Sentence examples for boundary value condition of from inspiring English sources

Thus the associated boundary value condition of (2.15) is as follows: (2.22).

The first aim of our paper is to probe how to give a suitable boundary value condition of (1.1).

By a direct calculation, we get that the solution u satisfies the periodic boundary value condition of problem (2.1).

The most significant feature of the paper is that the definition of the homogeneous boundary value condition of the above equation is given.

By the way, the author has been interested in the boundary value condition of a degenerate parabolic equation for some time, one may refer to [9].

It follows from the intermediate value theorem [11] that there exists a μ ∗ ∈ such that L = 0, that is y ( 1 ) = y μ ∗ ( 1 ) = 0, which satisfies the second boundary value condition of (1.2).

In this article, motivated by the references [15 18], we attempt to establish some sharper Lyapunov-type inequalities for (1.4) under the same boundary value conditions of Theorem 1.1.

A pair of functions ((u(t),v(t))in X=PC(J times PC(J)) is called a solution of (1.1) if ((u(t),v(t))) satisfy all the equations and boundary value conditions of system (1.1).

Moreover, when α → 2 in (1.2), the anti-periodic boundary value conditions in (1.2) are changed into u ( 0 ) = − u ( 1 ), u ′ ( 0 ) = − u ′ ( 1 ), which are coincident with anti-periodic boundary value conditions of second-order differential equations (see [33]).

Proof By Lemma 2.2, L 1 u = D 0 + α u ( t ) = 0 has the solution u ( t ) = c 0 + c 1 t, c 0, c 1 ∈ R. Combining it with the boundary value conditions of BVP (1.1), one has Ker L 1 = { u ∈ X | u = c 1 t, c 1 ∈ R }.

We consider a class of nonlinear fractional differential equations with nonlocal integral boundary value conditions of this form: left { textstylebegin{array}l} D^{alpha}_{0^u(t) +p(t)f t, u(t)) =0,quad 0< t< 1, u(0) =u^{prime}(0) =cdots=u^{ n-2)}(0) =u^{ n-2 =lambda I^{beta}_{0^u(eta) =lambdaint^{eta}_{0}frac {(eta-s)^{beta-1}u(s)}{Gamma(beta)},mathrm{d}s, end{array}displaystyle right.