Generalized Monotone Mapping with an Application for Solving a Variational Inclusion Problem

Abstract

The aim of this paper is to define a generalized monotone mapping, which is the sum of symmetric cocoercive mapping and symmetric monotone mapping. The resolvent operator associated with generalized monotone mapping is defined and some of its properties are shown. We solve a variational inclusion problem using these new concepts. For illustration, some examples are given.

Appendix: Verification (Calculations) of Example 4.1

Let X=ℝ² with usual inner product. Let A,B:ℝ²→ℝ² be defined by

Let H:ℝ²×ℝ²→ℝ² is defined by

Then

which implies that

i.e., H(A,B) is ${\frac{3}{10}}$-cocoercive with respect to A, and

which implies that

i.e., H(A,B) is 1-relaxed cocoercive with respect to B.

which implies that

i.e., A is ${\frac{\sqrt{1}0}{3}}$ -expansive, for n=3,4.

which implies that

i.e., B is ${\frac{1}{\sqrt{n}}}$ -Lipschitz continuous, for n=1,2.

which implies that

i.e., H(A,B) is mixed Lipschitz continuous with constant ${\frac {\sqrt{1}0}{n}}$, for n=5,6.

(v) Let P,R:ℝ²→ℝ² be defined by

Now

which implies that

i.e., H(A,B) is ${\frac{1}{n}}$-mixed strongly monotone with respect to P and R, for n=6,7.

(vi) Let f,g:ℝ²→ℝ² be defined by

Suppose that M:ℝ²×ℝ²→ℝ² is defined by

Then

which implies that

i.e., M(f,g) is 4-strongly monotone with respect to f, and

which implies that

i.e., M(f,g) is ${\frac{1}{4}}$-relaxed monotone with respect to g.

Moreover, for λ=1, M is H(⋅,⋅)-Co-monotone with respect to A,B,f and g.

(vii) Let R,S,T:ℝ²→ℝ², be the identity mappings, then R,S and T are $\mathcal{D}$-Lipschitz continuous mappings with constant 1.

(viii) Let P,F,G:ℝ²→ℝ² be defined by

Then

which implies that P is 1-Lipschitz continuous.

which implies that

i.e., F is ${\frac{\sqrt{5}}{n}}$-Lipschitz continuous, for n=3,4.

which implies that

i.e., G is ${\frac{5}{n}}$-Lipschitz continuous, for n=11,12.

which implies that

i.e., S is ${\frac{1}{n}}$-relaxed Lipschitz continuous with respect to F, for n=2,3.

which implies that

i.e., T is ${\frac{1}{n}}$-relaxed monotone with respect to G, for n=3,4.

(xi) For the constants

obtained from above conditions (i) to (x), one can easily verify, for λ=1, that the condition (xi) of Theorem 4.1, given by

holds.