Fixed Point And Coincidence Point Theorem In Partial Order Metric Spaces

Since the 20th century, with society’s, science’s and technology’s rapid develop-ment, nonlinear problems have aroused people’s attention day by day. Nowadays, the nonlinear functional analysis and its applications plays an important role in modern functional analysis theory and has become its one important research field, which has received the attention of scholars both at home and abroad. The fixed point theory is an important component of nonlinear functional analysis, which has been widely studied and applied in various fields. In recent years, the notion of coupled fixed point and coupled coincidence point have been introduced by some people. This theory has been developed extremely fast and has been extended in various directions by many authors. In this paper, we studies the existence and uniqueness of coupled coincidence point, tripled and N order fixed point, B-quadruple fixed and coincidence point in partially ordered metric spaces.The thesis is divided into four sections according to contents.Chapter 1 Preference, we introduce the research background, main contents and significance of this paper.Chapter 2 We introduce the coupled coincidence point theorems for compatible mappings in partially ordered metric spaces.The contractive condition is: d(F(x, y), F(u, v))+d(F(y, x), F(v,u))≤Ψ[d(g(x),g(u))+d(g(y),g(v))], where the function Ψ:[0,∞)→[0,∞), satisfies (ⅰ)Ψ(0)=0, (ⅱ)(?)t> 0, Ψ(t)<t, limΨ(r)<t, and (?)x,y,u,v∈X, g(x)≥g(u), g(y)≤g(v), F and g is compatible mappings.We get the existence and uniqueness of the coupled coincidence point for compatible mappings F and g by investigate.Chapter 3 We introduce the tripled and N order fixed point theorems in par-tially ordered metric spaces.The quasi-contractive conditions are: d(F(x, y, z), F(u, v, w))+d(F(y, z, x), F(v, w, u))+d(F(z, x, y), F(w, u, v)) ≤λ max{d(x, u) + d(y, v) + d(z, w), d(x, F(x, y, z)) + d(y, F(y, z, x)) + d(z, F(z, x, y)), d(u, F(u, v, w)) + d(v, F(v, w, u)) + d(w, F(w, u, v)), d(x, F(u, v, w)) + d(y, F(v, w, u)) + d(z, F(w, u, v)), d(u, F(x, y, z)) + d(v, F(y, z, x)) + d(w, F(z, x, y))}. where λ∈ [0,1), (?)x≥u, y＜v, z≥w.d(G(x1, x2,…, xN), G(x1,X2,… xN)) + d(G(x2,…, xN, x1), G(x2,…, xN, x1)) +…+d(G(xN, x1,…, xN-1), G(xN, x1,…,xN-1)) ≤λmax{d(x1, x1) + d(x2, x2) +…+ d(xN, xN), d(x1, G(x1,x2,…, xN)) + d(x2, G(x2,…,xN, x1))+… +d(xN, G(xN, x1,…, xN-1)), d(x1,G(x1,x2,…,xN)) + d(x2, G(x2,…,xN, x1))+… + d(xN, G(xN, x1,… , xN-1)); d(x1, G(x1,x2, …, xN)) + d(x2, G(x2,…, xN, x1)) + … + d(xN, G(xN, x1…, xN-1)), d(x1, G(x1,x2,…, xN)) + d(x2, G(x2,…,xN, x1)) +… + d(xN, G(xN, x1,…,xN-1))}. where λ∈ [0,1),(?)x1≥x1, x2≥ x2,…, xN≥ xN.We get the existence of the the tripled fixed point for mappings F and N order fixed point for mappings G respectively by discussed.Chapter 4 We introduce the B-quadruple fixed and coincidence point theorems in partially ordered metric spaces.The generalized contractive conditions are: d(F(x, y, z, t), F(u, v, w, r)) ≤ id(x, u) + jd(y, v) + kd(z, w) + ld(t, r), where i, j, k, l∈[0,1), i+j + k + l<1, (?)x≥u, y ≥v, z≤w, t≤r. d(F(x, y, z, t), F(u, v, w, r)) ≤id(g(x),g(u)) + jd(g(y),g(v)) + kd(g(z),g(w)) + ld(g(t),g(r)), where i,j,k,l∈[0,1),i+j+k+l<1,(?) x,y,z,u,v,w∈X,g(x)≥g(u),g(y)≥ g(v),g(z)≤g(w),g(t)≤g(r).This chapter didcuss the existence and uniqueness of the B-quadruple fixed point for mappings F. It also probe the existence and uniqueness of the B-quadruple coincidence point for mappings F and g.It is generalize the min conclusions in references of [28,29].