Sentence examples for exists a pair from inspiring English sources

By introducing a definition for the coupled lower and upper solutions of BVP (1.1) and (1.2), we obtain the existence of solutions of the problem based on the assumption that there exists a pair of coupled lower and upper solutions.

If there exists a pair of conjugate points on with respect to (1.12), then (1.14).

Moreover we have that is a negative eigenvalue of if and only if there exists a pair,, such that (D).

If the graph is not 3-connected, there exists a pair of vertices whose removal disconnects the graph.

Theorem 5 Assume that there exists a pair of lower and upper solutions of problem (1 - 2).

It is wellknown that if there exists a pair of conjugate points in, then the classical Lyapunov inequality (1.11).

Let J∗T v) denote the set of all integers s such that there exists a pair of disjoint S 2,4,v s intersecting in s triangles.

By reduction to absurdity, we assume that there exists a pair of constants (T>0) and (varepsilonin 0, 1)) such that P{tau_{+infty}< T}>varepsilon.

For system (2.9) and every, the trivial solution is exponentially stable in the mean-square if there exists a pair of positive constants and such that (2.12).

In general, a set A is finite and its cardinality is n if there exists a pairing of its elements with the set {1, 2, 3, …, n}.

(Because the empty set has no elements, its cardinality is defined as 0.) In general, a set A is finite and its cardinality is n if there exists a pairing of its elements with the set {1, 2, 3, …, n}.