Abstract
In this paper, we present a new modified inertial forward–backward algorithm for finding a common solution of the quasi-variational inclusion problem and the variational inequality problem in a q-uniformly smooth Banach space. The proposed algorithm is based on descent, splitting and inertial ideas. Under suitable assumptions, we prove that the sequence generated by the iterative algorithm converges strongly to the unique solution of the abovementioned problems. Numerical examples are also given to demonstrate our results.
Highlights
Many authors are studying algorithms for reckoning a zero point for monotone operators in a Hilbert space
We introduce a new modified inertial forward–backward algorithm for finding a common solution of the quasi-variational inclusion problem and the variational inequality problem in q-uniformly smooth Banach spaces
We present an algorithm for solving quasi-variational inclusion problems and variational inequality problems in Banach spaces
Summary
Many authors are studying algorithms for reckoning a zero point for monotone operators in a Hilbert space. This is the problem of finding a point x ∈ H such that 0 ∈ Tx, (1). The set of zero points of T is denoted by T−1(0) Diverse problems such as convex minimization, monotone variational inequalities over convex sets, equilibrium problems etc. The proximal point algorithm proposed by Martinet [1,2] and generalized by Rockafellar [3,4], xn+1 = (I + αkT)−1xn, ∀n ∈ N, (2). (I + αkT)−1 is the resolvent operator of maximal monotone operator T, I is the identity mapping and {αn} ⊂ (0, +∞) is a regularization sequence. Rockafellar [3] and Bruck and Reich [5] proved that the sequence
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