Existence results for a system of nonlinear operator equations and block operator matrices in locally convex spaces

The purpose of this paper is to prove some fixed point results dealing with a system of nonlinear equations defined in an angelic Hausdorff locally convex space ( X , { | ⋅ | p } p ∈ Λ ) (X,\{\lvert\,{\cdot}\,\rvert_{p}\}_{p\in\Lambda}) having the 𝜏-Krein–Šmulian property, where 𝜏 is a weaker Hausdorff locally convex topology of 𝑋. The method applied in our study is connected with a family Φ Λ τ \Phi_{\Lambda}^{\tau} -MNC of measures of weak noncompactness and with the concept of 𝜏-sequential continuity. As a special case, we discuss the existence of solutions for a 2 × 2 2\times 2 block operator matrix with nonlinear inputs. Furthermore, we give an illustrative example for a system of nonlinear integral equations in the space C ⁢ ( R + ) × C ⁢ ( R + ) C(\mathbb{R}^{+})\times C(\mathbb{R}^{+}) to verify the effectiveness and applicability of our main result.

Fixed point theorems in WC-Banach algebras and their applications to infinite systems of integral equations

The paper is devoted to prove a few fixed point theorems for operators acting in WC-Banach algebras and satisfying some conditions expressed in terms of a generalized Lipschitz continuity and measures of weak noncompactness. Moreover, the assumptions imposed on the mentioned operators are formulated with help of weak topology and weak sequential continuity. Our fixed point results will be illustrated by proving the existence of solutions of an infinite system of nonlinear integral equations.

Weakly demicompact linear operators and axiomatic measures of weak noncompactness

In this paper, we study the relationship between the class of weakly demicompact linear operators, introduced in [KRICHEN, B.—O’REGAN, D.: On the class of relatively weakly demicompact nonlinear operators, Fixed Point Theory 19 (2018), 625–630], and measures of weak noncompactness of linear operators with respect to an axiomatic one. Moreover, some Fredholm and perturbation results involving the class of weakly demicompact linear operators are investigated. Our results are then used to investigate the relationship between the relative essential spectrum of the sum of two linear operators and the relative essential spectrum of each of these operators.

Generalized form of fixed point theorems in Banach algebras under weak topology with an application

In this manuscript, by means of the technique of measures of weak noncompactness, we establish a generalized form of fixed point theorems for a 2 x 2 block operator matrix involving multivalued maps acting on suitable Banach algebras. The results obtained are then applied to a coupled system of nonlinear integral equations.

Weak solutions for nonlinear fractional differential equation with fractional separated boundary conditions in Banach spaces

This paper deals with nonlinear fractional differential equation with fractional separated boundary conditions. We investigate the existence of weak solutions in Banach spaces. To obtain such result we apply an appropriate fixed point theorem and the technique of measures of weak noncompactness. An example illustrating the theory is given.