On Caristi’s theorem and successive approximations

Mira Anisiu
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Mira-Cristiana Anisiu
Institutul de Matematica, Cluj-Napoca, Romania

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M.-C. Anisiu, On Caristi’s theorem and successive approximations, Seminar on Functional Analysis and Numerical Methods, 1-10, Preprint, 86-1, Univ. Babeş-Bolyai Cluj-Napoca, 1986 (pdf file here)

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[1] J.P. Aubin, J. Siegel, Fixed points and stationary points of dissipative multivalued mappings. Proc. Amer. Math. Soc. 78(3)(1980), 391-398
[2] F.E. Browder, On a theorem of Caristi and Kirk, Fixed point theory and its applications, Ed. Swaminathan, Academic Press 1976, 23-28
[3] J. Caristi, Fixed point theorems for mappings satisfying inwardness conditions. Trans. Amer. Math. Soc. 215(1976), 241-251
[4] J. Caristi, W.A. Kirk, Geometric fixed point theory and inwardness conditions, Proc. Conf. on Geometry of Metric and Linear Spaces, Michigan 1964, Lecture Notes in Mathematics 490, Springer Verlag
[5] J. Dugundji, A. Granas, Fixed Point Theory, I, Monografie Matematyczne, Tom 61, Warszawa 1982
[6] I. Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc. 1(3)(1979), 443-474
[7] W.A. Kirk, Caristiís fixed point theorem and metric convexity, Colloq. Math. 36(1976), 81-86
[8] M. Maschler, B. Peleg, Stable sets and stable points of set-valued dynamic systems with applications to game theory, SIAM J. Control Optim. 14(6)(1976), 985-995
[9] J. Siegel, A new proof of Caristi’s fixed point theorem, Proc. Amer. Math. Soc. 66(1)(1977), 54-56
[10] M.R. Taskoviµc, A monotone principle of fixed points, Proc. Amer. Math. Soc. 94(3)(1985), 427-432.

1986-Anisiu-OnCaristi

"BABES-BOLYAI" UNIVERSITY, Faculty of Mathematics Research Seminars
Seminar of Functional Analysis and Numerical Methods Preprint Nr.1, 1986, pp. 1-10.

ON CARISTI'S THEOREM AND SUCCESSIVE APPROXIMATIONS

Mira-Cristiana Anisiu

Caristi's theorem [4] is an interesting and powerful generalization of the contraction principle. The first proofs [ 2 , 3 , 7 2 , 3 , 7 2,3,72,3,72,3,7 ] have shown the existence of a fixed point for the given function lacking the constructive aspects. J. Siegel [9] has considered the problem of the approximation of the fixed point in Caristi's theorem by countable iterations of some functions related to the initial one.
In the paper [10], M. R. Taskovič gives general conditions for each sequence of successive approximations of the function T : X → X T : X → X T:X rarr XT: X \rightarrow XT:X→X, where X X XXX is a topological space, to have a subsequence which converges to a fixed point for T T TTT. Then Caristi's theorem is derived, considering that each function T T TTT which satisfies the hypotheses in Caristi's one satisfies also those in Taskovič's theorem. In order to present this theorem, we mention firstly some definitions.
Let X X XXX be a topological space and T : X → X T : X → X T:X rarr XT: X \rightarrow XT:X→X a function. Denote by F T = { x ∈ X : T x = x } F T = { x ∈ X : T x = x } F_(T)={x in X:Tx=x}F_{T}=\{x \in X: T x=x\}FT={x∈X:Tx=x} the fixed point set of the function T T TTT. The set o ( x ) = { x , T x , T 2 x , … } o ( x ) = x , T x , T 2 x , … o(x)={x,Tx,T^(2)x,dots}o(x)=\left\{x, T x, T^{2} x, \ldots\right\}o(x)={x,Tx,T2x,…} is called the orbit of x x xxx for each x x xxx in X X XXX. A function B : X → R B : X → R B:X rarrRB: X \rightarrow \mathbb{R}B:X→R is said T T TTT-orbitally lower semicontinuous (T-orbitally l.s.c.) at p p ppp if from { x n } n ∈ N x n n ∈ N {x_(n)}_(n inN)\left\{x_{n}\right\}_{n \in \mathbb{N}}{xn}n∈N sequence in o ( x ) o ( x ) o(x)o(x)o(x) and x → n p x → n p xrarr"n"px \xrightarrow{n} px→np it follows B ( p ) ≤ lim inf n B ( x n ) B ( p ) ≤ lim inf n   B x n B(p) <= l i m   i n f_(n)B(x_(n))B(p) \leq \liminf _{n} B\left(x_{n}\right)B(p)≤lim infnB(xn). If B B BBB is T T TTT-orbitally l.s.c. at each p p ppp in X X XXX, it is called T T TTT-orbitally l.s.c..
The topological space X X XXX is said to satisfy the condition of TCSconvergence if B ( T n x ) → n 0 B T n x → n 0 B(T^(n)x)rarr"n"0B\left(T^{n} x\right) \xrightarrow{n} 0B(Tnx)→n0 implies that { T n x } n ∈ N T n x n ∈ N {T^(n)x}_(n inN)\left\{T^{n} x\right\}_{n \in \mathbb{N}}{Tnx}n∈N has a convergent subsequence.
Then Theorem 2 in [10], with the conclusions completed with some results which follow from its proof, is stated like this.
Theorem 1 . Let X X XXX be a topological space, T : X → X , B : X → [ 0 , ∞ ) T : X → X , B : X → [ 0 , ∞ ) T:X rarr X,B:X rarr[0,oo)T: X \rightarrow X, B: X \rightarrow [0, \infty)T:X→X,B:X→[0,∞) a T T TTT-orbitally l.s.c. function such that X X XXX satisfies the condition of TCS-convergence and B ( x ) = 0 B ( x ) = 0 B(x)=0B(x)=0B(x)=0 implies T x = x T x = x Tx=xT x=xTx=x.
Let γ : [ 0 , ∞ ) → [ 0 , ∞ ) γ : [ 0 , ∞ ) → [ 0 , ∞ ) gamma:[0,oo)rarr[0,oo)\gamma:[0, \infty) \rightarrow[0, \infty)γ:[0,∞)→[0,∞) be a function such that γ ( t ) < t γ ( t ) < t gamma(t) < t\gamma(t)<tγ(t)<t and lim sup z → t + 0 γ ( z ) < t lim sup z → t + 0   γ ( z ) < t l i m   s u p_(z rarr t+0)gamma(z) < t\limsup _{z \rightarrow t+0} \gamma(z)<tlim supz→t+0γ(z)<t for each t > 0 t > 0 t > 0t>0t>0, the following condition being satisfied
(1) B ( T x ) ≤ γ ( B ( x ) ) for each x in X ∖ F T . (1) B ( T x ) ≤ γ ( B ( x ) )  for each  x  in  X ∖ F T .  {:(1)B(Tx) <= gamma(B(x))" for each "x" in "X\\F_(T)". ":}\begin{equation*} B(T x) \leq \gamma(B(x)) \text { for each } x \text { in } X \backslash F_{T} \text {. } \tag{1} \end{equation*}(1)B(Tx)≤γ(B(x)) for each x in X∖FT. 
Then for each x x xxx in X X XXX there exists a subsequence { T n j x } j ∈ N T n j x j ∈ N {T^(n_(j))x}_(j inN)\left\{T^{n_{j}} x\right\}_{j \in \mathbb{N}}{Tnjx}j∈N of successive approximations starting from x x xxx, which is convergent to a fixed point ξ ξ xi\xiξ of T T TTT.
For the sake of completeness we present the proof.
Proof of Theorem 1. Let x x xxx be an arbitrary element of X X XXX; if x x xxx is a fixed point of T T TTT (or if T n x T n x T^(n)xT^{n} xTnx is for some n ≥ 1 n ≥ 1 n >= 1n \geq 1n≥1 ), the conclusion holds. Let now T n + 1 x ≠ T n x T n + 1 x ≠ T n x T^(n+1)x!=T^(n)xT^{n+1} x \neq T^{n} xTn+1x≠Tnx for each n ≥ 0 n ≥ 0 n >= 0n \geq 0n≥0. The condition (1) gives then
B ( T n + 1 x ) ≤ γ ( B ( T n x ) ) < B ( T n x ) , for each n ≥ 0 . B T n + 1 x ≤ γ B T n x < B T n x ,  for each  n ≥ 0 . B(T^(n+1)x) <= gamma(B(T^(n)x)) < B(T^(n)x)," for each "n >= 0.B\left(T^{n+1} x\right) \leq \gamma\left(B\left(T^{n} x\right)\right)<B\left(T^{n} x\right), \text { for each } n \geq 0 .B(Tn+1x)≤γ(B(Tnx))<B(Tnx), for each n≥0.
The properties of γ γ gamma\gammaγ assure the fact that B ( T n x ) → n 0 B T n x → n 0 B(T^(n)x)rarr"n"0B\left(T^{n} x\right) \xrightarrow{n} 0B(Tnx)→n0. The space X X XXX satisfying the condition of T C S T C S TCST C STCS-convergence, it follows that there is a subsequence { T n j x } j ∈ N T n j x j ∈ N {T^(n_(j))x}_(j inN)\left\{T^{n_{j}} x\right\}_{j \in \mathbb{N}}{Tnjx}j∈N convergent to ξ ∈ X ξ ∈ X xi in X\xi \in Xξ∈X. The function B B BBB being T T TTT-orbitally l.s.c., we have
B ( ξ ) ≤ lim inf j B ( T n j x ) = lim inf n B ( T n ) = 0 , B ( ξ ) ≤ lim inf j   B T n j x = lim inf n   B T n = 0 , B(xi) <= l i m   i n f_(j)B(T^(n_(j))x)=l i m   i n f_(n)B(T^(n))=0,B(\xi) \leq \liminf _{j} B\left(T^{n_{j}} x\right)=\liminf _{n} B\left(T^{n}\right)=0,B(ξ)≤lim infjB(Tnjx)=lim infnB(Tn)=0,
hence B ( ξ ) = 0 B ( ξ ) = 0 B(xi)=0B(\xi)=0B(ξ)=0 and T ξ = ξ T ξ = ξ T xi=xiT \xi=\xiTξ=ξ, so the theorem is proved.
Then, in the paper [10], Caristi's theorem is presented as a consequence of this theorem; we show that this is not the case.
We recall Caristi's theorem.
Theorem 2 [3,4,6]. Let ( X , d X , d X,dX, dX,d ) be a complete metric space and T T TTT : X → X X → X X rarr XX \rightarrow XX→X a given function. Suppose that there is an lower semicontinuous (l.s.c.) function G : X → [ 0 , ∞ ) G : X → [ 0 , ∞ ) G:X rarr[0,oo)G: X \rightarrow[0, \infty)G:X→[0,∞) such that
(2) d ( x , T x ) ≤ G ( x ) − G ( T x ) for each x in X . (2) d ( x , T x ) ≤ G ( x ) − G ( T x )  for each  x  in  X . {:(2)d(x","Tx) <= G(x)-G(Tx)" for each "x" in "X.:}\begin{equation*} d(x, T x) \leq G(x)-G(T x) \text { for each } x \text { in } X . \tag{2} \end{equation*}(2)d(x,Tx)≤G(x)−G(Tx) for each x in X.
Then the function T T TTT has a fixed point in X X XXX.
A simple proof, based upon an idea of Brondsted, is given in the book [5, p.16] in the following way.
Proof of Theorem 2. Define a multifunction C : X → 2 X ∖ { ∅ } C : X → 2 X ∖ { ∅ } C:X rarr2^(X)\\{O/}C: X \rightarrow 2^{X} \backslash\{\emptyset\}C:X→2X∖{∅} by C x = { y ∈ X : G ( y ) − d ( x , y ) ≤ G ( x ) } C x = { y ∈ X : G ( y ) − d ( x , y ) ≤ G ( x ) } Cx={y in X:G(y)-d(x,y) <= G(x)}C x=\{y \in X: G(y)-d(x, y) \leq G(x)\}Cx={y∈X:G(y)−d(x,y)≤G(x)}. The function G G GGG being l.s.c., C x C x CxC xCx is closed for each x x xxx in X X XXX. Let x 0 x 0 x_(0)x_{0}x0 be an arbitrary element of X X XXX. We construct a sequence { x n } n ∈ N x n n ∈ N {x_(n)}_(n inN)\left\{x_{n}\right\}_{n \in \mathbb{N}}{xn}n∈N choosing x 1 ∈ C x 0 x 1 ∈ C x 0 x_(1)in Cx_(0)x_{1} \in C x_{0}x1∈Cx0 such that G ( x 1 ) ≤ 1 + inf G | C x 0 G x 1 ≤ 1 + inf G C x 0 G(x_(1)) <= 1+ i n f G|_(Cx_(0))G\left(x_{1}\right) \leq 1+\left.\inf G\right|_{C x_{0}}G(x1)≤1+infG|Cx0; after obtaining x 1 , x 2 , … , x n − 1 x 1 , x 2 , … , x n − 1 x_(1),x_(2),dots,x_(n-1)x_{1}, x_{2}, \ldots, x_{n-1}x1,x2,…,xn−1, we take x n ∈ C x n − 1 x n ∈ C x n − 1 x_(n)in Cx_(n-1)x_{n} \in C x_{n-1}xn∈Cxn−1 such that
G ( x n ) ≤ 1 / n + inf G | C x n − 1 G x n ≤ 1 / n + inf G C x n − 1 G(x_(n)) <= 1//n+ i n f G|_(Cx_(n-1))G\left(x_{n}\right) \leq 1 / n+\left.\inf G\right|_{C x_{n-1}}G(xn)≤1/n+infG|Cxn−1
The sequence C x 0 ⊇ C x 1 ⊇ … C x 0 ⊇ C x 1 ⊇ … Cx_(0)supe Cx_(1)supe dotsC x_{0} \supseteq C x_{1} \supseteq \ldotsCx0⊇Cx1⊇… is nonincreasing. For each x x xxx in C x n , n ≥ 1 C x n , n ≥ 1 Cx_(n),n >= 1C x_{n}, n \geq 1Cxn,n≥1, we have x ∈ C x n ⊆ C x n − 1 x ∈ C x n ⊆ C x n − 1 x in Cx_(n)sube Cx_(n-1)x \in C x_{n} \subseteq C x_{n-1}x∈Cxn⊆Cxn−1, hence G ( x ) ≥ inf G | C x n − 1 ≥ G ( x n ) − 1 / n G ( x ) ≥ inf G C x n − 1 ≥ G x n − 1 / n G(x) >= i n f G|_(Cx_(n-1)) >= G(x_(n))-1//nG(x) \geq\left.\inf G\right|_{C x_{n-1}} \geq G\left(x_{n}\right)-1 / nG(x)≥infG|Cxn−1≥G(xn)−1/n. Because x ∈ C x n , d ( x n , x ) ≤ G ( x n ) − G ( x ) ≤ 1 / n x ∈ C x n , d x n , x ≤ G x n − G ( x ) ≤ 1 / n x in Cx_(n),d(x_(n),x) <= G(x_(n))-G(x) <= 1//nx \in C x_{n}, d\left(x_{n}, x\right) \leq G\left(x_{n}\right)-G(x) \leq 1 / nx∈Cxn,d(xn,x)≤G(xn)−G(x)≤1/n. It follows that diam C x n ≤ 2 / n diam C x n ≤ 2 / n diam Cx_(n) <= 2//n\operatorname{diam} C x_{n} \leq 2 / ndiamCxn≤2/n for each n ≥ 1 n ≥ 1 n >= 1n \geq 1n≥1 and applying Cantor's theorem we obtain x ∗ ∈ X x ∗ ∈ X x^(**)in Xx^{*} \in Xx∗∈X such that ⋂ n = 0 ∞ C x n = { x ∗ } ⋂ n = 0 ∞   C x n = x ∗ nnn_(n=0)^(oo)Cx_(n)={x^(**)}\bigcap_{n=0}^{\infty} C x_{n}=\left\{x^{*}\right\}⋂n=0∞Cxn={x∗}. The condition (2) implies d ( x ∗ , T x ∗ ) ≤ G ( x ∗ ) − G ( T x ∗ ) d x ∗ , T x ∗ ≤ G x ∗ − G T x ∗ d(x^(**),Tx^(**)) <= G(x^(**))-G(Tx^(**))d\left(x^{*}, T x^{*}\right) \leq G\left(x^{*}\right)-G\left(T x^{*}\right)d(x∗,Tx∗)≤G(x∗)−G(Tx∗); but x ∗ ∈ C x n x ∗ ∈ C x n x^(**)in Cx_(n)x^{*} \in C x_{n}x∗∈Cxn, hence d ( x ∗ , x n ) ≤ G ( x n ) − G ( x ∗ ) d x ∗ , x n ≤ G x n − G x ∗ d(x^(**),x_(n)) <= G(x_(n))-G(x^(**))d\left(x^{*}, x_{n}\right) \leq G\left(x_{n}\right)-G\left(x^{*}\right)d(x∗,xn)≤G(xn)−G(x∗) for each n n nnn in N N N\mathbb{N}N. It follows that d ( T x ∗ , x n ) ≤ G ( x n ) − G ( T x ∗ ) d T x ∗ , x n ≤ G x n − G T x ∗ d(Tx^(**),x_(n)) <= G(x_(n))-G(Tx^(**))d\left(T x^{*}, x_{n}\right) \leq G\left(x_{n}\right)-G\left(T x^{*}\right)d(Tx∗,xn)≤G(xn)−G(Tx∗), i.e. T x ∗ ∈ ⋂ n = 0 ∞ C x n = { x ∗ } T x ∗ ∈ ⋂ n = 0 ∞   C x n = x ∗ Tx^(**)innnn_(n=0)^(oo)Cx_(n)={x^(**)}T x^{*} \in \bigcap_{n=0}^{\infty} C x_{n}=\left\{x^{*}\right\}Tx∗∈⋂n=0∞Cxn={x∗} and x ∗ x ∗ x^(**)x^{*}x∗ is a fixed point for T T TTT.
Remark 1 In this proof of Caristi's theorem, the fixed points of T T TTT are obtained as limits of sequences of successive approximations for the multifunction C C CCC defined above.
As it was shown in [2], the proof of Theorem 2 is obvious when T T TTT is continuous; in this case the sequence of successive approximations of the given function T T TTT starting from each x x xxx in X X XXX converges to a fixed point of T T TTT, even in the absence of the lower semicontinuity of G G GGG.
We formulate the mentioned result in [2] in two propositions.
Proposition 1 Let ( X , d X , d X,dX, dX,d ) be a complete metric space, T : X → X T : X → X T:X rarr XT: X \rightarrow XT:X→X and G : X → [ 0 , ∞ ) G : X → [ 0 , ∞ ) G:X rarr[0,oo)G: X \rightarrow[0, \infty)G:X→[0,∞) arbitrary functions that satisfy the condition (2). Then the sequence { T n x } n ∈ N T n x n ∈ N {T^(n)x}_(n inN)\left\{T^{n} x\right\}_{n \in \mathbb{N}}{Tnx}n∈N converges for each x x xxx in X X XXX.
Proof. Let x x xxx be a given element in X X XXX. Applying (2) for x , T x , … , T n x x , T x , … , T n x x,Tx,dots,T^(n)xx, T x, \ldots, T^{n} xx,Tx,…,Tnx we obtain
d ( x , T x ) ≤ G ( x ) − G ( T x ) d ( T x , T 2 x ) ≤ G ( T x ) − G ( T 2 x ) ⋯ d ( T n x , T n + 1 x ) ≤ G ( T n x ) − G ( T n + 1 x ) d ( x , T x ) ≤ G ( x ) − G ( T x ) d T x , T 2 x ≤ G ( T x ) − G T 2 x ⋯ d T n x , T n + 1 x ≤ G T n x − G T n + 1 x {:[d(x","Tx) <= G(x)-G(Tx)],[d(Tx,T^(2)x) <= G(Tx)-G(T^(2)x)],[ cdots],[d(T^(n)x,T^(n+1)x) <= G(T^(n)x)-G(T^(n+1)x)]:}\begin{aligned} & d(x, T x) \leq G(x)-G(T x) \\ & d\left(T x, T^{2} x\right) \leq G(T x)-G\left(T^{2} x\right) \\ & \cdots \\ & d\left(T^{n} x, T^{n+1} x\right) \leq G\left(T^{n} x\right)-G\left(T^{n+1} x\right) \end{aligned}d(x,Tx)≤G(x)−G(Tx)d(Tx,T2x)≤G(Tx)−G(T2x)⋯d(Tnx,Tn+1x)≤G(Tnx)−G(Tn+1x)
Summing up these inequalities we have
∑ k = 0 n d ( T k x , T k + 1 x ) ≤ G ( x ) − G ( T n + 1 x ) ≤ G ( x ) ∑ k = 0 n   d T k x , T k + 1 x ≤ G ( x ) − G T n + 1 x ≤ G ( x ) sum_(k=0)^(n)d(T^(k)x,T^(k+1)x) <= G(x)-G(T^(n+1)x) <= G(x)\sum_{k=0}^{n} d\left(T^{k} x, T^{k+1} x\right) \leq G(x)-G\left(T^{n+1} x\right) \leq G(x)∑k=0nd(Tkx,Tk+1x)≤G(x)−G(Tn+1x)≤G(x)
hence { T n x } n ∈ N T n x n ∈ N {T^(n)x}_(n inN)\left\{T^{n} x\right\}_{n \in \mathbb{N}}{Tnx}n∈N is a Cauchy sequence; the completeness of X X XXX guarantees the convergence of { T n x } n ∈ N T n x n ∈ N {T^(n)x}_(n inN)\left\{T^{n} x\right\}_{n \in \mathbb{N}}{Tnx}n∈N.
The next proposition shows that if T T TTT is continuous (or even orbitally continuous in the sense that T n x → n ξ T n x → n ξ T^(n)xrarr"n"xiT^{n} x \xrightarrow{n} \xiTnx→nξ implies T n + 1 x → n T ξ T n + 1 x → n T ξ T^(n+1)xrarr"n"T xiT^{n+1} x \xrightarrow{n} T \xiTn+1x→nTξ for each ξ ξ xi\xiξ in X X XXX ), the limit of the sequence of successive approximations of T T TTT starting from every point x x xxx in X X XXX is a fixed point for T T TTT.
Proposition 2 In the hypotheses of Proposition 1, if T T TTT is (orbitally) continuous, the sequence of successive approximations of T T TTT starting from every point x x xxx in X X XXX converges to a fixed point of T T TTT.
Proof. Let x x xxx be an arbitrary point in X X XXX; by Proposition 1, { T n x } n ∈ N T n x n ∈ N {T^(n)x}_(n inN)\left\{T^{n} x\right\}_{n \in \mathbb{N}}{Tnx}n∈N converges to ξ ∈ X ; T ξ ∈ X ; T xi in X;T\xi \in X ; Tξ∈X;T being (orbitally) continuous, T n + 1 x → n T ξ T n + 1 x → n T ξ T^(n+1)xrarr"n"T xiT^{n+1} x \xrightarrow{n} T \xiTn+1x→nTξ. But { T n + 1 x } n ∈ N T n + 1 x n ∈ N {T^(n+1)x}_(n inN)\left\{T^{n+1} x\right\}_{n \in \mathbb{N}}{Tn+1x}n∈N converges also to ξ ξ xi\xiξ, hence ξ ξ xi\xiξ is a fixed point of T T TTT.
In the absence of the (orbitally) continuity of T T TTT, Proposition 2 is no more true (even if G G GGG is continuous), as the following example shows.
Example 1 Let X X XXX be the complete metric space X = { 0 , − 1 } ∪ { 1 / n X = { 0 , − 1 } ∪ { 1 / n X={0,-1}uu{1//nX=\{0,-1\} \cup\{1 / nX={0,−1}∪{1/n : n ∈ N } n ∈ N } n inN}n \in \mathbb{N}\}n∈N} with the usual metric on R , T : X → X R , T : X → X R,T:X rarr X\mathbb{R}, T: X \rightarrow XR,T:X→X given by
T x = { 1 / ( n + 1 ) , x = 1 / n − 1 , x ∈ { 0 , − 1 } T x = 1 / ( n + 1 ) , x = 1 / n − 1 , x ∈ { 0 , − 1 } Tx={[1//(n+1)","x=1//n],[-1","x in{0","-1}]:}T x=\left\{\begin{array}{l} 1 /(n+1), x=1 / n \\ -1, x \in\{0,-1\} \end{array}\right.Tx={1/(n+1),x=1/n−1,x∈{0,−1}
and G : X → [ 0 , ∞ ) , G ( x ) = x + 1 G : X → [ 0 , ∞ ) , G ( x ) = x + 1 G:X rarr[0,oo),G(x)=x+1G: X \rightarrow[0, \infty), G(x)=x+1G:X→[0,∞),G(x)=x+1 for each x x xxx in X X XXX.
G G GGG is continuous on X X XXX and the condition (2) is satisfied, so Caristi's theorem applies; but the sequence of successive approximations of T T TTT starting from all the points of the form x = 1 / n , n ∈ N x = 1 / n , n ∈ N x=1//n,n inNx=1 / n, n \in \mathbb{N}x=1/n,n∈N, converges to 0 which is not a fixed point for T T TTT. There are only two points, namely 0 and -1 , having the property that the sequence of successive approximations converges to the fixed point of T T TTT. In this example the conclusion of Theorem 1 does not hold, hence we cannot find any functions B B BBB and γ γ gamma\gammaγ to fulfil the conditions in Theorem 1. It is clear then that Caristi's theorem is not a corollary of Theorem 1.
There are two questions related to the above considerations.
QUESTION 1. Does there exist a complete metric space X X XXX and the functions T : X → X , T ≠ i d X T : X → X , T ≠ i d X T:X rarr X,T!=id_(X)T: X \rightarrow X, T \neq i d_{X}T:X→X,T≠idX and G : X → [ 0 , ∞ ) G : X → [ 0 , ∞ ) G:X rarr[0,oo)G: X \rightarrow[0, \infty)G:X→[0,∞), satisfying (2) such that { T n x } n ∈ N T n x n ∈ N {T^(n)x}_(n inN)\left\{T^{n} x\right\}_{n \in \mathbb{N}}{Tnx}n∈N converges to a fixed point of T T TTT iff x x xxx is a fixed point?
QUESTION 2. In the conditions in Question 1, what can one say about the convergence of { T n x } n ∈ N T n x n ∈ N {T^(n)x}_(n inN)\left\{T^{n} x\right\}_{n \in \mathbb{N}}{Tnx}n∈N to a fixed point of T T TTT if X X XXX is also connected?
The problem of the approximation of the fixed points in the case of multifunctions satisfying conditions of Caristi type was considered in papers like [ 1 , 8 ] [ 1 , 8 ] [1,8][1,8][1,8].
For a multifunction A : X → 2 X A : X → 2 X A:X rarr2^(X)A: X \rightarrow 2^{X}A:X→2X, a fixed point is an element x x xxx in X X XXX such that x ∈ A x x ∈ A x x in Axx \in A xx∈Ax; a strict fixed point is an element y y yyy in X X XXX such that A y = { y } A y = { y } Ay={y}A y=\{y\}Ay={y}.
In the following we present the analogous of Propositions 1 and 2 for multifunctions, the sequences of successive approximations converging to some fixed point of A A AAA.
Proposition 3 Let ( X , d ) ( X , d ) (X,d)(X, d)(X,d) be a complete metric space, A : X → 2 X ∖ { ∅ } , G : X → [ 0 , ∞ ) A : X → 2 X ∖ { ∅ } , G : X → [ 0 , ∞ ) A:X rarr2^(X)\\{O/},G:X rarr[0,oo)A: X \rightarrow 2^{X} \backslash \{\emptyset\}, G: X \rightarrow[0, \infty)A:X→2X∖{∅},G:X→[0,∞) arbitrary functions which satisfy the condition (3)
for each x x xxx in X X XXX there exists y y yyy in A x A x AxA xAx such that d ( x , y ) ≤ G ( x ) − G ( y ) d ( x , y ) ≤ G ( x ) − G ( y ) d(x,y) <= G(x)-G(y)d(x, y) \leq G(x)-G(y)d(x,y)≤G(x)−G(y).
Then for each x x xxx in X X XXX there exists a sequence { x n } n ∈ N x n n ∈ N {x_(n)}_(n inN)\left\{x_{n}\right\}_{n \in \mathbb{N}}{xn}n∈N, with x 1 = x x 1 = x x_(1)=xx_{1}=xx1=x and x n + 1 ∈ A x n x n + 1 ∈ A x n x_(n+1)in Ax_(n)x_{n+1} \in A x_{n}xn+1∈Axn for each n n nnn in N N N\mathbb{N}N, which is a convergent one.
Proof. Using the condition (3), we obtain for each x x xxx in X X XXX an element y = T x ∈ A x y = T x ∈ A x y=Tx in Axy=T x \in A xy=Tx∈Ax such that d ( x , T x ) ≤ G ( x ) − G ( T x ) d ( x , T x ) ≤ G ( x ) − G ( T x ) d(x,Tx) <= G(x)-G(Tx)d(x, T x) \leq G(x)-G(T x)d(x,Tx)≤G(x)−G(Tx). Applying Proposition 1 , the sequence { T n x } n ∈ N T n x n ∈ N {T^(n)x}_(n inN)\left\{T^{n} x\right\}_{n \in \mathbb{N}}{Tnx}n∈N is convergent and it is obviously a sequence of successive approximation for the multifunction A A AAA.
As it was shown in the paper of J.P. Aubin and J. Siegel [1,T2.4], if A : X → 2 X ∖ { ∅ } A : X → 2 X ∖ { ∅ } A:X rarr2^(X)\\{O/}A: X \rightarrow 2^{X} \backslash\{\emptyset\}A:X→2X∖{∅} is closed (in the sense that Gr A = { ( x , y ) ∈ X × X : y ∈ A x } Gr A = { ( x , y ) ∈ X × X : y ∈ A x } Gr A={(x,y)in X xx X:y in Ax}\operatorname{Gr} A=\{(x, y) \in X \times X: y \in A x\}GrA={(x,y)∈X×X:y∈Ax} is a closed set in the space X × X X × X X xx XX \times XX×X ), the following Proposition similar to Proposition 2 can be easily proved.
Proposition 4 In the hypotheses of Proposition 3, if A A AAA is a closed multifunction, A A AAA has fixed points and for each x x xxx in X X XXX there exists a sequence of successive approximations for A A AAA which converges to a fixed point of A A AAA.
Proof. Using Proposition 3, we obtain a convergent sequence { x n } n ∈ N x n n ∈ N {x_(n)}_(n inN)\left\{x_{n}\right\}_{n \in \mathbb{N}}{xn}n∈N, such that lim n x n = z ∈ X lim n   x n = z ∈ X lim_(n)x_(n)=z in X\lim _{n} x_{n}=z \in Xlimnxn=z∈X. But ( x n , x n + 1 ) ∈ GrA x n , x n + 1 ∈ GrA (x_(n),x_(n+1))in GrA\left(x_{n}, x_{n+1}\right) \in \operatorname{GrA}(xn,xn+1)∈GrA, hence ( z , z ) ∈ G r A ― = G r A ( z , z ) ∈ G r A ¯ = G r A (z,z)in bar(GrA)=GrA(z, z) \in \overline{G r A}=G r A(z,z)∈GrA―=GrA and x ∈ A x x ∈ A x x in Axx \in A xx∈Ax.
It is also possible to obtain strict fixed points as limits of successive approximations, imposing suitable conditions on the multifunction A A AAA.
Proposition 5 Let ( X , d ) ( X , d ) (X,d)(X, d)(X,d) be a complete metric space, A : X → 2 X ∖ { ∅ } , G : X → [ 0 , ∞ ) A : X → 2 X ∖ { ∅ } , G : X → [ 0 , ∞ ) A:X rarr2^(X)\\{O/},G:X rarr[0,oo)A: X \rightarrow 2^{X} \backslash\{\emptyset\}, G: X \rightarrow[0, \infty)A:X→2X∖{∅},G:X→[0,∞) such that
(4) d ( x , y ) ≤ G ( x ) − G ( y ) d ( x , y ) ≤ G ( x ) − G ( y ) d(x,y) <= G(x)-G(y)d(x, y) \leq G(x)-G(y)d(x,y)≤G(x)−G(y) for each x x xxx in X X XXX and for each y y yyy in A x A x AxA xAx
(5) A 2 x ⊆ A x for each x in X (5) A 2 x ⊆ A x  for each  x  in  X {:(5)A^(2)x sube Ax" for each "x" in "X:}\begin{equation*} A^{2} x \subseteq A x \text { for each } x \text { in } X \tag{5} \end{equation*}(5)A2x⊆Ax for each x in X
A x A x AxA xAx is closed for each x x xxx in X X XXX or A A AAA is l.s.c. (i.e. from y ∈ A x y ∈ A x y in Axy \in A xy∈Ax and x n → n x x n → n x x_(n)rarr"n"xx_{n} \xrightarrow{n} xxn→nx it follows that there exists y n ∈ A x n , y n → n y y n ∈ A x n , y n → n y y_(n)in Ax_(n),y_(n)rarr"n"yy_{n} \in A x_{n}, y_{n} \xrightarrow{n} yyn∈Axn,yn→ny for each x x xxx in X X XXX ).
Then the multifunction A A AAA has at least one strict fixed point and for each x x xxx in X X XXX there is a sequence of successive approximations for A A AAA which converges to a strict fixed point.
Proof. Let x 0 = x x 0 = x x_(0)=xx_{0}=xx0=x be an arbitrary element of X X XXX : we obtain a sequence considering an element x 1 ∈ A x 0 x 1 ∈ A x 0 x_(1)in Ax_(0)x_{1} \in A x_{0}x1∈Ax0 such that G ( x 1 ) ≤ 1 + inf G | A x 0 G x 1 ≤ 1 + inf G A x 0 G(x_(1)) <= 1+ i n f G|_(Ax_(0))G\left(x_{1}\right) \leq 1+\left.\inf G\right|_{A x_{0}}G(x1)≤1+infG|Ax0, and, when x 1 , … , x n − 1 x 1 , … , x n − 1 x_(1),dots,x_(n-1)x_{1}, \ldots, x_{n-1}x1,…,xn−1 are known, choosing x n ∈ A x n − 1 x n ∈ A x n − 1 x_(n)in Ax_(n-1)x_{n} \in A x_{n-1}xn∈Axn−1 such that G ( x n ) ≤ 1 / n + inf G | A x n − 1 G x n ≤ 1 / n + inf G A x n − 1 G(x_(n)) <= 1//n+ i n f G|_(Ax_(n-1))G\left(x_{n}\right) \leq 1 / n+\left.\inf G\right|_{A x_{n-1}}G(xn)≤1/n+infG|Axn−1.
Using (4), we have
diam A x n ≤ sup { d ( y , x n ) + d ( x n , z ) : y , z ∈ A x n } = 2 sup { d ( x n , z ) : z ∈ A x n } ≤ 2 [ G ( x n ) − inf G | A x n ] diam A x n ≤ sup d y , x n + d x n , z : y , z ∈ A x n = 2 sup d x n , z : z ∈ A x n ≤ 2 G x n − inf G A x n {:[diam Ax_(n) <= s u p{d(y,x_(n))+d(x_(n),z):y,z in Ax_(n)}],[=2s u p{d(x_(n),z):z in Ax_(n)} <= 2[G(x_(n))- i n f G|_(Ax_(n))]]:}\begin{aligned} \operatorname{diam} A x_{n} & \leq \sup \left\{d\left(y, x_{n}\right)+d\left(x_{n}, z\right): y, z \in A x_{n}\right\} \\ & =2 \sup \left\{d\left(x_{n}, z\right): z \in A x_{n}\right\} \leq 2\left[G\left(x_{n}\right)-\left.\inf G\right|_{A x_{n}}\right] \end{aligned}diamAxn≤sup{d(y,xn)+d(xn,z):y,z∈Axn}=2sup{d(xn,z):z∈Axn}≤2[G(xn)−infG|Axn]
But from (5) it follows that A x n ⊆ A x n − 1 A x n ⊆ A x n − 1 Ax_(n)sube Ax_(n-1)A x_{n} \subseteq A x_{n-1}Axn⊆Axn−1 for each n n nnn in N N N\mathbb{N}N, hence
diam A x n ≤ 2 [ G ( x n ) − inf G | A x n − 1 ] ≤ 2 / n . diam A x n ≤ 2 G x n − inf G A x n − 1 ≤ 2 / n . diam Ax_(n) <= 2[G(x_(n))- i n f G|_(Ax_(n-1))] <= 2//n.\operatorname{diam} A x_{n} \leq 2\left[G\left(x_{n}\right)-\left.\inf G\right|_{A x_{n-1}}\right] \leq 2 / n .diamAxn≤2[G(xn)−infG|Axn−1]≤2/n.
It follows that ⋂ A x n ― = { x ∗ } ⋂ A x n ¯ = x ∗ nnn bar(Ax_(n))={x^(**)}\bigcap \overline{A x_{n}}=\left\{x^{*}\right\}⋂Axn―={x∗}. Because x n + p , x n + 1 ∈ A x n x n + p , x n + 1 ∈ A x n x_(n+p),x_(n+1)in Ax_(n)x_{n+p}, x_{n+1} \in A x_{n}xn+p,xn+1∈Axn, we have d ( x n + p , x n + 1 ) ≤ 2 / n d x n + p , x n + 1 ≤ 2 / n d(x_(n+p),x_(n+1)) <= 2//nd\left(x_{n+p}, x_{n+1}\right) \leq 2 / nd(xn+p,xn+1)≤2/n and { x n } n ∈ N x n n ∈ N {x_(n)}_(n inN)\left\{x_{n}\right\}_{n \in \mathbb{N}}{xn}n∈N is a Cauchy sequence in the complete metric space X X XXX, hence it is a convergent one. But d ( x n + 1 , x ∗ ) ≤ diam A x n ― ≤ 2 / n d x n + 1 , x ∗ ≤ diam A x n ¯ ≤ 2 / n d(x_(n+1),x^(**)) <= diam bar(Ax_(n)) <= 2//nd\left(x_{n+1}, x^{*}\right) \leq \operatorname{diam} \overline{A x_{n}} \leq 2 / nd(xn+1,x∗)≤diamAxn―≤2/n and it follows that x ∗ x ∗ x^(**)x^{*}x∗ is the limit of { x n } n ∈ N x n n ∈ N {x_(n)}_(n inN)\left\{x_{n}\right\}_{n \in \mathbb{N}}{xn}n∈N.
Using the condition (6) we show that x ∗ x ∗ x^(**)x^{*}x∗ is in fact a strict fixed point for A A AAA.
a) If A x A x AxA xAx is closed for each x x xxx in X X XXX, we have ⋂ n A x n = { x ∗ } ⋂ n   A x n = x ∗ nnn_(n)Ax_(n)={x^(**)}\bigcap_{n} A x_{n}=\left\{x^{*}\right\}⋂nAxn={x∗}. Since x ∗ ∈ A x n x ∗ ∈ A x n x^(**)in Ax_(n)x^{*} \in A x_{n}x∗∈Axn for each n n nnn in N N N\mathbb{N}N, the condition (5) implies A x ∗ ⊆ A x n A x ∗ ⊆ A x n Ax^(**)sube Ax_(n)A x^{*} \subseteq A x_{n}Ax∗⊆Axn for each n n nnn in N N N\mathbb{N}N, i.e. A x ∗ ⊆ ⋂ n A x n = { x ∗ } A x ∗ ⊆ ⋂ n   A x n = x ∗ Ax^(**)subennn_(n)Ax_(n)={x^(**)}A x^{*} \subseteq \bigcap_{n} A x_{n}=\left\{x^{*}\right\}Ax∗⊆⋂nAxn={x∗}. But A x ∗ ≠ ∅ A x ∗ ≠ ∅ Ax^(**)!=O/A x^{*} \neq \emptysetAx∗≠∅, hence A x ∗ = { x ∗ } A x ∗ = x ∗ Ax^(**)={x^(**)}A x^{*}=\left\{x^{*}\right\}Ax∗={x∗}.
b) Let now A A AAA be l.s.c.. We have x n → n x ∗ x n → n x ∗ x_(n)rarr"n"x^(**)x_{n} \xrightarrow{n} x^{*}xn→nx∗; then for each y y yyy in A x ∗ A x ∗ Ax^(**)A x^{*}Ax∗ there exists an element y n ∈ A x n ( n ∈ N ) y n ∈ A x n ( n ∈ N ) y_(n)in Ax_(n)(n inN)y_{n} \in A x_{n}(n \in \mathbb{N})yn∈Axn(n∈N) such that y n → n y y n → n y y_(n)rarr"n"yy_{n} \xrightarrow{n} yyn→ny. But
d ( y n , x ∗ ) ≤ d ( y n , x n + 1 ) + d ( x n + 1 , x ∗ ) ≤ diam A x n + d ( x n + 1 , x ∗ ) d y n , x ∗ ≤ d y n , x n + 1 + d x n + 1 , x ∗ ≤ diam A x n + d x n + 1 , x ∗ d(y_(n),x^(**)) <= d(y_(n),x_(n+1))+d(x_(n+1),x^(**)) <= diam Ax_(n)+d(x_(n+1),x^(**))d\left(y_{n}, x^{*}\right) \leq d\left(y_{n}, x_{n+1}\right)+d\left(x_{n+1}, x^{*}\right) \leq \operatorname{diam} A x_{n}+d\left(x_{n+1}, x^{*}\right)d(yn,x∗)≤d(yn,xn+1)+d(xn+1,x∗)≤diamAxn+d(xn+1,x∗)
so y n → n x ∗ y n → n x ∗ y_(n)rarr"n"x^(**)y_{n} \xrightarrow{n} x^{*}yn→nx∗. It follows that y = x ∗ y = x ∗ y=x^(**)y=x^{*}y=x∗, hence we obtain again A x ∗ = { x ∗ } A x ∗ = x ∗ Ax^(**)={x^(**)}A x^{*}=\left\{x^{*}\right\}Ax∗={x∗}.
Remark 2 It is easily seen that Caristi's theorem for multifunctions is a consequence of Proposition 5 (therefore Theorem 2 is a consequence too). Indeed, suppose that G : X → [ 0 , ∞ ) G : X → [ 0 , ∞ ) G:X rarr[0,oo)G: X \rightarrow[0, \infty)G:X→[0,∞) is l.s.c. and A A AAA satisfies (4); then the multifunction C : X → 2 X ∖ { ∅ } C : X → 2 X ∖ { ∅ } C:X rarr2^(X)\\{O/}C: X \rightarrow 2^{X} \backslash\{\emptyset\}C:X→2X∖{∅} defined in the proof of theorem 2 has closed values and it obviously verifies the condition (5). It follows that there exists an element x ∗ x ∗ x^(**)x^{*}x∗ in X X XXX such that C x ∗ = { x ∗ } C x ∗ = x ∗ Cx^(**)={x^(**)}C x^{*}=\left\{x^{*}\right\}Cx∗={x∗}. But the condition (4) implies A x ⊆ C x A x ⊆ C x Ax sube CxA x \subseteq C xAx⊆Cx for each x x xxx in X X XXX, and because of A x ∗ ≠ ∅ A x ∗ ≠ ∅ Ax^(**)!=O/A x^{*} \neq \emptysetAx∗≠∅ we have A x ∗ = { x ∗ } A x ∗ = x ∗ Ax^(**)={x^(**)}A x^{*}=\left\{x^{*}\right\}Ax∗={x∗}. In this case, the strict fixed point is also obtained as a limit of successive approximations, but these are considered for the multifunction C C CCC, as it was pointed out in the case of the initial theorem of Caristi.
To obtain the fixed points as limits of successive approximations for the given (multi)function, it is necessary to impose supplementary conditions on the functions T : X → X T : X → X T:X rarr XT: X \rightarrow XT:X→X, respectively A : X → 2 X ∖ { ∅ } A : X → 2 X ∖ { ∅ } A:X rarr2^(X)\\{O/}A: X \rightarrow 2^{X} \backslash\{\emptyset\}A:X→2X∖{∅}.

References

[1] J.P. Aubin, J. Siegel, Fixed points and stationary points of dissipative multivalued mappings. Proc. Amer. Math. Soc. 78(3)(1980), 391-398
[2] F.E. Browder, On a theorem of Caristi and Kirk, Fixed point theory and its applications, Ed. Swaminathan, Academic Press 1976, 23-28
[3] J. Caristi, Fixed point theorems for mappings satisfying inwardness conditions. Trans. Amer. Math. Soc. 215(1976), 241-251
[4] J. Caristi, W.A. Kirk, Geometric fixed point theory and inwardness conditions, Proc. Conf. on Geometry of Metric and Linear Spaces, Michigan 1964, Lecture Notes in Mathematics 490, Springer Verlag
[5] J. Dugundji, A. Granas, Fixed Point Theory, I, Monografie Matematyczne, Tom 61, Warszawa 1982
[6] I. Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc. 1(3)(1979), 443-474
[7] W.A. Kirk, Caristi's fixed point theorem and metric convexity, Colloq. Math. 36(1976), 81-86
[8] M. Maschler, B. Peleg, Stable sets and stable points of set-valued dynamic systems with applications to game theory, SIAM J. Control Optim. 14(6)(1976), 985-995
[9] J. Siegel, A new proof of Caristi's fixed point theorem, Proc. Amer. Math. Soc. 66(1)(1977), 54-56
[10] M.R. Taskovič, A monotone principle of fixed points, Proc. Amer. Math. Soc. 94(3)(1985), 427-432.

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