R. P. Pant

Note on common fixed points under strict contractive conditions

The first error occurs in lines 4–7 on page 329 which claim that the inequality d(f u,ff u) < max { d(gu,gf u), k[d(f u,gu)+ d(ff u,gf u)]/2, k[d(ff u,gu)+ d(f u,gf u)]/2= kd(f u,ff u) leads to a contradiction. However, this inequality does not lead to a contradiction unless condition (ii) is slightly modified. To overcome this problem we replace the condition (ii) of Theorem 2.1 by d(f x,fy) < max { d(gx,gy), k[d(f x,gx)+ d(fy,gy)]/2, [d(fy,gx)+ d(f x,gy)]/2, 1 k < 2. (ii′) With the above modification the theorem can be proved along the similar lines as given in the original one with minor changes in accordance with the replaced condition (ii′).

MEIR-KEELER TYPE FIXED POINT THEOREMS AND DYNAMICS OF FUNCTIONS

1. Generalized contractions The study of fixed points of mappings satisfying contractive type conditions has been a very active field of research activity during the last two decades. For a self-mapping f of a metric space (X, d) the most geneal type of contractive condition is either a Banach type condition (1) d ( f x , f y ) < k max{d(x, y), d(x, fx),d{y, f y , ) , [d{x, f y ) + d(y, fx)]/2}, 0 < k < 1, or a Meir-Keeler type (s, <5) contractive condition (2) given e > 0 there exists a 5 > 0 such that e < max{d(x, y), d(x, f x ) , d ( y , f y ) , [d(x, f y ) + d(y, fx)]/2} < e + S d ( f x , f y ) < £ or a (^-contractive condition of the form (3) d ( f x , f y ) < cj)(max{d(x, y),d(x, f x ) , d{y, f y ) , [d(x, f y ) + d{y, fx)]/2}) where : R + —> R + is such that {t) < t for each t > 0. It can be seen that condition (1) is a particular case of both (2) and (3). In the more general setting pertaining to common fixed points of four mappings, say A, B, S, T of a metric space (X, d) the conditions (2) and (3) respectively assume the form 1991 Mathematics Subject Classification: 54H25.

DUALITY IN VECTOR OPTIMIZATION UNDER TYPE 1 α-INVEXITY IN BANACH SPACES

The aim of this paper is to provide global optimality conditions and duality results for a class of nonconvex vector optimization problems posed on Banach spaces. In this paper, we introduce the concept of quasi type I α-invex, pseudo type I α-invex, quasi pseudo type I α-invex, and pseudo quasi type I α-invex functions in the setting of Banach spaces, and we consider a vector optimization problem with functions defined on Banach spaces. A few sufficient optimality conditions are given, and some results on duality are proved.

Applications of Fixed Point Theorems in Divisibility of Polynomials

Abstract The aim of the present paper is to study the dynamics and fixed points of a class of functions induced by integers and then apply the fixed point theorem to obtain general tests of divisibility of numbers. In the second part, we generalize the theorems on divisibility to obtain conditions of divisibility of polynomials by (x 4 + q) type polynomials. We not only obtain generalizations of the well-known remainder theorem and the factor theorem of algebra but also compute explicit value of the quotient. The results are computationally explicit and can be easily implemented on digital computers.