Cayley inclusion problem with its corresponding generalized resolvent equation problem in uniformly smooth Banach spaces

ABSTRACT

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.

Keywords:

• Algorithm
• Cayley
• Lipschitz
• solution
• uniformly

2010 AMS Subject Classifications:

• 47H09
• 49J40

Acknowledgments

All authors are thankful to the reviewers for their valuable comments which improved this manuscript a lot.

Figure 2. The convergence of $\left\{x_n\right\}$ and $\left\{s_n\right\}$ with initial values $s_0=1$ and $s_0=5$.

Table 2. Computational results for different initial values of $s_0$.

Disclosure statement

No potential conflict of interest was reported by the authors.