An adaptive splitting algorithm for the sum of two generalized monotone operators and one cocoercive operator

Splitting algorithms for finding a zero of sum of operators often involve multiple steps which are referred to as forward or backward steps. Forward steps are the explicit use of the operators and backward steps involve the operators implicitly via their resolvents. In this paper, we study an adaptive splitting algorithm for finding a zero of the sum of three operators. We assume that two of the operators are generalized monotone and their resolvents are computable, while the other operator is cocoercive but its resolvent is missing or costly to compute. Our splitting algorithm adapts new parameters to the generalized monotonicity of the operators and, at the same time, combines appropriate forward and backward steps to guarantee convergence to a solution of the problem.

On Strongly Generalized Preinvex Fuzzy Mappings

In this article, we introduce a new notion of generalized convex fuzzy mapping known as strongly generalized preinvex fuzzy mapping on the invex set. Firstly, we have investigated some properties of strongly generalized preinvex fuzzy mapping. In particular, we establish the equivalence among the strongly generalized preinvex fuzzy mapping, strongly generalized invex fuzzy mapping, and strongly generalized monotonicity. We also prove that the optimality conditions for the sum of G-differentiable preinvex fuzzy mappings and non-G-differentiable strongly generalized preinvex fuzzy mappings can be characterized by strongly generalized fuzzy mixed variational-like inequalities, which can be viewed as a novel and innovative application. Several special cases are discussed. Results obtained in this paper can be viewed as improvement and refinement of previously known results.

Generalized monotonicity in terms of differential inequalities

The classical notions of monotonicity and convexity can be characterized via the nonnegativity of the first and the second derivative, respectively. These notions can be extended applying Chebyshev systems. The aim of this note is to characterize generalized monotonicity in terms of differential inequalities, yielding analogous results to the classical derivative tests. Applications in the fields of convexity and differential inequalities are also discussed.