Solutions of three classes of constrained matrix inequalities

In this paper, the following three classes of constrained matrix inequalities are considered: \(\begin{aligned}&G^*X+X^*G\ge H \ (> H), \text{ s.t. } \ AXB=D; \\&G^*X+X^*G\ge H \ (> H), \text{ s.t. }\ AX=C, \ XB=D; \\&{{G}}^*XD+D^*X^*{{G}}\ge H \ (> H), \text{ s.t. }\ AX=B. \end{aligned}\) $$\begin{array}{cc}& G^{\ast }X+X^{\ast }G\ge H(>H),\text{s.t.}AXB=D;\hfill \\ \hfill & G^{\ast }X+X^{\ast }G\ge H(>H),\text{s.t.}AX=C,XB=D;\hfill \\ \hfill & G^{\ast }XD+D^{\ast }{X}^{\ast }G\ge H(>H),\text{s.t.}AX=B.\hfill \end{array}$$ The solvability conditions and the explicit representations for the general solutions are provided when the stated conditions hold.