotone convergence of algorithmic models, Applied Math. and Comput. 48, (2-3), (1992), 167-176. 108. Approximating Newton-like iterations in Banach space, P.U.J.M., Vol. XXV, (1992), 49-59. 109. On the approximation of quadratic equations in Banach space using finite rank operators, Rev. Academia de ciencias Zaragoza 47, (1992), 67-76. 110. Remarks on the convergence of Newton's method under Holder continuity conditions. Tamkang J. Math. Vol. 23, No. 4 (1992), 269-277. 111. On the solution of nonlinear operator equations in Banach space and their discretizations, Pure Mathematics and Applications, Ser. B, Vol. 3, No. 2-3-4 (1992), 157- 173. 112. On the convergence of optimization algorithms modeled by point-to-set mappings, Pure Mathematics and Applications, Ser. B, Vol. 3, No. 2-3-4, (1992), 77-86. 113. On an application of a modification of a Newton-Like method to the approximation of implicit functions, Bull. Malays Math. Soc., Vol 16, 1, (1993), 25-32. 114. On the convergence of inexact Newton-like methods, Public. Math. Debrecen, Vol. 43, 1-2, (1993), 79-85. 115. On some projection methods for the solution of nonlinear operator equations with nondifferentiable operators. Tamkang J. Math. Vol. 24, No. 1 (1993), 1-8. 116. An initial value method for solving singular perturbed two-point boundary value problems. Arabian Journ. Scienc. and Engineer. Vol. 18, 1 (1993), 3-5. 117. On the solution of nonlinear equations with a nondifferentiable term, Mathematica- Revue D'analyse Numerique et de theorie de l'approximation Tome 22, 2 (1993), 125-135. 118. Some methods for finding error bounds for Newton-like methods under mild differentiability conditions, Acta Math. Hungarica, 61, (3-4), (1993), 183-194. 119. On the secant method, Publicationes Mathematicae Debrecen, Vol. 43, 3-4, (1993), 223-238. 120. Improved error bounds for Newton's method under generalized Zabrejko-Nguen-type assumptions. Appl. Math. Letters Vol. 6, No. 3, (1993), 75-77. 121. A fourth order iterative method in Banach spaces. Appl. Math Letters Vol. 6, No. 4, (1993), 97-98. 122. Newton-like methods and nondiscrete mathematical induction, Studia Scientiarum Mathematicarum Hungarica, 28 (1993), 417-426. 123. Robust estimation and testing for general nonlinear regression models. Appl. Math. and Comp. 58, (1993), 85-101. 124. A mesh independence principle for nonlinear operator equations in Banach space and their discretizations, Studia Scientiarum Math. Hung. 28, (1993) 401-415. 125. Sharp error bounds for the secant method under weak assumptions. P.U.J.M. Vol. XXVI, (1993), 54-62. 126. An error analysis of Stirling's method in Banach spaces, Tamkang J. Math., Vol. 24, No. 2 (1993), 115-133. 127. New sufficient conditions for the approximation of distinct solutions of the quadratic equation in Banach space, Tamkang J. Math, Vol. 24, No. 4, (1993), 355-372. 128. On the convergence of inexact Newton methods. Chinese J. Math. Sept. Vol 21, No. 3 (1993) 227-234. 129. On the solution of equations with nondifferentiable operators, Tamkang J. Math., Vol. 24, No. 3, (1993), 237-249. 130. Sufficient conditions for the convergence of general iteration schemes, Chinese J. Math., Vol. 21, No. 2 (1993), 195-205. 131. On a two-point Newton method in Banach spaces of order four and applications, (1993), Proceedings of the 9th Annual Conference on Applied Mathematics, CAM 93, University of Central, Oklahoma, Edmond, (1993), 34-48, P.U.J.M. Vol. 27, (1994). 132. On a two-point Newton method in Banach spaces of order three and applications Proceedings of the 9th Annual Conference on Applied Mathematics, CAM 93, University of Central, Oklahoma, Edmond, (1993), 24-37, P.U.J.M., Vol. 27, (1994). 133. On a two point Newton-method in Banach spaces and the Ptak error estimates, Proceedings of the 9th Annual Conference on Applied Mathematics, CAM 93, University of Central Oklahoma, Edmond, (1993), 8-24. 134. Sufficient convergence conditions for iteration schemes modeled by point-to-set mappings, Proceedings of the 9th Annual Conference on applied Mathematics, CAM 93, University of Central Oklahoma, CAM 93, Edmond, OK (1993), 48-52. Also Applied Mathematics Letters, (1996). 135. On the convergence of a Chebysheff-Halley-type method under Newton-Kantorovich hypotheses. Appl. Math. Letters Vol. 6, No. 5, (1993), 71-74. 136. On an application of a variant of the closed graph theorem and the secant method. Tamkang J. Math. Vol. 24, No. 3 (1993), 251-267. 137. Newton-like methods in partailly ordered Banach spaces. Approx. Theory and Its Applic. 9 : 1 (1993), 1-10. 138. Results on the Chebyshev method in Banach spaces. Projecciones Revista Vol. 12, No. 2, (1993), 119-128. 139. On the convergence of an Euler-Chebysheff-type method under Newton-Kantorovich hypotheses, Pure Mathematics and Applications, Vol. 4, No. 3, (1993), 369-373. 140. A note on the Halley method in Banach spaces. Appl. Math. and Comp. 58 (1993), 215-224. 141. On the solution of underdetermined systems of nonlinear equations in Euclidean spaces, Pure Mathematics and Applications, Vol 4, No. 3, (1993), 199-209. 142. On the a posteriori error bounds for a certain iteration under Zabrejko-Ngyen assumptions, Rev. Academia de Ciencias, Zaragoza (48), (1993), 77-85. 143. Newton-like methods in generalized Banach spaces, Functiones et Approximatio, XXII, (1993), 107-114. 144. On S-order of convergence, Rev. Academia de Ciencias Zaragoza, 48, (1993), 69- 76. 145. A theorem on perturbed Newton-like methods in Banach spaces. Studia Scientiarum Mathematicarum Hungarica, 29, (1994), 295-305. 146. Some notes on nonstationary multistep iteration processes. Acta Mathematica Hungarica Vol. 64, 1, (1994), 59-64. 147. Stirling's method in generalized Banach spaces. Annales Univ. Sci. Budapest. Sect. Comp. 15, (1994), 1-15. 148. Improved a posteriori error bounds for Zincenko's iteration, Intern. J. Comp. Math., Vol. 51, (1994), 51-54. 149. The Jarrat method in a Banach space setting. J. Comp. Appl. Math. 51, (1994), 103-106. 150. The midpoint method in Banach spaces and Ptak-error estimates, Appl. Math. and Computation, 62, 1(1994), 1-16. 151. A convergence theorem for Newton-like methods under generalized Chen- Yamamoto-type assumptions, Appl. Math. and Comput. 61, 1, (1994). 152. On the convergence of some projection methods and inexact Newton-like iterations, Tamkang J. Math. Vol. 25, No. 4, (1994), 335-341. 153. On Newton's method and nonlinear operator equations, P.U.J.M. Vol 27, (1994). 154. On the midpoint iterative method for solving nonlinear operator equations and applications to the solution of integral equations, University of Arkansas, Research Report, Vol A-R-21, Mathematica-Revue D'analyse Numerique et de Theorie de l'approximation, Tome 23, fasc. 2, (1994), 139-152. 155. Parameter based algorithms for approximating local solutions of nonlinear complex equations, Proyecciones Vol. 13, No. 1, (1994), 53-61. 156. The Halley-Werner method in Banach spaces, Mathematica-Revue D'analyse Numerique et de Theorie de l'approximation, Tome 23, fasc. 2, (1994), 1-14. 157. On a multistep Newton method in Banach spaces and the Ptak error estimates, Proceeding of the Tenth Annual conference on Applied Mathematics, CAM 94, University of Central Oklahoma, Edmond, (1994), 1-15. 158. Error bound representations of Chebysheff-Halley-type methods in Banach spaces, Rev. Academia de Ciencias Zaragoza 49, (1994), 57-69. 159. On the monotone convergence of fast iterative methods in partially ordered topological spaces, Proceedings of the Tenth Annual Conference on Applied Mathematics, CAM 94, University of Central Oklahoma, Edmond, (1994), 16-19. 160. On the discretization of Newton-like methods, Internat. J. Computer. Math. Vol. 52, (1994), 161-170. 161. A local convergence theorem for the super-Halley method in a Banach space, Appl. Math. Lett. Vol. 7, No. 5, (1994), 49-52. 162. A convergence analysis for a rational method with a parameter in Banach space, Pure Mathematics and Applications 5, 1, (1994), 59-73. 163. On sufficient conditions of the convergence and an optimality of the error estimate for a high speed iterative algorithm for solving nonlinear algebraic systems, Chinese J. Math. Vol. 22, No. 4, (1994), 373-384. 164. On the convergence of modified contractions, Journ. Comput. Appl. Math., 55, (1994), 183-189. 165. A multipoint Jarratt-Newton-type approximation algorithm for solving nonlinear operator equations in Banach spaces, Functiones et Aproximatio Commentarii Matematiki, XXIII, (1994), 97-108. 166. Convergence results for the super-Halley method using divided differences, Functiones et Approximatio Commentari Mathematici, XXIII, (1994), 109-122. 167. On the aposteriori error estimates for Stirling's method. Studia Scientiarum Mathematicarum Hungarica 30, (1995), 205-216. 168. Sufficient conditions for the convergence of Newton-like methods under weak smoothness assumptions, Mathematics, CAM 95, Edmond, OK (1995), Proceeding of the eleventh conference on computational and applied mathematics, (1995), 1-13. 169. On Stirling's method, Tamkang J. Math. Vol 27, No. 1, (1995). 170. Stirling's method and fixed points of nonlinear operator equations in Banach space, Bulletin of the Institute of Mathematics Academia Sinica, Vol. 23, No. 1, (1995), 13-20. 171. Error bounds for fast two-point Newton methods of order four, Mathematics, CAM 95, Edmond, OK (1995), Proceedings of the Eleventh Annual Conference on Computational and Applied Mathematics, (1995), 14-18. 172. Improved error bounds for fast two-point Newton methods of order three, Mathematics, CAM 95, Edmond, OK (1995), Proceedings of the Eleventh Annual Conference on Computational and Applied Mathematics, (1995), 19-23. 173. A unified approach for constructing fast two-step Newton-like methods, Monatshefte fur Mathematik. 119, (1995), 1-22. 174. Results on Newton methods; Part I: A unified approach for constructing perturbed Newton-like methods in Banach space and their applications. Appl. Math. and Comp., (1995). 175. Results on Newton methods, Part II: Perturbed Newton-like methods in generalized Banach spaces, Appl. Math. and Comp., (1995). 176. An inverse-free Jarratt type approximation in a Banach space. Journal of Approximation theory and its Applications, 11, (1995). 177. An error Analysis for the secant method under generalized Zabrejko-Nguen-type assumptions, Arabian Journal of Science and Engineering 20:1, (1995), 197-206. 178. Optimal-order parameter identification in solving nonlinear systems in a Banach space. Journal of Computational Mathematics, 13, 2, (1995). 179. Nondifferentiable operator equations on Banach spaces with a convergence structure, Pure Mathematics and Applications, (PUMA), Vol. 6, 1, (1995). 180. Perturbed Newton-like methods and nondifferentiable operator equations on Banach spaces with a convergence structure, Southwest Journal of Pure and Applied Mathematics, Vol. 1, (1995), 1-12. 181. Results on controlling the residuals of perturbed Newton-like methods on Banach spaces with a convergence structure. (SWJPAM) Southwest Journal of Pure and Applied Mathematics, Vol. 1 (1995), 32-38. 182. On an application of a variant of the closed graph theorem to the solution of nonlinear equations, Pure Mathematics and Applications, (PUMA). 183. On the method of tangent hyperbolas, Journal of Approximation Theory and its Applications. 184. On the convergence of two-step methods generated by point-to-point operators, Appl. Math. and Comput. Submitted for Publication 185. A unified approach for constructing fast two-step methods in Banach space and their applications. 186. Convergence theorems for Newton-like methods under generalized Newton- Kantorovich conditions. 187. A simplified proof concerning the convergence and error bound for a rational cubic method in Banach spaces and applications to nonlinear integral equations. 188. The Chebyshev method in Banach spaces and the Ptak error estimates. 189. The Halley method in Banach spaces and the Ptak error estimates. 190. The Halley-Werner method in Banach spaces and the Ptak error estimates. 191. Error bounds for the midpoint method in Banach spaces. 192. Error bounds for the Chebyshev method in Banach spaces. 193. Error bounds for the Halley method in Banach spaces. 194. Error bounds for the Halley-Werner method in Banach spaces. 195. Improved error bounds for Newton-like iterations under Chen-Yamamoto assumptions. 196. An error analysis for Steffensen's method. 197. On the Secant method and the Ptak error estimates. 198. A convergence theorem for Steffensen's method and the Ptak error estimates. 199. An error analysis for the Steffensen method under generalized Zabrejko-Nguen-type assumptions. 200. On the solution of nonlinear equations under Holder continuity assumptions. 201. On the convergence of an Euler-Chebysheff-type method using divided differences of order one. 202. On the convergence of a Chebysheff-Halley-type method using divided differences of order one. 203. On the monotone convergence of an Euler-Chebysheff-typemethod in partially ordered topological spaces. 204. On the monotone convergence of a Chebysheff-Halley-type method in partially ordered topological spaces. 205. Improved error bounds for an Euler-Chebysheff-type method. 206. Improved error bounds for a Chebysheff-Halley-type method. 207. Convergence results for a fast iterative method in linear spaces. 208. Error bounds for an almost fourth order method under generalized conditions. 209. A unified approach for the construction of fast two-step Newton methods. 210. On the approximation of quadratic equations in Banach space using finite rank operators. 211. On the method of tangent parabolas. 212. A study on the order of convergece of a rational iteration for solving quadratic equations in a Banach space. 213. On the super-Halley method using divided differences. 214. Perturbed Newton methods in generalized Banach spaces. 215. A mesh independence principle for perturbed Newton-like methods and their discretizations. 216. On an extension of the mesh-independence principle for operator equations in Banach space. 217. Results involving nondifferentiable equations on Banach spaces with a convergence structure and Newton methods. 218. Inexact Newton methods and nondifferenctiable operator equations on Banach spaces with a convergence structure. 219. A mesh independence principle for inexact Newton-like methods and their discretizations under generalized Lipschitz conditions. 220. Generalized conditions for the convergence of inexact Newton methods on Banach spaces with a convergence structure and applications. 221. Generalized conditions for the convergence of inexact Newton-like methods on Banach spaces with a convergence structure and applications. 222. On a new Newton-Mysovskii-type theorem and applications. Part I: Inexact Newton- like methods and their discretizations. 223. On a new Newton-Mysovskii-type theorem and applications. Part II: The asymptotic mesh independence principle for inexact Newton-Galerkin-like methods. 224. Approximating solutions of operator equations using modified contractions and applications. 225. A unified approach for solving nonlinear operator equations and applications. (M). BOOKS 1. The Theory and Applications of Iteration Methods. ISBN 0-8493-8014-6, CRC Press, Inc. Boca Raton, Florida, U.S.A. (1993). This textbook was written for students in engineering, the physical sciences, mathematics, and economics as an upper undergraduate or graduate level. Prerequisits for using the text are calculus, linear algebra, elements of functional analysis, and the fundamentals in differential equations. Students with some knowledge of the principles of numerical analysis and optimization will have an advantage, since the general schemes and concepts can be easily followed if particular methods, special cases are already known. However such knowledge is not essential in understanding the material of this book. A large number of problems in applied mathematics and also in engineering are solved by finding the solutions of certain equations. For example, dynamic systems are mathematically modelled by difference or differential equations, and their solutions represent usually the states of the systems. For the sake of simplicity assume that a time-invariant system is driven by the equation x = f(x), where x is the state, then the equilibrium states are determined by solving the equation f(x) = 0. Similar equations are used in the case of discrete systems. The unknowns of engineering equations can be functions (difference, differential, integral equations), vectors (system of linear or nonlinear algebraic equations), or real or complex numbers (single algebraic equations with single unknowns). Except special cases the most commonly used solution methods are iterative, when starting from one or several initial approximations a sequence is constructed, which converges to a solution of the equation. Iteration methods are applied also for solving optimization problems. In such cases the iteration sequences converge to an optimal solution of the problem in hand. Since all of these methods have the same recursive structure, they can be introduced and discussed in a general framework. In recent years, the study of general iteration schemes has included a substantial effort to identify properties of iteration schemes that will guarantee their convergence in some sense. A number of these results have used an abstract iteration scheme that consists of the recursive application of a point-to-set mapping. In this book we are concerned with this type of results. Each Chapter contains several new theoretical results and important applications in engineering, in dynamic economic systems, in input-output systems, in the solution of nonlinear and differential equation, and optimization problems. Chapter 1 gives an outline of general iteration schemes, in which the convergence ofsuch schemes is examined. We also show that our conditions are very general: most classical results can be obtained as special cases, and if the conditions are weakened slightly then our results may not hold. In Chapter 2 the discrete time-scale Liapunov theory is extended to time dependent, higher order, nonlinear difference equations. In addition, the speed of convergence is estimated in most cases. The monotone convergence to the solution is examined in Chapter 3 and comparison theorems are proven in Chapter 4. It is also shown that our results generalize well-known classical theorems such as the contraction mapping principle, the lemma of Kantorovich, the famous Gronwall lemma, and the well known stability theorem of Uzawa. Chapter 5 examines conditions for the convergence of special single-step methods such as Newton's method, modified Newton's method, and Newton-like methods generated by point-to- point mappings in a Banach space setting. The speed of convergence of such sequences is examined using the theory of majorants and a method called "continuous induction", which builds on a special variant of Banach's closed graph theorem. Finally, Chapter 6 examines conditions for monotone convergence of special single-step methods such as Newton's method, Newton-like methods, and secant methods generated by point-to-point mappings in a partially ordered space setting. At the end of each chapter case studies and numerical examples are presented from different fields of engineering, and economy. 2. Approximate solutions of nonlinear operator equations in abstract spaces and applications (to appear). 3. The solution of polynomial operator equations in abstract spaces and applications (to appear). (N) EDITING I am the Editor in Chief of the Southwest Journal of Pure and Applied Mathematics or "SWJPAM" for short. This is a purely electronic Journal established in 1995. (O) OTHER The following professors have requested papers: 1. Etzio Venturino, Univer. Iowa, (Dept. Math.), U.S.A. 2. A.G. Kartsatos, Univer. South Florida, (Dept. Math.), U.S.A. 3. H. Jarchow, Institute fur Angewandte Mathematik der Universitat Zurich Ch-8001 Zurich, Switzerland. 4. M.S. Khan, King Abdulaziz Univer., (Dept. Math.), Saudi Arabia. 5. Manfred Knebusch, Universitat Regensburg Fakultat fur Mathematik 8400 Regensburg Universitatsstrabe 31, W. Germany. 6. Ernest J. Eckert. College of environmental sciences, The University of Wisconsin- Green Bay, 2420 Nicolet Dr., Green Bay, WI 54302, U.S.A. 7. Josef Danes, Mathematical Institute Charles University Sokolovska 83 18600 Prague 8-Karlin, Chechoslovakia. 8. Goral Reddy, Dept. Math. Univ. ST. Andrews, Scotland. 9. Jerzy Popenda, Dept. Math. T. Univ. Poznan, Poland. 10. Vlastimil Ptak, Chechoslovak Academy of Science, Praha, Chechoslovakia. 11. Alejandro Figueroa, Universidad de Magallanes, Punta Arenas-Chile. 12. Dragan Jucic, Osijek, Yugoslavia. 13. Ahmad B. Casdam, Multan, Pakistan. 14. Luis Saste Habana, Cuba. 15. S.N. Mishra, Lesotho, Africa. 16. Josef Kral, Prague, Chechoslovakia. 17. Juan J. Nieto, Santiago, Spain. 18. S.D. Chatterji, Lausanne, Switzerland. 19. Peter Madhe, Berlin, Germany. 20. Ioan Muntean, Cluj, Romania. 21. S.L. Singh, Xardwar, India. 22. P.D.N. Sriniras, India. 23. S. Grzegorskii, Lublin, Poland. 24. Toma's Arechaga, Aires, Argentina. 25. J.D. Deader, Salt Lake, Utah, U.S.A. 26. P. Drouet, Rhone, France. 27. J. Weber, The University of Wisconsin, Milwakee, U.S.A. 28. David C. Kurtz, Rollins College, U.S.A. 29. Jorge L. Quiroz, Colima Mexico. 30. Ming - Po Chen, Taiwan, Republic of China. 31. Mustafa Telci, Begtepe, Ankara Turkey. 32. Helmut Dietrich, Merseburg, Germany. 33. Dong Chen, Fayeteville, Arkansas, U.S.A. 34. Mohammad Tabatabai, Cameron University, OK, U.S.A. 35. M.S. Khan, Sultan Quboos University, Muscat, Sultanate of Oman. 36. Laszlo Mate, Technical University, Budapest, Hungary. 37. Pathak, H.K., Bhilai Nayar, INDIA. 38. Osvaldo, Pino Garcia, Habanna, Cuba. 39. B.K. Sharma, Ravishankor University, Raipur, India. 40. Aied Al-Knazi King Abdul Aziz Univ., Jeddah, Saudi Arabia. 41. Hassan-Qasin King Abdul Aziz Univ., Jeddah, Saudi Arabia. 42. Tadeusz Jankowski, Univ. Gdansk, Gdansk ,Poland. 43. K. Kurzak, University Teachers College, Dept. Chemistry, Siedlce, Poland. 44. R. Gonzalez, 2000 Rosario, Argentina. 45. Emad Fatemi, Ecole Polytechnique Federale de Lqusanne, Switzerland. 46. Prasad Balusu, Univ. Rochester, MI, U.S.A. 47. Dieter Schott, Rostolki, Germany. 48. J.M. Martinez, IMECC-UNICAMP, Brasil. 49. Prasad Balusu, India. 50. Qun-sheng Zhou, P.R. China. 51. W. Kliesch, Universitat Leipzig, Germany. 52. Adriana Kindybalyuk, Ukrain Academy of Sciences, Kiev, Ukraine. 53. Roman Brovsek, Ljubljana Slovenia. 54. D. Mathieu, L.M.R.E., France. 55. Donald Schaffner, Rutger University, N.J., U.S.A. 56. David Ward, Barron Associates, Charlottesville, VA, U.S.A. 57. Eugene Parker, Batton Associates, Charlottesville, VA, U.S.A. 58. Miguel Gomez, Habana Cuba. 59. L. Brueggemann, Leipzig-Halle Germany. 60. Fidel Delgado, Habana, Cuba. 61. B.C. Dhage, Maharashtra, India. 62. Leida Perea, Habana, Cuba. 63. David Ruch, Sam Houston Univ., Huntsville, Texas. 64. Patrick J. Van Fleet, Sam Houston Univ., Huntsville, Texas. 65. Tomas Arechaga, BS. Aires, Argentina. 66. M.A. Hernandez, Spain. 67. J. Illuateau, Romania. 68. Ioan A. Rus, Univ. of Cluj-Napoca, Romania. 69. V.K. Jain, Kharagpur, India. 70. Alan Lun, University of Melbourne, Victoria Australia. 71. A.M. Saddeek, Assiut University of Mathematics, Assiut, A.R. Egypt. (P) AWARDS, HONORS AND AFFILIATIONS Chairman Applied Mathematics Section Annual Meeting of the American Mathematical and Mathematical Association of America Meeting, held at San Franscisco, Jan. 1991. Book Reviewer "Elementary Numerical Analysis" by Kendall Atkinson, University of Iowa, Published by Wiley and Sons Inc. Publ. (1992). Scientific Papers Reviewer For (a) Journal of Computational and Applied Mathematics (b) P.U.J.M. (c) Mathematica Slovaca (d) Pure Mathematics and Applications (PUMA) (e) Southwest Journal of Pure and Applied Mathematics Author of Three Books (see) 4(O). Journal Editing I am the Editor in Chief of the Southwest Journal of Pure and Applied Mathematics or "SWJPAM" for sort. Outstanding Graduation Record I was able to finish both my M.S. and Ph.D. degrees at the University of Georgia at the record time of two years which has not been broken yet. International Recognition A total of 69 scientists from five continents have requested reprints of 91% of his published work so far. I have been asked to participate in the evaluation process for tenure and promotion by several U.S.A. and international universities. Member American Mathematical Society Member Mathematical Association of America Member Pi Mu Epsilon Nominated for the Distinguished Faculty Award for 1993 and 1995, Cameron University DEPARTMENTAL SERVICE (a) Member of the graduate studies committee (N.M.S.U.) (b) Member of the graduate faculty (N.M.S.U.) (c) I have been asked and provided input to the members of the departmental personnel committee concerning hiring, updating the math major and other matters. (d) I have been serving as a regular advisor to students and have helped some of them to present papers and give talks at conferences. (e) See also items 3 and 4. UNIVERSITY SERVICE (a) I have been participating in the Cameron Interscholastic service. (b) I have been serving some of the Cameron faculty as consultant. (c) Dean's representative (N.M.S.U.). (d) See also items 3 and 4. STUDENT SERVICE (a) I have been participating in many of our student activities including Mathematics, Pi Mu Epsilon and CS Club activities. (b) See also items 3 and 4. COMMUNITY SERVICE (a) I have been helping people from Lawton (Fort Sill, Goodyear plan and others) and surrounding areas with their mathematical problems. A PROGRESS REPORT ON DR. IOANNIS K. ARGYROS DEPARTMENT OF MATHEMATICAL SCIENCES Dr. Argyros started graduate studies in 1982 at the University of Georgia, and by 1984 he had finished both M.Sc and Ph.D. degrees in Mathematics. His record time for finishing these graduate degrees has not been broken yet. His work was based on problems suggested by Professor S. Chandrasekhar (University of Indiana, Nobel prize of Physics 1983). Because of his exceptional results he was asked by Professor Felix Browder (President of the American Mathematical Society) to give a talk, while still a student at the international conference of Functional Analysis and Applications held at the University of Berkeley on July 1983. Dr. Argyros worked for two years at the University of Iowa and four years at the New Mexico State University as an assistant professor of Mathematics before coming at Cameron during the Fall of 1990 as an associate professor of Mathematics. In 1993 he was granted tenure, and in 1994 he was promoted to the rank of full professor of mathematics. Dr. Argyros has produced two Ph.D students and seven M.Sc. students. He has also helped several students find a job or pursue a graduate degree at reputable universities. At the previous two universities, where he served he was involved in the recruitment of the new graduate students, preparation of their schedule and their comprehensive examinations as a member of the graduate committee. He was also serving as a Dean's representative in the evaluation of graduate students from other departments. According to the opinion of the Departments of Mathematics where he has worked so far his teaching skills rank at the top of the faculty. Dr. Argyros has published more than 200 scientific papers, approximately 7% of which jointly with professors from our as well as other universities. Approximately 68% of these papers are published with the name of Cameron University. In fact these papers have been published not only in U.S.A. but at the top refereed journals in 23 countries in four continents. A total of 69 scientists from five continents have requested reprints of 91% of his published work so far. More than a hundred scientific manuscripts have cited or used his results so far. During the Spring of 1995 he became Editor in Chief of a new reputable electronic journal entitled Southwest Journal of Pure and Applied Mathematics or SWJPAM. He has also been asked from several U.S.A. and international universities to participate in the evaluation process of several professors for tenure and promotion. His expertise are in several areas of applied Mathematics (Numerical-Real Analysis, Optimization, Differential-Integral equations, Mathematical Physics, Mathematical Economics and Approximation theory), but he has also published work in pure Mathematics (Functional Analysis). His results have applications in Astrophysics (neutron transport, radiative transfer and kinetic theory of gases), Mechanics (Electromechanical networks), Economics (dynamic economic systems), to the solution of nonlinear programming problems, and statistics (evaluation of t-estimators). Dr. Argyros has given talks and presented scientific papers at several local and international converences, in some of which he served as a chairman. Dr. Argyros is repeatedly mentioned in several U.S. and International publications and books. In fact he reviewed the book entitled "Elementary Numerical Analysis" authored by Professor Kendall Atkinson and published in November of 1992 (second edition) by Wiley and Sons Inc. This book is considered to be the best and most widespread book in the field. In this book the author praises Dr. Argyros for his teaching, as well as his research skills, some of which were incorporated for the further improvement of this book. During his career, Dr. Argyros has received several grants from universities and the U.S.A. army in order to pursue his teaching and research career. He is also the reviewer of several reputable U.S. and International journals. So far he has reviewed approximately 39 scientific papers. In 1993 and 1995 he was nominated for the Distinguished Faculty Award for those years. In 1993 he, jointly with professor Ferenc Szidarovszky from the University of Arizona published a graduate textbook entitled "The Theory and Applications of Iteration Models" (CRC Press Inc. Publishing Company, Boca Raton, Florida) to be used by students in Mathematics, Physics, Economics, Engineering and the applied sciences. This book received excellent reviews from the scientific community before publication. Here are some quotations about the book expressed by some of the reviewers: "This is an excellent deep book that was missing from the market", "Their treatment is lean and coherent yet remarkably comprehensive. Their careful explanations are fully supported by numerous examples and illustrations that make the material clear and comprehensible to students". During the Fall of 1993 he was contracted to write another graduate textbook entitled "Approximate Solutions of Nonlinear Operator Equations in Abstract Spaces and Applications" to be used by students in Mathematics, Physics, Astrophysics, Mechanics, Engineering and the applied sciences. This text is approximately 500 pages and will be published in 1996. During 1998, my third graduate book will be published entitled "The Solution of Polynomial Operator Equations in Abstract Spaces and Applications".