Fixed point theorems for mixed monotone operators and applications to integral equations

Common Coupled Fixed Point Theorem Under Generalized Mizoguchi-Takahashi Contraction for Hybrid Pair of Mappings Generalized Mizoguchi-Takahashi Contraction

“…It is to be noted that to find coupled coincidence point, we do not employ the condition of continuity of any mapping involved therein. We improve, extend and generalize the results of Amini-Harandi and O'Regan [3], Bhaskar and Lakshmikantham [6], Ciric et al [8], Du [17], Harjani et al [18] and Mizoguchi and Takahashi [24]. The effectiveness of our generalization is demonstrated with the help of an example.…”

“…Let (X, d) be a complete metric space and T : X → CB(X) be a multivalued mapping. Assume that [17] and Harjani et al [18].…”

Coupled coincidence points for mixed monotone operators in partially ordered metric spaces

“…(i) if a non-decreasing sequence {x n } → x, then x n x for all n, (ii) if a non-increasing sequence {y n } → y, then y y n for all n. Afterward, the theory of coupled fixed point in partially ordered metric spaces has developed rapidly (see [1,4,9,[11][12][13][14][15][16]22] and references therein). Luong and Thuan [15] proved the following result.…”

“…Harjani et al [9] proved some generalizations of the main results in [7] and discussed the existence and uniqueness of the solution of non-linear integral equations. Theorem 1.6 [9] Let (X, ) be a partially ordered set and suppose there exists a metric d on X such that (X, d) is a complete metric space.…”

“…Theorem 1.6 [9] Let (X, ) be a partially ordered set and suppose there exists a metric d on X such that (X, d) is a complete metric space. Let F : X × X → X be a mapping having the mixed monotone property on X such that …”