If T is strongly order preserving, then there exists a fixed point in ([!![a,b ]!!]) which is stable relative to ([!![a,b ]!!]).
Under, is satisfied, and is strongly order-preserving.
We know that is strongly order-preserving, so does given in Lemma 2.1.
Lemma 3.2 (see [20]). is a linear completely continuous operator and also a linear completely continuous operator from In addition, is strongly order-preserving.
Then ((mathbb{Y},mathbb{Y}^)) is a strongly ordered Banach space.
Define (X^=C overline{Omega}, mathbb{R}_^{2})); then ((X,X^)) is a strongly ordered space.
for some γ ( 0 < γ ≤ 1 ), then we say that f ( z ) is strongly convex of order γ in U, and we denote by K ˜ the class of all such functions.
Lemma 4.1 If g j , j = 1, …, m, are strongly convex of order m on X and ν T h is strongly convex of order m on X, then the same conclusion of Theorem 4.1 also holds.
Assume further that f i, i = 1, …, p, are strongly convex of order m on X, g j, j ∈ I ( x 0 ) are strongly quasiconvex of order m on X and ν T h is strongly quasiconvex of order m on X.
Since g j, j ∈ I ( x 0 ) are strongly quasiconvex of order m on X and ν T h is strongly quasiconvex of order m on X, it follows that there exist c j > 0, η j ∈ ∂ g j ( x 0 ), j ∈ I ( x 0 ), c > 0, and ζ ∈ ∂ ν T h ( x 0 ) such that 〈 η j, x − x 0 〉 + ∥ x − x 0 ∥ m c j ≦ 0, 〈 ζ, x − x 0 〉 + ∥ x − x 0 ∥ m c ≦ 0. (3.3).
