Before giving our main results, we first introduce the weak *P*-monotone property.

**Weak P-monotone property** Let (A,B) be a pair of nonempty subsets of a partially ordered metric space (X,d) with {A}_{0}\ne \mathrm{\varnothing}. Then the pair (A,B) is said to have the *weak* *P*-*monotone property* if for any {x}_{1},{x}_{2}\in {A}_{0} and {y}_{1},{y}_{2}\in {B}_{0},

\{\begin{array}{c}d({x}_{1},{y}_{1})=d(A,B)\hfill \\ d({x}_{2},{y}_{2})=d(A,B)\hfill \end{array}\Rightarrow \phantom{\rule{1em}{0ex}}d({x}_{1},{x}_{2})\le d({y}_{1},{y}_{2});

furthermore, {y}_{1}\ge {y}_{2} implies {x}_{1}\ge {x}_{2}.

Now we are in a position to give our main results.

**Theorem 2.1** *Let* (X,\le ) *be a partially ordered set*, *and let* (X,d) *be a complete metric space*. *Let* (A,B) *be a pair of nonempty closed subsets of* *X* *such that* {A}_{0}\ne \mathrm{\varnothing}. *Let* f:A\to B *be an increasing mapping with* f({A}_{0})\subseteq {B}_{0}, *and let there exist* \beta \in \Gamma *such that*

d(f(x),f(y))\le \beta (d(x,y))d(x,y),\phantom{\rule{1em}{0ex}}\mathrm{\forall}y\ge x.

*Assume that either* *f* *is continuous or that* {\overline{A}}_{0} *is such that if an increasing sequence* {x}_{n}\to x\in {\overline{A}}_{0}, *then* {x}_{n}\le x, ∀*n*. *Suppose that the pair* (A,B) *has the weak* *P*-*monotone property*. *And for some* {x}_{0}\in {A}_{0}, *there exists* {\stackrel{\u02c6}{x}}_{0}\in {B}_{0} *such that* d({x}_{0},{\stackrel{\u02c6}{x}}_{0})=d(A,B) *and* {\stackrel{\u02c6}{x}}_{0}\le f({x}_{0}). *Besides*, *if for each* x,y\in {\overline{A}}_{0}, *there exists* z\in {\overline{A}}_{0} *which is comparable to* *x* *and* *y*, *then there exists an* {x}^{\ast} *in* *A* *such that* d({x}^{\ast},f{x}^{\ast})=d(A,B).

*Proof* We first prove that {B}_{0} is closed. Let \{{y}_{n}\}\subseteq {B}_{0} be a sequence such that {y}_{n}\to q\in B. It follows from the weak *P*-monotone property that

d({y}_{n},{y}_{m})\to 0\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}d({x}_{n},{x}_{m})\to 0,

as n,m\to \mathrm{\infty}, where {x}_{n},{x}_{m}\in {A}_{0} and d({x}_{n},{y}_{n})=d(A,B), d({x}_{m},{y}_{m})=d(A,B). Then \{{x}_{n}\} is a Cauchy sequence so that \{{x}_{n}\} converges strongly to a point p\in A. By the continuity of a metric *d*, we have d(p,q)=d(A,B), that is, q\in {B}_{0}, and hence {B}_{0} is closed.

Let {\overline{A}}_{0} be the closure of {A}_{0}, we claim that f({\overline{A}}_{0})\subseteq {B}_{0}. In fact, if x\in {\overline{A}}_{0}\setminus {A}_{0}, then there exists a sequence \{{x}_{n}\}\subseteq {A}_{0} such that {x}_{n}\to x. By the continuity of *f* and the closeness of {B}_{0}, we have fx={lim}_{n\to \mathrm{\infty}}f{x}_{n}\in {B}_{0}. That is, f({\overline{A}}_{0})\subseteq {B}_{0}.

Define an operator {P}_{{A}_{0}}:f({\overline{A}}_{0})\to {A}_{0}, by {P}_{{A}_{0}}y=\{x\in {A}_{0}:d(x,y)=d(A,B)\}. Since the pair (A,B) has the weak *P*-monotone property and *f* is increasing, we have

d({P}_{{A}_{0}}fx,{P}_{{A}_{0}}fy)\le d(fx,fy)\le \beta (d(x,y))d(x,y),\phantom{\rule{1em}{0ex}}{P}_{{A}_{0}}fy\ge {P}_{{A}_{0}}fx,

for any y\ge x\in {\overline{A}}_{0}. Obviously, {P}_{{A}_{0}}f is increasing. Let {x}_{n}, x\in {\overline{A}}_{0}, {x}_{n}\to x, when *f* is continuous, then we have

d({P}_{{A}_{0}}f{x}_{n},{P}_{{A}_{0}}fx)\le d(f{x}_{n},fx)\to 0\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}{P}_{{A}_{0}}f{x}_{n}\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}{P}_{{A}_{0}}fx,\phantom{\rule{1em}{0ex}}\text{as}n\to \mathrm{\infty}.

Then {P}_{{A}_{0}}f is continuous. When {\overline{A}}_{0} is such that if an increasing sequence {x}_{n}\to x\in {\overline{A}}_{0}, then {x}_{n}\le x (∀*n*), we need not prove the continuity of {P}_{{A}_{0}}f. For some {x}_{0}\in {A}_{0}, there exists {\stackrel{\u02c6}{x}}_{0}\in {B}_{0} such that d({x}_{0},{\stackrel{\u02c6}{x}}_{0})=d(A,B) and {\stackrel{\u02c6}{x}}_{0}\le f({x}_{0}). That is,

d({x}_{0},{\stackrel{\u02c6}{x}}_{0})=d(A,B),\phantom{\rule{2em}{0ex}}d({P}_{{A}_{0}}f{x}_{0},f{x}_{0})=d(A,B),\phantom{\rule{1em}{0ex}}{\stackrel{\u02c6}{x}}_{0}\le f({x}_{0}).

By the weak *P*-monotone property, we have {P}_{{A}_{0}}f{x}_{0}\ge {x}_{0}.

This shows that {P}_{{A}_{0}}f:{\overline{A}}_{0}\to {\overline{A}}_{0} is a contraction satisfying all the conditions in Theorem 1.3. Using Theorem 1.3, we can get {P}_{{A}_{0}}f has a unique fixed point {x}^{\ast}. That is, {P}_{{A}_{0}}f{x}^{\ast}={x}^{\ast}\in {A}_{0}, which implies that

d({x}^{\ast},f{x}^{\ast})=d(A,B).

Therefore, {x}^{\ast} is the unique one in {A}_{0} such that d({x}^{\ast},f{x}^{\ast})=d(A,B). □

**Theorem 2.2** *Let* (X,\le ) *be a partially ordered set*, *and let* (X,d) *be a complete metric space*. *Let* (A,B) *be a pair of nonempty closed subsets of* *X* *such that* {A}_{0}\ne \mathrm{\varnothing}. *Let* f:A\to B *be a continuous and nondecreasing mapping with* f({A}_{0})\subseteq {B}_{0}, *and let* *f* *satisfy*

d(f(x),f(y))\le d(x,y)-\psi (d(x,y))\phantom{\rule{1em}{0ex}}\mathit{\text{for}}y\ge x,

*where* \psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) *is a continuous and nondecreasing function such that* *ψ* *is positive in* (0,\mathrm{\infty}), \psi (0)=0 *and* {lim}_{t\to \mathrm{\infty}}\psi (t)=\mathrm{\infty}. *Suppose that the pair* (A,B) *has the weak* *P*-*monotone property*. *If for some* {x}_{0}\in {A}_{0}, *there exists* {\stackrel{\u02c6}{x}}_{0}\in {B}_{0} *such that* d({x}_{0},{\stackrel{\u02c6}{x}}_{0})=d(A,B) *and* {\stackrel{\u02c6}{x}}_{0}\le f({x}_{0}), *then there exists an* {x}^{\ast} *in* *A* *such that* d({x}^{\ast},f{x}^{\ast})=d(A,B).

*Proof* In Theorem 2.1, we have proved that {B}_{0} is closed and f({\overline{A}}_{0})\subseteq {B}_{0}. Now we define an operator {P}_{{A}_{0}}:f({\overline{A}}_{0})\to {A}_{0} by {P}_{{A}_{0}}y=\{x\in {A}_{0}:d(x,y)=d(A,B)\}. Since the pair (A,B) has the weak *P*-monotone property, by the definition of *f*, we have

d({P}_{{A}_{0}}fx,{P}_{{A}_{0}}fy)\le d(fx,fy)\le d(x,y)-\psi (d(x,y)),\phantom{\rule{1em}{0ex}}{P}_{{A}_{0}}fy\ge {P}_{{A}_{0}}fx,

for any y\ge x\in {\overline{A}}_{0}. Obviously, {P}_{{A}_{0}}f is continuous and nondecreasing. For some {x}_{0}\in {A}_{0}, there exists {\stackrel{\u02c6}{x}}_{0}\in {B}_{0} such that d({x}_{0},{\stackrel{\u02c6}{x}}_{0})=d(A,B) and {\stackrel{\u02c6}{x}}_{0}\le f({x}_{0}). That is,

d({x}_{0},{\stackrel{\u02c6}{x}}_{0})=d(A,B),\phantom{\rule{2em}{0ex}}d({P}_{{A}_{0}}f{x}_{0},f{x}_{0})=d(A,B),\phantom{\rule{1em}{0ex}}{\stackrel{\u02c6}{x}}_{0}\le f({x}_{0}).

By the weak *P*-monotone property, we have {P}_{{A}_{0}}f{x}_{0}\ge {x}_{0}.

This shows that {P}_{{A}_{0}}f:{\overline{A}}_{0}\to {\overline{A}}_{0} is a contraction satisfying all the conditions in Theorem 1.4. Using Theorem 1.4, we can get {P}_{{A}_{0}}f has a fixed point {x}^{\ast}. That is, {P}_{{A}_{0}}f{x}^{\ast}={x}^{\ast}\in {A}_{0}, which implies that

d({x}^{\ast},f{x}^{\ast})=d(A,B).

Therefore, {x}^{\ast} is the one in {A}_{0} such that d({x}^{\ast},f{x}^{\ast})=d(A,B). □

**Theorem 2.3** *Let* *X* *be a partially ordered set*, *and let* (X,d) *be a complete metric space*. *Let* (A,B) *be a pair of nonempty closed subsets of* *X* *such that* {A}_{0}\ne \mathrm{\varnothing}. *Let* T:A\to B *be a continuous and nondecreasing mapping with* T({A}_{0})\subseteq {B}_{0}, *and let* *T* *satisfy*

\psi (d(Tx,Ty))\le \varphi (d(x,y)),\phantom{\rule{1em}{0ex}}\mathrm{\forall}y\ge x,

*where* *ψ* *is an altering distance function and* \varphi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) *is a continuous function with the condition* \psi (t)>\varphi (t) *for all* t>0. *Suppose that the pair* (A,B) *has the weak* *P*-*monotone property*. *If for some* {x}_{0}\in {A}_{0}, *there exists* {y}_{0}\in {B}_{0} *such that* d({x}_{0},{y}_{0})=d(A,B) *and* {y}_{0}\le T{x}_{0}, *then there exists an* {x}^{\ast} *in* *A* *such that* d({x}^{\ast},T{x}^{\ast})=d(A,B).

*Proof* In Theorem 2.1, we have proved that {B}_{0} is closed and f({\overline{A}}_{0})\subseteq {B}_{0}. Define an operator {P}_{{A}_{0}}:T({\overline{A}}_{0})\to {A}_{0} by {P}_{{A}_{0}}y=\{x\in {A}_{0}:d(x,y)=d(A,B)\}. Since the pair (A,B) has the weak *P*-monotone property and *ψ* is nondecreasing, by the definition of *T*, we have

\psi (d({P}_{{A}_{0}}Tx,{P}_{{A}_{0}}Ty))\le \psi (d(Tx,Ty))\le \varphi (d(x,y)),\phantom{\rule{1em}{0ex}}{P}_{{A}_{0}}Ty\ge {P}_{{A}_{0}}Tx,

for any y\ge x\in {\overline{A}}_{0}. Since

this shows that {P}_{{A}_{0}}T is continuous and nondecreasing. Because there exist {x}_{0}\in {A}_{0} and {y}_{0}\in {B}_{0} such that d({x}_{0},{y}_{0})=d(A,B) and {y}_{0}\le T{x}_{0}, by the weak *P*-monotone property, we have {x}_{0}\le {P}_{{A}_{0}}T{x}_{0}.

This shows that {P}_{{A}_{0}}T:{\overline{A}}_{0}\to {\overline{A}}_{0} is a contraction from a complete metric subspace {\overline{A}}_{0} into itself and satisfies all the conditions in Theorem 1.6. Using Theorem 1.6, we can get {P}_{{A}_{0}}T has a fixed point {x}^{\ast}. That is, {P}_{{A}_{0}}T{x}^{\ast}={x}^{\ast}\in {A}_{0}, which implies that

d({x}^{\ast},T{x}^{\ast})=d(A,B).

Therefore, {x}^{\ast} is the one in {A}_{0} such that d({x}^{\ast},T{x}^{\ast})=d(A,B). □

**Theorem 2.4** *Let* (X,\le ) *be a partially ordered set*, *and let* (X,d) *be a complete metric space*. *Let* (A,B) *be a pair of nonempty closed subsets of* *X* *such that* {A}_{0}\ne \mathrm{\varnothing}. *Let* T:A\to B *be a continuous and nondecreasing mapping with* T({A}_{0})\subseteq {B}_{0}, *and let* *T* *satisfy*

\psi \left(d(T(x),T(y))\right)\le \psi (d(x,y))-\varphi (d(x,y))\phantom{\rule{1em}{0ex}}\mathit{\text{for}}y\ge x,

*where* *ψ* *and* *ϕ* *are altering distance functions*. *Suppose that the pair* (A,B) *has the weak* *P*-*monotone property*. *If for some* {x}_{0}\in {A}_{0}, *there exists* {y}_{0}\in {B}_{0} *such that* d({x}_{0},{y}_{0})=d(A,B) *and* {y}_{0}\le T{x}_{0}, *then there exists an* {x}^{\ast} *in* *A* *such that* d({x}^{\ast},T{x}^{\ast})=d(A,B).

*Proof* In Theorem 2.1, we have proved that {B}_{0} is closed and f({\overline{A}}_{0})\subseteq {B}_{0}. Define an operator {P}_{{A}_{0}}:T({\overline{A}}_{0})\to {A}_{0} by {P}_{{A}_{0}}y=\{x\in {A}_{0}:d(x,y)=d(A,B)\}. Since the pair (A,B) has the weak *P*-monotone property and *ψ* is nondecreasing, by the definition of *T*, we have

\psi (d({P}_{{A}_{0}}Tx,{P}_{{A}_{0}}Ty))\le \psi (d(Tx,Ty))\le \psi (d(Tx,Ty))-\varphi (d(x,y)),\phantom{\rule{1em}{0ex}}{P}_{{A}_{0}}Ty\ge {P}_{{A}_{0}}Tx,

for any y\ge x\in {\overline{A}}_{0}. Since

This shows that {P}_{{A}_{0}}T is continuous and nondecreasing. Because there exist {x}_{0}\in {A}_{0} and {y}_{0}\in {B}_{0} such that d({x}_{0},{y}_{0})=d(A,B) and {y}_{0}\le T{x}_{0}, by the weak *P*-monotone property, we have {x}_{0}\le {P}_{{A}_{0}}T{x}_{0}.

This shows that {P}_{{A}_{0}}T:{\overline{A}}_{0}\to {\overline{A}}_{0} is a contraction from a complete metric subspace {\overline{A}}_{0} into itself and satisfies all the conditions in Theorem 1.7. Using Theorem 1.7, we can get {P}_{{A}_{0}}T has a fixed point {x}^{\ast}. That is, {P}_{{A}_{0}}T{x}^{\ast}={x}^{\ast}\in {A}_{0}, which implies that

d({x}^{\ast},T{x}^{\ast})=d(A,B).

Therefore, {x}^{\ast} is the one in {A}_{0} such that d({x}^{\ast},T{x}^{\ast})=d(A,B). □