Caristi , Nadler and H +-Type Contractive Mappings and Their Fixed Points in θ-Metric Spaces

A new proper generalization of metric called as θ-metric is introduced by Khojasteh et al. (Mathematical Problems in Engineering (2013) Article ID 504609). In this paper, first we prove the Caristi type fixed point theorem in an alternative and comparatively new way in the context of θ-metric. We also investigate two θ-metrics on CB(X) (family of nonempty closed and bounded subsets of a set X). Furthermore, using the obtained θ-metrics on CB(X), we prove two new fixed point results for multi-functions which generalize the results of Nadler and Lim type in the context of such spaces. In order to illustrate the usability of our results, we equipped them with competent examples.


Introduction
A wide range of pertinence made fixed point theory one of the most attractive areas of research in nonlinear analysis and hence mathematics.Fixed point results are the indispensable aid for showing the existence of solutions, not only in mathematical sciences, but also in game theory and economics.Kakutani [1] provided one such standard tool by means of a generalized form of Brouwer's fixed point theorem, which is used to prove the existence of Nash equilibrium in non-cooperative games.In order to study the applications of fixed point theorems and their equivalence to other results like intersection theorems, we refer the readers to a monograph of Border [2].
One of the most celebrated and applicable results in nonlinear analysis is Banach Contraction Principle (BCP), which inspired many mathematicians to work in fixed point theory.A number of generalizations of BCP have been obtained by many fixed point theorists in order to achieve the likelihood of more general fixed point results for mappings (both single and multivalued) in metric type spaces (cf.Boyd and Wong [3], Meir and Keeler [4], Geraghty [5], Lim [6], Khojasteh et al. [7], etc.).The famous version of BCP for multivalued mappings is obtained by Nadler [8] using the notion of the Pompeiu-Hausdorff metric.The fixed point theorem of Nadler is generalized by many authors in complete metric spaces, one of which is given by Pathak et al. in [9] using the notion of H + metric.
An interesting and fruitful generalization of the Banach Contraction Principle (BCP) on a complete metric space is the Caristi fixed point theorem (Caristi's FPT) [10].Caristi FPT is equivalent to the Ekeland's variational principle and Takahashi's nonconvex minimization theorem [11,12].Weston [13] proved the equivalence of the conclusion of Caristi's fixed point theorem with metric completeness.The result of Caristi has been extended and generalized in various ways (cf.[14,15] and references therein).Caristi's [10] fixed point theorem can be stated as follows: for a mapping T on a complete metric space (X, d), if there exists a lower semicontinuous function for every x ∈ X, then T has a fixed point.
On the other hand, in an effort to generalize BCP, which holds in all complete metric spaces, to a wide class of spaces, Khojasteh et al. [16] coined the notion of θ-metric.This proper generalization of metric is accomplished by replacing the triangular inequality with a weaker assumption.The authors in [16] also investigated the topology induced by θ-metric and presented some topological properties of this space.In addition to this, they characterized the BCP and Caristi type fixed point theorems in the setting of θ-metric space.
In this article, we prove the Caristi fixed point theorem in the θ-metric setting with a novel approach of proof.In addition to this, we investigate H θ and H + θ metrics along the lines of [9,17,18].Moreover, we prove some fixed point results for multivalued mappings along with illustrative examples.
The flow of work in this article is as follows: Section 2 presents some of the basic concepts.First, in Section 3, we prove the Caristi type fixed point theorems in an alternative and comparatively new way in the context of θ-metric.We also investigate two θ-metrics on CB(X) in Section 4. Furthermore, fixed point theorems for multifunction in the context of θ-metric spaces are proved in Section 5, which generalize various metric fixed point results.We equipped this article with competent examples.
The set of all B-actions is denoted by Υ.
Example 1 ([16]).The following functions are examples of B-action: In the following result, the notion of inverse B-action η is brought into focus.
In what follows, θ denotes B-actions.The authors of [16] formulated the concept of θ-metric spaces as follows: Definition 2 ([16]).A mapping d θ : X × X → R + is said to be θ-metric on a nonempty set X with respect to B-action θ ∈ Υ if the following hold true: For examples of θ-metric, readers are referred to [16].
In the following definition, the notions of convergence of a sequence, Cauchy sequence, completeness of θ-metric and continuity of mapping are discussed.Definition 3 ([16]).Let (X, d θ ) be a θ-metric space.Then, The authors of [16] observed that, in a θ-metric space (X, d θ ), every open ball is an open set and the topology is formed by the collection of open sets (denoted by τ d θ ).A pair (X, τ d θ ) is a Hausdorff topological space induced by a θ-metric on X.The set {B d θ (u, 1 n ) : n ∈ N} is a local base at u and the topology τ d θ is first countable.Remark 1.Recently, Brzdek et al. [19] introduced a notion of generalized d q metric which can be defined as: a function d : X × X → R + satisfying following axioms for all a, b, c ∈ X, If we compare the two functions θ and µ, it is observed that µ enjoys more freedom over θ, since continuity and symmetry are relaxed in case of µ.Prima facie, it appears that the concept of generalized d q -metric is more general than the θ-metric.It is also noteworthy here that, in order to generalize the notion of metric in the analogous form, the continuity and symmetry are necessary for θ function.
• A point x ∈ X is called a fixed point of a mapping f : X → X if and only if f (x) = x.

•
A point x ∈ X is called a periodic point of a mapping f : X → X if and only if there exists n ∈ N such that f n (x) = x.

Caristi Type Fixed Point Theorems
The following result is the restatement of Caristi type fixed point theorem presented by Khojasteh et al. in ([16], Theorem 34).We prove this theorem with a new and simple approach in the context of θ-metric space.Theorem 1.Let (X, d θ ) be a complete θ-metric space and µ ∈ P θ and ν ∈ Γ θ .Let T : X → X be a mapping satisfying for any x ∈ X.Then, T has a fixed point in X.
Proof.Let us define a multivalued map S : X → 2 X as Hence, S(u) is nonempty for every u ∈ X.
We now show that, for each v ∈ S(u), S(v) ⊆ S(u).
We define a sequence {u n } in X which starts from some arbitrary u 1 ∈ X. Suppose u n−1 is known and choose u n+1 ∈ S(u n ) such that, for û ∈ X, ( For any n ∈ N, since u n+1 ∈ S(u n ), by using Lemma 2, we have Taking the limit as n → ∞ and using continuity of ν, we have This yields us From (iv) of Definition 5, we get Now, using (i) of Definition 5, we have Taking the limit as n → ∞ on both sides of the above inequality, we get Therefore, by (iv) of Definition 5, we have Thus, {u n } is a Cauchy sequence in X.By the completeness of X, there exists some p ∈ X such that u n → p as n → ∞.
Since ν is continuous and µ is upper semicontinuous in the first variable, Thus, p ∈ ∩ ∞ n=1 S(u n ).Hence, ∩ ∞ n=1 S(u n ) is nonempty and S(p) ⊆ ∩ ∞ n=1 S(u n ).Now, for any q ∈ ∩ ∞ n=1 S(u n ) such that u n = q, by Definition 4, we have Therefore, by Inequality (1), we have Varying n over N, we get lim Therefore {u n } → q.The uniqueness of limit of a sequence ensures that p = q.Thus, we have, S(p) ⊆ ∩ ∞ n=1 S(u n ) = {p}.Thus, S(p) = {p}.In addition, from Inequality (1), we have ν(d θ (p, T p)) ≤ µ(p, T p).This yields T p ∈ S(p) = {p}.Thus, p = T p. Remark 2. The earlier proofs of Caristi fixed point theorem in metric space setting involve assigning a partial order on X.Then, they used Zorn's Lemma or the Brezis Browder order principle or transfinite induction.Even Khojasteh et al. [16] proved the above theorem using the same technique.
In our proof, we do not assume any partial order on X, so Zorn's Lemma or the Brezis-Browder theorem can not be applied.Thus, our proof is different from earlier proofs and comparatively new in the context of space as well as technique.
As a consequence, we obtain the following theorems.Theorem 2. Let (X, d θ ) be a complete θ-metric space and µ ∈ P θ and ν ∈ Γ θ .Let T : X → P(X) be a mapping satisfying for any x ∈ X and y ∈ Tx.Then, T has a fixed point in X.
Proof.The proof follows in the same manner as proof of Theorem 1.
Theorem 3. Let (X, d θ ) be a complete θ-metric space and µ ∈ P θ and ν ∈ Γ θ .Let f : X → X be a mapping satisfying for any x ∈ X.Then, T has a periodic point in X.
For the above ν and µ, T satisfies for every x ∈ X.Thus, T satisfies all the conditions of Theorem 1. Consequently, T has a fixed point 0.
Here, it is worth mentioning that T does not satisfy d(x, Tx) ≤ φ(x) − φ(Tx) when x = 1.Thus, T does not obey the Caristi fixed point theorem.
Proof.Clearly, due to non-negativity and symmetry of d θ , H θ is also non-negative and symmetric.Next, we show H θ (P, Q) = 0 if and only if P = Q.We only require to show that H θ (P, Q) = 0 =⇒ P = Q; the converse will be true due to property (i) of Definition 2. For this, suppose that H θ (P, Q) = 0 for any P, Q ∈ CB(X).This implies that sup{d θ (q, P)|q ∈ Q} = 0, which gives us d θ (q, P) = 0 for q ∈ Q.This yields q ∈ P. Thus, Now, it remains to prove that H θ (P, R) ≤ θ(H θ (P, Q), H θ (Q, R)) for any P, Q, R ∈ CB(X).Suppose P, Q, R ∈ CB(X).Let u ∈ P be arbitrary; there exists v ∈ Q and > 0 such that In addition, there exists w ∈ R such that Since u is arbitrary in P, we have Since is arbitrary, the above inequality yields Using the similar argument, we obtain Thus, from Inequalities ( 7) and ( 8), we get Proof.We only prove Other things follow in the same way as in the proof of Theorem 4. Suppose P, Q, R ∈ CB(X).Letting u ∈ P, there exists v ∈ Q and > 0 such that In addition, there exists w ∈ R such that Taking supremum in the above inequality, we get Since is arbitrary, this yields Using a similar argument, we get Adding Inequalities ( 9) and ( 10), we get Remark 3. H θ and H + θ defined above are equivalent metrics on CB(X), since It is worth mentioning here that the equivalence of the two θ-metric does not mean that the results proved with one are equivalent to others.This is shown by means of some examples in [9] in the case of metric spaces.

Fixed Point Results for Set-Valued Mappings
This section presents some fixed point results in θ-metric spaces for multivalued mappings.Firstly, we obtain a fixed point theorem using θ-Pompeiu-Hausdorff metric.Secondly, we prove some fixed point theorems of Pathak and Sahzad type [9] for the multivalued case using the H + θ metric.The results presented here generalize various results of the metric fixed point theory.

Lim-Nadler Type Fixed Point Theorems
Let Φ θ be a collection of mappings ϕ : R + → R + with (i) for each α > 0, there exists The following result is required to prove the fixed point theorem.Lemma 3. Let ϕ ∈ Φ θ such that for some u > 0, ϕ(u) ≤ u.Then, (i) ϕ(l) < l for every l > 0, (ii) for every sequence {l n } such that l n → l as n → ∞, l n ≥ l > 0, we have Proof.(i) Suppose there exists c > 0 such that ϕ(c) = c.Let c ∈ Im(θ) then for every u ∈ Im(θ) such that u < c, we can find v ∈ Im(θ) by (iii) of Definition 1 such that θ(u, v) = c. Therefore, This implies that ϕ(u) = u for every u ≤ c.Since ϕ ∈ Φ θ , for α = c 2 , there exists which is a contradiction.Thus, ϕ(l) < l for every l > 0.
(ii) For every s, t ∈ Im(θ), we have Thus, ϕ ∈ Φ θ .Theorem 6.Let (X, d θ ) be a complete θ-metric space and T : X → CB(X) be a multivalued mapping such that there exists ϕ ∈ for all a, b ∈ X.Then, T has a fixed point.
Proof.Let us take arbitrary a 0 in X and Fix a 1 ∈ Ta 0 .We choose a 2 ∈ Ta 1 such that In general, if a n is chosen such that a n / ∈ Ta n , then we can choose a n+1 ∈ Ta n such that Then, {d n = d θ (a n , a n+1 )} is a strictly decreasing sequence.Thus, there exists some d ≥ 0 such that d n → d.Suppose d > 0 .
Then, from Inequality (11), we have Tending n to ∞, we get Supposing that {a n } is not Cauchy, then there exist two subsequences of {a n } say {a n(k) }, {a m(k) } and > 0 such that Thus, we have Tending k to ∞, we get lim k→∞ d θ (a n(k) , a m(k) ) = .
We now show that H θ (Ta, Tb) ≤ ϕ(d θ (a, b)) holds for all a, b ∈ X.For this, let b ≥ a ≥ 0, then Ta ⊂ Tb, so we have

Pathak and Sahzad Type Fixed Point Theorem
We require the following concepts to prove our results in this section.
Definition 7. Let (X, d θ ) be a θ-metric space.A mapping T : X → CB(X) is an H + θ -contraction if (i) there exists L ∈ (0, 1) such that H + θ (Ta, Tb) ≤ Ld θ (a, b), for every a, b ∈ X, (ii) for every a ∈ X, b ∈ Ta and k > 0, there exists c ∈ Tb such that

HRemark 4 .
θ (Ta, Tb) = ln(1 + b) − ln(1 + a) = ln 1 + b 1 + a .Since a ≤ b, we have 1+b 1+a ≤ 1 + b − a, which implies ln 1+b 1+a ≤ ln(1 + |b − a|).Thus, we get H θ (Ta, Tb) ≤ ϕ(d θ (a, b)).All of the requirements of Theorem 6 are fulfilled.Hence, T has a fixed point a = 0.In 1969, Meir and Keeler[4] obtained an interesting generalization of BCP on a complete metric space.In 2001, Lim[6] characterized the Meir-Keeler contraction by introducing an L-function ϕ which satisfies the condition (i) of class Φ θ in metric context.Thus, our Theorem 6 characterizes Lim type fixed point results.Consequently, Theorem 6 generalizes various fixed point results of Lim-Nadler type in metric spaces in the context of both space and contractive conditions.