In this paper, we introduce a Yosida inclusion problem as well as a generalized Yosida approximation operator. Using the graph convergence of \(H(\cdot ,\cdot )\)-accretive operator and resolvent operator convergence discussed in Li and Huang (Appl Math Comput 217:9053–9061, 7 ), we establish the convergence for generalized Yosida approximation operator. As an application, we solve a Yosida inclusion problem in q-uniformly smooth Banach spaces. An example is constructed, and through MATLAB programming, we show some graphics for the convergence of generalized Yosida approximation operator.

In this article, we introduce a new type of generalized mixed ordered quasi-variational inclusion problem in the setting of real ordered Hilbert spaces. Further, we propose a perturbed three-step iterative algorithm for solving generalized mixed ordered quasi-variational inclusion problem. By using the relaxed resolvent operator method with XOR technique, we prove the existence of solution of generalized mixed ordered quasi-variational inclusion problem and discuss the convergence analysis and stability of the proposed algorithm. The iterative algorithm and results presented in this paper generalize, improve and significantly refine many previously known results of this area. Moreover, we give a numerical example to illustrate and show the convergence of the proposed algorithm in support of our main result.

In this paper, we introduce a system of generalized implicit variational inclusions which consists of three variational inclusions. We design an iterative algorithm with error terms based on relaxed resolvent operator due to Ahmad et al. (Stat Optim Inf Comput 4:183–193, 2016) for approximating the solution of our system. The convergence of the iterative sequences generated by the iterative algorithm is also discussed. An example is given which satisfy all the conditions of our main result.

In this paper, we introduce a new resolvent operator and we call it relaxed η-proximal operator. We demonstrate some of the properties of relaxed η- proximal operator. By applying this concept, we consider and study a variational -like inclusion problem with a nonconvex functional. Based on relaxed η-proximal operator, we define an iterative algorithm to approximate the solutions of a variational-like inclusion problem and the convergence of the iterative sequences generated by the algorithm is also discussed. Our results can be treated as refinement of many previously known results. An example is constructed in support of Theorem 1.