Abstract
In this paper, we give some generalizations of the functional type Caristi-Kirk theorem (see Functional Type Caristi-Kirk Theorems, 2005) for two mappings on metric spaces. We investigate the existence of some fixed points for two simultaneous projections to find the optimal solutions of the proximity two functions via Caristi-Kirk fixed point theorem.
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1 Introduction
Recall that a real-valued function ϕ defined on a metric space X is said to be lower (upper) semi-continuous if for any sequence \((x_{n})_{n}\) of X which converges to \(x\in X\), we have \(\phi(x) \le \liminf_{n} \phi(x_{n}) \) (\(\phi(x) \ge\limsup_{n} \phi(x_{n}) \)).
In 1976, Caristi (see [2]) obtained the following fixed point theorem on complete metric spaces, known as the Caristi fixed point theorem.
Theorem 1.1
Let \((X,d)\) be a complete metric space, \(T : X\rightarrow X\) be a mapping and \(\psi: X\rightarrow \mathbb{R}^{+}\) be a lower semi-continuous function such that, for all \(x\in X\),
Then T has a fixed point in X.
Let M be a nonempty set partially ordered by ≤. We will say that \(x\in M\) is a maximal element of M if and only if \(( x\leq y, y\in M \Rightarrow x=y)\).
Theorem 1.2
(I. Ekeland [3])
Let \((X,d)\) be a complete metric space and \(\phi: X\rightarrow \mathbb{R}^{+}\) be a lower semi-continuous function. Define a relation ≤ by for all \(x, y\in X\),
Then \((X,\le)\) is partially ordered and it has a maximal element.
It is noted that Theorems 1.1 and 1.2 are equivalent.
In 1994, Bae, Cho, and Yeom (see [4]) proved some functional versions of the Caristi-Kirk fixed point theorem; each of these including Theorem 1.1 as a particular case.
Let \(c: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}\) be some function. Denote, for \(\alpha\in \mathbb{R}^{+}\),
Clearly \({ \liminf_{t\to\alpha^{+}} c(t)\leq c(\alpha) \leq\limsup_{t\to\alpha^{+}} c(t) }\). And we say that c is right lower (upper) semi-continuous at α if \(\liminf_{t\to\alpha ^{+}} c(t)=c(\alpha)\) (\(\limsup_{t\to\alpha^{+}} c(t)=c(\alpha )\)).
We obtain a sequential characterization of these local properties:
Proposition 1.3
c is right lower (upper) semi-continuous at α if and only if for all sequence \((t_{n})_{n}\) such that \(t_{n} \to\alpha\) and \(t_{n} \ge\alpha \) for all n, we have:
Proposition 1.4
If c is right lower (upper) semi-continuous at α then it is right locally bounded below (above) at α: \(\exists\lambda=\lambda(\alpha) >0\), such that \(\inf(c([\alpha ,\alpha+\lambda])) > -\infty\) \((\sup(c([\alpha,\alpha+\lambda])) < \infty)\).
Theorem 1.5
(see [4])
Let \(\phi:X\rightarrow \mathbb{R}^{+}\) be a lower semi-continuous function and \(c: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}\) be a upper semi-continuous function from the right such that, for all \(x\in X\),
Then T has a fixed point in X.
If \(H : \mathbb{R}^{+}\times \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}\), let us consider the functional Caristi-Kirk type contraction
Theorem 1.6
Let \(\phi:X\rightarrow \mathbb{R}^{+}\) be a lower semi-continuous function. If \({c: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}}\) be a right locally bounded from above and \(H: \mathbb{R}^{+}\times \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}\) be a locally bounded function such that, for all \(x\in X\),
Then T has at least one fixed point in X.
For \(H(s,t)=s\), we obtain the following.
Theorem 1.7
(see [5])
Let \(\phi:X\rightarrow \mathbb{R}^{+}\) be a lower semi-continuous function and \(c: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}\) be a right locally bounded from above such that, for all \(x\in X\),
Then T has at least one fixed point in X.
The following definitions (see [6]) will be needed.
Let H be a Hilbert space, \(C_{i}\) be a nonempty closed convex subset of H where \(i\in I=\{1,\ldots,m\}\),
and \(P_{C_{i}} : H\rightarrow C_{i}\), \(1\leq i\leq m\), the metric projection onto \(C_{i}\).
Definition 1.8
(A Cegielski [6])
-
1.
The operator \({ T= \sum_{i\in I} w_{i}P_{C_{i}} }\), where \((w_{1},\ldots, w_{m})\in\Delta_{m}\) and \(I=\{1,\ldots,m\}\), is called a simultaneous projection.
-
2.
The function \(f:H\rightarrow \mathbb{R}^{+}\) defined by
$$ f(x)= \frac{1}{2} { \sum_{i\in I} w_{i} \Vert P_{C_{i}}x-x\Vert ^{2} },\quad x\in H, $$(3)called the proximity function.
-
3.
The set defined by
$$\mathop{\operatorname{Argmin}}_{x\in C} f(x) = \bigl\{ z\in C ; f(z)\leq f(x) \text{ for all } x \in C \bigr\} , $$where \(C \subset H\) and \(f: C \rightarrow \mathbb{R}\), is called a subset of minimizers of f.
The set of all fixed points of self mapping T of a metric space X will be denoted by \(\operatorname{Fix}(T)\).
Recently, Farskid Khojasteh and Erdal Karapinar (see [7]) proved the following result.
Theorem 1.9
Let \({ T= \sum_{i\in I} w_{i}P_{C_{i}} }\) be a simultaneous projection, where \(w\in\Delta_{m}\) and a proximity function \(f:H\rightarrow \mathbb{R}\) defined by equation (3).
Then we have
Moreover, if \(\Vert x-Tx\Vert \geq1\), for all \(x\in K\), where
then \(\operatorname{Fix}(T)\neq\emptyset\).
2 Main results
We prove a functional version of Caristi-Kirk theorem for two pairs of mappings on metric spaces.
Theorem 2.1
Let \((X,d)\) be a complete metric space, \(\phi:X\rightarrow \mathbb{R}^{+}\) be a lower semi-continuous function and \(T,S : X\rightarrow X\) two mappings such that, for all \(x\in X\),
where \(c: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}\) be a right locally bounded from above and H a locally bounded function from \(\mathbb{R}^{+}\times \mathbb{R}^{+}\) to \(\mathbb{R}^{+}\). Then there exists an element \(x^{\ast}\in X\) such that \(Tx^{\ast}=x^{\ast}=Sx^{\ast}\).
Proof
First step. Let \(\alpha= \inf\phi(X)\); as c is locally bounded from above, there exists \(\lambda=\lambda(\alpha )>0\) such that \(\mu= \sup c([\alpha,\alpha+\lambda]) < \infty\). It follows that there exists \(\nu=\nu(\mu)>0\) such that \(H(t,s)\le \nu\) for all \(s,t\in[0,\mu]\).
For some \(x_{0} \in X\) such that \(\alpha\leq\phi(x_{0})\leq\alpha +\lambda\), we define the set \(X_{0}\) by
\(X_{0}\) is a nonempty closed subset of X. By (4), we have
for all \(x\in X_{0}\); and consequently, \(T(X_{0})\subset X_{0}\) and \(S(X_{0})\subset X_{0}\). And since \(\phi(x), \phi(Tx) , \phi(Sx) \in [\alpha,\alpha+\lambda]\), for all \(x\in X_{0}\), we obtain
and then
Second step. We define a partial order ≤ on \(X_{0}\) as follows: for \(x,y\in X_{0}\)
Since ϕ is lower semi-continuous function on the complete metric space \((X_{0},d)\), we see by the Ekeland theorem (see [3]) that \((X_{0},\le)\) has a maximal element \(x^{\ast}\) such that
If \(\phi(Sx^{\ast}) \leq\phi(Tx^{\ast})\), we obtain \(d(x^{\ast},Sx^{\ast})\leq v(\phi(x^{\ast})-\phi(Sx^{\ast}))\); then \(x^{\ast}\leq Sx^{\ast}\), which implies \(Sx^{\ast}=x^{\ast}\) and \(Tx^{\ast}=x^{\ast}\).
The same conclusion holds in the case \(\phi(Tx^{\ast}) \le\phi(Sx^{\ast})\). □
Corollary 2.2
Let \((X,d)\) be a complete metric space, \(\phi:X\rightarrow \mathbb{R}^{+}\) be a lower semi-continuous function and \(T,S : X\rightarrow X\) two mappings such that, for all \(x\in X\),
where \(c: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}\) be a right locally bounded from above. Then there exists an element \(x^{\ast}\in X\) such that \(Tx^{\ast}=x^{\ast}=Sx^{\ast}\).
Corollary 2.3
Let \((X,d)\) be a complete metric space, \(\phi :X\rightarrow \mathbb{R}^{+}\) a lower semi-continuous function and \(T,S : X\rightarrow X\) two mappings such that, for all \(x\in X\),
Then there exists an element \(x^{\ast}\in X\) such that \(Tx^{\ast}=x^{\ast}=Sx^{\ast}\).
Let \(g:\mathbb{R}^{+} \rightarrow \mathbb{R}^{+}\) be locally bounded above in the sense that g is bounded above on each \([0,a]\), (\(a>0\)).
Corollary 2.4
Let \((X,d)\) be a complete metric space, \(\phi :X\rightarrow \mathbb{R}^{+}\) be a lower semi-continuous function and \(T,S : X\rightarrow X\) two mappings such that, for all \(x\in X\),
Then there exists an element \(x^{\ast}\in X\) such that \(Tx^{\ast}=x^{\ast}=Sx^{\ast}\).
Proof
We define a function c on \(\mathbb{R}^{+}\) by \(\forall t\in \mathbb{R}^{+}\), \(c(t)= \sup g([0,t])\). c is increasing and then it is right locally bounded above. By (5), we have, for all \(x\in X\),
which implies
for all \(x\in X\). By Corollary 2.2, T and S have a common fixed point. □
Theorem 2.5
Let \((X,d)\) be a complete metric space, \(\phi, \psi :X\rightarrow \mathbb{R}^{+}\) be a lower semi-continuous functions and \(T,S : X\rightarrow X\) two continuous mappings such that, for all \(x\in X\),
where \(c: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}\) be a right locally bounded from above and \(H: \mathbb{R}^{+}\times \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}\) be a locally bounded function.
Assume that there exists \(x_{0} \in X\) such that \(\psi(Tx_{0})\leq\psi (Sx_{0})\) and \(\phi(Sx_{0})\leq\phi(Tx_{0})\). Then there exists an element \(x^{\ast}\in X\) such that \(Tx^{\ast}=x^{\ast}=Sx^{\ast}\).
Proof
The set \(X_{0} =\{ x\in X ; \psi(Tx)\leq\psi(Sx) \text{ and }\phi(Sx)\leq\phi(Tx)\}\) is nonempty (\(x_{0} \in X_{0}\)) and closed (hence complete), because \(\phi\circ T\), \(\phi\circ S\), \(\psi \circ T\), and \(\psi\circ S\) are lower semi-continuous.
First case. Let \(\alpha= \inf(\phi+\psi)(X_{0})\); since the function c is locally bounded from above, there exists \(\lambda =\lambda(\alpha)>0\) such that \(\mu= \sup c([\alpha,\alpha +\lambda])< \infty\). Also there exists \(\nu=\nu(\mu)>0\) with \(H(t,s)\leq\nu\), whenever \((t,s)\in[0,\mu]^{2}\).
Let \(x_{1} \in X_{0}\) such that \(\alpha\leq(\phi+\psi)(x_{1})\leq\alpha +\lambda\). And let
\(X_{1}\) is nonempty (\(x_{1} \in X_{1}\)) and closed since \(\phi+\psi\) is lower semi-continuous.
By (6), we obtain
Hence, for all \(x\in X_{1}\), \(Tx, Sx\in X_{1}\). For all \(x\in X_{1}\), we have
then \(\max\{c((\phi+\psi)(x)), c((\phi+\psi)(Tx)), c((\phi +\psi)(Sx))\} \leq\mu\); and, consequently,
Second case. We introduce the partial order ≤ on \(X_{1}\) by
Since \(\psi+\phi\) are lower semi-continuous functions, \((X_{1},\leq)\) has a maximal element \(x^{\ast}\), by the Ekeland theorem. If \(d(x^{\ast},Sx^{\ast})\le d(x^{\ast},Tx^{\ast})\), we obtain
It follows that \(x^{\ast}\leq Sx^{\ast}\) and then \(Sx^{\ast}=x^{\ast}\). And since
we conclude \(\phi(x^{\ast})= \psi(Sx^{\ast})\), and \(d(x^{\ast},Tx^{\ast})=0\) i.e. \(Tx^{\ast}=x^{\ast}\).
If \(d(x^{\ast},Tx^{\ast})\leq d(x^{\ast},Sx^{\ast})\), we obtain \(d(x^{\ast},Tx^{\ast})=0\) and \(d(x^{\ast},Sx^{\ast})=0\) by the same arguments. □
Example 2.6
Consider the space \(X=[0,+\infty[\) with the usual metric d and define T, S, ψ, and ϕ by
Let \(H(t,s)= \max\{t,s\}\), \((t,s)\in \mathbb{R}^{+}\times \mathbb{R}^{+}\), and \(c(y)=1\), for each \(y\in \mathbb{R}^{+}\).
For all \(x\in X\), we have
For \(x_{0} = \frac{1}{2}\), we have \(\psi(Tx_{0})= \frac{1}{2}< \psi (Sx_{0})\) and \(\phi(Sx_{0})= \frac{1}{2}= \phi(Tx_{0})\). Note that \(x^{\ast}=1\) is a common fixed point of T and S.
Theorem 2.7
Let d and δ be two metrics on a nonempty set X. Assume that \((X,d)\) is complete. Let \((T_{n})_{n}\) be a sequence of lower semi-continuous self mappings on X such that, for all \(x\in X\) and for all \(n,m\in \mathbb{N}^{\ast}\), we have
where \(c: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}\) be a right locally bounded from above and H a locally bounded function from \(\mathbb{R}^{+}\times \mathbb{R}^{+}\) to \(\mathbb{R}^{+}\). Then there exists an element \(x^{\ast}\in X\) such that, for all \(n\in \mathbb{N}^{*}\), \(T_{n} x^{\ast}=x^{\ast}\).
Proof
As in the proof of Theorem 2.1, there exists a complete subset \(X_{o}\) of X such that, for all \(n,m\in \mathbb{N}^{\ast}\), \(T_{n}X_{0} \subset X_{0}\), and for all \(x\in X_{0}\),
where \(\nu\in \mathbb{R}^{+}\). By (8), we have
for each \(x\in X\) and \(n,m\mathbb{N}^{\ast}\). since ϕ is lower semi-continuous and \((X,d)\) is complete, the Caristi fixed point theorem implies that there exists \(x_{n,m} \in X\) such that \(T_{n}T_{m}x_{n,m}=x_{n,m}\).
We have
Then \(\phi(x_{n,m})=\phi(T_{m}x_{n,m})\). By (8), we obtain
which leads to \(T_{n}x_{n,m}=x_{n,m}\). By the second relation of (8), we have
which leads to \(T_{m}x_{n,m}=x_{n,m}\).
Hence, there exists \(x_{n,m}\in X\) such that \(T_{n}(x_{n,m})=x_{n,m}=T_{m}(x_{n,m})\).
Let \(m_{o}\in \mathbb{N}^{*}\). For each \(n,m\in \mathbb{N}^{*}\),
Consequently, for \(n=m_{o}\) and for all \(m\in \mathbb{N}^{*}\), we obtain \(T_{m}x_{m_{o},m_{o}}= x_{m_{o},m_{o}}\). □
Theorem 2.8
Let \((X,d)\) and \((Y,\delta)\) be two complete metric spaces. Let \(T : X\rightarrow Y\), \(S : Y\rightarrow X\) be two mappings and \(\psi: X\rightarrow \mathbb{R}^{+}\), \(\phi: Y\rightarrow \mathbb{R}^{+} \) two lower semi-continuous functions such that, for all \((x,y)\in X\times Y\),
where \(c: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}\) is a right locally bounded from above and \(H: \mathbb{R}^{+}\times \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}\) is a locally bounded function. Then there exists a couple \((x^{\ast},y^{*})\in X\times Y\) such that \(STx^{\ast}=x^{\ast}\) and \(TSy^{*}=y^{*}\). Also, then \(Tx^{\ast}=y^{*}\) and \(Sy^{*}=x^{\ast}\).
Proof
First case. Let \(\alpha= \inf(\psi(X)+\phi (Y))\). The function c is locally bounded from above, there exists \(\lambda=\lambda(\alpha)>0\) in such a way that \(\beta=\sup([\alpha ,\alpha+\lambda])< \infty\) and there exists \(\nu=\nu(\beta)>0\) with \(H(t,s)\le\nu\) for each \(s,t\in[0,\beta]\).
By definition of α, there exists \((x_{0},y_{0}) \in X\times Y\) such that
The set \(A = \{(x,y)\in X\times Y ; \psi(x)+\phi(y)\leq\psi (x_{0})+\phi(y_{0}) \} \) is nonempty and closed.
By (9), we obtain
For all \((x,y)\in A\), we have
Then \(c(\psi(x)+\phi(y)), c(\psi(Sy)+\phi(Tx)) \le\beta\), and hence
Second case. Let \((x,y)\in A\); we have
Since \((Sy,Tx) \in A\), we have
and then
Define the partial order ≤ in A as follows: for \((x,y),(x',y')\in A\)
Let \((x_{\alpha},y_{\alpha})_{\alpha}\) be some chain of A;
\((\psi(x_{\alpha})_{\alpha})\) and \((\phi(y_{\alpha})_{\alpha})\) are increasing bounded and thus convergent sequences.
Let \({ \gamma= \lim_{\alpha}\psi(x_{\alpha})}\) and \({ \eta= \lim_{\alpha}\phi(y_{\alpha}) }\).
For \(\varepsilon>0\), there exists \(\alpha_{o} \) such that, for all \(\beta\ge\alpha\ge\alpha_{o}\),
which implies that \(((x_{\alpha},y_{\alpha}))_{\alpha}\) is a Cauchy sequence in the complete space \((A,d_{\infty})\) where \(d_{\infty}\) is defined by \(d_{\infty}((x,y),(x',y'))= \max\{ d(x,x'),\delta(y,y') \} \). It follows that there exists \((x^{\ast},y^{*})\in A\) such that \({ \lim_{\alpha} x_{\alpha} =x^{\ast}}\) and \({ \lim_{\alpha} y_{\alpha} =y^{\ast}}\).
We obtain
Hence, \((x_{\alpha},y_{\alpha})\leq(x^{\ast},y^{*})\). And by the Ekeland theorem, \((A,\leq)\) has a maximal element \((\bar{x},\bar{y})\). By (10), we have
And since \((A, d_{\infty})\) is complete and \((STx,TSy)\in A\), for all \((x,y)\in A\), there exists \((\bar{x},\bar{y})\in A\) such that \((ST\overline{x},TS\overline{y})= (\overline{x},\overline{y})\). □
For \(h(t,s)=1\), for all \((t,s)\in \mathbb{R}^{+}\times \mathbb{R}^{+}\), we have the following.
Theorem 2.9
Let \((X,d)\) and \((Y,\delta)\) be two metric spaces such that \((X,d)\) is complete. Let \(T : X\rightarrow Y\), \(S : Y\rightarrow X\) be two mappings and \(\psi: X\rightarrow \mathbb{R}^{+}\), \(\phi: Y\rightarrow \mathbb{R}^{+} \) two lower semi-continuous functions such that, for all \((x,y)\in X\times Y\),
Then there exists an unique couple \((x^{\ast},y^{\ast})\in X\times Y\) such \(STx^{\ast}=x^{\ast}\), \(TSy^{\ast}=y^{\ast}\); and then \(Tx^{\ast}=y^{\ast}\) and \(Sy^{\ast}=x^{\ast}\).
Proof
Let \(x\in X\), \(y=Tx\), and \(y'= TSTx\); we have
It follows that \(\phi(TSTx) \geq\psi(STx)\) and \(d(x,STx)\leq\psi (x)-\psi(STx)\), for all \(x\in X\). By the Caristi theorem, there exists \(x^{\ast}\in X\) such that \(STx^{\ast}=x^{\ast}\).
Let \(y^{\ast}= Tx^{\ast}\); for \(x= STx^{\ast}\) and \(y= y^{\ast}\), we have
Then \(\phi(y^{\ast})=\psi(x^{\ast})\) and \(x^{\ast}=Sy^{\ast}\). Hence, \(TSy^{\ast}=y^{\ast}\).
For uniqueness, let \((x,y)\in X\times Y\) such that \(STx=x\) end \(TSy=y\). We have
Similarly
So, \(\psi(x^{\ast})=\phi(y)\) and \(\phi(y^{\ast})-\psi(x)\). Then \(x=x^{\ast}\) and \(y=y^{\ast}\). □
Theorem 2.10
Let \((X,d)\) and \((Y,\delta)\) be metric spaces. Assume that \((X,d)\) is complete. Let \(T : X\rightarrow Y\), \(S : Y\rightarrow X\) be two mappings and \(\psi: X\rightarrow \mathbb{R}^{+}\), \(\phi: Y\rightarrow \mathbb{R}^{+} \) two lower semi-continuous functions such that, for all \((x,y)\in X\times Y\),
Then there exists an unique \((x^{\ast},y^{\ast})\in X\times Y\) such that \(STx^{\ast}=x^{\ast}\) and \(TSy^{\ast}=y^{\ast}\). And then \(Tx^{\ast}=y^{*}\) and \(Sy^{*}=x^{\ast}\).
Proof
For \(y=Tx\), \(x\in X\), we have
So, for all \(x\in X\), \(d(x,STx)\leq\psi(x)-\psi(STx)\). By the Caristi theorem, there exists \(x^{\ast}\in X\) such that \(STx^{\ast}=x^{\ast}\).
Let \(y^{\ast}= Tx^{\ast}\). For \(y=y^{\ast}\) and \(x= x^{\ast}\), we have
which leads to \(\phi(y^{\ast})=\psi(x^{\ast})\) and \(x^{\ast}=Sy^{\ast}\). Hence, \(TSy^{\ast}=y^{\ast}\).
For uniqueness, \((x,y)\in X\times Y\) such that \(STx=x\) and \(TSy=y\); we have
So \(\psi(x^{\ast})=\phi(y^{\ast})\). We obtain \(x^{\ast}= Sy\) and \(y^{\ast}=Tx\). Thereby, \(y= TSy=Tx^{\ast}=y^{\ast}\) and \(x=STx= Sy^{\ast}=x^{\ast}\). □
Example 2.11
Let \(X=[0,+\infty[\) and \(Y=[0,\frac{1}{2}]\cup\{1\}\); we use the usual metric d and the metric δ given by
We define \(T: X\longrightarrow Y\) and \(S:Y\longrightarrow X\) by
and \(Sy=1\), for all \(y\in Y\).
Let ψ and ϕ be defined, respectively, on X and Y by
The functions c and H are defined by \(c(t)=6t\) and \(H(t,s)=t\), for all \(s, t\in[0,+\infty[\).
We have \(STx=1\) and \(TSy=\frac{1}{2}\), for all \((x,y)\in X\times Y\).
We discuss the following cases:
Case 1: \(x\in[0,1[\) and \(y\in[0,\frac{1}{2}[\).
We obtain \(d(x,STx)= 1-x \), \(\delta(y,TSy)= y+\frac{1}{2}\), \(\psi (x)+\phi(y)= y+2\), \(\psi(x)- \phi(Tx)=\frac{1}{2}\) and \(\phi(y) - \psi(Sy)= y+1\). So
Case 2: \(x\in[0,1[\) and \(y=\frac{1}{2}\).
We obtain \(d(x,STx)= 1-x\), \(\delta(\frac{1}{2},TS\frac{1}{2})= 0\), \(\psi(x)+\phi(\frac{1}{2})=1\), \(\psi(x)- \phi(Tx)=\frac{1}{2}\), and \(\phi(\frac{1}{2})- \psi(S\frac{1}{2})=0\). So
Case 3: \(x\in[0,1[\) and \(y=1\).
We obtain \(d(x,STx)= 1-x\), \(\delta(1,TS1)= \frac{3}{2}\), \(\psi (x)+\phi(1)= \frac{3}{2}\), \(\psi(x)- \phi(Tx) = \frac{1}{2}\), and \(\phi(1) - \psi(S1)=\frac{1}{2} \). So
Case 4: \(x=1\) and \(y\in[0,\frac{1}{2}[\).
We obtain \(d(1,ST1)= 0 \), \(\delta(y,TSy)= y+\frac{1}{2}\), \(\psi (1)+\phi(y)= y+1\), \(\psi(1)- \phi(T1) = 0\) and \(\phi(y) - \psi(Sy)= y+1\). So
Case 5: \(x=1\) and \(y=\frac{1}{2}\).
We obtain \(d(1,ST1)= 0 \), \(\delta(\frac{1}{2},TS\frac{1}{2})=0\), \(\psi(1)+\phi(\frac{1}{2})= 0\), \(\psi(1)- \phi(T1)=0 \), and \(\phi(\frac{1}{2}) - \psi(S\frac{1}{2})= 0\). So
Case 6: \(x\in\mathopen{]}1,+\infty[ \) and \(y\in[0,\frac{1}{2}[ \).
We obtain \(d(x,STx)= x -1\), \(\delta(y,TSy)= y+\frac{1}{2}\), \(\psi (x)+\phi(y)= x+y+2\), \(\psi(x)- \phi(Tx)= x -\frac{1}{x+1}\), and \(\phi(y)- \psi(Sy)=y+1\). So
Case 7: \(x\in\mathopen{]}1,+\infty[ \) and \(y=\frac{1}{2} \).
We obtain \(d(x,STx)= x -1\), \(\delta(\frac{1}{2},TS\frac{1}{2})=0\), \(\psi(x)+\phi(\frac{1}{2})= x+1\), \(\psi(x)- \phi(Tx)= x -\frac {1}{x+1}\), and \(\phi(\frac{1}{2})- \psi(S\frac{1}{2})=0\). So
Case 8: \(x\in\mathopen{]}1,+\infty[ \) and \(y=1\).
We obtain \(d(x,STx)= x-1\), \(\delta(1,TS1)= \frac{3}{2} \), \(\psi (x)+\phi(1)= x+\frac{3}{2}\), \(\psi(x)- \phi(Tx)= x -\frac {1}{x+1} \), and \(\phi(1) - \psi(S1)= \frac{1}{2} \). So
Case 9: \(x =y=1\).
We have \(d(1,ST1)= 0\), \(\delta(1,TS1)= \frac{3}{2} \), \(\psi(1)+\phi (1)= \frac{1}{2}\), \(\psi(1)- \phi(T1)=0 \), and \(\phi(1) - \psi (S1)= \frac{1}{2} \). So
Note that \(T1= \frac{1}{2}\) and \(S\frac{1}{2}=1\).
Example 2.12
Let \(X=[0,1] \) and \(Y=[0,1]\cup\{2,3,\ldots,p \}\), where \(p\in \mathbb{N}\setminus\{0,1\}\); we consider the following metrics:
and
We define \(T : X\rightarrow Y\) and \(S: Y\rightarrow X\) by \(Tx = 1\) and \(Sy =1\). We define ψ and ϕ on X and Y resp. by
and
We have \(STx= 1\) and \(TSy=1 \), for all \((x,y) \in X\times Y\).
Case 1: \(x,y\in[0,1[\). We have
Case 2: \(x\in[0,1[\) and \(y\in\{2,\ldots,p \}\)
Case 3: \(x=1\) and \(y\in[0,1[\).
Case 4: \(x=1 \) and \(y\in\{2,\ldots,p \}\).
Case 5: \(x\in[0,1[ \) and \(y=1\).
Case 6: \(x=y=1\).
Note that \(T1= 1\) and \(S1 =1\).
3 Application
Theorem 3.1
Let \((X,d)\) and \((Y,\delta)\) be two metric spaces such that \((X,d)\) is complete. Let \(\psi: X\rightarrow \mathbb{R}^{+}\), \(\phi: Y\rightarrow \mathbb{R}^{+} \) be two lower semi-continuous functions. Assume that, for \((u,v)\in X\times Y \) such that \({ \psi(u)> \inf_{x\in X} \psi (x)}\) and \({ \phi(v)> \inf_{y\in Y} \phi(y) }\), there exists \((u',v')\in X\times Y\), \((u',v')\neq(u,v)\) such that
then there exists \((u_{o},v_{o})\in X\times Y\) such that
Proof
Assume \({ \psi(u) > \inf_{x\in X} \psi (x) }\) and \({ \phi(v) > \inf_{y\in Y} \phi(y)}\) for all \((u,v)\in X\times Y\).
For each \((u,v) \in X\times Y\). There exists \((u',v') \in X\times Y\) such that
Define the set
For all \((u,v)\in X\times Y\), we have \(E(u,v) \neq\emptyset\) and \((u,v)\notin E(u,v) \).
We define the mappings \(T: X\rightarrow Y\) and \(S: Y\rightarrow Y\) \(Tu=v'\) and \(Sv=u'\) where \((u',v')\in E(u,v)\). For all \((u,v)\in X\times Y\), we have
By Theorem 2.9, there exists \((u^{*},v^{*})\in X\times Y\) such that \(Tu^{*}=v^{*}\) and \(Sv^{*}=u^{*}\). Hence, \((Sv^{*},Tu^{*})=(u^{*},v^{*}) \in E(u^{*},v^{*})\) which is absurd. □
4 Caristi’s fixed point theorem for two pairs of mappings in Hilbert space
In this section, we prove the existence of fixed points for two simultaneous projections to find the optimal solutions for some proximity functions via the Caristi fixed point theorem.
Let H be a Hilbert space, \(I= \{1,\ldots,m \}\) and \(J= \{1,\ldots,p \}\); for each \((i,j)\in I\times J\), we consider two nonempty closed convex subsets \(C_{i}\) and \(D_{j}\) of H and we define the metric projections \(P_{C_{i}} : H\rightarrow C_{i}\) and \(P_{D_{j}} : H\rightarrow D_{j}\).
For \(k\in \mathbb{N}^{\ast}\), we define \(\Delta_{k}\) by
For each \(u=(u_{1}, \ldots, u_{m})\in\Delta_{m} \) and \(w=(w_{1}, \ldots, w_{p})\in\Delta_{p}\), we define the proximity functions \(f:H\rightarrow \mathbb{R}^{+}\) and \(g:H\rightarrow \mathbb{R}^{+}\) by
The set of all minimizers of f and g is defined by
Theorem 4.1
Let \({ T= \sum_{i\in I} u_{i}P_{C_{i}} }\) and \(S= \sum_{j\in I} w_{j}P_{D_{j}}\) be simultaneous projections, where I and J are defined as above, and define \(f:H\rightarrow \mathbb{R}\) and \(g: H \rightarrow \mathbb{R}\) by (12).
Then we have
Moreover, if
-
1.
\(\Vert x-Tx\Vert \geq1\), for all \(x\in K\), where
$$K= \bigl\{ x\in H ; T^{n+1}x\neq T^{n}x \textit{ for all } n \in \mathbb{N}^{*} \bigr\} , $$ -
2.
there exists \(x_{0} \in H\) such that, for all \(n\in \mathbb{N}\), \(g(T^{n+1}x_{0})\leq g(ST^{n}x_{0})\),
then \(\operatorname{Fix}(T)\cap \operatorname{Fix}(S) \neq\emptyset\).
Proof
f, g are convex and differentiable functions. Moreover, for all \(x\in H\),
Therefore, the sufficient and necessary optimality yields
Let
The set \(X_{0}\) is nonempty (\(x_{0} \in X_{0}\)) and closed (complete).
First step. \(\overline{K}\cap X_{0} = \emptyset\), there exists \(z\in X_{0}\) such that \(z\notin\overline{K}\), so there exists \(p\in \mathbb{N}^{*}\) such that \(T^{p+1}z=T^{p}z\).
Since \(\Vert P_{D_{j}}(ST^{p}z) -ST^{p}z\Vert \leq \Vert P_{D_{j}}(T^{p}z) -ST^{p}z\Vert \), we have
We obtain
Thus,
Second step. \(\overline{K}\cap X_{0} \neq\emptyset\). Prove that \(T(\overline{K}\cap X_{0}) \subset\overline{K}\cap X_{0}\).
Let \(x\in K\); for all \(n\in \mathbb{N}^{*}\), we have
which gives \(T(K)\subset K\); and since T is continuous, we obtain \(T(\overline{K}) \subset\overline{T(K)} \subset\overline{K}\).
For any \(x\in\overline{K}\cap X_{0} \), there exists a sequence \((z_{n})_{n\geq0}\) of K such that \({ \lim_{n} z_{n} = x }\). Let \(n\in \mathbb{N}\). Since \(\Vert z_{n}-Tz_{n}\Vert \geq1\), we have
This leads, for all \(n\in \mathbb{N}\), to
We make n to +∞, which gives
Since \(\overline{K}\cap X_{0}\) is complete, so by the first inequality of equation (13), there exists \(x^{\ast}\in\overline{K}\cap X_{0}\) such that \(Tx^{\ast}=x^{\ast}\). And since
we conclude \(\operatorname{Fix}(T)\cap \operatorname{Fix}(S) \neq\emptyset\). □
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Chaira, K., Marhrani, E. Functional type Caristi-Kirk theorem on two metric spaces and applications. Fixed Point Theory Appl 2016, 96 (2016). https://doi.org/10.1186/s13663-016-0586-4
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DOI: https://doi.org/10.1186/s13663-016-0586-4