Variational inclusion problem and total asymptotically nonexpansive mapping: graph convergence, algorithms and approximation of common solutions

. In this paper, under some new appropriate conditions imposed on the parameter and mappings involved in the resolvent operator associated with an ( H,η )-monotone operator, its Lipschitz continuity is proved and an estimate of its Lipschitz constant is computed. This paper is also concerned with the establishment of a new equivalence relationship between the graph convergence of a sequence of ( H,η )-monotone operators and their associated resolvent operators, respectively, to a given ( H,η )-monotone operator and its associated resolvent operator. A new iterative scheme for approximating a common element of the set of solutions of a variational inclusion problem and the set of fixed points of a given total asymptotically nonexpansive mapping is constructed. As an application of the obtained equivalence conclusion concerning graph convergence, under some suitable conditions, the strong convergence of the sequence generated by our suggested iterative algorithm to a common element of the above-mentioned two sets is proved. Our results improve and generalize the corresponding results of recent works.