Sentence examples for non-decreasing operator from inspiring English sources

Thus G is a non-decreasing operator with respect to the order ⪯ on X ˜.

Let N be a real Banach space with normal order cone K. Suppose that there exist α ≤ β such that T : ⊂ N → N is a completely continuous monotone non-decreasing operator with α ≤ T α and T β ≤ β.

As f is non-decreasing with respect to the second variable, for x, y ∈ P with y ≥ x and t ∈ [ 0, 1 ], we have ( T y ) ( t ) = ∫ 0 1 G ( t, s ) f ( s, y ( s ) ) d s ≥ ∫ 0 1 G ( t, s ) f ( s, x ( s ) ) d s ≥ ( T x ) ( t ), and this proves that T is a non-decreasing operator.

Suppose G : X → X is a non-decreasing operator with respect to the order ⪯ on X. Assume (i) in Theorem  2.1 and one of following conditions hold: (a) G is a continuous operator;   (b) if a monotone sequence x n in X tends to x ¯, then x n and x ¯ are comparable for all n.  .

For non-decreasing operators, in the case of an upper solution, it was proved in [4] the following result which is an improvement of those in [2, 3].

Assume that (H : Yrightarrow Y ) is a differentiable convex function and (circ,star: Y timesmu(Sigma rightarrow Y ) are non-decreasing operators satisfying the following conditions: 1. (a star0 = a circ0 = 0);   2.

If (H : Yrightarrow Y ) is a differentiable convex function with (H (0 ) = 0) and (H ^{prime} y geq1 ) for all (y in Y), (lambda_{i}= mu ({x_{1}, ldots, x_{i}} )) for all (i in{1, ldots, n}) and (circ,star: Y timesmu(Sigma rightarrow Y ) are non-decreasing operators satisfying the following conditions: 1. (a star0 = a circ0 = 0),   2.

In Section 2 we prove two results concerning non-decreasing and non-increasing operators in a shell, in presence of an upper or of a lower solution; in Remark 2.4 we present a comparison with previous results in this direction.

Thus, if F has the mixed monotone property on X, then the operator G is non-decreasing monotone for the order ⪯.

there is a φ ∈ Φ such that G : X 2 → X 2 satisfying ρ ( G ( x ˜ ), G ( y ˜ ) ) ≤ φ ( ρ ( x ˜, y ˜ ) ) for each x ˜, y ˜ ∈ X 2 with x ˜ ⪯ y ˜ ; there exists an x ˜ 0 ∈ X 2 such that x ˜ 0 ⪯ G ( x ˜ 0 ) ; one of (a) and (b) holds: G is a continuous operator; if a non-decreasing monotone sequence x ˜ n in X 2 tends to x ¯, then x ˜ n ⪯ x ¯ for all n.

Considering the operator F ˜ is non-decreasing monotone for the order ⪯ and x ˜ 0 ⪯ G ( x ˜ 0 ), we have x ˜ 0 ⪯ x ˜ 1 ⪯ x ˜ 2 ⪯ ⋯ ⪯ x ˜ n ⪯ ⋯.