Sentence examples for boundary value problems given from inspiring English sources

Here, we have successfully addressed the problem involving the existence and uniqueness of positive and nondecreasing solutions of a family of fractional q-difference boundary value problems given by Eqs.

In a similar manner, we now prove that the transformed boundary value problems given in Theorem 4.1 have eigenvalues, that is, the spectrum increases by one in each case.

The following paragraph needs to be inserted immediately after Theorem 4.2: It is important to note that the spectral parameter in the original boundary value problems given in cases (1 - 9) of Table 1 for Theorem 4.2 must first, without loss of generality, be shifted so as to ensure that all the eigenvalues are greater than zero.

Let us now combine the idea of a non-binary-valued confidence function c with the mixed boundary value problem given in (2).

The boundary value problem given by Eqs.

Consider the boundary value problem given by satisfying (2.5) with the following boundary conditions: (5.2). (5.3).

Consider the boundary value problem given by (1.1) for together with boundary conditions (4.1).

Consequently, we have the following unique solution of the boundary value problem given by Eqs.

This section is devoted to the solvability of the fractional boundary value problem given in (5).

Hence, the boundary value problem given by obeying (2.2) with (4.1) has eigenvalues.

Remark 1 For a i = − 1, i = 0, 1, 2, the boundary value problem given in (5) becomes anti-periodic.