Let X be an ordered Banach space, let P be a cone of X, let ≤ be a relation defined by the cone P in Definition 2.1 iii).
Let X be an ordered Banach space, P be a cone of X, and ≤ be a relation defined by the cone P in Definition 2.1 iii).
Proof Let r ⊂ X × X be a relation defined as follows: For any x, y ∈ X, x r y ⇔ F ( x, y ) + ω ( x, y ) D ⊆ − C. We will first show that r is transitive.
Proof Let X be a real ordered Banach space, let P be a normal cone with normal constant N in X, and let ≤ be an ordered relation defined by the cone P. Let A be a γ-order non-extended mapping and M be an α-weak-non-ordinary difference mapping with respect to A, it follows that J M, λ A exists from Lemma 3.5.
Theorem 3.7 Let X be a real ordered Banach space, let P be a normal cone with normal constant N in X, and let ≤ be an ordered relation defined by the cone P. Let A be a γ-order non-extended mapping, and let M be an ordered ( α A, λ ) -weak-ANODD mapping with respect to J M, λ A. If α A > 1 λ > 0, then the resolvent operator J M, λ A is continuous.
Proof Let X be a real ordered Banach space, let P be a normal cone with normal constant N in X, let ≤ be an ordered relation defined by the cone P. Let A be a γ-order non-extended mapping, and let M X X → 2 X be an ordered ( α A, λ ) -weak-ANODD mapping with respect to J M, λ A. If α A > 1 λ > 0, then J M, λ A ( x ) ⊕ J M, λ A ( y ) ≤ 1 γ ( α A λ − 1 ) ( x ⊕ y ).
Let (mathcal{R}) be a binary relation defined on a non-empty set X. Then a sequence ({x_{n}} subset X) is called (mathcal{R} -preserving if (x_{n},x_{n+1})inmathcal{R} -preservingl nif mathbb{N}_{0}.
Let (mathcal{R}) be a binary relation defined on a non-empty set X. Then any pair of points x, y in X is said to be (mathcal{R} -comparative if eitheR} -comparativecal{R}) or ((y,x)ifmathcal{R}), which is togeitherwritten as ([x,y inmathcal{R}).
Let (mathcal{R}) be a binary relation defined on a non-empty set X and a pair of points x, y in X.
Lemma 3.5 Let X be a real ordered Banach space, let P be a normal cone with normal constant N in X, let ≤ be an ordered relation defined by the cone P, let the operator ⊕ be an XOR operator.
Theorem 3.2 Let X be an ordered Banach space, let P be a normal cone with the normal constant N in the X, let ≤ be an ordering relation defined by the cone P, the operator ⊕ be a X O R operator.
