个人简介
Prof. Choonkil Park
Prof. Choonkil Park
Hanyang University, Republic of Korea
标题: Pata’s contraction on a metric space with a graph and its application
摘要: 

Let (X,d) be a metric space endowed with a graph G such that the set V(G) of vertices of G coincides with X. In this talk, we define the notion of Pata-G-contraction type mappings and obtain some fixed point theorems for such mappings. As some extends and subsumes many recent results which were obtained for other contractive type mappings on a partially ordered metric space. As an application of our main results, we present a theorem on the convergence of successive approximations for some linear operators in Banach spaces. 

Let (X,d) be a metric space and let D denote the diagonal of the Cartesian product X x X. Consider a directed graph G such that the set V(G) of its vertices coincides with X and the set E(G) of its edges contains all loops. We assume that G has no parallel edges and so we can identify G with the pair (V(G), E(G)). Moreover, we may treat G as a weighted graph by assigning to each edge the distance between its vertices. 

Definition 1. A mapping f : X →X is a Banach G-contraction (or simply a G-contraction} if f preserves edges of G and f decreases weights of edges of G in the following way: there exists k ∈(0,1) such that (x,y) in E(G) implies d(f(x),f(y)) ≤k d(x,y) for all x,y ∈X. 

Definition 2. A mapping f : X →X is a Pata-G-contraction if f preserves edges of G and f decreases weights of edges of G in the following way: d(fx, fy) ≤ (1 -ρ)d(x, y) + Λρ^k ψ(ρ) [1 + ||x|| + ||y||]^{β} for all ρ ∈ [0, 1] and x, y ∈X, where Λ ≥ 0, k ≥1 and β∈[0,] are fixed constants. We obtain the following main results.

Theorem 1. Let (X, d) be a metric space endowed with a graph G and f :  X → X be a Pata-G-contraction such that the graph G is weakly connected. For all x, y in X, the sequences {f^{n}(x)} and {f^{n} (y)} are Cauchy equivalent.
Theorem 2. Let (X, d) be a complete metric space endowed with a graph G and f : X →X be a Pata-G-contraction. Let X_{f} := {x ∈X : (x, f(x))∈ E(G)}. Assume that the following property holds:
for any sequence {x_{n}} in X, if x_{n} →x and (x_{n}, x_{n+1})∈ E(G) for all n ∈ N, then there exists a subsequence {x_{k_n}} of {x_{n}} with (x_{k_{n}}, x)∈ E(G) for all n ∈ N.
Then the following statements hold:
 (1) |F(f)|= | \[x]_{\tilde{G}} : x ∈X_f};
 (2) For any x ∈X_{f}, f|_{[x]_{\tilde{G}}} is a PO;
 (3) If X’ :=∪{ [x]_{\tilde{G}} :x ∈X_{f}}, then f|_{X’} is a WPO;
 (4) If f E(G), then f is a WPO.
Theorem 3. Let X be a Banach space and X_{0} be a closed subspace of X. Let T:  X → X be a linear operator such that ||T|_{X_0}|| <1. If the corresponding field I - T is such that (I - T)(X) , which is a subset X_{0}, then T is a WPO. Moreover, | F(T) |= | X- X_{0}| and
(x + X_{0}) ∩Fix T = {lim_{n →∞ } T^{n}x} for all x ∈X.

简介: 
Chun-Gil Park. He has accomplished his doctoral degree in Mathematics from the University of Maryland and is currently working as a professor at Hanyang University. He is working for several journals such as Journal of Nonlinear Science and Applications and Journal of Computational Analysis and Applications as the associate editors and Journal of Nonlinear Analysis and Applications as the Editor-in-Chief. His main research topics include operator algebras, functional inequalities, functional equations, non-commutative geometry, fixed point theory and fuzzy mappings. He has published a number of academic articles on international journals related to operator algebras, functional inequalities, functional equations, fixed point results related to graph, non-commutative geometry, soft and rough set, fixed point theory and fuzzy mappings. Within the last twenty years, he has successfully published more than 500 articles on SCI-E journals.