A CLASS OF γ-INVEX EQUILIBRIUM PROBLEMS
VASILE PREDA and MIRUNA BELDIMAN
We consider two new classes of (γ, µ)-preinvex functions and ξ-skew symmetric functions, respectively. Also, two new concepts of generalized pseudomonotonicity and generalized partially relaxed monotonicity of mappings are introduced. By applying the auxiliary principle technique, two predictor-corrector algorithms for solving mixed quasiγ-invex equilibrium problems are proposed and analyzed. The convergence of these methods is obtained under (σ, η)-pseudomonotonicity with respect toξ-skew symmetry and (τ, η) partially relaxed monotonicity assumptions.
AMS 2000 Subject Classification: 65K05, 47J20, 91B50.
Key words: quasiequilibrium problem, auxiliary principle technique, generalized pseudomonotonicity, generalized partially relaxed monotonicity, gene- ralized skew-symmetry.
1. INTRODUCTION
Equilibrium theory plays an important role in many fields such as me- chanics, physics, nonlinear programming, economics and engineering sciences [1, 6, 9, 10, 16, 17].
Because of their wide applicability, interesting and intensively studied subjects are the analysis of solution existence and the development of an effi- cient iterative algorithm to compute the approximate solution [5, 18, 21].
The most known techniques, like projection method with its variant forms and Wiener-Hopf equations method require strongly monotonicity, Lips- chitz continuity or differentiability conditions [2, 3, 8, 14]. This fact motivated the introduction of the auxiliary principle technique, first used for solving a variational inequality [7], then for many classes of variational inequalities and equilibrium problems [4, 12, 15]. The existence of this method encouraged the appearence of some extended properties of functions, such as convexity, monotonicity, coercivity [11, 13, 19, 20].
In this paper we introduce new concepts of generalized preinvexity and generalized skew symmetry. We also, consider a class of quasi γ-invex equi- librium problems, which involves (σ, η)-pseudomonotonicity with respect to a
REV. ROUMAINE MATH. PURES APPL.,53(2008),2–3, 227–237
ξ-skew symmetric function or (τ, η)-partially relaxed monotonicity. Here, we present strong and weak versions of these properties and show, by examples, the relations between them.
In order to solve the problems, we consider an auxiliaryγ-invex equilib- rium problem and prove the convergence of the iterative sequence for some new predictor-corrector methods.
2. DEFINITIONS AND SOME PRELIMINARY RESULTS LetHbe a real Hilbert space with normk·kand inner producth·,·i.Let K ⊂H be a nonempty convex subset, f :K →H and η(·,·) :K×K →H two continuous functions, and γ :K×K →Ra real function.
Definition 2.1 (see [12]). Let u∈ K.Then K is said to be γ-invex at u with respect to η ifu+tγ(v, u)η(v, u)∈K for allu,v∈K and t∈[0,1],
If K is γ-invex at u with respect to η for any u ∈ K, then K is called γ-invex with respect toη or (γ, η)-connected.
Remark 2.1. For γ ≡1 we say that K is an invex set with respect to η (see [20]).
In what follows we suppose thatK is a nonempty closed γ-invex subset in H with respect toη.
Definition 2.2 (see [12]). A function f :K →H is said to beγ-preinvex with respect to η if
f(u+tγ(v, u)·η(v, u))≤(1−t)f(u) +tf(v) for all u, v∈K and t∈[0,1].
Remark 2.2. For γ ≡ 1 we say that K is preinvex with respect to η (see [20]).
Definition 2.3. Letµbe a real number. We say thatf is a (γ, µ)-preinvex function with respect to η if
f(u+tγ(v, u)·η(v, u))≤(1−t)f(u) +tf(v)−t(1−t)µ· kη(v, u)k2 for all u, v∈K and t∈[0,1].
According asµ >0,µ= 0 orµ <0, we say thatf isγ-strongly preinvex, γ-preinvex orγ-weakly preinvex with respect to η.
Remark 2.3. Forµ= 0 we obtain Definition 2.2.
Remark2.4. It is obvious thatγ-strong preinvexity impliesγ-preinvexity, which in turn impliesγ-weak preinvexity, but the opposite implications do not hold.
Remark 2.5. In the differentiable case, the condition of (γ, µ)-preinvexity is equivalent to the inequality
f(v)−f(u)≥γ(v, u)
f0(u), η(v, u)
+µkη(v, u)k2, ∀u, v∈K.
Definition 2.4. Letϕ:H×H→R∪ {+∞}be a bifunction and ξ a real number. We say that ϕis ξ skew-symmetric with respect toη if
ϕ(u, u)−ϕ(u, v)−ϕ(v, u) +ϕ(v, v)≥ξkη(v, u)k2 for all u, v∈H.
A bifunction ϕ is called strongly skew-symmetric, skew-symmetric or weakly skew-symmetric with respect to η ifξ >0,ξ= 0 orξ <0.
Thus, we extend the concept of skew-symmetry obtained forξ= 0.
Remark2.6. It is clear that strong skew-symmetry implies skew-symmetry, which implies weak skew-symmetry with respect to η,but the converse is not true. In order to show that, we give the example below.
Example 2.1. LetH =R,ϕi :R×R→R,i= 1,2,3,ϕ1(u, v) = 2u2v2, ϕ2(u, v) = 2u2 −v2, ϕ3(u, v) = −u2v2 be three continuous bifunctions and η :R×R→R, η(u, v) =u2−v2. Obviously, ϕ1 is strongly skew-symmetric for any ξ with 0 < ξ ≤ 2, ϕ2 is skew-symmetric, but there is no ξ >0 such that ϕ2 is strongly skew-symmetric, ϕ3 is weakly skew-symmetric if ξ ≤ −1, but not skew-symmetric, with respect to the sameη.
Remark 2.7. In the case whereϕis bilinear, for all u, v∈H we have ϕ(u, u)≥ξkη(v, u+v)k2.
In what follows we consider a continuous functionF :K×K →R and a continuous bifunction ϕ:K×K→R∪ {+∞}. We are looking for a u∈K such that
(2.1) F(u, v) +ϕ(v, u)−ϕ(u, u)≥0, ∀v∈K.
This problem is called the mixed quasiγ-invex equilibrium problem. Forγ ≡1 we get the mixed quasi invex equilibrium problem (see [15]).
IfF(u, v) =hT u, η(v, u)i, whereT :H→Handηas above, this problem is equivalent to the mixed quasi variational-like problem, namely, find u∈K such that hT u, η(v, u)i+ϕ(v, u)−ϕ(u, u)≥0, ∀v∈K.
Now, we define some classes of operators.
Definition 2.5. Letσ be a real number,F :K×K→R a real function and ϕ:K×K → R∪ {+∞}. We say that F is (σ, η)-pseudomonotone with respect to ϕif
F(u, v) +ϕ(v, u)−ϕ(u, u)≥0⇒ −F(v, u) +ϕ(v, u)−ϕ(u, u)≥σkη(v, u)k2
for all u, v ∈ K. If σ > 0, σ = 0 or σ < 0, F is said to be strongly η- pseudomonotone, η-pseudomonotone or weakly η-pseudomonotone with re- spect to ϕ. The case σ = 0 is known as pseudomonotonicity with respect toϕ.
Remark 2.8. It is clear that strong η-pseudomonotonicity implies η- pseudomonotonicity, which implies weak η-pseudomonotonicity with respect to a bifunction, but the converse is not true.
Example 2.2. Let K = R, Fi : R×R → R, i = 1,2,3, F1(u, v) =
− u2−v22
,F2(u, v) =u−v,F3(u, v) = 14 u2−v22
,ϕi,i= 1,2,3 andη as in Example 2.1. It is easy to show thatF1 isη-strongly pseudomonotone with respect to ϕ3 for anyσ1 such that 0< σ1 ≤2, whileF2 is η-pseudomonotone with respect to ϕ2, but there is no ϕ:R×R → R such that F2 is strongly η-pseudomonotone with respect to ϕ. We can also prove thatF3 isη-weakly pseudomonotone with respect to ϕ1 for any σ3 ≤ −1, but it is not pseu- domonotone with respect to anyϕ:R×R→R.
Definition 2.6. Let τ be a real number. A function F is called (τ, η)- partially relaxed monotone if
F(u, v) +F(v, z)≤τkη(z, u)k2
for all u, v, z ∈ K. If τ < 0, τ = 0 or τ > 0 we say that F is strongly η- partially relaxed monotone, partially relaxed monotone or weakly η-partially relaxed monotone. Forτ = 0 andz=u withη(u, u) = 0, we get the definition of monotonicity (see [15]):
F(u, v) +F(v, u)≤0, ∀u, v∈K.
Remark 2.8. It is easy to show that strong η-partially relaxed mono- tonicity implies partially relaxed monotonicity, which implies weak η-partially relaxed monotonicity, but the opposite implications do not hold.
Example 2.3. Let K = R, Fi, ηi : R×R → R, i = 1,2,3, F1(u, v) = u4 − v4, F2(u, v) = −u2 −v2, F3(u, v) = −u4 −v4, η1(u, v) = u2 +v2, η2(u, v) = v−u,η3(u, v) = u2−v2. It is clear that F1 is weakly η1-partially relaxed monotone for τ1 = 1, but it is not partially relaxed monotone, F2 is partially relaxed monotone but it is not strongly partially relaxed monotone for any τ <0, while F3 is strongly η3-partially relaxed monotone for τ =−1.
Definition 2.7. A function F :K×K → R is called hemicontinuous if the mapping Ψ(t) : [0,1]→R, Ψ(t) =F(u+tη(v, u), v) is continuous att= 0 for all u, v∈K.
Relative to problem (2.1) we state
Lemma2.1. Let K, γ, η, F and ϕ as above and µ1, µ2, σ be three real numbers. Assume that
(i1) F is (σ, η)-pseudomonotone with respect to ϕ;
(i2) the function u→F(v, u) is (γ, µ1)-preinvex for any v∈K;
(i3) the function v→ϕ(v, u) is (γ, µ2)-preinvex for any u∈K;
(i4) K is a γ-invex set;
(i5) kη(vt, u)k=tkη(v, u)kfor any vt=u+tγ(v, u)η(v, u), where u, v∈ K and t∈[0,1];
(i6) lim
t&0
F(vt,vt)
t = 0for any u∈K solution of (2.1)and all v∈K; (i7) F is hemicontinuous;
(i8) µ1+µ2 ≥0.
Then problem (2.1)is equivalent to finding u∈K such that
(2.2) F(v, u) +ϕ(u, u)−ϕ(v, u)≤σkη(v, u)k2 for allv∈K.
Proof. Letu∈K be a solution of theγ-invex equilibrium problem (2.1).
Then for all v∈K we have
F(u, v) +ϕ(v, u)−ϕ(u, u)≥0.
By (i1), F is (γ, η)-pseudomonotone with respect to ϕ and it follows from (2.1) that
(2.3) −F(v, u) +ϕ(v, u)−ϕ(u, u)≥σkη(v, u)k2, for any v∈K, sou is a solution of (2.2).
Conversely, letu∈K be a solution of problem (2.2). Hence (2.4) −F(v, u) +ϕ(v, u)−ϕ(u, u)≥σkη(u, v)k2.
SinceKis aγ-invex set with respect toη, we havevt=u+tγ(v, u)·η(v, u)∈K for all u, v∈K and t∈[0,1].Taking v=vt in (2.4),for any vt∈K we have
F(vt, u) +ϕ(u, u)−ϕ(vt, u)≤ −σkη(vt, u)k2. Now, by (i3) and (i5) we have
F(vt, u)+ϕ(u, u)≤tϕ(v, u)+(1−t)ϕ(u, u)−t(1−t)µ2kη(v, u)k2−σt2kη(v, u)k2. From this relation we get
(2.5) F(vt, u)≤t{ϕ(v, u)−ϕ(u, u)−(1−t)µ2kη(v, u)k2−tσkη(v, u)k2}.
Using (i2) we deduce that
F(vt, vt)≤tF(vt, v) + (1−t)F(vt, u)−t(1−t)µ1kη(v, u)k2
and now, by (2.5), we have
F(vt, vt)≤tF(vt, v) + (1−t)t{ϕ(v, u)−ϕ(u, u)−(1−t)µ2kη(v, u)k2−
−σtkη(v, u)k2} −t(1−t)µ1kη(v, u)k2, which is equivalent to
F(vt, vt)≤tF(vt, v) + (1−t)t{ϕ(v, u)−ϕ(u, u)−(1−t)µ2kη(v, u)k2−
−σtkη(v, u)k2−µ1kη(v, u)k2}.
Dividing by t, we obtain F(vt, vt)
t ≤F(vt, v) + (1−t){ϕ(v, u)−ϕ(u, u)−(1−t)µ2kη(v, u)k2−
−σtkη(v, u)k2−µ1kη(v, u)k2}.
From (i6) and (i7), for anyv ∈K we get,
F(u, v) +{ϕ(v, u)−ϕ(u, u)− kη(v, u)k2(µ1+µ2)} ≥0.
Since µ1 +µ2 ≥ 0 by (i8), u ∈ K is a solution of (2.1). Thus the lemma is proved.
Corollary 2.1 (see [15]). Let F be pseudomonotone, hemicontinuous and preinvex in the second argument and let ϕ be preinvex in the first argu- ment. Then problems (2.1)and (2.2)are equivalent.
Proof. Take µ1=µ2 =σ = 0 in Lemma 2.1.
Remark 2.9. Problem (2.2) is also a mixed γ-invex quasi equilibrium problem, called the dual problem of (2.1).
Remark 2.10. It follows from (i1) that F(u, u)≤0 for any u∈K. Thus, assumption (i6) is justified.
Remark 2.11. If in Lemma 2.1 we replace (i2) and (i3) by F strongly preinvex andϕweakly preinvex or byF weakly preinvex andϕstrongly prein- vex respectively, then the result still holds.
3. ALGORITHMS AND CONVERGENCE
In this section we consider two classes of iterative methods for solving (2.1) by using the auxiliary principle technique, and study the convergence under generalized pseudomonotonicity and preinvexity assumptions.
Let u∈K, E :K →H be a differentiable and (γ, µ)-preinvex function with respect to η, let µ and β be real numbers, and ρ a positive constant.
Consider the auxiliary γ-invex equilibrium problem: find a unique w ∈ K such that
ρF(w, v)+γ(v, w)
E0(w)−E0(u), η(v, w)
≥ρ{ϕ(w, w)−ϕ(v, w)}+βkη(w, u)k2 for all v∈K.
Algorithm 3.1. For a given u0 ∈ H, calculate the approximate solution un+1 by the iterative scheme
ρF(un+1, v) +γ(v, un+1)
E0(un+1)−E0(un), η(v, un+1) + +ρ{ϕ(v, un+1)−ϕ(un+1, un+1)} ≥βkη(un+1, un)k2, for all v∈K.
Remark 3.1. In the case where β = 0 or η(u, u) = 0 for any u ∈K we retrieve Algorithm 3.1 from [15]. We also get [15] forη(v, u) =v−uandβ= 0, or for F(v, u) = hT u, η(v, u)i and β = 0. In these cases, the method can be applied for solving mixed quasi-equilibrium problems and quasi variational-like inequalities, respectively.
Theorem3.1. Let K, ϕ, γ,η,F,E, µ,β and ρ be as above, and let σ, ξ be two real numbers. Suppose that
(i1) F is (σ, η)-pseudomonotone with respect to ϕ;
(i2) ϕis ξ-skew symmetric with respect to η;
(i3) γ(u, v)η(u, v) =γ(u, z)η(u, z) +γ(z, v)η(z, v) for allu, v, z∈H;
(i4) µ+β >0;
(i5) ξ−σ ≥0.
Then the approximate solution un+1 obtained from Algorithm 3.1converges to a solution u∈K of the mixed quasi γ-invex equilibrium problem (2.1).
Proof. Let u ∈K be a solution of (2.1). Then for any v ∈ K from (i1) we have
(3.1) −F(v, u) +ϕ(v, u)−ϕ(u, u)≥σkη(u, v)k2. If we take v=un+1, in (3.1) we get
(3.2) −F(un+1, u) +ϕ(un+1, u)−ϕ(u, u)≥σkη(u, un+1)k2. As in [22], we consider the function
B(u, z) =E(u)−E(z)−γ(u, z)
E0(z), η(u, z) . Using the (γ, µ)-preinvexity and differentiability ofE we deduce that
(3.3) B(u, z)≥µkη(u, z)k2.
Now, by (i2), (i3), (3.2) and (3.3) we get
B(u, un)−B(u, un+1) =E(un+1)−E(un)−γ(u, un)
E0(un), η(u, un) +
+γ(u, un+1)
E0(un+1), η(u, un+1)
=
=E(un+1)−E(un)−γ(u, un+1)
E0(un)−E0(un+1), η(u, un+1)
−
−γ(un+1, un)
E0(un), η(un+1, un)
≥
≥µkη(un+1, un)k2+γ(u, un+1)
E0(un+1)−E0(un), η(u, un+1)
≥
≥µkη(un+1, un)k2−ρF(un+1, u) +ρ[ϕ(un+1, un+1)−ϕ(u, un+1)]+
+βkη(un+1, un)k2≥µkη(un+1, un)k2−ρ[ϕ(un+1, u)−ϕ(u, u)+σkη(un+1, u)k2]+
+ρ[ϕ(un+1, un+1)−ϕ(u, un+1)] +βkη(un+1, un)k2 ≥
≥µkη(un+1, un)k2+ρξkη(un+1, u)k2−ρσkη(un+1, u)k2+ +βkη(un+1, un)k2= (µ+β)kη(un+1, un)k2+ρ(ξ−σ)kη(un+1, u)k2. Thus,
(3.4) B(u, un)−B(u, un+1)≥(µ+β)kη(un+1, un)k2+ρ(ξ−σ)kη(un+1, u)k2. From (3.4), (i2),(i4) and (i5) we obtain
(3.5) B(u, un)−B(u, un+1)≥0.
Obviously, if un+1 =un thenunis a solution of (2.1). If this is not true, then it follows from (i4) and (i5) that B(u, un)−B(u, un+1) is strictly positive and
n→∞limkη(un+1, u)k= 0.
Now, we can prove the convergence of the sequence (un)n≥1 from Al- gorithm 3.1 by using the technique of Zhu and Marcotte [22]. The proof is complete.
Forγ = 1, we get the following result.
Corollary3.1 (see [17]). Suppose that one of the conditions from Re- mark 3.1is satisfied and
(i01) K is an invex set;
(i02) F is pseudomonotone;
(i03) µ >0;
(i04) ϕis skew-symmetric;
(i05) η(u, v) =η(u, z) +η(z, v) for all u, v, z∈K.
Then the approximate solutionun+1from Algorithm3.1converges to a solution u∈K of the corresponding invex equilibrium problem (2.1).
Another γ-invex auxiliary equilibrium problem which can be used for solving (2.1) is: find a uniquew∈K such that
ρF(u, v)+γ(v, w)hE0(w)−E0(u), η(v, w)i ≥ρ{ϕ(w, w)−ϕ(v, w)}+βkη(w, u)k2 for all v∈K.
Based on this remark, we can give
Algorithm 3.2. For a given u0 ∈H, calculate the approximate solution un+1 by the iterative scheme
ρF(un, v) +γ(v, un+1)
E0(un+1)−E0(un), η(v, un+1) + +ρ[ϕ(v, un+1)−ϕ(un+1, un+1)]≥βkη(un+1, u)k2 for all v∈K.
Remark 3.3. In the case where β = 0 or η(u, u) = 0 for any u ∈K, we retrieve Algorithm 3.6 from [15].
Relative to Algorithm 3.2 we have the following result.
Theorem3.2. Assume assumptions(i2)–(i3)from Theorem3.1hold and (j1) F is (τ, η)-partially relaxed monotone, where τ is a real number;
(j2) µ−ρτ >0;
(j3) β+ξ≥0. Then the approximate solution un+1 from Algorithm 3.2 converges to a solution u ∈ K of the mixed quasi γ-invex equilibrium prob- lem (2.1).
Proof. As in Theorem 3.1, we get
B(u, un)−B(u, un+1)≥µkη(un+1, un)k2+ +γ(u, un+1)
E0(un+1)−E0(un), η(u, un+1)
≥µkη(un+1, un)k2−
−ρF(un, u) +ρ[ϕ(un+1, un+1)−ϕ(u, un+1)] +βkη(un+1, u)k2. It follows from (i2) and (j1) that
B(u, un)−B(u, un+1)≥µkη(un+1, un)k2+βkη(un+1, u)k2+ +ρ[ξkη(un+1, u)k2−ϕ(u, u) +ϕ(un+1, u)−τkη(un+1, un)k2+F(u, un+1)].
But u∈K is a solution of (2.1) and by takingv=un+1, in (2.1) we get F(u, un+1) +ϕ(un+1, u)−ϕ(u, u)≥0.
Hence
B(u, un)−B(u, un+1)≥(µ−ρτ)kη(un+1, un)k2+ρ(β+ξ)kη(un+1, un)k2. Obviously, if un+1 = un, un is a solution for (2.1). If un and un+1 are different, then it follows from (j2) and (j3) thatB(u, un)−B(u, un+1) is strictly positive and, as in Theorem 3.1, we can prove the convergence of the iterative sequence (un)n≥1 using the technique of Zhu and Marcotte [22]. Thus, the theorem is proved.
Corollary3.2 (see [17]). Suppose that one of the conditions from Re- mark 3.1 is satisfied, conditions (i01), (i03)–(i05) from Corollary 3.1 hold, and
(j01) F is partially relaxed strong monotone with τ >0;
(j02) µ >0;
(j03) α∈ 0,µρ .
Then the approximate solution un+1 from Algorithm 3.6 in [15] converges to a solution u∈K of the invex equilibrium problem (2.1).
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Received 26 March 2007 University of Bucharest
Faculty of Mathematics and Computer Science Str. Academiei 14
010014 Bucharest, Romania and
Romanian Academy Institute of Mathematical Statistics
and Applied Mathematics Calea 13 Septembrie no. 13 050711 Bucharest, Romania
