Existence of fixed point in **partially** **ordered** **sets** has been considered recently in [11-25]. Tarski’s theorem is used in [26] to show the existence of solutions for fuzzy equations and in [27] to prove existence theorems for fuzzy differential equations. In [24,25,27] some applications to matrix equations and to ordinary differential equations are presented. In [28-32], it is proved some fixed point theorems for a mixed mono- tone mapping in a metric space endowed with partial order and the authors apply their results to problems of existence and uniqueness of solutions for some boundary value problems [26,32]. We begin by stating the result of Rhoades [2] after the follow- ing definition.

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question are not the conclusions which maybe known but the proofs as developed in the metric setting. Maybe one of the most beautiful results known in the hyperconvex metric spaces is the intersection property discovered by Baillon [10]. The boundedness assumption in Baillon’s result is equivalent to the complete lattice structure in our set- ting. Indeed, any nonempty subset of a complete lattice has an infimum and a supre- mum. We have the following result in **partially** **ordered** **sets**.

**Partially** **ordered** **sets** (or poset) are generalizations of **ordered** **sets**. A **partially** **ordered** set is a set together with a binary relation that indicates that, for some pairs of elements in the set, one of the elements precedes the other. A set X is a **partially** **ordered** set if it has a binary relation x y defined on it that satisfies

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Abstract This work is devoted to the study of global solution for initial value problem of interval fractional integro-diﬀerential equations involving Caputo-Fabrizio fractional derivative without singular kernel admitting only the existence of a lower solution or an upper solution. Our method is based on ﬁxed point in **partially** **ordered** **sets**. In this study, we guaranty the existence of special kind of interval H-diﬀerence that we will be faced it under weak conditions. The method is illustrated by an example.

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The aim of this paper is to extend the ﬁxed point results of contraction mappings in vectorial **partially** **ordered** **sets** by using vectorial norms introduced by Agarwal [, ] which will allow us to investigate the existence of periodic solutions of vectorial delay diﬀerential equations. For more on ﬁxed point theory, the reader may consult the books [, ]. To be able to prove such results, we will need some monotonicity result of the ﬂow. The main diﬃculty encountered comes from the fact that we are not working on a classical normed vector space. In fact, consider the example of the evolution of a burning zone. The velocity of its evolution is not the same if the burning area is narrow or if it is wide. In this case we cannot study the trajectory of one point without taking into account the others. Therefore we need to consider mappings deﬁned on the subsets of R n which

satisfied, one needs to verify only the first condition. In any continuous poset P, the way below relation has the interpolation property, that is, for x, y ∈ P with x y, there exists an element z ∈ P, such that x z y. For more information about continuous lattices, and continuous domains, the standard reference is 1. A subset S of a poset P is called up-complete if for any directed subset D of S, for which sup D exists in P, sup D is contained in S. We use the fol- lowing notations: ↓ A {x : x ≤ a for some a ∈ A}; ⇓ A {x : x a for some a ∈ A}; and ⇓ x ⇓ {x}. A subset J of a poset P is called a lower set if and only if J ↓ J. Recall that a subset U of a poset P is Scott open if and only if U is an upper set satisfying the property that if the supremum of any directed set D is in U, then D itself intersects U. The lower topology on a poset P has subbasic closed **sets** of the form ↑ x, x ∈ P. The join of the lower topology and the Scott topology on P is the Lawson topology and is denoted by λP .

as n —» oo. This is enough to prove that EA(Q; Pn>p) —» A(Q\ Sc) as n —^ oo for all finite partial orders Q, as follows. Let Pn be the atomless **partially** **ordered** measure space ([0,1], B , /iL, ~<n), and suppose Q is any finite partial order with \Q\ = k. By the definitions of A {Q\Pn,p) and A(Q;Pn), the difference EA(Q;Pn)P) — EA(Q;Pn) is non-zero only because of the positive probability th at in a random sample of k elements from Pn some of the elements are in the same interval [i/n, (i + 1 )/n), for some i. Since the measure of these intervals tends to zero as n —► oo, we have that EA(Q; Pn>p) — EA(Q; Pn) —► 0 as n —* oo. So, it is enough to show that EA(Q; Pn) —► A(Q; Sc), which follows if P ( ^ n induces different partial order from -<) —> 0 as n —>

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In [] and [], several ﬁxed point theorems are proved for set-valued mappings on china- complete posets and some applications have been provided. In the theorems in [] and [], the condition A requires that the values of the considered mapping must be inductive with a ﬁnite number of maximal elements. In this paper, we ﬁrst show that this condition is necessary for the considered mappings to have a ﬁxed point. Then we will show that the set of ﬁxed points of some order-preserving set-valued mappings is inductive, which provides some useful properties for applications. We will also develop more ﬁxed point theorems on both **partially** **ordered** **sets** and **partially** **ordered** topological spaces. More- over, we will investigate some conditions for the considered mappings to substitute the chain-completeness of the underlying spaces.

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In this paper, we consider extended Nash equilibriums of nonmonetized noncooperative games. By using a modiﬁed ﬁxed point theorem of set-valued mappings on **partially** **ordered** **sets**, we prove an existence theorem of extended Nash equilibriums of the nonmonetized noncooperative game. Finally, an example is given to illustrate the advantages of our results.

The mapping g is -comonotonic with all the elements of L, but g is not -monotone. To conclude we should point out that Theorem could be applied to obtain maximal generators of integral stochastic orders which have been generated by means of **partially** **ordered** **sets**, since the maximal generator is the unique generator which satisﬁes the con- dition required in Theorem .

Sorting algorithms have attracted a great deal of attention and study, as they have numerous applications to Mathematics, Computer Science and related fields. In this thesis, we first deal with the mathematical analysis of the Quick sort algorithm and its variants. Specifically, we study the time complexity of the algorithm and we provide a complete demonstration of the variance of the number of comparisons required, a known result but one whose detailed proof is not easy to read out of the literature. We also examine variants of Quick sort, where multiple pivots are chosen for the partitioning of the array. The rest of this work is dedicated to the analysis of finding the true order by further pair wise comparisons when a partial order compatible with the true order is given in advance. We discuss a number of cases where the **partially** **ordered** **sets** arise at random. To this end, we employ results from Graph and Information Theory. Finally, we obtain an alternative bound on the number of linear extensions when the **partially** **ordered** set arises from a random graph, and discuss the possible application of Shell sort in merging chains.

This thesis discusses the topic of Boolean algebras. In order to build intuitive understanding of the topic, research began with the investigation of Boolean algebras in the area of Abstract Algebra. The content of this initial research used a particular nota- tion. The ideas of **partially** **ordered** **sets**, lattices, least upper bounds, and greatest lower bounds were used to define the structure of a Boolean algebra. From this fundamental understanding, we were able to study atoms, Boolean algebra isomorphisms, and Stone’s Representation Theorem for finite Boolean algebras. We also verified and proved many properties involving Boolean algebras and related structures.

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in E such that x n → x ∗ as n → ∞, then x n x ∗ (resp. x n x ∗ ) for all n ∈ N . Clearly, the **partially** **ordered** Banach space C(J, R ) is regular and the conditions guaranteeing the regularity of any **partially** **ordered** normed linear space E may be found in Heikkil¨a and Lakshmikantham [?] and the references therein.

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The ﬁxed point theory for monotone operators in **ordered** Banach spaces has been inves- tigated extensively in the past years [–]. Many new ﬁxed point theorems have been proved under the nonlinear contractive condition by using the theorem of cone and mono- tone iterative technique. These results have been applied to study the ordinary diﬀerential equations, partial diﬀerential equations, and integral equations.

Proof Since C is a nonempty bounded closed and convex subset of the reﬂexive Banach space X, it is a weakly compact subset of (X, ω, ). From Proposition ., it is also a **partially** **ordered** vector space with respect to the weak topology ω on X. By Theorem ., (C, ) is a chain complete poset. The condition A in this corollary implies that F(x) is a weakly compact subset of the bounded closed and convex subset C, for every x ∈ P. Then this corollary immediately follows from Lemma . and Theorem ..

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Causal linearizability is defined in terms of an execution structure [32], tak- ing two different relations over operations into account: a “precedence order” (describing operations that are **ordered** in real time) and a “communication re- lation”. Execution structures allow one to infer the additional precedence orders from communication orders (see Definition 3(A5)). Applied to Fig. 2, for a weak memory execution in which the assert fails, the execution restricted to stack S would not be causally linearizable in the first place (see Section 3 for full details). Execution structures are generic, and can be constructed for any weak memory execution that includes method invocation/response events. We develop one such scheme for mapping executions to execution structures based on the happens-before relation of the C11 memory model.

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1. Introductions. In [8], Katsaras combine the concepts of [ 0 , 1 ] -topology and order structure to bring out the so-called **ordered** fuzzy topological spaces. Several authors have continued on the work of Katsaras in the area of [ 0 , 1 ] -topology and order [3, 4, 10].

The concept of **ordered** g-contraction is introduced, and some ﬁxed and common ﬁxed point theorems for g-nondecreasing **ordered** g-contraction mapping in **partially** **ordered** metric spaces are proved. We also show the uniqueness of the common ﬁxed point in the case of an **ordered** g-contraction mapping. The theorems presented are generalizations of very recent ﬁxed point theorems due to Golubovi´c et al. (Fixed Point Theory Appl. 2012:20, 2012).

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Abstract. The existence of best proximity point is an important aspect of optimization theory. We define the concept of proximally monotone Lipschitzian mappings on a **partially** **ordered** metric space. Then we obtain sufficient conditions for the existence and uniqueness of best proximity points for these mappings in **partially** **ordered** CAT(0) spaces. This work is a continuation of the work of Ran and Reurings [Proc. Amer. Math. Soc. 132 (2004), 1435–1443] and Nieto and Rodr´ıguez-L´opez [Order, 22 (2005), 223–239] for the new class of mappings introduced herein.

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Theorem 1.1. Let (X, ≤, d) be a **partially** **ordered** set and suppose there is a metric d on X such that (X, d) is a complete metric space. Suppose F : X × X × X → X such that F has the mixed monotone property and there exist j, r, l ≥ 0 with j + r + l < 1 such that

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