A new technique for convergence theorem of fixed point problem of quasi-nonexpansive mapping

Abstract

Abstract For the purpose of this paper, we use the method different from the relaxed extragradient method for finding a common element of the set of fixed points of a quasi-nonexpansive mapping, the set of solutions of equilibrium problems, and the set of solutions of a modified system of variational inequalities without demiclosed condition of W and $W_{\omega}:= (1-\omega )I+\omega W$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>W</mml:mi> <mml:mi>ω</mml:mi> </mml:msub> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:mi>ω</mml:mi> <mml:mo>)</mml:mo> <mml:mi>I</mml:mi> <mml:mo>+</mml:mo> <mml:mi>ω</mml:mi> <mml:mi>W</mml:mi> </mml:math> , where W is a quasi-nonexpansive mapping and $\omega\in (0,\frac{1}{2} )$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo>)</mml:mo> </mml:math> in the framework of Hilbert space. By using our main result, we obtain a strong convergence theorem involving a finite family of nonspreading mappings and another corollary. Moreover, we give a numerical example to encourage our main theorem.