On a System of Nonlinear Variational Inclusions with Hh,η-Monotone Operators

and Applied Analysis 3 4 strictly η-monotone if 〈 u − v, ηx, y > 0, ∀x, y ∈ H, x / y, u ∈ M x , v ∈ M ( y ) ; 2.5 5 strongly monotone if there exists a constant r > 0 satisfying 〈 u − v, x − y ≥ r∥x − y∥2, ∀x, y ∈ H, u ∈ M x , v ∈ My; 2.6 6 strongly η-monotone if there exists a constant r > 0 satisfying 〈 u − v, ηx, y ≥ r∥x − y∥2, ∀x, y ∈ H, u ∈ M x , v ∈ My; 2.7 7 maximal monotone resp., maximally strictly monotone, maximally strongly monotone if M is monotone resp., strictly monotone, strongly monotone and I λM H H for any λ > 0; 8 maximal η-monotone resp., maximally strictly η-monotone, maximally strongly ηmonotone if M is η-monotone resp., strictly η-monotone, strongly η-monotone and I λM H H for any λ > 0; 9 Hh-monotone resp., strictly Hh-monotone, strongly Hh-monotone if M is monotone resp., strictly monotone, strongly monotone and h λM H H for any λ > 0; 10 Hh,η-monotone resp., strictly Hh,η-monotone, strongly Hh,η-monotone if M is ηmonotone resp., strictly η-monotone, strongly η-monotone and h λM H H for any λ > 0. Definition 2.2. Let g : H → H and η : H ×H → H be operators. The operator g is called 1 Lipschitz continuous if there exists a constant r > 0 satisfying ∥g x − gy∥ ≤ r∥x − y∥, ∀x, y ∈ H; 2.8 2 monotone if 〈 g x − gy, x − y ≥ 0, ∀x, y ∈ H; 2.9 3 η-monotone if 〈 g x − gy, ηx, y ≥ 0, ∀x, y ∈ H; 2.10 4 strongly monotone if there exists a constant r > 0 satisfying 〈 g x − gy, x − y ≥ r∥x − y∥2, ∀x, y ∈ H; 2.11 4 Abstract and Applied Analysis 5 strongly η-monotone if there exists a constant r > 0 satisfying 〈 g x − gy, ηx, y ≥ r∥x − y∥2, ∀x, y ∈ H; 2.12 6 relaxed Lipschitz if there exists a constant r > 0 satisfying 〈 g x − gy, x − y ≤ −r∥x − y∥2, ∀x, y ∈ H. 2.13 Definition 2.3. Let N : H ×H → H and a, g : H → H be operators. The operator N is said to be 1 strongly monotone with respect to a and g in the first argument if there exists a constant r > 0 satisfying 〈 N a x , z −Nay, z, g x − gy ≥ r∥x − y∥2, ∀x, y, z ∈ H; 2.14 2 Lipschitz continuous in the first argument if there exists a constant r > 0 satisfying ∥N x,w −Ny,w∥ ≤ r∥x − y∥, ∀x, y,w ∈ H. 2.15 Similarly, we could define the strong monotonicity ofN with respect to a and g in the second argument and the Lipschitz continuity ofN in the second argument. Remark 2.4. For h I, the definition of the Hh-monotone resp., strictly Hh-monotone, strongly Hh-monotone operator reduces to the definition of the maximal monotone resp., maximally strictly monotone, maximally strongly monotone operator. Remark 2.5. Notice that M is maximal monotone resp., maximally strictly monotone, maximally strongly monotone if and only if M is monotone resp., strictly monotone, strongly monotone and there is no other monotone resp., strictly monotone, strongly monotone operator whose graph contains strictly the graph Graph M ofM. Lemma 2.6 see 20 . Let {αn}n≥0, {βn}n≥0 and {γn}n≥0 be nonnegative sequences satisfying αn 1 ≤ 1 − λn αn βnλn γn, ∀n ≥ 0, 2.16 where {λn}n≥0 ⊂ 0, 1 , ∑∞ n 0 λn ∞, limn→∞βn 0 and ∑∞ n 0 γn < ∞. Then limn→∞αn 0. 3. The Properties of Strictly Hh,η-Monotone Operators and Strongly Hh,η-Monotone Operators In this section, we discuss some properties of the set-valued strictlyHh,η-monotone operators and set-valued strongly Hh,η-monotone operators, respectively, dealing with a η-monotone operator h in Hilbert spaces. Abstract and Applied Analysis 5 Theorem 3.1. Let H be a real Hilbert space and let h : H → H, η : H ×H → H and M : H → 2 be operators such that h is η-monotone and M is strictlyHh,η-monotone. Then i M is maximally strictly η-monotone; ii h λM −1 : H → H is a single-valued operator for each λ > 0. Proof. i Since M is strictly Hh,η-monotone, it follows that M is strictly η-monotone. Suppose that there exists a strictly η-monotone set-valued operator A : H → 2 satisfying Graph A / Graph M , that is, there exists u0, x0 ∈ Graph A \Graph M such that 〈 x0 − y, η u0, v 〉 > 0, ∀v, y ∈ Graph M with v / u0. 3.1and Applied Analysis 5 Theorem 3.1. Let H be a real Hilbert space and let h : H → H, η : H ×H → H and M : H → 2 be operators such that h is η-monotone and M is strictlyHh,η-monotone. Then i M is maximally strictly η-monotone; ii h λM −1 : H → H is a single-valued operator for each λ > 0. Proof. i Since M is strictly Hh,η-monotone, it follows that M is strictly η-monotone. Suppose that there exists a strictly η-monotone set-valued operator A : H → 2 satisfying Graph A / Graph M , that is, there exists u0, x0 ∈ Graph A \Graph M such that 〈 x0 − y, η u0, v 〉 > 0, ∀v, y ∈ Graph M with v / u0. 3.1 Notice that h u0 λx0 ∈ H h λM H for any λ > 0. Thus there exists v0, y0 ∈ Graph M satisfying h v0 λy0 h u0 λx0, ∀λ > 0. 3.2 Suppose that u0 v0. Equation 3.2 means that x0 y0. Therefore, Graph M v0, y0 ) u0, x0 ∈ Graph A \Graph M , 3.3 which is impossible. Suppose that u0 / v0. It follows from 3.1 , 3.2 , and the η-monotonicity of h and strict η-monotonicity of A that 0 < 〈 x0 − y0, η u0, v0 〉 −λ−1h u0 − h v0 , η u0, v0 〉 ≤ 0, 3.4 which is a contradiction. Hence M is maximally strictly η-monotone. ii Suppose that there exists some u ∈ H such that h λM −1 u contains at least two different elements x and y. Since M is strictly Hh,η-monotone, h is η-monotone, and λ−1 u − h x ∈ M x , λ−1 u − h y ∈ M y , it follows that 0 < 〈 λ−1 u − h x − λ−1u − hy, ηx, y 〉 − λ−1h x − hy, ηx, y ≤ 0, 3.5 which is a contradiction. Consequently, the operator h λM −1 is single valued. This completes the proof. Definition 3.2. Let H be a real Hilbert space and let h : H → H, η : H × H → H and M : H → 2 be operators such that h is η-monotone andM is strictlyHh,η-monotone. Then for each λ > 0, the resolvent operator J M,λ : H → H is defined by J h,η M,λ x h λM −1 x , ∀x ∈ H. 3.6 6 Abstract and Applied Analysis Theorem 3.3. Let H be a real Hilbert space and let η : H ×H → H be Lipschitz continuous with constant Lη. Assume that h : H → H is η-monotone and M : H → 2 is stronglyHh,η-monotone with constant r. Then for every λ > 0, the resolvent operator J M,λ : H → H is Lipschitz continuous with constant Lη/λr. Proof. Since M is strongly Hh,η-monotone with constant r, it follows that M is strictly Hh,ηmonotone. Let x, y be in H. In view of λ−1 x − h J M,λ x ∈ M J h,η M,λ x and λ −1 y − h J M,λ y ∈ M J M,λ y and the strong monotonicity of M, we deduce that λ−1 〈 x − y, η ( J h,η M,λ x , J h,η M,λ ( y ))〉 − λ−1 〈 h ( J h,η M,λ x ) − h ( J h,η M,λ ( y )) , η ( J h,η M,λ x , J h,η M,λ ( y ))〉 〈 λ−1 ( x − h ( J h,η M,λ x )) − λ−1 ( y − h ( J h,η M,λ ( y ))) , η ( J h,η M,λ x , J h,η M,λ ( y ))〉 ≥ r ∥∥∥J h,η M,λ x − J h,η M,λ ( y )∥∥∥ 2 . 3.7