Multivalued quasi variational inclusions and implicit resolvent equations

Three-Step Iterative Algorithms for Multivalued Quasi Variational Inclusions

“…This equivalence is used to suggest an iterative method for solving the quasi variational inclusions. For this purpose, we need the following well known result; see Noor [20,21]. However, we include its proof for the sake of completeness.…”

“…For γ n = 0, Algorithm 3.1 is the Ishikawa iterative scheme for solving multivalued quasi variational inclusions; see Noor [15]. For β n = 0 = γ n and α n = λ Algorithm 3.1 has been studied by Noor [20,21].…”

“…Let A · · H × H −→ H be a maximal monotone operator with respect to the first argument. For a given nonlinear operator N · · H × H −→ H, consider the problem of finding u ∈ H, w ∈ T u , y ∈ V u such that 0 ∈ N w y + A g u u (2.1) which is called the multivalued quasi variational inclusions; see Noor [15,20,21,23,24]. Some special cases of (2.1) have been studied by Noor [14][15][16][17][18][19][20][21][22], Ding [4], Noor and Noor [25], and Noor et al [26] recently.…”

“…Equations (2.6) are called the implicit resolvent equations introduced and studied by Noor [21,23]. In particular, if A g u u ≡ A u then J A u = I + ρA −1 = J A and the implicit resolvent equations (2.6) are equivalent to finding z u ∈ H w ∈ T u y ∈ V u such that…”

“…Multivalued quasi variational inclusion, which was introduced and studied by Noor [15,20,21,23,24], is a useful and important extension of the variational principles with a wide range of applications in industry, physical, regional, social, pure, and applied sciences. Some special cases have been studied by many authors including Ding [4], Noor [12][13][14][15][16], and Noor and Noor [25].…”