초록 ▼

연구목표
새로운 개념을 도입하여 부동점이론, 최소화이론, Ishikawa iteration, 변분부등식, 함수방정식에 관한 기존의 유명한 연구결과들을 확장발전 시킴으로서 이들 이론들의 활용범위를 더 넓혀 비선형함수해석학의 발전에 기여 한다.
연구내용
1. 새로운 개념을 도입하여 일반화된 완비거리공간 상에서 Ekeland의 변분원리, Takahashi의 최소화정리를 비롯하여 Carisi, Ciric, Kada-Suzuki-Takahashi, Ume, Browder등에 의해서 얻어진 기존의 유명한 부동점정리들을 확장 발

연구목표
새로운 개념을 도입하여 부동점이론, 최소화이론, Ishikawa iteration, 변분부등식, 함수방정식에 관한 기존의 유명한 연구결과들을 확장발전 시킴으로서 이들 이론들의 활용범위를 더 넓혀 비선형함수해석학의 발전에 기여 한다.
연구내용
1. 새로운 개념을 도입하여 일반화된 완비거리공간 상에서 Ekeland의 변분원리, Takahashi의 최소화정리를 비롯하여 Carisi, Ciric, Kada-Suzuki-Takahashi, Ume, Browder등에 의해서 얻어진 기존의 유명한 부동점정리들을 확장 발전, 통합하고, 부동점이론과 관련하여 Ishikawa iteration의 수렴성도 연구하였다.
2. 부동점을 이용하여 Multistage decision processes의 동태계획법에서 도출한 함수 방정식의 해의 존재성 및 유일성 증명하고 최근 Liu 교수가 제시한 open problem도 해결하였다.
3. 새로운 종류의 general variational inclusion과 general resolvent equation 을 도입하여 연구하였다. 즉 general variational inclusion과 general resolvent equation의 동치관계를 증명하였고 resolvent equation technique을 사용하여 general variational inclusion과 general resolvent equation의 해를 구하는데 사용할 수 있는 새로운 iterative algorithm들을 개발하였으며, 이 iterative algorithm에 의해서 얻어지는 iterative sequence의 수렴성도 증명하였다.
4. 수학의 여러분야에서 응용할 수 있는 부등식도 얻었다.
연구성과
1. Fixed point theorems in generalizing spaces of quasi-metric family and applications, Indian Journal of Pure and Applied Mathematics, 33(7) (2002), 1041-1051. (SCI)
2. On properties of solutions for a calss of functional equations arising in dynamic programming, Journal of Oprimization Theory and Applications, 117(3) (2003), 533-551. (SCI)
3. On the convergence of the Ishikawa iterates associated with a pair of multi-valued mappings, ACTA Mathematica Hungarica, 98 (1-2), (2003), 1-8. (SCI)
4. Resolvent equations technique for general variational inclusions, Proceedings of the Japan Academy, Series A, 78(10) (2002), 188-193.(SCI)
5. An inequality for a positive real function, Mathematical Inequalities & Applications, 5(4) (2002), 693-696. (SCI)
6. A Minimization theorem in quasi metric spaces and its applications, Inter. J. Math. & Math. Sci., 31(7) (2002), 443-447.
7. Some results on fixed point theorems for multivalued mappings in complete metric spaces, Inter. J. Math. & Math. Sci., 30(6) (2002), 319-325.
8. A general lemma for fixed point theorems involving more than two maps in D-metric spaces with applications, Inter. J. Math. & Math. Sci., 2003(11) (2003), 661-672.
9. Coincidence and fixed point theorems in Metric and Banach spaces, Inter. J. Math. & Math. Sci., 26(6) (2001), 331-339.

Abstract ▼

Purpose of Research
The purpose of this project extend, improve and unify the previously many well-known results in the field of fixed point theory, minimization theory, iterative method, variational inequality and functional equation by using new concept and method
Contents of Research
1.

Purpose of Research
The purpose of this project extend, improve and unify the previously many well-known results in the field of fixed point theory, minimization theory, iterative method, variational inequality and functional equation by using new concept and method
Contents of Research
1. We improved and unified Takahashi's nonconvex minimization theorems, Ekelands's variational principle, Caristi's fixed point theorem, Ciric's fixed point theorem and other well known fixed point theorems on generalized complete metric spaces, and studied convergence of the Ishikawa iteration.
2. We proved the existance, uniqueness and iterative approximation of solutions for a class of functional equations arising in dynamic programming of multistage decision process, and solved an open problem posed by Liu
3. We introduced and studied a new class of variational inclusions and resolvent equations, respectively, and established the equivalence between the general variational inclusions and general resolvent equations. Using the resolvent equation technique, we construct some new iterative algorithms for solving the classe of variational inclusions and resolvent equations. Under suitable conditions, the convergence analyses of the iterative algorithms were also studied.
Effectiveness of Research
1. Fixed point theorems in generalizing spaces of quasi-metric family and applications, Indian Journal of Pure and Applied Mathematics, 33(7) (2002), 1041-1051. (SCI)
2. On properties of solutions for a calss of functional equations arising in dynamic programming, Journal of Oprimization Theory and Applications, 117(3) (2003), 533-551. (SCI)
3. On the convergence of the Ishikawa iterates associated with a pair of multi-valued mappings, ACTA Mathematica Hungarica, 98 (1-2), (2003), 1-8. (SCI)
4. Resolvent equations technique for general variational inclusions, Proceedings of the Japan Academy, Series A, 78(10) (2002), 188-193.(SCI)
5. An inequality for a positive real function, Mathematical Inequalities & Applications, 5(4) (2002), 693-696. (SCI)
6. A Minimization theorem in quasi metric spaces and its applications, Inter. J. Math. & Math. Sci., 31(7) (2002), 443-447.
7. Some results on fixed point theorems for multivalued mappings in complete metric spaces, Inter. J. Math. & Math. Sci., 30(6) (2002), 319-325.
8. A general lemma for fixed point theorems involving more than two maps in D-metric spaces with applications, Inter. J. Math. & Math. Sci., 2003(11) (2003), 661-672.
9. Coincidence and fixed point theorems in Metric and Banach spaces, Inter. J. Math. & Math. Sci., 26(6) (2001), 331-339.

목차 Contents

• Ⅰ. 연구계획 요약문 ...3
• 1. 국문요약문 ...3
• Ⅱ. 연구결과 요약문 ...4
• 1. 국문요약문 ...4
• 2. 영문요약문 ...5
• Ⅲ. 연구내용 ...6
• 1. 서론 ...6
• 2. 연구방법 및 이론 ...6
• 3. 결과 및 고찰 ...7
• 4. 결론 ...10
• 5. 인용문헌 ...12