Abstract
Abstract The aim of this paper is to introduce a new H ( ⋅ , ⋅ ) -η-cocoercive operator and its resolvent operator. We study some of the properties of H ( ⋅ , ⋅ ) -η-cocoercive operator and prove the Lipschitz continuity of resolvent operator associated with H ( ⋅ , ⋅ ) -η-cocoercive operator. Finally, we apply the techniques of resolvent operator to solve a generalized set-valued variational-like inclusion problem in Banach spaces. Our results are new and generalize many known results existing in the literature. Some examples are given in support of definition of H ( ⋅ , ⋅ ) -η-cocoercive operator. MSC:47H19, 49J40.
Highlights
Variational inclusion problems are interesting and intensively studied classes of mathematical problems and have wide applications in the field of optimization and control, economics and transportation equilibrium, and engineering sciences, etc., see for example [ – ]
M is H(·, ·)-η-cocoercive with respect to A and B
We show that the mapping M need not be H(·, ·)-η-cocoercive with respect to A and B
Summary
Variational inclusion problems are interesting and intensively studied classes of mathematical problems and have wide applications in the field of optimization and control, economics and transportation equilibrium, and engineering sciences, etc., see for example [ – ]. Ahmad et al [ ] introduced H(·, ·)cocoercive operators and apply them to solve a set-valued variational inclusion problem in Hilbert spaces. ∀x, y ∈ X, j(η(x, y)) ∈ J(η(x, y)); (ii) H(·, B) is said to be η-relaxed cocoercive with respect to B, if there exists a constant γ > such that
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