In this paper, we establish some new variants of Leray–Schauder-type fixed point theorems for a
2
×
2
block operator matrix defined on nonempty, closed, and convex subsets
Ω
of Banach spaces. Note here that
Ω
need not be bounded. These results are formulated in terms of weak sequential continuity and the technique of De Blasi measure of weak noncompactness on countably subsets. We will also prove the existence of solutions for a coupled system of nonlinear equations with an example.
Let ℋ be a Krein space with fundamental symmetry J. Starting with a canonical block-operator matrix representation of J, we study the regular subspaces of ℋ. We also present block-operator matrix representations of the J-self-adjoint projections for the regular subspaces of ℋ, as well as for the regular complements of the isotropic part in a pseudo-regular subspace of ℋ.
An element a in a Banach algebra A has p-Drazin inverse provided that there
exists b ? comm(a) such that b = b2a,ak-ak+1b?J(A) for some k ? N. In
this paper, we present new conditions for a block operator matrix to have
p-Drazin inverse. As applications, we prove the p-Drazin invertibility of
the block operator matrix under certain spectral conditions.
In this paper, we will establish some results on perturbation theory of block
operator matrices acting on Xn, where X is a Banach space. These results are
exploited to investigate the M-essential spectra of a general class of
operators defined by a 3x3 block operator matrix acting on a product of
Banach spaces X3.
Let A, B and be closed range operators. The explicit matrix expressions
for various generalized inverses are obtained by using block operator matrix
methods. Some subtle relationships between the properties of sub-blocks in
operator matrices A, B and their range relations are built. New necessary
and sufficient conditions for the equivalent relations, inclusion relations
and mixed-type generalized inverses relations are presented. Some recent
mixed-type reverse-order laws results are covered and many new mixed-type
generalized inverses relations are established by using this block-operator
matrix technique.
In this paper, we study the existence of solutions for a system of quadratic
integral equations of Chandrasekhar type by applying fixed point theorem of
a 2 x 2 block operator matrix defined on a nonempty bounded closed convex
subsets of Banach algebras where the entries are nonlinear operators.
Leray–Schauder Fixed Point Theorems for Block Operator Matrix with an Application
