In this paper, we propose a class of embedded solutions of Einstein’s field equations with the de Sitter cosmological function \(\varLambda (u)\). This class of solutions describes Vaidya-Bonnor-monopole-de Sitter space-time with variable \(\varLambda (u)\). It may also be interpreted as Vaidya-Bonnor black hole in monopole-de Sitter space with variable \(\varLambda (u)\). It is shown the natural modification of Einstein’s field equations with the de Sitter cosmological function \(\varLambda (u)\). In the energy-momentum tensor of the gravitational field in the embedded solution it is also seen the interaction of electromagnetic field with monopole and de Sitter fields having different equations of state parameters. It is also established that the time like vector field of the matter distribution in the embedded space-time geometry is expanding, accelerating and shearing, but non-rotating. We have also discussed the areas, entropies, surface gravities and temperatures for the different horizons for the solution. From the embedded solution we may also recover possible solutions with variable \(\varLambda (u)\) such as (i) Vaidya-Bonnor-de Sitter, (ii) Vaidya-Bonnor-monopole, (iii) Vaidya-monopole, (iv) charged monopole-de Sitter and (v) uncharged monopole-de Sitter. From the study of these exact solutions, it is found that the physical properties of an embedded black hole are depended on the nature of the background spaces. It is also true in the case of the cosmological de Sitter space with variable \(\varLambda (u)\) that the charged monopole-de Sitter and the uncharged monopole-de Sitter spaces have different properties depending upon the charged spaces.

在本文中，我们提出了一类具有 de Sitter 宇宙学函数\(\varLambda (u)\)的爱因斯坦场方程的嵌入解。此类解决方案描述了具有变量\(\varLambda (u)\) 的Vaidya-Bonnor-monopole-de Sitter 时空。它也可以解释为单极-德西特空间中具有变量\(\varLambda (u)\) 的Vaidya-Bonnor 黑洞。它显示了爱因斯坦场方程的自然修正与德西特宇宙学函数\(\varLambda (u)\). 在嵌入解中引力场的能量-动量张量中，还可以看到电磁场与具有不同状态参数方程的单极场和德西特场的相互作用。还确定嵌入的时空几何中物质分布的类似时间的矢量场正在膨胀、加速和剪切，但不旋转。我们还讨论了解决方案不同层级的面积、熵、表面重力和温度。从嵌入式解决方案中，我们还可以使用变量\(\varLambda (u)\)恢复可能的解决方案例如（i）Vaidya-Bonnor-de Sitter，（ii）Vaidya-Bonnor-monopole，（iii）Vaidya-monopole，（iv）带电的monopole-de Sitter和（v）不带电的monopole-de Sitter。通过对这些精确解的研究，发现嵌入黑洞的物理特性取决于背景空间的性质。在具有变量\(\varLambda (u)\)的宇宙学德西特空间的情况下也是如此，带电单极-德西特空间和不带电单极-德西特空间根据带电空间具有不同的性质。