Fixed Point Theorems of Set-Valued Mappings in Partially Ordered Hausdorff Topological Spaces

and Applied Analysis 3 (A 4 ). Set [x, +∞) ⪯1 = {z ∈ X : x⪯ 1 z} and (−∞, x] ⪯1 = {z ∈ X : z⪯ 1 x}. Obviously, [x, +∞) ⪯1 = (−∞, x], (−∞, x] ⪯1 = [x, +∞), Tx 0 ∩ [x 0 , +∞) ⪯1 ̸ =Ø, and T is increasing and has compact value on [x 0 , +∞) ⪯1 . Applying Theorem 1 on (X, ⪯ 1 ), we find that T has a maximal fixed point x ∈ [x 0 , +∞) ⪯1 = (−∞, x 0 ] corresponding to ⪯ 1 . Let x ∈ (−∞, x 0 ] be a fixed point of T. If x ⪯ x, then x⪯ 1 x and hence x = x by the maximality of x corresponding to ⪯ 1 ; that is, x is a minimal fixed point of T in (−∞, x 0 ] corresponding to ⪯. The proof is complete. Example 3. LetX = {0}∪ {1/n : n = 1, 2, 3, . . .}with the usual metric d(x, y) = ‖x − y‖ for each x, y ∈ X and the usual order x ⪯ y ⇔ x ≤ y for each x, y ∈ X. It is easy to check that (A 1 )–(A 5 ) are satisfied. Let T : X → 2 be defined by

Since then, Kirk's method has been widely used in the generalizations of primitive Caristi's result and the study of fixed point theorems of monotone mappings with respect to a partial order introduced by a functional and many satisfactory fixed point results have been obtained in metric spaces (see [3][4][5][6][7][8][9][10]).
The purpose of this paper is to generalize the results of [3][4][5][6][7][8][9][10] to general topological spaces.Under suitable assumptions, we proved several fixed point theorems of set-valued monotone mappings and set-valued Caristi-type mappings in partially ordered Hausdorff topological spaces, which indeed extend and improve many recent results in the setting of metric spaces.
Let  be a nonempty subset of  and  :  → 2  , where 2  denote the family of all nonempty subset of . is increasing on , if for each ,  ∈  with  ⪯  and each  ∈ , there exists V ∈  such that  ⪯ V;  is quasiincreasing on , if for each ,  ∈  with  ⪯  and each V ∈ , there exists  ∈  such that  ⪯ V.  has compact value on , if  is compact for each  ∈ . is a Caristitype mapping, if for each  ∈ , there exists  ∈  such that  ⪯ .
Let ⪯ 1 be the inverse partial order of ⪯.It is clear that  :  → 2  is increasing on  with respect to ⪯ 1 if  is quasiincreasing on  with respect to ⪯.
Step 2. We show that each increasing sequence {  } ⊂  has an upper bound in .Since ( 3 ) holds on  by Step 1, there exists a subsequence {   } ⊂ {  } and some  ∈  such that Note that for arbitrary given ,   ⪯    for all   ≥ , and then by ( 9) and ( 5 ) we have   ⪯ .Moreover the arbitrary property of  forces   ⪯  for each ; that is,  is an upper bound of {  }.
Step 3. We show that each totally ordered set {  } ∈Γ ⊂  has an upper bound in , where Γ is a directed set.
If there exists  ∈ {  } such that  = sup ∈Γ   , then  is an upper bound of {  } ∈Γ and hence the proof is finished.Thus we may assume that  ̸ = sup ∈Γ   for each  ∈ {  }.By ( 1 ), there is a sequence {   } ⊂ {  } such that there exists Set Note that {  } is totally ordered, and then {  } is well defined and is an increasing sequence such that    ⪯   for all .By step 2, {  } has an upper bound in ; denote it by .Moreover by (10) and (11), which implies that  is an upper bound of {  }.
Let  :  → 2  be defined by Clearly,  is a quasi-increasing mapping and has compact value on .Note that 1 ⪯ 1 and (−∞, 1] = ; then by Theorem 2,  has a minimal fixed point  = 0. The following theorem extends primitive Caristi's result to Hausdorff topological spaces.Theorem 4. Let  be a Hausdorff topological space and let ⪯ be a partial order on .Assume that ( 1 ), ( 3 ), and ( 6 ) are satisfied.Then each set-valued Caristi-type mapping  :  → 2  has fixed point in .

Proof. In analogy to
Step 2 in the proof of Theorem 1, by ( 3 ) and ( 6 ), we can prove that each increasing sequence {  } ⊂  has an upper bound in .Thus following Step 3 in the proof of Theorem 1, each totally ordered chain of  has an upper bound by ( 1 ).Moreover by Zorn's lemma, (, ⪯) has a maximal element; denote it by  * .Note that there exists some  * ∈  * such that  * ⪯  * ; then  * =  * by the maximality of  * and hence  * is a fixed point of .The proof is complete.
The following lemma shows that the conditions ( 1 ) and ( 2 ) are not hard to be satisfied.
Proof.We only show ( 1 ) is satisfied, and the proof of the other case is similar.
Let {  } ∈Γ be a totally ordered set of , where Γ is a directed set, and set  = {(  ) :  ∈ Γ}.Note that  is bounded below; then inf  exists, and so there exists a subset Let  ∈ Γ be an element such that   ̸ = sup ∈Γ   ; then (  ) ̸ = inf  by ( 7 ).Suppose that    ⪯   for each .By ( 7 ), (  ) ≤ (   ) for each , and consequently, we have (  ) = inf  by (15).This is a contradiction, and so there exists some  0 such that This shows that ( 1 ) is satisfied.The proof is complete.

Applications to Metric Spaces
In this section, we shall show that most of the fixed point results in the setting of metric spaces of [3][4][5][6][7][8][9][10] could be derived from Theorems 1-4.Let (, ) be a metric space and let ⪯ be a partial order on .We list the conditions used in [3,6,9,10]