Extending Kannan and Chatterjea Theorems in 2-Banach Spaces by Using Sequentially Convergent Mapping

Abstract

S. Gähler ([8]), defined a 2-normed and A. White ([1]) a 2-banach space Further, the contractive mapping in 2-normed space in [4] is defined by in P. K. Hatikrishnan, K. T. Ravindran. In [2] and [10], there are given generalizations of Kannan ([5]) and Chatterjea ([9]) fixed points theorems in 2-Banach spaces. In this paper by using sequentially convergent mappings, the results given in [2] and [10] will be generalized in 2-Banach spaces. They might also be considered as generalizations of the results given in [11]. ______________________________________________________ 2010 Mathematics Subject Classification. 46J10, 46J15, 47H10

Some New Fixed Point Theorems in 2-Banach Spaces

“…for all x, y, z ∈ L for λ ∈ (0, 1 2 ), then it exists a unique fixed point for S in L. In [11] further generalizations for these results are proven, and in [1], by using a sequentially convergent mapping, are proved the generalized M. Kir and H. Kiziltunc theorems. In our further considerations, by using the class Θ of monotony increasing continuous functions f : [0, +∞) → R such that f −1 (0) = {0}, we will elaborate the generalizations for some of the stated results.…”

“…Consequence 5. ( [1]). Let (L, ||·, ·||) be a 2-Banach space, S : L → L and a mapping T : L → L is such that it is continuous, injection and sequentially convergent mapping.…”

“…Let (L, ||·, ·||) be a 2-Banach space, S : L → L and a mapping T : L → L is such that it is continuous, injection and sequentially convergent mapping. If there exists α ∈ (0, 1 2 ) such that ||T Sx − T Sy, z|| 2 ≤ α(||T x − T Sx, z|| 2 + ||T y − T Sy, z|| 2 ), for all x, y, z ∈ L, holds true, then there is a unique fixed point for S and for each x 0 ∈ X the sequence {S n x 0 } converges to the fixed point.…”

“…Let (L, ||·, ·||) be a 2-Banach space, S : L → L and f ∈ Θ be such that f (a + b) ≤ f (a) + f (b), for all a, b ≥ 0. If there exists α ∈ (0, 1 2 ) so that f (||Sx − Sy, z||) ≤ α[f (||x − Sy, z||) + f (||y − Sx, z||)],…”

“…Let (L, ||·, ·||) be a 2-Banach space, S : L → L and the mapping T : L → L be such that it is continuous, injection and sequentially convergent. If there exists α ∈ (0, 1 2 ) so that ||T Sx − T Sy, z|| ≤ α(||T x − T Sy, z|| + ||T y − T Sx, z||),…”