On maximality principles related to Ekeland’s theorem

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

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M.-C. Anisiu, On maximality principles related to Ekeland’s theorem, Seminar on Functional Analysis and Numerical Methods, 1-8, Preprint, 87-1, Univ. Babeş-Bolyai Cluj-Napoca, 1987 (pdf file here)

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[6] I. Ekeland, Noncovex minimization problems, Bull. Amer. Math. Soc. 1(3)(1979), 443-474
[7] S. Park, Characterizations of metric completeness, Colloq. Math. XLIX(1984), 21-26
[8] M. Turinici, Di§erential inequalities via maximal element techniques, Nonlinear Analysis 5(1981), 757-763
[9] M. Turinici, Di§erential Lipschitzianness tests on abstract quasimetric spaces, Acta. Math. Hung. 41(1983), 93-100.

1987-Anisiu-OnEkeland

"BABEŞ-BOLYAI" UNIVERSITY FACULTY OF MATHEMATICS AND PHYSICS

Research Seminars
Seminar on Functional Analysis and Numerical Methods
Preprint nr.1, 1987, pp. 1-8

ON MAXIMALITY PRINCIPLES RELATED TO EKELAND'S THEOREM

Mira-Cristiana Anisiu

The large area of applications of the variational principle of I. Ekeland [5, 6] has determined in the last years a great interest in the generalizations of this principle. These are usually formulated as maximality principles. In the following we refer to the relations between such maximality principles.
The full statement of the theorem of I. Ekeland is the following.
Theorem 1 [ 5 , 6 ] 1 [ 5 , 6 ] 1[5,6]1[5,6]1[5,6] Let ( X , d X , d X,dX, dX,d ) be a complete metric space and f : X → R ∪ { + ∞ } f : X → R ∪ { + ∞ } f:X rarrRuu{+oo}f: X \rightarrow \mathbb{R} \cup\{+\infty\}f:X→R∪{+∞} a lower semicontinuous (l.s.c.) lower bounded function nonidentically equal to + ∞ + ∞ +oo+\infty+∞ (i.e. a proper function). Then for each ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0 and x 1 ∈ X x 1 ∈ X x_(1)in Xx_{1} \in Xx1∈X such that
f ( x 1 ) < inf { f ( x ) : x ∈ X } + ε , f x 1 < inf { f ( x ) : x ∈ X } + ε , f(x_(1)) < i n f{f(x):x in X}+epsi,f\left(x_{1}\right)<\inf \{f(x): x \in X\}+\varepsilon,f(x1)<inf{f(x):x∈X}+ε,
and for each λ > 0 λ > 0 lambda > 0\lambda>0λ>0 there exists x 0 ∈ X x 0 ∈ X x_(0)in Xx_{0} \in Xx0∈X such that
f ( x 0 ) ≤ f ( x 1 ) d ( x 0 , x 1 ) ≤ λ and f ( x ) > f ( x 0 ) − 1 λ d ( x 0 , x ) for each x in X ∖ { x 0 } . f x 0 ≤ f x 1 d x 0 , x 1 ≤ λ  and  f ( x ) > f x 0 − 1 λ d x 0 , x  for each  x  in  X ∖ x 0 . {:[f(x_(0)) <= f(x_(1))],[d(x_(0),x_(1)) <= lambda" and "],[f(x) > f(x_(0))-(1)/(lambda)d(x_(0),x)" for each "x" in "X\\{x_(0)}.]:}\begin{aligned} & f\left(x_{0}\right) \leq f\left(x_{1}\right) \\ & d\left(x_{0}, x_{1}\right) \leq \lambda \text { and } \\ & f(x)>f\left(x_{0}\right)-\frac{1}{\lambda} d\left(x_{0}, x\right) \text { for each } x \text { in } X \backslash\left\{x_{0}\right\} . \end{aligned}f(x0)≤f(x1)d(x0,x1)≤λ and f(x)>f(x0)−1λd(x0,x) for each x in X∖{x0}.
In the paper [3] there are given several equivalent variants of the Ekeland's principle. Denoting dom f = { x ∈ X : f ( x ) < + ∞ } f = { x ∈ X : f ( x ) < + ∞ } f={x in X:f(x) < +oo}f=\{x \in X: f(x)<+\infty\}f={x∈X:f(x)<+∞}, one of the simplest is the following one (known as Brondsted's Lemma).
Theorem 2 [4] Let ( X , d X , d X,dX, dX,d ) be a complete metric space and f : X → R ∪ { + ∞ } f : X → R ∪ { + ∞ } f:X rarrRuu{+oo}f: X \rightarrow \mathbb{R} \cup\{+\infty\}f:X→R∪{+∞} a l.s.c. lower bounded proper function. Then the relation defined by
x ≲ y iff x = y or x , y ∈ dom f , d ( x , y ) ≤ f ( x ) − f ( y ) x ≲ y  iff  x = y  or  x , y ∈ dom f , d ( x , y ) ≤ f ( x ) − f ( y ) x≲y" iff "x=y" or "x,y in dom f,quad d(x,y) <= f(x)-f(y)x \lesssim y \text { iff } x=y \text { or } x, y \in \operatorname{dom} f, \quad d(x, y) \leq f(x)-f(y)x≲y iff x=y or x,y∈domf,d(x,y)≤f(x)−f(y)
is on order relation and for each x 0 x 0 x_(0)x_{0}x0 in dom f f fff there is a maximal element x ∗ ∈ X x ∗ ∈ X x^(**)in Xx^{*} \in Xx∗∈X such that x 0 ≲ x ∗ x 0 ≲ x ∗ x_(0)≲x^(**)x_{0} \lesssim x^{*}x0≲x∗.
By order relation we mean a relation which is reflexive, transitive and antisymmetric.
It is clear that for every relation on X X XXX one can define a multivalued mapping F : X → 2 X F : X → 2 X F:X rarr2^(X)F: X \rightarrow 2^{X}F:X→2X such that y ∈ F ( x ) y ∈ F ( x ) y in F(x)y \in F(x)y∈F(x) iff x ≲ y x ≲ y x≲yx \lesssim yx≲y. The relation ≲ ≲ ≲\lesssim≲ is reflexive iff x ∈ F ( x ) x ∈ F ( x ) x in F(x)x \in F(x)x∈F(x) for each x ∈ X x ∈ X x in Xx \in Xx∈X, antisymmetric iff ( y ∈ F ( x ) , x ∈ F ( y ) ( y ∈ F ( x ) , x ∈ F ( y ) (y in F(x),x in F(y)(y \in F(x), x \in F(y)(y∈F(x),x∈F(y) implies x = y x = y x=yx=yx=y ) and transitive iff ( y ∈ F ( x ) y ∈ F ( x ) y in F(x)y \in F(x)y∈F(x) implies F ( y ) ⊆ F ( x ) F ( y ) ⊆ F ( x ) F(y)sube F(x)F(y) \subseteq F(x)F(y)⊆F(x) for each x , y x , y x,yx, yx,y in X X XXX ). The transitivity condition has the equivalent form F 2 ( x ) ⊆ F ( x ) F 2 ( x ) ⊆ F ( x ) F^(2)(x)sube F(x)F^{2}(x) \subseteq F(x)F2(x)⊆F(x) for each x x xxx in X X XXX, where F 2 ( x ) = { t ∈ X F 2 ( x ) = { t ∈ X F^(2)(x)={t in XF^{2}(x)=\{t \in XF2(x)={t∈X : there exists y ∈ F ( x ) y ∈ F ( x ) y in F(x)y \in F(x)y∈F(x) such that t ∈ F ( y ) } t ∈ F ( y ) } t in F(y)}t \in F(y)\}t∈F(y)}.
In the papers [3,8] one gives the following theorem, which is also restated in the terms of the multivalued mapping F F FFF.
Theorem 3 [3] Let ( X , d X , d X,dX, dX,d ) be a metric space and ≲ ≲ ≲\lesssim≲ an order relation on X X XXX such that
(1) for each monotone sequence { x n } n ∈ N x n n ∈ N {x_(n)}_(n inN)\left\{x_{n}\right\}_{n \in \mathbb{N}}{xn}n∈N (i.e. x n ≲ x m , ∀ n ≤ m ) x n ≲ x m , ∀ n ≤ m ) x_(n)≲x_(m),AA n <= m)x_{n} \lesssim x_{m}, \forall n \leq m)xn≲xm,∀n≤m) we have d ( x n , x n + 1 ) → n 0 d x n , x n + 1 → n 0 d(x_(n),x_(n+1))rarr"n"0d\left(x_{n}, x_{n+1}\right) \xrightarrow{n} 0d(xn,xn+1)→n0
(2) the set ( x , ≲ ) = { y ∈ X : x ≲ y ( x , ≲ ) = { y ∈ X : x ≲ y (x,≲)={y in X:x≲y(x, \lesssim)=\{y \in X: x \lesssim y(x,≲)={y∈X:x≲y ) is closed for each x x xxx in X X XXX.
Then for each x 0 x 0 x_(0)x_{0}x0 in X X XXX there is a maximal element x ¯ x ¯ bar(x)\bar{x}x¯ in X X XXX such that x 0 ≲ x ¯ x 0 ≲ x ¯ x_(0)≲ bar(x)x_{0} \lesssim \bar{x}x0≲x¯.
A generalization of Theorem 2 was given in the paper [9] in the following way.
Theorem 4 [9] Let ( X , d X , d X,dX, dX,d ) be a quasimetric space (that is the function d : X 2 → R + d : X 2 → R + d:X^(2)rarrR_(+)d: X^{2} \rightarrow \mathbb{R}_{+}d:X2→R+satisfies all the requirements of a metric except sufficiency) and ≲ ≲ ≲\lesssim≲ a reflexive and transitive relation on X X XXX which satisfies the conditions
(1') for each monotone sequence { x n } n ∈ N x n n ∈ N {x_(n)}_(n inN)\left\{x_{n}\right\}_{n \in \mathbb{N}}{xn}n∈N, inf d ( x n , x n + 1 ) = 0 d x n , x n + 1 = 0 d(x_(n),x_(n+1))=0d\left(x_{n}, x_{n+1}\right)=0d(xn,xn+1)=0 and
(3) for each monotone sequence { x n } n ∈ N x n n ∈ N {x_(n)}_(n inN)\left\{x_{n}\right\}_{n \in \mathbb{N}}{xn}n∈N which is a Cauchy one there exists z z zzz in X X XXX such that x n ≲ z x n ≲ z x_(n)≲zx_{n} \lesssim zxn≲z for each n n nnn in N N N\mathbb{N}N.
Then for each x 0 x 0 x_(0)x_{0}x0 in X X XXX there exists a d d ddd-maximal element x ¯ x ¯ bar(x)\bar{x}x¯ in X X XXX (i.e. x ¯ ≲ x x ¯ ≲ x bar(x)≲x\bar{x} \lesssim xx¯≲x implies d ( x ¯ , x ) = 0 d ( x ¯ , x ) = 0 d( bar(x),x)=0d(\bar{x}, x)=0d(x¯,x)=0 ) such that x 0 ≲ x ¯ x 0 ≲ x ¯ x_(0)≲ bar(x)x_{0} \lesssim \bar{x}x0≲x¯.
Remark 1 It is clear that any relation which satisfies the conditions in Theorem 3 satisfies also those in Theorem 4. Indeed, let ( X , d X , d X,dX, dX,d ) be a complete metric space and ≲ ≲ ≲\lesssim≲ an order relation which verifies (1) and (2). It remains to verify that ≲ ≲ ≲\lesssim≲ satisfies (3). Let { x n } n ∈ N x n n ∈ N {x_(n)}_(n inN)\left\{x_{n}\right\}_{n \in \mathbb{N}}{xn}n∈N be a Cauchy sequence in X X XXX; the completeness of X X XXX implies that { x n } n ∈ N x n n ∈ N {x_(n)}_(n inN)\left\{x_{n}\right\}_{n \in \mathbb{N}}{xn}n∈N converges to an element z z zzz in X X XXX. But x m ∈ ( x n , ≲ ) x m ∈ x n , ≲ x_(m)in(x_(n),≲)x_{m} \in\left(x_{n}, \lesssim\right)xm∈(xn,≲) for each m > n m > n m > nm>nm>n and ( x n , ≲ x n , ≲ x_(n),≲x_{n}, \lesssimxn,≲ ) is closed, hence z ∈ ( x n , ≲ ) z ∈ x n , ≲ z in(x_(n),≲)z \in\left(x_{n}, \lesssim\right)z∈(xn,≲) for each n n nnn in N N N\mathbb{N}N. It follows x n ≲ z x n ≲ z x_(n)≲zx_{n} \lesssim zxn≲z for each n n nnn in N N N\mathbb{N}N and (3) is satisfied.
Remark 2 In a metric space ( X , d X , d X,dX, dX,d ), the antisymmetry of ≲ ≲ ≲\lesssim≲ follows from the transitivity and the condition (1), respectively (1').
Let x , y ∈ X x , y ∈ X x,y in Xx, y \in Xx,y∈X such that x ≲ y x ≲ y x≲yx \lesssim yx≲y and y ≲ x y ≲ x y≲xy \lesssim xy≲x. It follows that the sequence { x n } n ∈ N , x 2 n − 1 = x , x 2 n = y x n n ∈ N , x 2 n − 1 = x , x 2 n = y {x_(n)}_(n inN),x_(2n-1)=x,x_(2n)=y\left\{x_{n}\right\}_{n \in \mathbb{N}}, x_{2 n-1}=x, x_{2 n}=y{xn}n∈N,x2n−1=x,x2n=y is monotone. Using (1) or (1') it follows d ( x , y ) = 0 d ( x , y ) = 0 d(x,y)=0d(x, y)=0d(x,y)=0, hence x = y x = y x=yx=yx=y and ≲ ≲ ≲\lesssim≲ is antisymmetric.
In the following we give a generalization of the above theorems.
Theorem 5 Let ( X , d X , d X,dX, dX,d ) be a quasimetric space and ≲ ≲ ≲\lesssim≲ a transitive relation such that (3) holds and
(4) for each x x xxx in X X XXX and ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0, there is y = y ε , x y = y ε , x y=y_(epsi,x)y=y_{\varepsilon, x}y=yε,x such that x ≤ y x ≤ y x <= yx \leq yx≤y and d ( y ) ≤ ε d ( y ) ≤ ε d(y) <= epsid(y) \leq \varepsilond(y)≤ε, where d ( y ) = sup { d ( z , y ) : y ≲ z } d ( y ) = sup { d ( z , y ) : y ≲ z } d(y)=s u p{d(z,y):y≲z}d(y)=\sup \{d(z, y): y \lesssim z\}d(y)=sup{d(z,y):y≲z}.
Then for each x 0 x 0 x_(0)x_{0}x0 in X X XXX there is a d d ddd-maximal element x ¯ x ¯ bar(x)\bar{x}x¯ in X X XXX such that x 0 ≲ x ¯ x 0 ≲ x ¯ x_(0)≲ bar(x)x_{0} \lesssim \bar{x}x0≲x¯.
Remark 3 The condition (4) implies the fact that ( x , ≲ x , ≲ x,≲x, \lesssimx,≲ ) is nonvoid for each x x xxx in X X XXX; it appears in [9] as condition (1) in the proof of Theorem 1 [9].
Proof of Theorem 5. Let x 0 ∈ X x 0 ∈ X x_(0)in Xx_{0} \in Xx0∈X be given. Using (4), we obtain inductively 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 such that
for x 0 x 0 x_(0)x_{0}x0 and ε = 1 / 2 ε = 1 / 2 epsi=1//2\varepsilon=1 / 2ε=1/2, there is x 1 ∈ X , x 0 ≲ x 1 x 1 ∈ X , x 0 ≲ x 1 x_(1)in X,x_(0)≲x_(1)x_{1} \in X, x_{0} \lesssim x_{1}x1∈X,x0≲x1 and d ( x 1 ) ≤ 1 / 2 d x 1 ≤ 1 / 2 d(x_(1)) <= 1//2d\left(x_{1}\right) \leq 1 / 2d(x1)≤1/2;
for x n − 1 x n − 1 x_(n-1)x_{n-1}xn−1 already obtained and ε = 1 / 2 n ε = 1 / 2 n epsi=1//2^(n)\varepsilon=1 / 2^{n}ε=1/2n, there exists x n ∈ X , x n − 1 ≲ x n x n ∈ X , x n − 1 ≲ x n x_(n)in X,x_(n-1)≲x_(n)x_{n} \in X, x_{n-1} \lesssim x_{n}xn∈X,xn−1≲xn and d ( x n ) ≤ 1 / 2 n d x n ≤ 1 / 2 n d(x_(n)) <= 1//2^(n)d\left(x_{n}\right) \leq 1 / 2^{n}d(xn)≤1/2n.
The sequence { x n } n ∈ N x n n ∈ N {x_(n)}_(n inN)\left\{x_{n}\right\}_{n \in \mathbb{N}}{xn}n∈N is monotone and it has the property that d ( x n ) ≤ 1 / 2 n d x n ≤ 1 / 2 n d(x_(n)) <= 1//2^(n)d\left(x_{n}\right) \leq 1 / 2^{n}d(xn)≤1/2n for each n n nnn in N N N\mathbb{N}N. But x n ≲ x n + 1 x n ≲ x n + 1 x_(n)≲x_(n+1)x_{n} \lesssim x_{n+1}xn≲xn+1 and, applying the transitivity, x n ≲ x n + p x n ≲ x n + p x_(n)≲x_(n+p)x_{n} \lesssim x_{n+p}xn≲xn+p; it follows that d ( x n + 1 , x n + p ) ≤ 2 d ( x n ) ≤ 1 / 2 n − 1 d x n + 1 , x n + p ≤ 2 d x n ≤ 1 / 2 n − 1 d(x_(n+1),x_(n+p)) <= 2d(x_(n)) <= 1//2^(n-1)d\left(x_{n+1}, x_{n+p}\right) \leq 2 d\left(x_{n}\right) \leq 1 / 2^{n-1}d(xn+1,xn+p)≤2d(xn)≤1/2n−1, hence { x n } x n {x_(n)}\left\{x_{n}\right\}{xn} is Cauchy sequence. The condition (3) implies the existence of x ¯ x ¯ bar(x)\bar{x}x¯ in X X XXX such that x n ≲ x ¯ x n ≲ x ¯ x_(n)≲ bar(x)x_{n} \lesssim \bar{x}xn≲x¯ for each n n nnn in N N N\mathbb{N}N. It is obvious that x 0 ≲ x ¯ x 0 ≲ x ¯ x_(0)≲ bar(x)x_{0} \lesssim \bar{x}x0≲x¯.
But x n ≲ x ¯ x n ≲ x ¯ x_(n)≲ bar(x)x_{n} \lesssim \bar{x}xn≲x¯ for each n n nnn in N N N\mathbb{N}N, hence d ( x ¯ , x n ) ≤ 1 / 2 n d x ¯ , x n ≤ 1 / 2 n d(( bar(x)),x_(n)) <= 1//2^(n)d\left(\bar{x}, x_{n}\right) \leq 1 / 2^{n}d(x¯,xn)≤1/2n. We show that x ¯ x ¯ bar(x)\bar{x}x¯ is d d ddd-maximal. Let x ∈ X , x ¯ ≲ x x ∈ X , x ¯ ≲ x x in X, bar(x)≲xx \in X, \bar{x} \lesssim xx∈X,x¯≲x. It follows that x n ≲ x x n ≲ x x_(n)≲xx_{n} \lesssim xxn≲x for each n n nnn in N N N\mathbb{N}N, so d ( x , x n ) ≤ d ( x n ) ≤ 1 / 2 n d x , x n ≤ d x n ≤ 1 / 2 n d(x,x_(n)) <= d(x_(n)) <= 1//2^(n)d\left(x, x_{n}\right) \leq d\left(x_{n}\right) \leq 1 / 2^{n}d(x,xn)≤d(xn)≤1/2n. We obtain d ( x , x ¯ ) ≤ d ( x , x n ) + d ( x n , x ¯ ) ≤ 1 / 2 n − 1 d ( x , x ¯ ) ≤ d x , x n + d x n , x ¯ ≤ 1 / 2 n − 1 d(x, bar(x)) <= d(x,x_(n))+d(x_(n),( bar(x))) <= 1//2^(n-1)d(x, \bar{x}) \leq d\left(x, x_{n}\right)+d\left(x_{n}, \bar{x}\right) \leq 1 / 2^{n-1}d(x,x¯)≤d(x,xn)+d(xn,x¯)≤1/2n−1, hence d ( x , x ¯ ) = 0 d ( x , x ¯ ) = 0 d(x, bar(x))=0d(x, \bar{x})=0d(x,x¯)=0 and the theorem is proved.
Theorem 4, and, following Remark 1, Theorem 3 too, is a corollary of Theorem 5. Indeed, it remains to prove that, the conditions in Theorem 4 being satisfied, the condition (4) holds. Suppose that there exists x x xxx in X X XXX and ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0 such that for each y y yyy in X , x ≲ y X , x ≲ y X,x≲yX, x \lesssim yX,x≲y we have d ( y ) > ε d ( y ) > ε d(y) > epsid(y)>\varepsilond(y)>ε. The relation ≲ ≲ ≲\lesssim≲ being reflexive, ( x , ≲ ) ( x , ≲ ) (x,≲)(x, \lesssim)(x,≲) is nonvoid for each x x xxx in X X XXX. Let x 1 ∈ X x 1 ∈ X x_(1)in Xx_{1} \in Xx1∈X such that x ≲ x 1 x ≲ x 1 x≲x_(1)x \lesssim x_{1}x≲x1; but d ( x 1 ) > ε d x 1 > ε d(x_(1)) > epsid\left(x_{1}\right)>\varepsilond(x1)>ε, and we obtain x 2 ∈ X x 2 ∈ X x_(2)in Xx_{2} \in Xx2∈X, x 1 ≲ x 2 x 1 ≲ x 2 x_(1)≲x_(2)x_{1} \lesssim x_{2}x1≲x2 such that d ( x 1 , x 2 ) ≥ ε − ε / 2 d x 1 , x 2 ≥ ε − ε / 2 d(x_(1),x_(2)) >= epsi-epsi//2d\left(x_{1}, x_{2}\right) \geq \varepsilon-\varepsilon / 2d(x1,x2)≥ε−ε/2. We have x ≲ x 2 x ≲ x 2 x≲x_(2)x \lesssim x_{2}x≲x2, so d ( x 2 ) > ε d x 2 > ε d(x_(2)) > epsid\left(x_{2}\right)>\varepsilond(x2)>ε and there exists x 3 ∈ X , x 2 ≲ x 3 , d ( x 2 , x 3 ) ≥ ε − ε / 2 2 x 3 ∈ X , x 2 ≲ x 3 , d x 2 , x 3 ≥ ε − ε / 2 2 x_(3)in X,x_(2)≲x_(3),d(x_(2),x_(3)) >= epsi-epsi//2^(2)x_{3} \in X, x_{2} \lesssim x_{3}, d\left(x_{2}, x_{3}\right) \geq \varepsilon-\varepsilon / 2^{2}x3∈X,x2≲x3,d(x2,x3)≥ε−ε/22. Inductively, for
x n ∈ X , x ≲ x n x n ∈ X , x ≲ x n x_(n)in X,x≲x_(n)x_{n} \in X, x \lesssim x_{n}xn∈X,x≲xn with d ( x n ) > ε d x n > ε d(x_(n)) > epsid\left(x_{n}\right)>\varepsilond(xn)>ε we obtain x n + 1 ∈ X , x n ≲ x n + 1 x n + 1 ∈ X , x n ≲ x n + 1 x_(n+1)in X,x_(n)≲x_(n+1)x_{n+1} \in X, x_{n} \lesssim x_{n+1}xn+1∈X,xn≲xn+1 and d ( x n , x n + 1 ) ≥ ε − ε / 2 n d x n , x n + 1 ≥ ε − ε / 2 n d(x_(n),x_(n+1)) >= epsi-epsi//2^(n)d\left(x_{n}, x_{n+1}\right) \geq \varepsilon-\varepsilon / 2^{n}d(xn,xn+1)≥ε−ε/2n. We have inf ( x n , x n + 1 ) ≥ inf { ε − ε / 2 n } = ε / 2 inf x n , x n + 1 ≥ inf ε − ε / 2 n = ε / 2 i n f(x_(n),x_(n+1)) >= i n f{epsi-epsi//2^(n)}=epsi//2\inf \left(x_{n}, x_{n+1}\right) \geq \inf \left\{\varepsilon-\varepsilon / 2^{n}\right\}=\varepsilon / 2inf(xn,xn+1)≥inf{ε−ε/2n}=ε/2, contradicting (1') in Theorem 4. It follows that (4) is satisfied and using Theorem 5 the conclusion of Theorem 4 holds.
In the following example, Theorem 5 applies, but Theorem 4 doesn't.
Example 1 Let X = [ 0 , 1 ] ∪ [ 2 , 3 ] X = [ 0 , 1 ] ∪ [ 2 , 3 ] X=[0,1]uu[2,3]X=[0,1] \cup[2,3]X=[0,1]∪[2,3] with the usual metric on R R R\mathbb{R}R and the reflexive and transitive relation given by
x ≲ y if x ≤ y or ( x = 2 , y = 1 ) x ≲ y  if  x ≤ y  or  ( x = 2 , y = 1 ) x≲y" if "x <= y" or "(x=2,y=1)x \lesssim y \text { if } x \leq y \text { or }(x=2, y=1)x≲y if x≤y or (x=2,y=1)
≤ ≤ <=\leq≤ being the natural order relation on R R R\mathbb{R}R.
Condition (1) fails to be satisfied, because the sequence x 1 = 1 x 1 = 1 x_(1)=1x_{1}=1x1=1, x 2 = 2 , x 3 = 1 , … x 2 = 2 , x 3 = 1 , … x_(2)=2,x_(3)=1,dotsx_{2}=2, x_{3}=1, \ldotsx2=2,x3=1,… is monotone and d ( x n , x n + 1 ) ↛ 0 d x n , x n + 1 ↛ 0 d(x_(n),x_(n+1))↛0d\left(x_{n}, x_{n+1}\right) \nrightarrow 0d(xn,xn+1)↛0. The relation ≲ ≲ ≲\lesssim≲ is not antisymmetric.
It is worth mentioning that the theorem of Brézis-Browder [1] and Ekeland are simple consequences of Theorem 4, as it was shown in [9].
In the following we give a characterization of the completeness of a metric space in the terms of maximal elements for a relation, respectively strict fixed points for a mutivalued mapping.
In the paper [7], S. Park gives seven characterizations of metric completeness related to the conditions in theorems of Caristi-Ekeland type [2,5]. We propose one more characterization to be added to the mentioned list.
Let ( X , d X , d X,dX, dX,d ) be a metric space. Among the equivalent statements given in the theorem in [7], there are the following two (denoted there by (iii), respectively (iv)).
(a) For every sequence { F n } n ∈ N F n n ∈ N {F_(n)}_(n inN)\left\{F_{n}\right\}_{n \in \mathbb{N}}{Fn}n∈N of nonempty closed subsets of X X XXX such that F n + 1 ⊆ F n , n ∈ N F n + 1 ⊆ F n , n ∈ N F_(n+1)subeF_(n),n inNF_{n+1} \subseteq F_{n}, n \in \mathbb{N}Fn+1⊆Fn,n∈N, and the sequence { diam F n } n ∈ N diam F n n ∈ N {diamF_(n)}_(n inN)\left\{\operatorname{diam} F_{n}\right\}_{n \in \mathbb{N}}{diamFn}n∈N converges to 0 , we have ⋂ n = 1 ∞ F n ≠ ∅ ⋂ n = 1 ∞   F n ≠ ∅ nnn_(n=1)^(oo)F_(n)!=O/\bigcap_{n=1}^{\infty} F_{n} \neq \emptyset⋂n=1∞Fn≠∅.
(b) Every lower semicontinuous function h : X → ( − ∞ , + ∞ ) h : X → ( − ∞ , + ∞ ) h:X rarr(-oo,+oo)h: X \rightarrow(-\infty,+\infty)h:X→(−∞,+∞) which is bounded from below has a d d ddd-point p p ppp in X X XXX, that is h ( p ) − h ( x ) < d ( p , x ) h ( p ) − h ( x ) < d ( p , x ) h(p)-h(x) < d(p,x)h(p)-h(x)< d(p, x)h(p)−h(x)<d(p,x) for every point x x xxx in X , x ≠ p X , x ≠ p X,x!=pX, x \neq pX,x≠p.
We consider now the following statement.
Every multivalued mapping F : X → 2 X F : X → 2 X F:X rarr2^(X)F: X \rightarrow 2^{X}F:X→2X such that there exists a sequence { x n } n ∈ N , x n + 1 ∈ F ( x n ) , n ∈ N x n n ∈ N , x n + 1 ∈ F x n , n ∈ N {x_(n)}_(n inN),x_(n+1)in F(x_(n)),n inN\left\{x_{n}\right\}_{n \in \mathbb{N}}, x_{n+1} \in F\left(x_{n}\right), n \in \mathbb{N}{xn}n∈N,xn+1∈F(xn),n∈N and
  1. F ( z ) = F ( z ) ― F ( z ) = F ( z ) ¯ F(z)= bar(F(z))F(z)=\overline{F(z)}F(z)=F(z)― for each z z zzz in { x n } n ∈ N x n n ∈ N {x_(n)}_(n inN)\left\{x_{n}\right\}_{n \in \mathbb{N}}{xn}n∈N
  2. F ( z ) ≠ ∅ F ( z ) ≠ ∅ F(z)!=O/F(z) \neq \emptysetF(z)≠∅ for each z z zzz in { x n } n ∈ N x n n ∈ N {x_(n)}_(n inN)\left\{x_{n}\right\}_{n \in \mathbb{N}}{xn}n∈N
  3. z 2 ∈ F ( z 1 ) ⇒ F ( z 2 ) ⊆ F ( z 1 ) z 2 ∈ F z 1 ⇒ F z 2 ⊆ F z 1 z_(2)in F(z_(1))=>F(z_(2))sube F(z_(1))z_{2} \in F\left(z_{1}\right) \Rightarrow F\left(z_{2}\right) \subseteq F\left(z_{1}\right)z2∈F(z1)⇒F(z2)⊆F(z1) for each z 1 , z 2 z 1 , z 2 z_(1),z_(2)z_{1}, z_{2}z1,z2 in { x n } n ∈ N ― x n n ∈ N ¯ bar({x_(n)}_(n inN))\overline{\left\{x_{n}\right\}_{n \in \mathbb{N}}}{xn}n∈N―
  4. diam F ( x n ) → n 0 diam F x n → n 0 diam F(x_(n))rarr"n"0\operatorname{diam} F\left(x_{n}\right) \xrightarrow{n} 0diamF(xn)→n0,
    has a strict fixed point x ∗ x ∗ x^(**)x^{*}x∗ (i.e. F ( x ∗ ) = { x ∗ } F x ∗ = x ∗ F(x^(**))={x^(**)}F\left(x^{*}\right)=\left\{x^{*}\right\}F(x∗)={x∗} ) and x ∗ = lim n x n x ∗ = lim n   x n x^(**)=lim_(n)x_(n)x^{*}=\lim _{n} x_{n}x∗=limnxn.
    Theorem 6 The following implications hold:
( a ) ⇒ ( ∗ ) ⇒ ( b ) ( a ) ⇒ ( ∗ ) ⇒ ( b ) (a)=>(**)=>(b)(a) \Rightarrow(*) \Rightarrow(b)(a)⇒(∗)⇒(b)
Proof. ( a ) ⇒ ( ∗ ) ( a ) ⇒ ( ∗ ) (a)=>(**)(a) \Rightarrow(*)(a)⇒(∗). Let { x n } n ∈ N x n n ∈ N {x_(n)}_(n inN)\left\{x_{n}\right\}_{n \in \mathbb{N}}{xn}n∈N be as in the hypothesis of (*). Then the condition 3) implies F ( x n + 1 ) ⊆ F ( x n ) , n ∈ N F x n + 1 ⊆ F x n , n ∈ N F(x_(n+1))sube F(x_(n)),n inNF\left(x_{n+1}\right) \subseteq F\left(x_{n}\right), n \in \mathbb{N}F(xn+1)⊆F(xn),n∈N and using 1), 2) and 4) we can apply (a), hence ⋂ n F ( x n ) = { x ∗ } ⋂ n   F x n = x ∗ nnn_(n)F(x_(n))={x^(**)}\bigcap_{n} F\left(x_{n}\right)=\left\{x^{*}\right\}⋂nF(xn)={x∗}. From x n + 1 , x ∗ ∈ F ( x n ) x n + 1 , x ∗ ∈ F x n x_(n+1),x^(**)in F(x_(n))x_{n+1}, x^{*} \in F\left(x_{n}\right)xn+1,x∗∈F(xn) and b) we obtain x ∗ = lim n x n x ∗ = lim n   x n x^(**)=lim_(n)x_(n)x^{*}=\lim _{n} x_{n}x∗=limnxn. But x ∗ ∈ F ( x n ) x ∗ ∈ F x n x^(**)in F(x_(n))x^{*} \in F\left(x_{n}\right)x∗∈F(xn) and using 3) we get F ( x ∗ ) ⊆ F ( x n ) , n ∈ N F x ∗ ⊆ F x n , n ∈ N F(x^(**))sube F(x_(n)),n inNF\left(x^{*}\right) \subseteq F\left(x_{n}\right), n \in \mathbb{N}F(x∗)⊆F(xn),n∈N, so F ( x ∗ ) ⊆ { x ∗ } F x ∗ ⊆ x ∗ F(x^(**))sube{x^(**)}F\left(x^{*}\right) \subseteq\left\{x^{*}\right\}F(x∗)⊆{x∗}. It follows F ( x ∗ ) = { x ∗ } F x ∗ = x ∗ F(x^(**))={x^(**)}F\left(x^{*}\right)=\left\{x^{*}\right\}F(x∗)={x∗}, because F ( x ∗ } ≠ ∅ F x ∗ ≠ ∅ F(x^(**)}!=O/F\left(x^{*}\right\} \neq \emptysetF(x∗}≠∅.
( ∗ ) ⇒ ( b ) ( ∗ ) ⇒ ( b ) (**)=>(b)(*) \Rightarrow(b)(∗)⇒(b). For the given function h h hhh, we define F : X → 2 X F : X → 2 X F:X rarr2^(X)F: X \rightarrow 2^{X}F:X→2X by F ( x ) = { y ∈ X : d ( x , y ) ≤ h ( x ) − h ( y ) } F ( x ) = { y ∈ X : d ( x , y ) ≤ h ( x ) − h ( y ) } F(x)={y in X:d(x,y) <= h(x)-h(y)}F(x)=\{y \in X: d(x, y) \leq h(x)-h(y)\}F(x)={y∈X:d(x,y)≤h(x)−h(y)}. We have x ∈ F ( x ) x ∈ F ( x ) x in F(x)x \in F(x)x∈F(x) for x x xxx in X X XXX, hence F ( x ) ≠ ∅ F ( x ) ≠ ∅ F(x)!=O/F(x) \neq \emptysetF(x)≠∅; the condition 1) holds on X X XXX because h h hhh is lower semicontinuous. To verify 3), let z 1 , z 2 ∈ X , z 2 ∈ F ( z 1 ) z 1 , z 2 ∈ X , z 2 ∈ F z 1 z_(1),z_(2)in X,z_(2)in F(z_(1))z_{1}, z_{2} \in X, z_{2} \in F\left(z_{1}\right)z1,z2∈X,z2∈F(z1) and z ∈ F ( z 2 ) z ∈ F z 2 z in F(z_(2))z \in F\left(z_{2}\right)z∈F(z2); it means
d ( z 1 , z 2 ) ≤ h ( z 1 ) − h ( z 2 ) and d ( z 2 , z ) ≤ h ( z 2 ) − h ( z ) then d ( z 1 , z ) ≤ h ( z 1 ) − h ( z ) , i.e. ∈ F ( z 1 ) and F ( z 2 ) ⊆ F ( z 1 ) . d z 1 , z 2 ≤ h z 1 − h z 2  and  d z 2 , z ≤ h z 2 − h ( z )  then  d z 1 , z ≤ h z 1 − h ( z ) ,  i.e.  ∈ F z 1  and  F z 2 ⊆ F z 1 . {:[d(z_(1),z_(2)) <= h(z_(1))-h(z_(2))" and "],[d(z_(2),z) <= h(z_(2))-h(z)],[" then "d(z_(1),z) <= h(z_(1))-h(z)","" i.e. "in F(z_(1))" and "F(z_(2))sube F(z_(1)).]:}\begin{aligned} & d\left(z_{1}, z_{2}\right) \leq h\left(z_{1}\right)-h\left(z_{2}\right) \text { and } \\ & d\left(z_{2}, z\right) \leq h\left(z_{2}\right)-h(z) \\ & \text { then } d\left(z_{1}, z\right) \leq h\left(z_{1}\right)-h(z), \text { i.e. } \in F\left(z_{1}\right) \text { and } F\left(z_{2}\right) \subseteq F\left(z_{1}\right) . \end{aligned}d(z1,z2)≤h(z1)−h(z2) and d(z2,z)≤h(z2)−h(z) then d(z1,z)≤h(z1)−h(z), i.e. ∈F(z1) and F(z2)⊆F(z1).
Let x 0 ∈ X x 0 ∈ X x_(0)in Xx_{0} \in Xx0∈X be given; we obtain inductively 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 such that x n + 1 ∈ F ( x n ) , h ( x n + 1 ) < inf h ( F ( x n ) ) + 1 / n x n + 1 ∈ F x n , h x n + 1 < inf h F x n + 1 / n x_(n+1)in F(x_(n)),h(x_(n+1)) < i n f h(F(x_(n)))+1//nx_{n+1} \in F\left(x_{n}\right), h\left(x_{n+1}\right)<\inf h\left(F\left(x_{n}\right)\right)+1 / nxn+1∈F(xn),h(xn+1)<infh(F(xn))+1/n. It follows that for x x xxx in F ( x n + 1 ) ⊆ F ( x n ) F x n + 1 ⊆ F x n F(x_(n+1))sube F(x_(n))F\left(x_{n+1}\right) \subseteq F\left(x_{n}\right)F(xn+1)⊆F(xn) we have
d ( x , x n + 1 ) ≤ h ( x n + 1 ) − h ( x ) ≤ h ( x n + 1 ) − inf h ( F ( x n ) ) < 1 / n , d x , x n + 1 ≤ h x n + 1 − h ( x ) ≤ h x n + 1 − inf h F x n < 1 / n , d(x,x_(n+1)) <= h(x_(n+1))-h(x) <= h(x_(n+1))-i n f h(F(x_(n))) < 1//n,d\left(x, x_{n+1}\right) \leq h\left(x_{n+1}\right)-h(x) \leq h\left(x_{n+1}\right)-\inf h\left(F\left(x_{n}\right)\right)<1 / n,d(x,xn+1)≤h(xn+1)−h(x)≤h(xn+1)−infh(F(xn))<1/n,
hence diam F ( x n ) → n 0 F x n → n 0 F(x_(n))rarr"n"0F\left(x_{n}\right) \xrightarrow{n} 0F(xn)→n0. Applying ( ∗ ) ( ∗ ) (**)(*)(∗) we obtain p = lim n x n , F ( p ) = { p } p = lim n   x n , F ( p ) = { p } p=lim_(n)x_(n),F(p)={p}p=\lim _{n} x_{n}, F(p)= \{p\}p=limnxn,F(p)={p}. Then for each x x xxx in X ∖ { p } X ∖ { p } X\\{p}X \backslash\{p\}X∖{p} we have x ∉ F ( p ) x ∉ F ( p ) x!in F(p)x \notin F(p)x∉F(p), hence d ( p , x ) > h ( p ) − h ( x ) d ( p , x ) > h ( p ) − h ( x ) d(p,x) > h(p)-h(x)d(p, x)> h(p)-h(x)d(p,x)>h(p)−h(x).
It follows that condition (*) can be included in the list in [7] as (iii'), being equivalent to the completeness of the space X X XXX.
We also mention that ( ∗ ) ( ∗ ) (**)(*)(∗) can be expressed in the terms of a relation in the following way.
For each relation defined on the metric space 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 such that x n ≲ x n + 1 , n ∈ N x n ≲ x n + 1 , n ∈ N x_(n)≲x_(n+1),n inNx_{n} \lesssim x_{n+1}, n \in \mathbb{N}xn≲xn+1,n∈N and
  1. ( x n , ≲ ) = { y ∈ X : x n ≲ y } x n , ≲ = y ∈ X : x n ≲ y (x_(n),≲)={y in X:x_(n)≲y}\left(x_{n}, \lesssim\right)=\left\{y \in X: x_{n} \lesssim y\right\}(xn,≲)={y∈X:xn≲y} is a closed set for each n n nnn in N N N\mathbb{N}N
  2. ( z , ≲ ) ≠ ∅ ( z , ≲ ) ≠ ∅ (z,≲)!=O/(z, \lesssim) \neq \emptyset(z,≲)≠∅ for each z ∈ { x n } n ∈ N ― z ∈ x n n ∈ N ¯ z in bar({x_(n)}_(n inN))z \in \overline{\left\{x_{n}\right\}_{n \in \mathbb{N}}}z∈{xn}n∈N―
  3. ≲ ≲ ≲\lesssim≲ is transitive on the set { x n } n ∈ N ― x n n ∈ N ¯ bar({x_(n)}_(n inN)){\overline{\left\{x_{n}\right\}_{n \in \mathbb{N}}}}{xn}n∈N―
  4. diam ( x n , ≲ ) → n 0 x n , ≲ → n 0 (x_(n),≲)rarr"n"0\left(x_{n}, \lesssim\right) \xrightarrow{n} 0(xn,≲)→n0
    there exists a maximal element x ∗ x ∗ x^(**)x^{*}x∗ (i.e. x ∗ ≲ x x ∗ ≲ x x^(**)≲xx^{*} \lesssim xx∗≲x implies x ∗ = x x ∗ = x x^(**)=xx^{*}=xx∗=x ).

References

[1] H. Brézis, F.E. Browder, A general ordering principle in nonlinear functional analysis, Advances in Math. 21(1976), 355-364
[2] J. Caristi, Fixed point theorems for mappings satisfying inwardness conditions, Trans. Amer. Math. Soc. 215(1976), 241-251
[3] S. Dáncs, H. Hegedüs, P. Medvegyev, A general ordering and fixed-point principle in complete metric space, Acta. Sci. Math. 46(1983), 381-388
[4] J. Dugundji, A. Granas, Fixed Point Theory, I. Monografie Matematyczne, Tom 61, Warszawa 1982
[5] I. Ekeland, Sur les problemès variationnels, C. R. Acad. Sci. Paris Sér. A-B 275(1972), 1057-1059
[6] I. Ekeland, Noncovex minimization problems, Bull. Amer. Math. Soc. 1(3)(1979), 443-474
[7] S. Park, Characterizations of metric completeness, Colloq. Math. XLIX(1984), 21-26
[8] M. Turinici, Differential inequalities via maximal element techniques, Nonlinear Analysis 5(1981), 757-763
[9] M. Turinici, Differential Lipschitzianness tests on abstract quasimetric spaces, Acta. Math. Hung. 41(1983), 93-100.
Institutul de Matematică
Oficiul Poştal 1, C.P. 68
3400 Cluj-Napoca, România
This paper is in final form and no version of it will be submitted for publication elsewhere.
1987

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