Mohd. Ishtyak

The aim of this work is to study a system of variational inclusions involving generalized Yosida and Cayley operators through inertial extrapolation scheme in real Banach space. To obtain faster convergence of the sequences generated by algorithm, we use one inertial extrapolation scheme, although we have established some more iterative schemes. To achieve our goal, we prove an important Lemma ensuring the convergence of sum of two sequences. We provide a numerical example.

We focus our study on a co-variational inequality problem involving two generalized Yosida approximation operators in real uniformly smooth Banach space. We show some characteristics of a generalized Yosida approximation operator, which are used in our main proof. We apply the concept of nonexpansive sunny retraction to obtain a solution to our problem. Convergence analysis is also discussed.

In this paper, we study a co-variational inequality problem involving Yosida approximation operators in real uniformly smooth Banach space and define an iterative method to obtain its solution. We prove some properties of Yosida approximation operator for our purpose, that is strongly accretivity and Lipschitz continuity. An existence as well as convergence result is proved for co-variational inequality problem involving Yosida approximation operators using the concept of nonexpansive sunny retraction.

In this paper, we study a Yosida variational inclusion problem with its corresponding Yosida resolvent equation problem. We mention some schemes to solve both the problems, but we focus our study on discussing convergence criteria for the Yosida variational inclusion problem in real Banach space and for the Yosida resolvent equation problem in q-uniformly smooth Banach space. For faster convergence, we apply an inertial extrapolation scheme for both the problems. An example is provided.

In this paper, we solve a system of mixed variational inclusions involving a generalized Cayley operator and the generalized Yosida approximation operator. An iterative algorithm is suggested to discuss the convergence analysis. We have shown that our system admits a unique solution by using the properties of q-uniformly smooth Banach space, and we discuss the convergence criteria for sequences generated by iterative algorithm. Two examples are constructed, and an application is provided.

In this article, we introduce a new kind of split monotone variational inclusion problem involving Cayley operator in the setting of infinite-dimensional Hilbert spaces. We develop a general iterative method to approximate the solution of the split monotone variational inclusion problem involving Cayley operator. Under some suitable conditions, a convergence theorem for the sequences generated by the proposed iterative scheme is established, which also solves certain variational inequality problems related to strongly positive linear operators. Finally, a numerical example is presented to study the efficiency of the proposed algorithm through MATLAB programming.

A mixed variational inequality problem involving generalized Yosida approximation operator is considered and studied in -uniformly smooth Banach space. We have shown that mixed variational inequality problem involving generalized Yosida approximation operator is equivalent to a fixed-point equation. The fixed-point formulation is applied to establish an algorithm to obtain the solution of mixed variational inequality problem involving generalized Yosida approximation operator. Convergence criteria are also discussed. In support of our main result, we provide an example using Matlab program together with a computation table and convergence graphs. To check the validity of mixed variational inequality problem involving generalized Yosida approximation operator and its fixed-point formulation, we construct one more example.

A new inclusion problem is introduced using generalized Cayley operator and we call it Cayley inclusion problem. We also study its corresponding resolvent equation problem. By using a generalized resolvent operator and generalized Yosida approximation operator, first we establish a fixed point formulation for Cayley inclusion problem. An algorithm is defined to find the solution of Cayley inclusion problem. An existence and convergence result is proved. Secondly, we have shown the equivalence of Cayley inclusion problem with a resolvent equation. We define an iterative algorithm with some of its equivalent forms for solving resolvent equation problem. A numerical example is constructed and a convergence graph is shown by using MATLAB program.

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.