In this paper, we study an absolutely new problem, namely, the Cayley inclusion problem which involves the Cayley operator and a multi-valued mapping with XOR-operation. We have shown that the Cayley operator is a single-valued comparison and it is Lipschitz-type-continuous. A fixed point formulation of the Cayley inclusion problem is shown by using the concept of a resolvent operator as well as the Yosida approximation operator. Finally, an existence and convergence result is proved. An example is constructed for some of the concepts used in this work.

In this paper, we consider and study a system of multi-valued mixed variational inclusions with XOR-operation ⊕ in real ordered uniformly smooth Banach spaces. This system consists of bimappings, multi-valued mappings and Cayley operators. An iterative algorithm is suggested to find the solution to a system of multi-valued mixed variational inclusions with XOR-operation ⊕ and consequently an existence and convergence result is proved. In support of our main result, an example is constructed.

In this paper, we consider a resolvent operator which depends on the composition of two mappings with ⊕ operation. We prove some of the properties of the resolvent operator, that is, that it is single-valued as well as Lipschitz-type-continuous. An existence and convergence result is proven for a generalized implicit set-valued variational inclusion problem with ⊕ operation. Some special cases of a generalized implicit set-valued variational inclusion problem with ⊕ operation are discussed. An example is constructed to illustrate some of the concepts used in this paper.

In this paper, we consider the parametric mixed Cayley inclusion problem with Exclusive or (XOR)-operation and show its equivalence with the parametric resolvent equation problem with XOR-operation. Since the sensitivity analysis, Cayley operator, inclusion problems, and XOR-operation are all applicable for solving many problems occurring in basic and applied sciences, such as financial modeling, climate models in geography, analyzing “Black Box processes”, computer programming, economics, and engineering, etc., we study the sensitivity analysis of the parametric mixed Cayley inclusion problem with XOR-operation. For this purpose, we use the equivalence of the parametric mixed Cayley inclusion problem with XOR-operation and the parametric resolvent equation problem with XOR-operation, which is an alternative approach to study the sensitivity analysis. In support of some of the concepts used in this paper, an example is provided.