Abstract
AbstractIn this paper, we introduce the concepts of "Equation missing"-compatible mappings, b-coupled coincidence point and b-common coupled fixed point for mappings F, G : X × X → X, where (X, d) is a cone metric space. We establish b-coupled coincidence and b-common coupled fixed point theorems in such spaces. The presented theorems generalize and extend several well-known comparable results in the literature, in particular the recent results of Abbas et al. [Appl. Math. Comput. 217, 195-202 (2010)]. Some examples are given to illustrate our obtained results. An application to the study of existence of solutions for a system of non-linear integral equations is also considered.2010 Mathematics Subject Classifications: 54H25; 47H10.
Highlights
Ordered normed spaces and cones have applications in applied mathematics, for instance, in using Newton’s approximation method [1,2,3,4] and in optimization theory [5]
K-metric and K-normed spaces were introduced in the mid-20th century ([2]; see [3,4,6]) by using an ordered Banach space instead of the set of real numbers, as the codomain for a metric
A subset P of E is called a cone if and only if: (a) P is closed, non-empty and P ≠ {0E}, (b) a, b Î R, a, b ≥ 0, x, y Î P imply that ax + by Î P, (c) P ∩ (-P) = {0E}, where 0E is the zero vector of E
Summary
Ordered normed spaces and cones have applications in applied mathematics, for instance, in using Newton’s approximation method [1,2,3,4] and in optimization theory [5]. Let (X, d) be a cone metric space with a cone P having non-empty interior, F : X × X ® X and g : X ® X be mappings satisfying d(F(x, y), F(u, v)) α1(d(F(x, y), gx) + d(gu, gx)) + α2(d(F(y, x), gy) +d(F(v, u), gv)) + α3(d(F(u, v), gx) + d(F(x, y), gu)) + α4(d(F(v, u), gy) +d(F(y, x), gv)) + α5(d(F(u, v), gu) + d(gv, gy)), for all x, y, u, v Î X, where ai, i = 1, ..., 5 are nonnegative real numbers such that
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