Journal of the Egyptian Mathematical Society (2014) xxx, xxx-xxx

Egyptian Mathematical Society Journal of the Egyptian Mathematical Society

www.etms-eg.org www.elsevier.com/locate/joems

Auxiliary problem and algorithm for a generalized mixed equilibrium problem

Suhel Ahmad Khan

Department of Mathematics, BITS-Pilani, Dubai Campus, Dubai 345055, United Arab Emirates Received 30 January 2014; accepted 8 March 2014

KEYWORDS

Generalized mixed equilibrium problems; Lipschitz continuous; Strongly monotone; Simultaneous hemicontinu-ous mapping

Abstract In this paper, we consider generalized mixed equilibrium problem and its auxiliary problem in Hilbert space. Further, we establish an existence and uniqueness theorem for the auxiliary problem. Furthermore, using this theorem we construct an algorithm for generalized mixed equilibrium problem and discuss the convergence analysis of the algorithm and existence of solution of generalized mixed equilibrium problem.

2000 MATHEMATICS SUBJECT CLASSIFICATION: 49J30; 47H10; 47H17; 90C99

© 2014 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society.

1. Introduction

One of the most important and interesting problems in the theories of equilibrium problems and variational inequalities is to develop the methods which give efficient and implementable algorithms for solving equilibrium problems and variational inequalities. These methods include projection method and its variant forms, linear approximation, descent and Newton's methods, and the method based on auxiliary principle technique.

It is well known that the projection method and its variants cannot be extended for mixed equilibrium problems involving

E-mail address: khan.math@gmail.com

Peer review under responsibility of Egyptian Mathematical Society.

non-differentiable term. To overcome this drawback, one uses usually the auxiliary principle technique. This technique deals with finding a suitable auxiliary problem and prove that the solution of an auxiliary problem is the solution of original problem by using fixed-point approach which was used by [1]. Recent work in [2-8], is an extension of this technique to suggest and analyze a number of iterative methods for solving various classes of variational inequalities and equilibrium problems.

Motivated by recent work going in this direction, in this paper, we extend auxiliary principle technique to a generalized mixed equilibrium problem (for short, GMEP) in Hilbert space. We prove existence of the unique solution of an auxiliary problem related to GMEP, which enable us to construct an algorithm for finding the approximate solution of GMEP. Further we prove that the approximate solution is strongly convergent to the unique solution of GMEP. The algorithms and results of this paper are new and different from the algorithms and results of [5]. The results presented here generalize the techniques and results of [2,3].

1110-256X © 2014 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society. http://dx.doi.Org/10.1016/j.joems.2014.03.003

2. Preliminaries

Let H be a real Hilbert space whose inner product and norm are denoted by {.,.) and ||.||, respectively, and let K be nonempty, closed and convex subset of H. Given the single-valued mappings T, S : H ! H, N, g : H x H ! H and a bifunction f: H x H ! R such that f(x, x) — 0Vx 2 H, then we consider the generalized mixed equilibrium problem (GMEP) of finding x 2 K such that

f(x, y) + (N(Tx, Sx), g(y, x))+ b(x, y)—b(x, x) p 0, Vy 2 K,

where the bifunction b : H x H ! R, which is not necessarily differentiable, satisfies the following properties:

(i) b is linear in the first argument;

(ii) b is bounded, that is, there exists a constant y > 0 such that b(x,y) 6 y||x||||y||, Vx,y 2 H;

(iii) b(x,y)- b(x,z) 6 b(x,y — z), Vx,y,z 2 H;

(iv) b is convex in the second argument.

Some special cases:

(I) If N (Tx, Sx)—B(x), b(x, y)—0 and g(y, x)—y — x Vx, y 2 K, where B : K ! K, then GMEP (2.1) reduces to the mixed equilibrium problem of finding x 2 K such that

f(x, y) + (Bx, y — x) p 0, Vy 2 K, (2.2)

which has been studied in [9].

(II) Iff(x, y) —0; b(x, y) —0and N (Tx, Sx) — B(x) Vx, y 2 K, where B : K ! K, then GMEP (2.1) reduces to the vari-ational-like inequality problem of finding x 2 K such that

(Bx, g(y, x)) p 0, Vy 2 K. (2.3)

This problem has been studied in [10].

(III) If N(Tx, Sx) — 0 Vx 2 K, then GMEP (2.1) reduces to the generalized equilibrium problem of finding x 2 K such that

f(x,y) + b(x,y) — b(x, x) p 0, Vy 2 K. (2.4)

This problem has been studied in [5].

(IV) If, in (III), b(x,y) — 0 Vx,y 2 K, then GMEP (2.1) reduces to the equilibrium problem of finding x 2 K such that

f(x,y) p 0, Vy 2 K. (2.5)

This problem has been studied in [11].

(V) If N (Tx, Sx) — B(x), b(x, y) — /(y) — /(x) Vx, y 2 K, where / : K ! R and f (x, y) — 0Vx, y 2 K, then GMEP (2.1) reduces to the problem of finding x 2 K such that

(Bx, g(y, x)) + /(y) — /(x) p 0, Vy 2 K. (2.6)

This problem has been studied in [12] in Rn.

(VI) If, in (V), g(y, x)—y — x Vx, y 2 K, then GMEP (2.1) reduces to the variational inequality problem of finding x 2 K such that

(Bx,y — x)+ /(y) — /(x) p 0, Vy 2 K. (2.7)

This problem has been studied in [13].

3. Auxiliary problem and existence of solutions

First related to GMEP (2.1), we consider the auxiliary problem and then establish an existence theorem for the auxiliary problem:

Auxiliary problem (AP). Given x 2 K, find z 2 K such that

pf(z,y) + (Az — Ax + pN(Tx, Sx), g(y,z))

+ p[b(x,y) b(x,z)] p 0, Vy 2 K, (3.1)

where p > 0 is a constant and A : K ! H is not necessarily a linear mapping.

We observe that if z — x, clearly z is a solution of GMEP (2.1).

Now, we give the following definitions and concepts.

Definition 3.1 14. Let K be a subset of a topological vector space X. A set-valued mapping T: K ! 2X is called Knaster-Kuratowski-Mazurkiewieg mapping (KKM mapping), if for each nonempty finite subset fx1, x2,..., xng c K, we have Cofx1,..., xng c Un=1T(x,).

Lemma 3.1 [14]. Let K be a subset of a topological vector space X and let T: K! 2X be a KKM mapping. If for each x 2 K, T(x) is closed and for atleast one x 2 K, T(x) is compact, then

f|T(x)-0.

Definition 3.2. Let f: K x K ! R; N : H x H ! H; T, S : K ! K and g : H x H ! H. Then:

(a) T is said to be a-strongly monotone if there exists a constant a > 0 such that

f(x,y)+f(y, x) + a||x — y||2 < 0, Vx,y 2 H;

(b) g is said to be d-Lipschitz continuous if there exists a constant d > 0 such that

l|g(x,y)|| 6 d||x — y|, Vx,y 2 H;

(c) A is said to be s-strongly g-monotone if there exists a constant s > 0 such that

(Ax — Ay, g(x,y)) p s||x — y|2, Vx,y 2 H;

(d) N is said to be e-strongly mixed g-monotone with respect to T and S, if there exists a constant e > 0 such that

(N(Tx, Sx) — N(Ty, Sy), g(x,y)) p e||x — y|2, Vx,y 2 H;

(e) N is said to be (^, fi2)-Lipschitz continuous if there exist constants bi, > 0 such that

||N(x,,y1)— N(x2,y2)| 6 bJ|x1 — x2|+b2|y1 — y21|, Vx1, x2, y1, y2 2 H;

(f) f and A are said to be simultaneously hemicontinuous if for k 2 [0,1],yk :— ky +(1 — k)z,y,z 2 K, we have

f(yk, p) + (A(y) p)—!F(z, p) + (A(z), p)

as k ! 0+ for any p 2 K.

Theorem 3.1. Let K be a nonempty, closed and convex subset in H. Let g : K x K ! K be affine in the first argument and continuous in the second argument such that g(y, x)+g(x, y) — 08x, y 2 K; let b : K x K ! R be convex in the second argument and continuous; letf: K x K ! R be convex and lower semicon-tinuous in the second argument and f(x, x) — 08x 2 K; let A : K ! H be g-monotone and let f and A be simultaneously hemicontinuous. If there exists a nonempty compact subset D of H and z0 2 D n K such that for any z 2 K \ D, we have

-pf(y, z) + iAy, g(y,z)) P pf(z,y) + iAz, g(y, z)).

From (3.5) and (3.6), we have

- pf(z0, z) + {Az0 - Ax + pN(Tx, Sx), g(z0, z)) + p[b(x, z0)- b(x,z)\ < 0,

for given x 2 K. Then AP (3.1 ) has a solution. Moreover, in addition, if A is s-strongly g-monotone then the solution is unique.

Proof. Define the set-valued mappings P, Q : K ! 2K as

P(y) — fz 2 K : pf(z,y) + {Az - Ax + pN(Tx, Sx), g(y, z))

+ p[b(x,y)-b(x,z)\ P 0g, (3.3)

Q(y) — fz 2 K : pf(y, z) + {Ay - Ax + pN(Tx, Sx), g(y, z)) + p[b(x,y)-b(x,z)\ P 0g, (3.4)

for y 2 K, respectively.

We claim that P is a KKM-mapping. Indeed, let fz1, z2,..., zmg be a finite subset of K and let ki P 0,1 6 i 6 m with 1ki — 1. Suppose that

z—s mm 1 kz ¿u mm №. Then

pf(z, zi) + {Az - Ax + pN(Tx, Sx), g(zu z)) + p[b(x, z,)- b(x,z)\ < 0,8i.

Since f and b is convex in the second argument and g is affine in the first argument, using above inequality we have

0 — pf(z,z) + {Az - Ax + pN(Tx, Sx), g(z, z))

+ p[b(x,z)- b(x,z)\ — pf\ z,Y^kz

{Az - Ax + pN(Tx, Sx), g i^^kizi, z I )p

b k^i) kib(x,z.

J^ki[pf(z, z,) + {Az - Ax + pN(Tx, Sx), g(z„ z))

+ p(b(x,zi)-b(x,z))] < 0,

which is absurd. Thus z 2 |J™ \ P{zi). Since z was an arbitrary element of Cofz^ z2..., zmg, hence

Cofzi, z2..., zm} ^{jm=lP{zi). Thus P is a KKM mapping. Now, we claim that P(y) c Q(y) for every y 2 K. Indeed, let

z 2 P(y), we have

pf(z,y) + {Az, g(y, z)) P {Ax - pN(Tx, Sx), g(y, z))

- p[b(x,y) — b(x,z)]. (3.5)

Since f and A are monotone, then we have

- pf(y, z) + {Ay - Ax + pN(Tx, Sx), g(y, z)) + p[b(x,y)-b(x,z)\ P 0,

that is, z 2 Q(y). Thus Q is also a KKM mapping.

Since f is lower semicontinuous and g is continuous in the second argument, and b is continuous, it follows that Q(y) is closed for each y 2 K.

Finally, we claim that, for z0 2 D n K, Q(z0) is compact. Indeed suppose that there exists z 2 Q(z0) such that z £ D. Since z0 2 D n K and z 2 Q(z0), we have

- pf(z0, z) + {Az0 - Ax + pN(Tx, Sx), g(z0, z))

+ p[b(x, za)-b(x, z)\ P 0. (3.7)

Since z £ D, by hypothesis (3.2), we have

- pf(z0, z) + {Az0 - Ax + pN(Tx, Sx), g(z0, z)) + p[b(x, z0)- b(x,z)\ < 0,

which is contradiction to (3.7). Hence Q(z0) c D. Since D is compact and Q(z0) is closed, Q(z0) is compact.

Hence by Lemma 3.1, it follows that |~|z^KQ(y)—;, that is, there exists a z 2 K such that

- pf(y, z) + {Ay - Ax + pN(Tx, Sx), g(y, z)) + p[b(x,y)-b(x,z)\ P 0,8y 2 K.

Since K is convex, for any k 2 (0,1\ and any y, z 2 K we have yk :— ky +(1 - k)z 2 K. Hence for given x 2 K, we have

- Pf(Уk, z) + {Ayk - Ax + PN(Tx, Sx), g(y^ z))

+ p[b(x,yk)-b(x,z)\ P 0.

Since b is convex in the second argument and g is affine in the first argument, preceding inequality reduces to

- Pf(Уk, z) + k{Ayk - Ax + PN(Tx, Sx), g(У, z)) + pk[b(x,y)-b(x, z)\ P 0,

where we have used g(z,z) — 0. Now, using (3.8), we have

0 — pf(yk-! yk) 6 pkf(yk'/ y)+ p(1 - f(yX; z)

6 pkf(yk,y) + (1 - k.)[k{Ayx - Ax + pN(Tx, Sx), g(y, z)) + pk[(b(x, y)-b(x, z))\ Dividing by k, we have

pkf(yk, y) + (1 - k){Ayk - Ax + pN(Tx, Sx), g (y, z)) + (1 - k)p[b(x, y)- b(x, z)\ P 0.

Since f and A are simultaneous hemicontinuous, then letting k ! 0+, we have

pf(z,y) + {Az - Ax + pN(Tx, Sx), g(y,z)) + p[b(x,y)-b(x,z)\ P 0, 8y 2 K.

Therefore z 2 K is a solution of AP (3.1).

Uniqueness of solution Let z1 and z2 be two solutions of AP (3.1). Then, for all y 2 K,

p/(zi,y) + {Azi - Ax + pN(Tx, Sx), g(y, zi)) + p[b(x,y)-b(x,zi)] P 0,

p/(z2,y) + {Az2 - Ax + pN(Tx, Sx), g(y, z2)) + p[b(x,y)-b(x,Z2)] P 0.

(3.10)

Taking y — z2 in (3.9), y — z1 in (3.10) and adding these inequalities, we have

p(f(zU Z2 )+f(z2, Zl)) -(Azi - Az2, g(zi, Z2)) P 0,

since g(x, y)+ g(y, x) — 0 for all x, y 2 K.

Since f is monotone and A is s-strongly g-monotone, then it follows from preceding inequality that

s\\z1 — z2||2 6 0.

Since s > 0, we have z1 — z2. This completes the proof. □

4. Algorithms and convergence analysis

Based on Theorem 3.1, we construct an algorithm for GMEP (2.1). Further, we prove the existence of solutions for GMEP (2.1) and discuss the convergence criteria for the sequence generated by our algorithm.

For given x0 2 K, we know from Theorem 3.1 that the AP (3.1) has a solution, say, x1 2 K, that is,

pf(x1,y) + (Ax1 — Ax0 + pN(Tx0, Sx0), g(y, x^)

+ p[b(x0,y) — b(x0, x0] P 0, Vy 2 K.

By Theorem 3.1, again, for x2 2 K, AP (3.1) has a solution

x2, that is,

pf(x2, y) + (Ax2 — Ax1 + pN( Tx 1, Sx1), g(y, x2)) + p[b(x1,y) —b(x1, x2)] P 0, Vy 2 K.

Hence by induction, we have:

Algorithm 4.1. For a given x0 2 K, compute an approximate solution xn 2 K satisfies the following condition:

pF(x„+1, y) + (Ax„+1 — Axn + pN(Txn, Sx„), g(y, xn+1)) + p[b(x„,y) — b(xn, xn+1)\ P 0 Vy 2 K, n — 0,1,2,...,

where p > 0 is a constant and A : K ! H is not necessarily a linear mapping.

Some special cases:

(I) If g(y, x)—y — x and b(x, y)—0, N(Tx, Sx)—B(x) Vx, y 2 K, where B : K ! H, then Algorithm 4.1 reduces to the following algorithm for problem (2.2).

Algorithm 4.2. For a given x0 2 K, compute an approximate solution xn 2 K satisfy

pf(xn+1, y) + (Axn+1 — Axn + pBxn,y — xn+1)) P 0, Vy 2 K,

n — 0,1,2,..., where p > 0 is a constant and A : K ! H is not necessarily a linear mapping.

If A — h', where h' is the derivative of a given strictly convex function h on K, then Algorithm 4.2 reduces to the algorithm studied in [9].

(II) If N(Tx, Sx) — 0 and g(y, x)— y — xVx, y 2 K, then Algorithm 4.1 reduces to the following algorithm for problem (2.4).

Algorithm 4.3. For a given x0 2 K, compute an approximate solution xn 2 K satisfy

pf(x„+1, y) + (Ax„+1 — Ax„, y — x„+1)) + p[b(x„, y) — b(xn, xn+1)] p 0, Vy 2 K,

n — 0,1,2,..., where p > 0 is a constant and A : K ! H is not necessarily a linear mapping. Algorithm 4.3 is different from one considered in [5].

(III) If N(Tx, Sx) — 0, b(x,y) — 0 and g(y, x) — y — x Vx, y 2 K, then Algorithm 4.1 reduces to the following algorithm for Problem 2.5.

Algorithm 4.4. For a given x0 2 K, compute an approximate solution xn 2 K satisfy

pf(x„+1, y) + (Ax„+1 — Ax„,y — x„+1)) p 0, Vy 2 K, n — 0,1,2,...,

where p > 0 is a constant and A : K ! H is not necessarily a linear mapping.

Theorem 4.1. Let K be a nonempty, closed and convex subset in H. Let g : K x K ! Kbe d-Lipschitz continuous and be such that g is affine in the second argument and g(x, y)+ g(y, x) — 0 for all x, y 2 K. Let T, S : K ! H be t-Lipschitz continuous and s-Lips-chitz continuous mappings, respectively; let N : H x H ! H be e-strongly mixed g-monotone with respect to T and S, and (b1, b2)-Lipschitz continuous; let f: K x K ! R be convex and lower semicontinuous in the second argument and a-strongly monotone; let A : K ! H be s-strongly g-monotone and r-Lips-chitz continuous; let f and A be simultaneously hemicontinuous and letb : H x H ! R satisfies properties (i)-(iv). If hypothesis (3.2) of Theorem 3.1 holds and p > 0 satisfy

e — /(y — a)(/s — k)

b2 — /2(y — a)

yj[e — /(y — a)(/s — k)]2 — [b2 — /2(y — a)2][d2 — (/s — k)f

b2 — /2(y — a)2

e > l(y — a)(ls — k)^[b2 — l2(y — a)2][d2 — (ls — k)2] b2 > l2(y — a)2; d2 > (ls — k)2; y > a; ls > k; k < l

k :— Vd2 — 2s + r2; l — 1; b — b11 + b2s. (4.2)

Then the sequence {x„} generated by Algorithm 4.1 converges strongly to x 2 K, where x is the unique solution of GMEP (2.1).

Proof. For any y 2 K, it follows from Algorithm 4.1 that

pf (xn,y) + {Ax„ — Ax„—i + pN(Txn—i, Sxn—i), g(y, xn))

+ p[b(xn—i,y)—b(xn—i,xn)] P 0, (4.3)

pf (xn+i,y) + {Axn+i — Axn + pN(Txn, Sx„), g(y, xn+i))

+ p[b(xn,y)—b(xn,xn+i)] P 0. (4.4)

Taking y — xn+i in (4.3) and y — xn in (4.4), respectively, we have

pf (xn, x„+i) + {Ax„ — Axn—i + pN(Txn—i, Sxn—i), g(x„+u xn)) + p[b(xn—i, xn+i)—b(xn—i, xn)\ P 0, and (4.5)

pf (xn+i, xn)+{Axn+i — Axn + pN(Txn, Sxn), g(x„, xn+i))

+ p[b(xn, xn) — b(xn, xn+i) P 0. (4.6)

Adding (4.5) and (4.6), we have

—p[f(xn+i, xn) +F(x„, xn+i) + {Axn+i — Axn, g(xn+i, xn)) 6 {Axn — Axn—i — p[N(Txn, Sxn)

— N(Txn—i, Sxn—i)], g(xn+i, xn)) + p[b(xn—i, xn+i)

— b(xni xn+i) + (b(xm xn)— b(xn—i, xn)\.

Since f is a-strongly monotone, A is s-strongly g-monotone and b is linear in the first argument, preceding inequality becomes

pa\\xn+1 — xn\\ + s\\x„+l — xn\\ 6 [WAxn — Axn—i — g(xn, xn—l)\\

+ \\g(x„, xn—i)— p[N(Tx„, Sx„) — N{Txn—i, Sxn—i^WWgixn+i, x„)\\

+ p[b(xn — xn—i, xn) — b(x„ — xn — i, xn+i )] ■

Using properties (ii) and (iii) of b and d-Lipschitz continuity of g, we have

(s + pa)\\xn+i — xnf 6 [d\\Axn — Axn—i — g(xn, xn—i)\\ + d\\g(xn, xn—i)— p[N(Txn, Sxn) — N(Txn—i, Sxn—i)]\\] + pc\\xn — xn—iW] x\\xn+i — xn\\ (4.7)

Since N is e-strongly mixed g-monotone with respect to T and S, and (bi, b2)-Lipschitz continuous, and T and S are t-Lipschitz continuous and s-Lipschitz continuous, respectively, we estimate

\\g(xn, xn—i) — p[N(Txn, Sxn) — N(Txn—i, Sxn—1)]\\2 — \\g(xn, xn—i) \\2 — 2p{N(Txn, Sxn)

— N(Txn—i, Sxn—i), g(xn, xn—i)) + p2 \\N(Txn, Sxn)

— N(Txn—i, Sxn—i)\\2

6 d2 \ \ xn — xn—1 \ \ 2 — 2pe \ \ xn — xn—1 \ \ 2

+ p2 11 N(Txn, Sxn)-N(Txn-u Sxn-Vn ,

\ I N(Txn, Sxn)- N(Txn-1, Sxn-1) I I

6 b1 \1 Txn - Txn-1 \1 + fb2 \1 Sxn - Sxn-1 \

6 (fit + f2s)\ \ xn xn-1 \ \ .

From (4.8) and (4.9), we have

\ \ g(xn, xn-1 )- p[N(Txn, Sxn)- N(Txn-1, Sxn-1) 6 (d2 - 2pe + p2(f1t + f2s)2f \\xn - xn-1 \\.

(4.10)

Since A is s-strongly g-monotone and r-Lipschitz continuous, we estimate

\\ Axn - Axn-1 - g(xn, xn-1) \\ 6 (d2 - 2s + r2)2 \\xn - xn-1 \\.

(4.11)

From (4.7), (4.10) and (4.11), we have

(s + pa)\\x„+1 -x„\\2 6 Id ^(d2 -2s + r2)2 + (d2 -2pe + p2(flt + f2s)2)'J + py

(4.12)

e ■:=

s + pa

(d2 - 2s + r2)1 + (d2 - 2pe + p2 (f 11 + f 2s)2)2 + py

By assumption (4.2), 6 < 1 and hence it follows from (4.12) that {xn} is a Cauchy sequence in K c H. Let xn ! x 2 H as n !i. x 2 K as K is closed. Thus by the continuity of f, A, T, S, N, g it follows from (4.1) that

pf(x,y) + {Ax — Ax + pN(Tx, Sx), g(y, x)) + p[b(x,y)— b(x, x)] P 0, 8y 2 K.

Since p > 0, we have

f(x,y) + {N(Tx, Sx), g(y,x))+ b(x,y)— b(x,x) P 0, 8y 2 K,

that is, x is the unique solution of GMEP (2.1). This completes the proof. □

We have the following consequences of Theorem 4.1.

Corollary 4.1. Let Kbe a nonempty, closed and convex subset in H. Let B : K! H be e-strongly monotone and ft-Lipschitz continuous; let f: K x K ! R be convex and lower semicontin-uous in the second argument, f(x, x) — 08x 2 K, and a-strongly monotone; let A : K! H be s-strongly g-monotone and r-Lipschitz continuous; let f and A be jointly hemicontinuous. If there exists a nonempty compact subset D of H and z0 2 D n K such that for any z 2 K\ D, we have

—pf(z0, z) + {Az0 — Ax + pBx, z0 — z)+ p[b(x, z0)— b(x, z)] < 0,

for given x 2 K, and if p > 0 satisfy

^J[e + a(s — k)]2 — (b2 — a2)[1 — (s — k)2] b2 — a2

■-k)

f2 - a2

e + a(s - k) >\/(f2 - a2)[1 -(s - k)2\; f > a; s > k, k < 1,

where k :— \/1 — 2s + r2. Then the sequence {xn} generated by Algorithm 4.2 converges strongly to x 2 K, where x is the unique solution of problem (2.2).

Corollary 4.2. Let K,f, A be same as Corollary 4.1, and let b : H x H ! R be satisfy properties (i)-(iv). If there exists a nonempty compact subset D of H and zQ 2 D \ K such that for any z 2 K \ D, we have

—pf(z0,z) + (Az0 — Ax,z0 — z)+ p[b(x,zq)— b(x, z)] < 0,

for given x 2 K and if p > 0 satisfy k + py < s + pa, where k — V1 — 2s + r2. Then the sequence {xn} generated by Algorithm 4.3 converges strongly to x 2 K, where x is the unique solution of problem (2.4).

Corollary 4.3. Let K, f, A be same as Corollary 4.1. If there exists a nonempty compact subset D of H and zQ 2 D \ K such that for any z 2 K \ D, we have

—pf(zo, z) + (Azo — Ax, zq — z) < 0,

for given x 2 K and if p > 0 satisfy k < s + pa, where k — V1 — 2s + r2. Then the sequence {xn} generated by Algorithm 4.4 converges strongly to x 2 K, where x is the unique solution of Problem 2.5.

We remark that the technique presented in this paper can be applied for the mixed equilibrium problems involving set-valued mappings. Such problem will be the generalization of problems considered by [4,6-8].

References

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