^{1, 2}

^{2, 3}

^{4}

^{1}

^{2}

^{3}

^{4}

We introduce a Halpern-type iteration for a generalized mixed equilibrium problem in uniformly smooth and uniformly convex Banach spaces. Strong convergence theorems are also established in this paper. As applications, we apply our main result to mixed equilibrium, generalized equilibrium, and mixed variational inequality problems in Banach spaces. Finally, examples and numerical results are also given.

Let

One classical way often used to approximate a fixed point of a nonlinear self-mapping

Subsequently, motivated by Halpern [

Let

If

If

If

If

If

If

The problem (

For solving the generalized mixed equilibrium problem, let us assume the following [

for all

for all

The purpose of this paper is to investigate strong convergence of Halpern-type iteration for a generalized mixed equilibrium problem in uniformly smooth and uniformly convex Banach spaces. As applications, our main result can be deduced to mixed equilibrium, generalized equilibrium, mixed variational inequality problems, and so on. Examples and numerical results are also given in the last section.

In this section, we need the following preliminaries and lemmas which will be used in our main theorem.

Let

Let

We know the following: for any

Let

Let

In fact, we have the following result.

Let

Let

Let

Let

We make use of the following mapping

Let

Let

Following [

Let

It is known that

The following lemmas give us some nice properties of real sequences.

Assume that

Let

In this section, we prove our main theorem in this paper. To this end, we need the following proposition.

Let

Let

Let

From Lemma

We next show that if there exists a subsequence

We next consider the following two cases.

Finally, we show that

There exists a subsequence

As a direct consequence of Theorem

Let

Let

Let

In this section, we give examples and numerical results for our main theorem.

Let

It is easy to check that

Let

We next give two numerical results for algorithm (

Let

See Table

1 | 1.0000 |

2 | 0.0856 |

3 | 0.0111 |

4 | 0.0047 |

5 | 0.0033 |

261 | 0.0001 |

262 | 0.0000 |

Let

See Table

1 | −1.0000 |

2 | −0.0481 |

3 | −0.0074 |

4 | −0.0038 |

5 | −0.0027 |

217 | −0.0001 |

218 | 0.0000 |

Tables

In the view of computation, our algorithm is simple in order to get strong convergence for generalized mixed equilibrium problems.

The first and the second authors wish to thank the Thailand Research Fund and the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand. The third author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant no. 2011-0021821).