In this paper, we introduce three new inertial-like Bregman projection methods with a nonmonotone adaptive step-size for solving quasi-monotone variational inequalities in real Hilbert spaces. Under some suitable conditions, the weak convergence of these methods is proved without the prior knowledge of the Lipschitz constant of the operator and the strong convergence of some proposed methods under a strong quasi-monotonicity assumption of the mapping is also provided. Finally, several numerical experiments and applications in image restoration problems are provided to illustrate the performance of the proposed methods.

Let E be a uniformly convex and q-uniformly smooth real Banach space. Let $A:E\to E$ be an α- inverse strongly accretive mapping of order q, $B:E⊸E$ be a set-valued m- accretive mapping and $S:E\to E$ be a nonexpansive mapping. In this article, a viscosity-type forward-backward splitting method for approximating a zero of (A + B) which is also a fixed point of S is introduced studied. Strong convergence theorem of the method is proved under suitable conditions. Furthermore, the convergence result obtained is applied to convex minimization and image restoration problems. Finally, numerical illustrations are presented to compare the convergence of the sequence of our algorithm and that of some recent important algorithms.

LetBbe a uniformly convex and uniformly smooth real Banach space with dual space $B^{*}$B∗ . Let $F:B\to B^{*}$F:B→B∗ , $K:B^{*} \to B$K:B∗→B be maximal monotone mappings. An iterative algorithm is constructed and the sequence of the algorithm is proved to converge strongly to a solution of the Hammerstein equation $u+KFu=0$u+KFu=0 . This theorem is a significant improvement of some important recent results which were proved in real Hilbert spaces under the assumption thatFandKare maximal monotonecontinuousandbounded. The continuity and boundedness restrictions onKandFhave been dispensed with, using our new method, even in the more general setting considered in our theorems. Finally, numerical experiments are presented to illustrate the convergence of the sequence of our algorithm.