Approximate Equilibrium Problems and Fixed Points

Abstract

We find a common element of the set of fixed points of a map and the set of solutions of an approximate equilibrium problem in a Hilbert space. Then, we show that one of the sequences weakly converges. Also we obtain some theorems about equilibrium problems and fixed points.

1. Introduction

Equilibrium is “everywhere:” in economics, physics, engineering, chemistry, biology, and so forth. From the mathematical modelling point of view equilibrium can be described in fixed point theorems, optimisation problems, variational inequalities, complementarity problems, and so forth. Equilibrium systems can be studied from several points of view: existence of solutions; existence of nontrivial solutions; number of solutions; properties of the solution set; and the numerical approximation of solutions. In the first a short description of what is a mathematical equilibrium system in general, we will present several particular classes of approximate equilibrium systems and the relations between then.

closed, bounded, and convex subset of 𝐻.

Condition 1. The following condition appears implicitly in [1].
We assume that the map 𝜑∶𝐶×𝐶→𝐑 satisfies the following conditions:(i)𝜑(𝑥,𝑥)=0,forall𝑥∈𝐶; (ii)𝜑ismonotone,thatis,𝜑(𝑥,𝑦)+𝜑(𝑦,𝑥)≤0,forall𝑥,𝑦∈𝐶; (iii)forall𝑥,𝑦∈𝐶, lim_𝑡→0𝜑𝑥𝑡+(1−𝑡)𝑥,𝑦≤𝜑𝑥,𝑦; (1)(iv)foreachfixed𝑥∈𝐶, thefunction𝑦↦𝜑(𝑥,𝑦)isconvexandlowersemicontinuous.

Condition 2. Assume that the map 𝜑∶𝐶×𝐶→𝐑 for 𝜖>0 satisfies the following conditions.(𝐴₁)𝜑(𝑥,𝑥)=𝜖,forall𝑥∈𝐶. (𝐴₂)𝜑isapproximatemonotone,thatis,𝜑(𝑥,𝑦)+𝜑(𝑦,𝑥)≤𝜖,forall𝑥,𝑦∈𝐶. (𝐴₃)Forall𝑥,𝑦∈𝐶, lim_𝑡→0𝜑𝑥𝑡+(1−𝑡)𝑥,𝑦≤𝜑𝑥,𝑦. (2)(𝐴₄)Foreachfixed𝑥∈𝐶, thefunction𝑦↦𝜑(𝑥,𝑦)isconvexandlowersemicontinuous.

Definition 1 (see [1]). We say that 𝑥^∗∈𝐶 is an equilibrium point of 𝜑∶𝐶×𝐶→𝐑 if there exists a 𝑥^∗∈𝐶, such that 𝜑𝑥^∗,𝑦≥0∀𝑦∈𝐶; (3) the set of such 𝑥^∗∈𝐶 is denoted by EP(𝜑); that is, EP𝜑=𝑥^∗∈𝐶∶𝜑𝑥^∗,𝑦≥0,∀𝑦∈𝐶. (4)

Definition 2. Suppose 𝜖>0, we say that 𝑥^∗∈𝐶 is an approximate equilibrium point of 𝜑∶𝐶×𝐶→𝐑 if there exists a 𝑥^∗∈𝐶, such that 𝜑𝑥^∗,𝑦≥𝜖∀𝑦∈𝐶. (5) In this paper, the set of such an 𝑥^∗∈𝐶 is denoted by AEP(𝜑), that is, AEP𝜑=𝑥^∗∈𝐶∶𝜑𝑥^∗,𝑦≥𝜖,∀𝑦∈𝐶, (6)
and we set (𝑇)={()},𝐹(𝑇)={}.AF𝑥∈𝐶∶𝑑𝑥,𝑇𝑥<𝜖forsome𝜖>0𝑥∈𝐶∶𝑇𝑥=𝑥 (7)

2. Preliminaries

In the following we will present a known lemma which is needed in the proof of some results (see [2]).

Lemma 3. Let 𝜑∶𝐶×𝐶→𝐑 be a map satisfies Condition 1. Let 𝑟>0, and let 𝑥∈𝐻. Then there exists a 𝑧∈𝐶, such that 𝜑𝑧,𝑦+⟨𝑦−𝑧,𝑧−𝑥⟩𝑟≥0∀𝑦∈𝐶. (8)

Lemma 4. Let 𝜑∶𝐶×𝐶→𝐑 be a map satisfies Condition 2. Let 𝑟>0, 𝜖>0, and let 𝑥∈𝐻. Then there exists a 𝑧∈𝐶, such that 𝜑𝑧,𝑦+⟨𝑦−𝑧,𝑧−𝑥⟩𝑟≥𝜖∀𝑦∈𝐶. (9)

Proof. Consider the map 𝜓∶𝐶×𝐶→𝐑 by 𝜓(𝑥,𝑦)=𝜑(𝑥,𝑦)−𝜖. The map 𝜓 satisfies Condition 1. Then by Lemma 3, there exists a 𝑧∈𝐶, such that 𝜓𝑧,𝑦+⟨𝑦−𝑧,𝑧−𝑥⟩𝑟≥0∀𝑦∈𝐶. (10)
Thus 𝜑𝑧,𝑦+⟨𝑦−𝑧,𝑧−𝑥⟩𝑟≥𝜖∀𝑦∈𝐶. (11)

Lemma 5. Let 𝜑∶𝐶×𝐶→𝐑 be a map satisfies Condition 2. For 𝑟>0, 𝑥∈𝐻, 𝜖>0 we defined 𝑇_𝑟∶𝐻→𝐶, such that 𝑇_𝑟(𝑥)=𝑧∈𝐶∣𝜑𝑧,𝑦+⟨𝑦−𝑧,𝑧−𝑥⟩𝑟≥𝜖. (12)
Then(a)𝑇_𝑟 is single valued.(b)𝑇_𝑟 is firmly nonexpansive; that is, 𝑇_𝑟(𝑥)−𝑇_𝑟𝑦²≤𝑇_𝑟(𝑥)−𝑇_𝑟𝑦,𝑥−𝑦∀𝑥,𝑦∈𝐻. (13)(c)𝐹(𝑇_𝑟)=𝐴𝐸𝑃(𝜑). (d)𝐴𝐸𝑃(𝜑) is nonempty, closed, and convex.

Proof. (a) For 𝑥∈𝐻 and 𝑟>0, let 𝑧₁,𝑧₂∈𝑇_𝑟(𝑥). Then, 𝜑𝑧₁,𝑦+⟨𝑦−𝑧₁,𝑧₁−𝑥⟩𝑟𝜑𝑧≥𝜖∀𝑦∈𝐶,₂,𝑦+⟨𝑦−𝑧₂,𝑧₂−𝑥⟩𝑟≥𝜖∀𝑦∈𝐶. (14) Then 𝜑𝑧₁,𝑧₂≥⟨𝑧₂−𝑧₁,𝑥−𝑧₁⟩𝑟𝜑𝑧+𝜖,₂,𝑧₁≥⟨𝑧₁−𝑧₂,𝑥−𝑧₂⟩𝑟+𝜖. (15) Since 𝜑 is approximate monotone, 𝜖≥𝜑𝑧₁,𝑧₂+𝜑𝑧₂,𝑧₁≥⟨𝑧₂−𝑧₁,𝑥−𝑧₁⟩𝑟+⟨𝑧₁−𝑧₂,𝑥−𝑧₂⟩𝑟≥+2𝜖⟨𝑧₂−𝑧₁,𝑧₁−𝑧₂⟩𝑟+𝜖. (16) Now, since 𝜖>0 and 𝑟>0, we have ⟨𝑧₂−𝑧₁,𝑧₁−𝑧₂⟩≤0. (17) So, we have 𝑧₁=𝑧₂.
(b) Now we claim that 𝑇_𝑟 is a firmly nonexpansive. Indeed, if 𝑥,𝑦∈𝐻, 𝜑𝑇_𝑟(𝑥),𝑇_𝑟𝑦+𝑇_𝑟𝑦−𝑇_𝑟(𝑥),𝑇_𝑟(𝑥)−𝑥𝑟𝜑𝑇≥𝜖,_𝑟𝑦,𝑇_𝑟(𝑥)+𝑇_𝑟(𝑥)−𝑇_𝑟𝑦,𝑇_𝑟𝑦−𝑦𝑟≥𝜖. (18) Adding the two inequalities, we have 𝜑𝑇_𝑟(𝑥),𝑇_𝑟𝑦+𝜑𝑇_𝑟𝑦,𝑇_𝑟(𝑥)+𝑇_𝑟𝑦−𝑇_𝑟(𝑥),𝑇_𝑟(𝑥)−𝑇_𝑟𝑦−𝑥+𝑦𝑟≥2𝜖. (19) With (𝐴₂), we have 𝑇_𝑟𝑦−𝑇_𝑟(𝑥),𝑇_𝑟(𝑥)−𝑇_𝑟𝑦−𝑥−𝑦𝑟≥𝜖. (20) Now since 𝜖>0 and 𝑟>0, then ⟨𝑇_𝑟𝑦−𝑇_𝑟(𝑥),𝑇_𝑟(𝑥)−𝑇_𝑟𝑦−𝑥−𝑦⟩≥0 (21) so 𝑇_𝑟(𝑥)−𝑇_𝑟(𝑦)²≤⟨𝑇_𝑟(𝑥)−𝑇_𝑟𝑦,𝑥−𝑦⟩. (22) (c) Take 𝑥∈𝐾. Then 𝑥∈𝐹𝑇_𝑟→𝑥=𝑇_𝑟(𝑥)→𝜑𝑥,𝑦+𝑥−𝑥,𝑦−𝑥𝑟𝜑.≥𝜖∀𝑦∈𝐶→𝜑𝑥,𝑦≥𝜖∀𝑦∈𝐶→𝑥∈AEP (23) (d) At last, we claim that AEP(𝜑) is closed and convex. Indeed, since 𝑇_𝑟 is firmly nonexpansive, 𝑇_𝑟 is also nonexpansive, and since the fixed-point set of a nonexpansive operator is closed and convex [3, proposition 1.5.3]. Therefore follows from (b), (c).

In the following we will present a known theorem which is needed in the proof of some results (see [4]).

Theorem 6. Let 𝜑∶𝐶×𝐶→𝐑 be a map satisfies Condition 1, and let 𝑆∶𝐶→𝐶 be a nonexpentive mapping such that 𝐹(𝑆)∩𝐸𝑃(𝜑)≠∅. Let {𝑥_𝑛} and {𝑢_𝑛} be sequences generated initially by an arbitrary element 𝑥₁∈𝐻 and then by 𝜑𝑢_𝑛,𝑦+𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑟_𝑛𝑥≥0∀𝑦∈𝐶,_𝑛+1=𝛼_𝑛𝑢_𝑛+1−𝛼_𝑛𝑆𝑢_𝑛∀𝑛≥1, (24) where {𝛼_𝑛} and {𝑟_𝑛} satisfy the following conditions:(i){𝛼_𝑛}⊂[𝛼,𝛽] for some 𝛼,𝛽∈(0,1);(ii){𝑟_𝑛}⊂(0,∞) and liminf_𝑛𝑟_𝑛>0.
Then, the sequences {𝑥_𝑛} and {𝑢_𝑛} converge weakly to an element of 𝐹(𝑆)∩EP(𝜑)≠∅.

Lemma 7. Let 𝜓∶𝐶×𝐶→𝐑 be a map by 𝜓(𝑥,𝑦)=𝜑(𝑥,𝑦)−𝜖 that satisfies Condition 1, and let 𝑆∶𝐶→𝐶 be a nonexpentive mapping, then 𝐹(𝑆)∩𝐴𝐸𝑃(𝜑)≠∅ if and only if 𝐹(𝑆)∩𝐸𝑃(𝜓)≠∅.

In the following we will present a theorem which is extended Theorem 6.

Theorem 8. Let 𝜑∶𝐶×𝐶→𝐑 be a map satisfies Condition 2, and let 𝑆∶𝐶→𝐶 be a map such that 𝐹(𝑆)∩𝐴𝐸𝑃(𝜑)≠∅. Let {𝑥_𝑛} and {𝑢_𝑛} be sequences generated initially by an arbitrary element 𝑥₁∈𝐻 and then by 𝜑𝑢_𝑛,𝑦+𝑦−𝑢_𝑛,𝑢_𝑛−𝑥_𝑛𝑟_𝑛𝑥≥𝜖∀𝑦∈𝐶,_𝑛+1=𝛼_𝑛𝑢_𝑛+1−𝛼_𝑛𝑆𝑢_𝑛∀𝑛≥1, (25) where {𝛼_𝑛} and {𝑟_𝑛} satisfy the following conditions:(i){𝛼_n}⊂[𝛼,𝛽] for some 𝛼,𝛽∈(0,1);(ii){𝑟_𝑛}⊂(0,∞) and liminf_𝑛𝑟_𝑛>0.Then, the sequences {𝑥_𝑛} and {𝑢_𝑛} converge weakly to an element of 𝐹(𝑆)∩𝐴𝐸𝑃(𝜑).

Proof. Suppose the nap 𝜓∶𝐶×𝐶→𝐑 by 𝜓(𝑥,𝑦)=𝜑(𝑥,𝑦)−𝜖 that satisfies Condition 1. Since by 𝐹(𝑆)∩AEP(𝜑)≠∅, Lemma 9 implies that 𝐹(𝑆)∩EP(𝜓)≠∅ and holds conditions theorem (2.8), therefore the sequences {𝑥_𝑛} and {𝑢_𝑛} converge weakly to an element of 𝑧∈𝐹(𝑆)∩EP(𝜓). By Lemma 9 the sequences {𝑥_𝑛} and {𝑢_𝑛} converge weakly to 𝑧∈𝐹(𝑆)∩AEP(𝜑).

Lemma 9. Let 𝜑∶𝐶×𝐶→𝐑 be a map satisfies Condition 1. For 𝑥∈𝐶, define a mapping 𝑇∶𝐶→𝐶 such that is contraction. If 𝜑(𝑇𝑥,𝑦)≥0  for all 𝑦∈𝐶, then there exists 𝑥₀∈𝐸𝑃(𝑇).

Proof. Since 𝐶 is a nonempty closed, bounded, and convex subset of 𝐻, and 𝑇 is continues, then by Browde’s theorem there exists a 𝑥₀∈𝐶∶𝑇𝑥₀=𝑥₀. Since 𝜑(𝑇𝑥₀,𝑦)≥0 for all  𝑦∈𝐶, thus 𝜑(𝑥₀,𝑦)≥0  for all  𝑦∈𝐶. So ∃𝑥₀∈𝐶∶𝑥₀∈EP(𝑇).

Lemma 10. Let 𝜑∶𝐶×𝐶→𝐑 be a map satisfies Condition 1. For 𝑥∈𝐶, define a mapping 𝑇∶𝐶→𝐶 such that is contraction. If 𝜑(𝑇^𝑛𝑥,𝑦)≥0,  for all 𝑦∈𝐶, then 𝑥₀∈𝐸𝑃(𝑇).

Proof. Since 𝐶 is a nonempty closed, bounded, and convex subset of 𝐻, and 𝑇 is contraction, now by Theorem 2.1 of [4], since {𝑇^𝑛𝑥} converges to fixed point 𝑇, then there exists a 𝑥₀∈𝐶∶𝑇^𝑛𝑥₀=𝑥₀. Since for all 𝑦∈𝐶, 𝜑(𝑇^𝑛𝑥₀,𝑦)≥0, thus 𝜑(𝑥₀,𝑦)≥0 for all  𝑦∈𝐶. So there exists a 𝑥₀∈𝐶, such that 𝑥₀∈EP(𝑇).

Lemma 11. Let 𝜑∶𝐶→𝐑 be a real-valued function, let 𝜑∶𝐶×𝐶→𝐑 be a map satisfies Condition 2, and let 𝑇∶𝐶→𝐶 be a nonlinear onto mapping and satisfying ‖𝑥−𝑇𝑥‖+𝜑(𝑥,𝑇𝑥)<𝜖∀𝑥∈𝐶. (26) If   𝐸𝑃(𝜑)≠∅, then 𝐴𝐹(𝑇)≠∅.

Proof. If 𝑧∈EP(𝜑), then 0≤𝜑𝑧,𝑦∀𝑦∈𝐶. (27) Since 𝑧∈𝐶, there exists 𝑥∈𝐶, such that 𝑇𝑥=𝑧. Therefore 𝜑(𝑇𝑥,𝑥)≥0, and so 𝑑(𝑥,𝑇𝑥)+𝜑(𝑥,𝑇𝑥)<𝜖. (28) It follows that 𝑑(𝑥,𝑇𝑥)<𝜖 and 𝑥∈AF(𝑇).

Lemma 12. Let the map 𝜑∶𝐶×𝐶→𝐑 satisfy Condition 2. For 𝑥∈𝐶, define a mapping 𝑇∶𝐶→𝐶 such that is contraction. If 𝜑(𝑥,𝑇𝑥)<𝑑(𝑥,𝑇𝑥),  for all  𝑥∈𝐶, then there exists a 𝑥₀∈𝐴𝐸𝑃(𝑇).

Proof. Since 𝑇 is contraction by Theorem 2.1. of [1] AF(𝑇)≠∅, forall𝜖>0, and since for all 𝑥∈𝐶𝜑(𝑥,𝑇𝑥)<‖𝑥−𝑇𝑥‖, therefore for all 𝑥∈𝐶 and for some 𝜖>0, 𝜑(𝑥,𝑇𝑥)<𝜖,  for all   𝑥∈𝐶. Thus𝑥₀∈𝐴𝐸𝑃(𝑇).