tailieunhanh - Báo cáo hóa học: " Research Article A Fixed Point Approach to the Stability of a Volterra Integral Equation"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article A Fixed Point Approach to the Stability of a Volterra Integral Equation | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2007 Article ID 57064 9 pages doi 2007 57064 Research Article A Fixed Point Approach to the Stability of a Volterra Integral Equation Soon-Mo Jung Received 13 April 2007 Accepted 23 May 2007 Recommended by Jean Mawhin We will apply the fixed point method for proving the Hyers-Ulam-Rassias stability of a Volterra integral equation of the second kind. Copyright 2007 Soon-Mo Jung. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction In 1940 Ulam 1 gave a wide ranging talk before the mathematics club of the University of Wisconsin in which he discussed a number of important unsolved problems. Among those was the question concerning the stability of group homomorphisms. Let G1 be a group and let G2 be a metric group with the metric d - . Given e 0 does there exist a s 0 such that if a function h G1 G2 satisfies the inequality d h xy h x h y s for all x y e G1 then there exists a homomorphism H G1 - G2 with d h x H x e for all x e G1 The case of approximately additive functions was solved by Hyers 2 under the assumption that G1 and G2 are Banach spaces. Indeed he proved that each solution of the inequality II f x y - f x - f y II e for all x and y can be approximated by an exact solution say an additive function. In this case the Cauchy additive functional equation f x y f x f y is said to have the Hyers-Ulam stability. Rassias 3 attempted to weaken the condition for the bound of the norm of the Cauchy difference as follows f x y - f x - f y A HxH II yip 2 Fixed Point Theory and Applications and proved the Hyers theorem. That is Rassias proved the Hyers-Ulam-Rassias stability of the Cauchy additive functional equation. Since then the stability of several functional equations has been extensively investigated 4-10 .