In this paper, we study the solvability of the operator equations A * X + X * A = C and A * XB + B * X * A = C for general adjointable operators on Hilbert C * -modules whose ranges may not be closed. Based on these results we discuss the solution to the operator equation AXB = C, and obtain some necessary and sufficient conditions for the existence of a real positive solution, of a solution X with B * (X * + X)B ≥ 0, and of a solution X with B * XB ≥ 0. Furthermore in the special case that R(B) ⊆ R(A * ) we obtain a necessary and sufficient condition for the existence of a positive solution to the equation AXB = C. The above results generalize some recent results concerning the equations for operators with closed ranges. Mathematics Subject Classification (2000). Primary 46L08; Secondary 47A05.

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In this paper, we study the existence of weak solutions for differential equations of divergence formin coupled with a Dirichlet or Neumann boundary condition in separable Musielak-Orlicz-Sobolev spaces where a 1 satisfies the growth condition, the coercive condition, and the monotone condition, and a 0 satisfies the growth condition without any coercive condition or monotone condition. The right-hand side f : × R × R N → R is a Carathéodory function satisfying a growth condition dependent on the solution u and its gradient Du. We prove the existence of weak solutions by using a linear functional analysis method. Some sufficient conditions guarantee the existence enclosure of weak solutions between sub-and supersolutions. Our method does not require any reflexivity of the Musielak-Orlicz-Sobolev spaces.

We prove the existence results in the setting of Orlicz spaces for the following nonlinear elliptic equation:

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It is proved that for adjointable operators A and B between Hilbert C * -modules, certain majorization conditions are always equivalent without any assumptions on R(A * ), where A * denotes the adjoint operator of A and R(A * ) is the norm closure of the range of A * . In the case that R(A * ) is not orthogonally complemented, it is proved that there always exists an adjointable operator B whose range is contained in that of A, whereas the associated equation AX = B for adjointable operators is unsolvable.2010 Mathematics Subject Classification. 46L08, 47A05.