DEMONSTRATE MATHEMATICA Vol. XXIX No 3 1996 Ismat Beg, Abdul Latif, Tahira Yasmeen Minhas SOME FIXED POINT THEOREMS IN TOPOLOGICAL VECTOR SPACES 1. A fixed point theorem for nonself mappings Let A be a subset of a sequentially complete Hausdorff locally convex topological vector space E (over the field J?) with calibration I\ By the terminology of R.T. Moore [6], a calibration T for E means a collection of continuous seminorms p on E which induce the topology of E. Let / , g be nonself mappings from A into E. Let a p , b p , c p , d p and e p be nonnegative real numbers such that a p + b p + c p + d p + e p < 1 and for any x, y in A, and per (1) p(f(x) - g(y)) < a p p(x - y) + b p p(x - f(x)) + c p p(y - g{y)) + d p p(x - g(y)) + e p p(y - f(x)). Wlodarczyk [9] proved that / has a unique fixed point if / = g. In this section, we prove that /, g have a unique common fixed point if b p = c p and d p = e p . When / = g, because of p(x - y) — p(y — x), one can, without loss of generality, assume b p = c p and d p = e p . So our result generalizes the result of Wlodarczyk [9]. Since our Theorem includes Theorem 3.3 of Wlodarczyk [9], it also includes the corresponding theorems in: Hardy and Rogers [2], Goebel, Kirk and Shimi [1], Kannan [4], Nova [7] and Wong [10]. DEFINITION. Let R 0 C T, R 0 / {0}. A subset A of E is said to be of type To with respect to XQ 6 A, if the inequality p(y) < p(x), for some x £ A-x 0 and for all p 6 To implies that y G A — xo- THEOREM 1. Let E be a sequentially complete Hausdorff locally convex topological vector space with calibration T, let A be a subset of E and let f : A —> E, g : A E be two nonself mappings. Assume A is of type To (r<) C T), with respect to XQ 6 A,f and g satisfy (1), such that a p ,b p ,c p ,d p ,e p are non-negative real-valued functions on E X E for p 6 T. // (i) 7 = sup XtyeE {a p (x,y) + b p (x,y) + c p (x,y) + 2d p (x,y)} < 1; for p € T. Unauthenticated Download Date | 2/25/20 10:15 PM