Abstract

We investigate a class of new <svg style="vertical-align:-2.3205pt;width:85.949997px;" id="M2" height="15.0875" version="1.1" viewBox="0 0 85.949997 15.0875" width="85.949997" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x1D43B"/></g><g transform="matrix(.017,-0,0,-.017,14.97,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,20.852,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,26.734,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,30.609,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,37.307,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,41.182,12.138)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,47.064,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,53.779,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,59.66,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,63.536,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,70.233,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,74.109,12.138)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,79.991,12.138)"><use xlink:href="#x29"/></g> </svg>-mixed cocoercive operators in Hilbert spaces. We extend the concept of resolvent operators associated with <svg style="vertical-align:-2.3205pt;width:41.25px;" id="M3" height="15.0875" version="1.1" viewBox="0 0 41.25 15.0875" width="41.25" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x1D43B"/></g><g transform="matrix(.017,-0,0,-.017,14.97,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,20.852,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,24.728,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,31.425,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,35.301,12.138)"><use xlink:href="#x29"/></g> </svg>-cocoercive operators to the <svg style="vertical-align:-2.3205pt;width:85.949997px;" id="M4" height="15.0875" version="1.1" viewBox="0 0 85.949997 15.0875" width="85.949997" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x1D43B"/></g><g transform="matrix(.017,-0,0,-.017,14.97,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,20.852,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,26.734,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,30.609,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,37.307,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,41.182,12.138)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,47.064,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,53.779,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,59.66,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,63.536,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,70.233,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,74.109,12.138)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,79.991,12.138)"><use xlink:href="#x29"/></g> </svg>-mixed cocoercive operators and prove that the resolvent operator of <svg style="vertical-align:-2.3205pt;width:85.949997px;" id="M5" height="15.0875" version="1.1" viewBox="0 0 85.949997 15.0875" width="85.949997" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x1D43B"/></g><g transform="matrix(.017,-0,0,-.017,14.97,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,20.852,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,26.734,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,30.609,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,37.307,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,41.182,12.138)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,47.064,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,53.779,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,59.66,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,63.536,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,70.233,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,74.109,12.138)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,79.991,12.138)"><use xlink:href="#x29"/></g> </svg>-mixed cocoercive operator is single valued and Lipschitz continuous. Some examples are given to justify the definition of <svg style="vertical-align:-2.3205pt;width:85.949997px;" id="M6" height="15.0875" version="1.1" viewBox="0 0 85.949997 15.0875" width="85.949997" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x1D43B"/></g><g transform="matrix(.017,-0,0,-.017,14.97,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,20.852,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,26.734,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,30.609,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,37.307,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,41.182,12.138)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,47.064,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,53.779,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,59.66,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,63.536,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,70.233,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,74.109,12.138)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,79.991,12.138)"><use xlink:href="#x29"/></g> </svg>-mixed cocoercive operators. Further, by using resolvent operator technique, we discuss the approximate solution and suggest an iterative algorithm for the generalized mixed variational inclusions involving <svg style="vertical-align:-2.3205pt;width:85.949997px;" id="M7" height="15.0875" version="1.1" viewBox="0 0 85.949997 15.0875" width="85.949997" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x1D43B"/></g><g transform="matrix(.017,-0,0,-.017,14.97,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,20.852,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,26.734,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,30.609,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,37.307,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,41.182,12.138)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,47.064,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,53.779,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,59.66,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,63.536,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,70.233,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,74.109,12.138)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,79.991,12.138)"><use xlink:href="#x29"/></g> </svg>-mixed cocoercive operators in Hilbert spaces. We also discuss the convergence criteria for the iterative algorithm under some suitable conditions. Our results can be viewed as a generalization of some known results in the literature.

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