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Applications of Caristi's fixed point results
Journal of Inequalities and Applications volume 2012, Article number: 40 (2012)
Abstract
In the setup of locally convex spaces, applying Caristi's results we prove some fixed point results for nonself multivalued maps and common fixed point theorems for Caristi type maps utilizing two different techniques. We apply our results to obtain some common fixed points for a Banach operator pair from the set of best approximations. Consequently, we either improve or extend a number of known results in the fixed point theory.
2000 Mathematics Subject Classification: 47H10; 54H25.
1 Introduction
One of the most useful generalizations of the Banach contraction principle in the setting of metric spaces is known as Caristi's fixed point theorem. In the past decades, Caristi's fixed point theorem has been generalized and extended in several directions (see, [1, 2] and references therein). Applying this classical result, Massa [3], Yi and Zhao [4], Zhang [5], and Zhong et al. [6] and others proved fixed point theorems for nonself multivalued contraction maps in the setting of Banach spaces. There are spaces which are not normable (for example, see [7]). So there is natural and essential to study existence of fixed points in the setting of locally convex spaces. In fact, study of known fixed points results of Banach spaces to the case of locally convex spaces is neither trivial and nor easy. However, several interesting fixed point results for single valued and multivalued contraction and nonexpansive maps in the setting of locally convex spaces appeared in the literature, for example; see [8–15] and references there in.
In [16], Fang has introduced a notion of Ftype topological spaces and generalized the Caristi's fixed point theorem to such topological spaces. Recently, Cammaroto et al. [17] observed that each Hausdorff locally convex topological vector space is an Ftype topological space.
In [18], Chen and Li introduced the class of Banach operator pairs, as a new class of noncommuting maps and it has been further studied by Hussain [19, 20], Hussain et al. [21], Khan and Akbar [22, 23], and Pathak and Hussain [24].
In this article, applying Caristi's fixed point results and following the techniques in [5, 6], we prove some fixed point theorems for nonself multivalued contraction maps in the setup of locally convex spaces (see, Section 2). Consequently, Our results either improve or extend a number of known fixed point results including the corresponding results due to Massa [3], Yi and Zhao [4], Zhang [5], and Zhong et al. [6]. Section 3 contains some general common fixed point theorems for Caristi type maps. Applying our theorems we derive some results on the existence of common fixed points for a Banach operator pair from the set of best approximations. Our results of Section 3 extend and unify the study of AlThagafi [25], Chen and Li [18], Hussain and Khan [11], Jungck and Sessa [26], Khan and Akbar [23], Pathak and Hussain [24] and many others.
2 Fixed points for nonself multivalued maps
In this section, E denotes a complete Hausdorff locally convex topological vector space, \mathcal{P} is the family of continuous seminorms generating the topology of E, and K(E) is the family of nonempty compact subsets of E. For each p\in \mathcal{P} and A, B ∈ K(E), define
where d_{ p }(x, A) = inf{p(x  y) : y ∈ A} for any x ∈ E. It is known that D_{ p }is a metric on K(E) even though p is a seminorm, see [9, 12]. Let T : M ⊂ E → K(E) be a multivalued map. We recall the following notions: (a) T is called \mathcal{P}contraction if for each p\in \mathcal{P} there exists a constant k_{ p }, 0 ≤ k_{ p }< 1, such that D_{ p }(T(x),T(y)) ≤ k_{ p }p(x  y), for all x, y ∈ M. (b) A point x ∈ M is called a fixed point of T if and only if x ∈ T(x). (c) We say T satisfy the boundary condition (α) if for all x ∈ M and all y\in T\left(x\right),\phantom{\rule{0.3em}{0ex}}\left(x,y\right]\cap M\ne \mathrm{0\u0338}, where (x, y] = {(1  λ)x + λy : 0 < λ ≤ 1}. (d) T is called weakly inward if for each x\in M,T\left(x\right)\subset \overline{{I}_{M}\left(x\right)}, where I_{ M }(x) = {z : z = x + λ(y  x), y ∈ M, λ ≥ 1} is known as the inward set.
Now, we state the Caristi's fixed point result in the setting of Hausdorff locally convex topological vector space, see [17, 16].
Theorem 2.1 Let f : E → E be any arbitrary map. Suppose there exists a lower semicontinuous function φ : E → [0, +∞) such that for each x ∈ E and for eachp\in \mathcal{P},
Then f has a fixed point.
Another generalization of the Caristi's fixed point result is the following which is a variant of Lemma 1.2 [6].
Theorem 2.2 Let φ: E → [0, +∞) be a bounded below lower semicontinuous function and h : [0, +∞) → [0, +∞) be a continuous nondecreasing function such that\underset{0}{\overset{+\infty}{\int}}\frac{dr}{1+h\left(r\right)}=+\infty. Let f : E → E be a map such that for any given x_{0} ∈ E and for all x ∈ E,
Then f has a fixed point.
Applying Theorem 2.1, first we prove the following fixed point result.
Theorem 2.3. Let M be a nonempty closed subset of E and T : M → K(E) be a\mathcal{P}contraction map satisfying the boundary condition (α). Then T has a fixed point.
Proof. Let p\in \mathcal{P} be arbitrary and fixed. For each x ∈ M, choose y ∈ T(x) such that p(xy) = d_{ p }(x, T(x)). Set z_{ xp }as a farthest point from x in [x, y] ∩ M, that is;
Then we have
Since T is a compact valued \mathcal{P}contraction map, then for k_{ p },0 ≤ k_{ p }< 1, we have
Thus,
Define a self map f on M by f_{ p }(x) = z_{ xp }, x ∈ M, and define a nonnegative real valued function φ_{ p }by φ_{ p }(x) = (1  k_{ p })^{1}d_{ p }(x,T(x)), x ∈ M. Then we have
Since M is a closed subset of a complete space, so it is complete and hence by Theorem 2.1, f has a fixed point u ∈ M. Note that f_{ p }u = u = z_{ up }. Since, z_{ up }is the farthest point from u in [u, y] ∩ M and u = z_{ up }, so it follows that
and hence u ∈ T(u).
Remark 2.4. Theorem 2.3 extends the fixed point result of Massa [[3], Theorem 2] to Hausdorff locally convex spaces.
Another application of Theorem 2.1, is the following fixed point result.
Theorem 2.5. Let M be a closed subset of E and T : M → K(E) be a\mathcal{P}contraction map such that for all x ∈ M,
Then T has a fixed point.
Proof. Suppose that T has no fixed point. Then, d_{ p }(x, T(x)) > 0 for all x ∈ M. Choose q ∈ (0,1) such that {k}_{p}=k<\frac{1q}{1+q}, where k ∈ (0, 1). Since T is a compact valued map, there exists z\in T\left(x\right)\cap \overline{{I}_{M}\left(x\right)} such that
Then, there is some t ∈ (0,1] such that
Put w = (1  t)x + tz. Then there exists some y ∈ M, such that
Since
it follows that
and thus
Note that
Since T is a compact valued \mathcal{P}contraction, we can choose u ∈ T(x) and v ∈ T(y) such that p(w  u) = d_{ p }(w,T(x)) and
Now,
and thus we have
where c=\frac{1q}{1+q}k. Hence,
Define f : M → M by f(x) = y and define the φ : M → ℝ by \phi \left(x\right)=\frac{{d}_{p}\left(x,T\left(x\right)\right)}{c}. Clearly,
By Theorem 2.1, f has a fixed point x_{0} ∈ M. Thus, f(x_{0}) = x_{0}. On the other hand, we have
which is impossible. Hence, T has a fixed point.
Corollary 2.6. Let M be a closed subset of E and let T : M → K(E) be a weakly inward\mathcal{P}contraction map. Then T has a fixed point.
Remark 2.7. (a) Theorem 2.5 extends the fixed point result of Zhang [[5], Theorem 3.3].

(b)
It is worth to mention that in general, the contraction and weakly inward conditions of Theorem 2.5 can not be replaced with somewhat weaker conditions, namely, nonexpansive and T\left(x\right)\cap \overline{{I}_{M}\left(x\right)}\ne \mathrm{0\u0338}, even in the setting of Banach spaces, see [5].

(c)
Corollary 2.6 contains the fixed point result of Yi and Zhao [[4], Theorem 2.1].
Now, using Theorem 2.2 and a contractive condition basically due to [5], we prove the following fixed point result in the setting of Hausdorff locally convex topological vector spaces. Let h : [0, +∞) → [0, +∞) be a continuous nondecreasing function satisfying \underset{0}{\overset{+\infty}{\int}}\frac{dr}{1+h\left(r\right)}=+\infty.
Theorem 2.8. Let M be a closed subset of E and Let T : M → K(E) be a weakly inward map, x_{0} ∈ M, a given point and σ ∈ (0,1] a constant. If for each x, y ∈ M,
Then T has a fixed point.
Proof. Suppose that T has no fixed point. Then, d_{ p }(x, T(x)) > 0 for all x ∈ M. Choose c, 0 < c < σ and q\left(x\right)=\frac{\sigma c}{2\left(1+h\left(p\left({x}_{0}x\right)\right)\right)}, then 0 < q(x) < 1. Since T is a compact valued map, there exists z ∈ T(x) such that
Since T is weakly inward, there exist y ∈ M and λ ≥ 1 such that z = x + λ(y  x). Then,
Set t=\frac{1}{\lambda} and w = (1  t)x + tz. Note that
thus we get
Also, we have
and thus
because 0 < q(x) < 1. Since T is a compact valued map, we can choose u ∈ T(x) and v ∈ T(y) such that p(wu) = d_{ p }(w,T(x)) and
Now, using the above facts and the definition of T, we have
and hence,
For any x ∈ M, define f(x) = y and \phi \left(x\right)=\frac{{d}_{p}\left(x,T\left(x\right)\right)}{c}. Then, f is a selfmap of M and φ is a nonnegative real valued continuous function on M. Also, note that
Applying Theorem 2.2, f has a fixed point. But, due to the fact
it follows that f has no fixed point. This is a contradiction and hence T has a fixed point.
Remark 2.9. a) Theorem 2.8 extends the fixed point result of [[6], Theorem 2.5] to the setting of Hausdorff locally convex spaces.

b)
Theorem 2.8 is not true for multivalued nonexpansive maps, even in the setting of Banach spaces, see [6].
3 Banach operator pair and Caristi type maps
In this section, (E, τ) will be a Hausdorff locally convex topological vector space. A family {p_{ α }: α ∈ I} of seminorms defined on E is said to be an associated family of seminorms for τ if the family {γU : γ > 0}, where U={\bigcap}_{i=1}^{n}{U}_{{\alpha}_{i}} and {U}_{{\alpha}_{i}}=\left(x:{p}_{{\alpha}_{i}}\left(x\right)<1\right\}, forms a base of neighborhoods of zero for τ. A family {p_{ α }: α ∈ I} of seminorms defined on E is called an augmented associated family for τ if {p_{ α }: α ∈ I} is an associated family with property that the seminorm max{p_{ α }, p_{ β }} ∈ {p_{ α }: α ∈ I} for any α, β ∈ I. The associated and augmented associated families of seminorms will be denoted by A(τ) and A*(τ), respectively. It is wellknown that given a locally convex space (E,τ), there always exists a family {p_{ α }: α ∈ I} of seminorms defined on E such that {p_{ α }: α ∈ I} = A*(τ) (see [27, 28]).
The following construction will be crucial. Suppose that M is a τbounded subset of E. For this set M we can select a number λ_{ α }> 0 for each α ∈ I such that M ⊂ λ_{ α }U_{ α }where U_{ α }= {x : p_{ α }(x) ≤ 1}. Clearly B = ∩_{ α }λ_{ α }∩_{ α }is τbounded, τclosed, absolutely convex and contains M. The linear span E_{ B }of B in E is {\bigcup}_{n=1}^{\infty}nB. The Minkowski functional of B is a norm ‖ ⋅ ‖_{ B }on E_{ B }. Thus (E_{ B }, ‖⋅‖_{ B }) is a normed space with B as its closed unit ball and sup_{ α }p_{ α }(x/λ_{ α }) = ‖x‖_{ B }for each x ∈ E_{ B }(for details see [27–30]).
In [31], Ciric introduced the following generalization of continuity for selfmaps.
Definition 3.1. A mapping T of a topological space X into itself is said to be orbitally continuous if x_{0}, x ∈ X such that \underset{n\to \infty}{\text{lim}}{T}^{{i}_{n}}\left(x\right)={x}_{0}, then \underset{n\to \infty}{\text{lim}}T\left({T}^{{i}_{n}}\left(x\right)\right)=T{x}_{0}. (We shall say T is orbitally continuous at x_{0} (o.c.) if x_{0} is such a point.)
Jungck [32] generalized the above definition as follows.
Definition 3.2. A mapping T of a topological space X into itself is said to be almost orbitally continuous (a.o.c.) at x_{0} ∈ X if whenever \underset{n\to \infty}{\text{lim}}{T}^{{i}_{n}}\left(x\right)={x}_{0} for some x ∈ X and subsequence \left\{{T}^{{i}_{n}}\left(x\right)\right\} of {T^{n}(x)}, there exists a subsequence \left\{{T}^{{j}_{n}}\left(x\right)\right\} of {T^{n}(x)} such that \underset{n\to \infty}{\text{lim}}{T}^{{j}_{n}}\left(x\right)=T{x}_{0}. (If T is a.o.c. at all x ∈ M ⊂ X, we say T is (a.o.c.) on M; if M = X, T is a.o.c.)
In [32], Jungck proved the following generalization of Caristi's Theorem (see, Theorem 1.2 [33]) which will be needed in the sequel.
Theorem 3.3. Let (X, d) be a complete metric space and S, T be two a.o.c. mappings of X into itself. Suppose that there are a finite number of functions {ϕ_{ i }: 1 ≤ i ≤ n_{0}} of X into [0, ∞) such that
for all x, y ∈ X and some k ∈ [0,1). Then S and T have a common fixed point x_{0} ∈ X. Further, if x ∈ X, then S^{n}x → x_{0} and T^{n}x → x_{0} as n → ∞.
The pair (T, f) of selfmaps of M is called a Banach operator pair, if the set F(f) is Tinvariant, namely T(F(f)) ⊆ F(f). Obviously, commuting pair (T, f) is a Banach operator pair but converse is not true, in general; see [18, 24]. A mapping T : M → E is called demiclosed at 0 if {x_{ α }} converges weakly to x and {Tx_{ α }} converges to 0, then we have Tx = 0.
The aim of this section is to extend the above mentioned result of Jungck to locally convex spaces and establish general common fixed point theorems for Caristi type maps in the setting of a locally convex space. We apply our theorems to derive some results on the existence of common fixed points for a Banach operator pair from the set of best approximations. Our results extend and unify the study of AlThagafi [25], Chen and Li [18], Hussain and Khan [11], Jungck and Sessa [26], Khan and Akbar [23], and Pathak and Hussain [24] and many others.
We observe in the following example that the almost orbital continuity of a selfmap T on a metric space depends on the choice of the metric. Here, ω = N ∪ {0} and N is the set of positive integers.
Example 3.4. Let X=\left\{0\right\}\bigcup \left\{\frac{1}{{2}^{n}}:n\in \omega \right\} and T\left(0\right)=1,T\left(\frac{1}{{2}^{n}}\right)=\frac{1}{{2}^{n+1}} for n ∈ ω. Then T is not (a.o.c.) under usual metric on X([32], Example 4.2) but T is continuous, and therefore (a.o.c.) under the discrete metric on X.
Next, we establish a positive result in this direction in the context of linear topologies utilizing Minkowski functional.
Lemma 3.5. Let T be (a.o.c.) selfmap of a τbounded subset M of a Hausdorff locally convex space (E, τ). Then T is (a.o.c.) on M with respect to ‖ ⋅ ‖_{ B }.
Proof. By hypothesis, there exists a subsequence \left\{{T}^{{j}_{n}}\left(x\right)\right\} of {T^{n}(x)} such that \underset{n\to \infty}{\text{lim}}{p}_{\alpha}\left({T}^{{j}_{n}}\left(x\right)T{x}_{0}\right)=0 for each p_{ α }∈ A*(τ) for some x ∈ X, whenever \underset{n\to \infty}{\text{lim}}{p}_{\alpha}\left({T}^{{i}_{n}}\left(x\right){x}_{0}\right)=0 for each p_{ α }∈ A*(τ) for some x ∈ X and subsequence \left\{{T}^{{i}_{n}}\left(x\right)\right\} of {T^{n}(x)}. Taking supremum on both sides, we get
whenever
This implies that
whenever
Hence there exists a subsequence \left\{{T}^{{j}_{n}}\left(x\right)\right\} of {T^{n}(x)} such that \underset{n\to \infty}{\text{lim}}{\u2225{T}^{{j}_{n}}\left(x\right)T{x}_{0}\u2225}_{B}=0, whenever \underset{n\to \infty}{\text{lim}}{\u2225{T}^{{i}_{n}}\left(x\right){x}_{0}\u2225}_{B}=0 for some x ∈ X and subsequence \left\{{T}^{{i}_{n}}\left(x\right)\right\} of {T^{n}(x)} as desired.
An application of Lemma 3.5 provides the following general Caristi's Theorem in the setting of locally convex space.
Theorem 3.6. Let M be a nonempty τbounded, τsequentially complete subset of a Hausdorff locally convex space (E, τ) and S, T be two almost orbitally continuous mappings of M into itself. Suppose that there are a finite number of functions {ϕ_{ i }: 1 ≤ i ≤ n_{0}} of M into [0, ∞) such that
for all x, y ∈ X, p_{ α }∈ A*(τ) and some k ∈ [0,1). Then S and T have a common fixed point x_{0} ∈ X. Further, if x ∈ X, then S^{n}x → x_{0} and T^{n}x → x_{0} as n → ∞.
Proof. Since the norm topology on E_{ B }has a base of neighborhoods of 0 consisting of τclosed sets and M is τsequentially complete, therefore M is ‖ ⋅ ‖_{ B } sequentially complete in (E_{ B }, ‖ ⋅ ‖_{ B }) (see, [11, 29, 30]). By Lemma 3.5, S, T are ‖ ⋅ ‖_{ B } almost orbitally continuous mappings of M. From (4.1) we obtain for any x, y ∈ M,
Thus,
A comparison of our hypothesis with that of Theorem 3.3 tells that we can apply it to M as a subset of (E_{ B }, ‖⋅‖_{ B }) to conclude that there exists a point z in M such that Tz = Sz = z and if x ∈ M, then S^{n}x → z and T^{n}x → z as n → ∞.
Lemma 3.7. Let M be a nonempty τbounded subset of Hausdorff locally convex space (E, τ), S, T and f be selfmaps of M and S, T be (a.o.c). Suppose that there are a finite number of functions {ϕ_{ i }: 1 ≤ i ≤ n_{0}} of M into [0, ∞) such that
for all x, y ∈ X, p_{ α }∈ A*(τ) and some k ∈ [0,1). If F(f) is nonempty and τsequentially complete and τ  cl(T(F(f))) ⊆ F(f) and τ  cl(S(F(f))) ⊆ F(f). Then, M\cap F\left(S\right)\cap F\left(T\right)\cap F\left(f\right)\ne \mathrm{0\u0338}.
Proof. Note that for all x, y ∈ F(f), we have,
Hence S, T satisfy (3.1) on F(f) and τ  cl(T(F(f))) ⊆ F(f), and τ  cl(S(F(f))) ⊆ F(f). By Theorem 3.6, S, T have a fixed point z in F(f) and consequently F\left(S\right)\cap F\left(T\right)\cap F\left(f\right)\ne \mathrm{0\u0338}.
Corollary 3.8. Let M be a nonempty τbounded subset of Hausdorff locally convex space (E, τ), S, T, and f be selfmaps of M and S, T be (a.o.c). Suppose that there are a finite number of functions {ϕ_{ i }: 1 ≤ i ≤ n_{0}} of M into [0, ∞) such that
for all x, y ∈ X, p_{ α }∈ A*(τ) and some k ∈ [0,1). If F(f) is nonempty τsequentially complete and (T, f) is a Banach operator pair. Then, M\cap F\left(S\right)\cap F\left(T\right)\cap F\left(f\right)\ne \mathrm{0\u0338}.
The following result generalizes [[18], Theorems 3.2, 3.3] and improves [[25], Theorem 2.2], and [[26], Theorem 6]. Notice that [q, Tx] = {(1  k) q + kT x : k ∈ [0,1]}.
Theorem 3.9. Let M be a nonempty τbounded subset of Hausdorff locally convex [resp., complete] space (E, τ) and T, f be selfmaps of M. Suppose that T is continuous, F(f) is qstarshaped, τclosed [resp., τweakly closed], τ  cl(T(F(f))) ⊆ F(f) [resp., τ  wcl (T(F(f))) ⊆ F(f)], M is τcompact [resp., M is weakly τcompact, IT is demiclosed at 0, where I stands for identity map]. Assume that there are a finite number of functions {ϕ_{ i }: 1 ≤ i ≤ n_{0}} of M into [0, ∞) such that
for all x, y ∈ M, q_{ x }∈ [q, Tx], q_{ y }∈ [q, Ty] and k ∈ (0,1). Then M\cap F\left(T\right)\cap F\left(f\right)\ne \mathrm{0\u0338}.
Proof. Define T_{ n }: F(f) → F(f) by T_{ n }x = (1  k_{ n })q + k_{ n }Tx for all x ∈ F(f) and a fixed sequence of real numbers k_{ n }(0 < k_{ n }< 1) converging to 1. Since F(f) is qstarshaped and τ  cl(T(F(f))) ⊆ F(f) [resp., τ  wcl(T(F(f))) ⊆ F(f)], so τ  cl(T_{ n }(F(f))) ⊆ F(f)] [resp., τ  wcl(T_{ n }(F(f))) ⊆ F(f)] for each n ≥ 1.
Also by (3.3),
for each x, y ∈ F(f) and some 0 < k_{ n }< 1.
If M is τcompact so is τsequentially complete. By Lemma 3.7, for each n ≥ 1, there exists x_{ n }∈ F(f) such that x_{ n }= fx_{ n }= T_{ n }x_{ n }. The compactness of τ  cl(M) implies that there exists a subsequence {Tx_{ m }} of {Tx_{ n }} such that Tx_{ m }→ z ∈ cl(M) as m → ∞. Since {Tx_{ m }} is a sequence in T(F(f)) and τ  cl(T(F(f))) ⊆ F(f), therefore z ∈ F(f). Further, x_{ m }= T_{ m }x_{ m }= (1  k_{ m })q + k_{ m }Tx_{ m }→ z. By the continuity of T, we obtain Tz = z. Thus, M\cap F\left(T\right)\cap F\left(f\right)\ne \mathrm{0\u0338} proves the first case.
By Lemma 3.7, for each n ≥ 1, there exists x_{ n }∈ F(f) such that x_{ n }= fx_{ n }= T_{ n }x_{ n }. Moreover, we have p_{ α }(x_{ n }Tx_{ n }) → 0 as n → ∞. The weak τcompactness of M implies that there is a subsequence {Tx_{ m }} of {Tx_{ n }} converging weakly to y ∈ M as m → ∞. Since {Tx_{ m }} is a sequence in T(F(f)), therefore y ∈ τ  wcl(T(F(f))) ⊆ F(f). Also we have, x_{ m } Tx_{ m }→ 0 as m → ∞. If I  T is demiclosed at 0, then y = Ty. Thus, M\cap F\left(T\right)\cap F\left(f\right)\ne \mathrm{0\u0338}.
If {\sum}_{i=1}^{{n}_{0}}\left[{\varphi}_{i}\left(fx\right){\varphi}_{i}\left({q}_{x}\right)+{\varphi}_{i}\left(fy\right){\varphi}_{i}\left({q}_{y}\right)\right]\ge 0 in (3.4), then every fnonexpansive map T satisfies (3.4). Thus we obtain the following result.
Corollary 3.10. Let M be a nonempty τbounded subset of Hausdorff locally convex [resp., complete] space (E, τ) and T, f be selfmaps of M. Suppose that T is continuous, F(f) is qstarshaped, τ closed [resp, τ weakly closed], τ  cl(T(F(f))) ⊆ F(f) [resp, τ  wcl(T(F(f))) ⊆ F(f)], M is τcompact [resp, M is weakly τcompact, I  T is demiclosed at 0]. Assume that there are a finite number of functions {ϕ_{ i }: 1 ≤ i ≤ n_{0}} of M into [0, ∞) such that {\sum}_{i=1}^{{n}_{0}}\left[{\varphi}_{i}\left(fx\right){\varphi}_{i}\left({q}_{x}\right)+{\varphi}_{i}\left(fy\right){\varphi}_{i}\left({q}_{y}\right)\right]\ge 0 for all x, y ∈ M, q_{ x }∈ [q,Tx], q_{ y }∈ [q, Ty]. If T is fnonexpansive, then M\cap F\left(T\right)\cap F\left(f\right)\ne \mathrm{0\u0338}.
Corollary 3.11. Let M be a nonempty τbounded subset of Hausdorff locally convex [resp., complete] space (E, τ) and T, f be selfmaps of M. Suppose that T is continuous, F(f) is qstarshaped, and τ closed [resp., τ weakly closed], M is τcompact [resp., M is weakly τcompact, IT is demiclosed at 0], (T, f) is a Banach operator pair and satisfy (3.4) for all x, y ∈ M. Then M\cap F\left(T\right)\cap F\left(f\right)\ne \mathrm{0\u0338}.
We define {P}_{M}\left(u\right)=\left\{y\in M:{p}_{\alpha}\left(yu\right)={d}_{{p}_{\alpha}}\left(u,M\right),\text{for}\phantom{\rule{2.77695pt}{0ex}}\text{all}\phantom{\rule{2.77695pt}{0ex}}{p}_{\alpha}\in {A}^{*}\left(\tau \right)\right\} and denote by {\Im}_{0} the class of closed convex subsets of E containing 0. For M\in {\Im}_{0}, we define M_{ u }= {x ∈ M : p_{ α }(x) ≤ 2p_{ α }(u) for each p_{ α }∈ A*(τ)}. It is clear that {P}_{M}\left(u\right)\subset {M}_{u}\in {\Im}_{0}.
The following result extends [[25], Theorem 4.1] and [[34], Theorem 2.14] and corresponding results in [24].
Theorem 3.12. Let f, T be selfmaps of a a Hausdorff locally convex space E. If u ∈ E and M\in {\Im}_{0} such that T(M_{ u }) ⊆ M, τ  cl(T(M_{ u })) is compact and ‖Tx  u‖ ≤ ‖x  u‖ for all x ∈ M_{ u }, then P_{ M }(u) is nonempty, closed and convex with T(P_{ M }(u)) ⊆ P_{ M }(u). If, in addition, D ⊆ P_{ M }(u), D_{0} := D ∩ F(f) is qstarshaped, τ closed, τ  cl(T(D_{0})) ⊆ D_{0}, T is continuous on D and (4.3) holds for all x,y ∈ D, then {P}_{M}\left(u\right)\cap F\left(T\right)\cap F\left(f\right)\ne \mathrm{0\u0338}.
Proof. Follows the lines of proof of Theorem 3.9 [8], so is omitted.
Remark 3.13. It is worth to mention that our results are nontrivial generalizations of the corresponding known fixed point results in the setting of Banach spaces because there are plenty of spaces which are not normable (see, [[7], p. 113]). So it is natural to consider fixed point and approximation results in the context of locally convex spaces.
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The authors gratefully acknowledge the financial support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) represented by the Unit of Research Groups through the grant number (11/31/Gr) for the group entitled "Nonlinear Analysis and Applied Mathematics".
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Latif, A., Hussain, N. & Kutbi, M.A. Applications of Caristi's fixed point results. J Inequal Appl 2012, 40 (2012). https://doi.org/10.1186/1029242X201240
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DOI: https://doi.org/10.1186/1029242X201240