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.

$$\omega ^\omega $$-Base and infinite-dimensional compact sets in locally convex spaces

A locally convex space (lcs) E is said to have an $$\omega ^{\omega }$$ ω ω -base if E has a neighborhood base $$\{U_{\alpha }:\alpha \in \omega ^\omega \}$$ { U α : α ∈ ω ω } at zero such that $$U_{\beta }\subseteq U_{\alpha }$$ U β ⊆ U α for all $$\alpha \le \beta $$ α ≤ β . The class of lcs with an $$\omega ^{\omega }$$ ω ω -base is large, among others contains all (LM)-spaces (hence (LF)-spaces), strong duals of distinguished Fréchet lcs (hence spaces of distributions $$D^{\prime }(\Omega )$$ D ′ ( Ω ) ). A remarkable result of Cascales-Orihuela states that every compact set in an lcs with an $$\omega ^{\omega }$$ ω ω -base is metrizable. Our main result shows that every uncountable-dimensional lcs with an $$\omega ^{\omega }$$ ω ω -base contains an infinite-dimensional metrizable compact subset. On the other hand, the countable-dimensional vector space $$\varphi $$ φ endowed with the finest locally convex topology has an $$\omega ^\omega $$ ω ω -base but contains no infinite-dimensional compact subsets. It turns out that $$\varphi $$ φ is a unique infinite-dimensional locally convex space which is a $$k_{\mathbb {R}}$$ k R -space containing no infinite-dimensional compact subsets. Applications to spaces $$C_{p}(X)$$ C p ( X ) are provided.

The Fréchet Schwartz Algebra of Uniformly Convergent Dirichlet Series

The algebra of all Dirichlet series that are uniformly convergent in the half-plane of complex numbers with positive real part is investigated. When it is endowed with its natural locally convex topology, it is a non-nuclear Fréchet Schwartz space with basis. Moreover, it is a locally multiplicative algebra but not aQ-algebra. Composition operators on this space are also studied.

A CONVOLUTION-INDUCED TOPOLOGY ON THE ORLICZ SPACE OF A LOCALLY COMPACT GROUP

Let $G$ be a locally compact group with a fixed left Haar measure. In this paper, given a strictly positive Young function ${\rm\Phi}$, we consider $L^{{\rm\Phi}}(G)$ as a Banach left $L^{1}(G)$-module. Then we equip $L^{{\rm\Phi}}(G)$ with the strict topology induced by $L^{1}(G)$ in the sense of Sentilles and Taylor. Some properties of this locally convex topology and a comparison with weak$^{\ast }$, bounded weak$^{\ast }$ and norm topologies are presented.

Krein-Smulian-Type Theorems

Let (E, ℱ) be a weakly compactly generated Frechet space and let ℱ0 be another weaker Hausdorff locally convex topology on E. Let X be an ℱ-bounded compact subset of (E, ℱ0). The ℱ0-closed convex hull of X in E is then ℱ0-compact. We also give a new proof, without using Riemann–Lebesgue-integrable (Birkoff-integrable) functions, with the result that if (E, ∥ · ∥) is any Banach space and ℱ0 is fragmented by ∥ · ∥, then the same result holds. Furthermore, the closure of the convex hull of X in ℱ0-topology and in the original topology of E is the same.

HYPERGROUP ALGEBRAS AS TOPOLOGICAL ALGEBRAS

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}K$ be a locally compact hypergroup endowed with a left Haar measure and let $L^1(K)$ be the usual Lebesgue space of $K$ with respect to the left Haar measure. We investigate some properties of $L^1(K)$ under a locally convex topology $\beta ^1$. Among other things, the semireflexivity of $(L^1(K), \beta ^1)$ and of sequentially$\beta ^1$-continuous functionals is studied. We also show that $(L^1(K), \beta ^1)$ with the convolution multiplication is always a complete semitopological algebra, whereas it is a topological algebra if and only if $K$ is compact.

STRICT TOPOLOGY AS A MIXED TOPOLOGY ON LEBESGUE SPACES

LetXbe a locally compact space, and 𝔏∞0(X,ι) be the space of all essentially boundedι-measurable functionsfonXvanishing at infinity. We introduce and study a locally convex topologyβ1(X,ι) on the Lebesgue space 𝔏1(X,ι) such that the strong dual of (𝔏1(X,ι),β1(X,ι)) can be identified with$({\frak L}_0^\infty (X,\iota ),\|\cdot \|_\infty )$. Next, by showing thatβ1(X,ι) can be considered as a natural mixed topology, we deduce some of its basic properties. Finally, as an application, we prove thatL1(G) , the group algebra of a locally compact Hausdorff topological groupG, equipped with the convolution multiplication is a complete semitopological algebra under theβ1(G) topology.

On the Generalized Weighted Lebesgue Spaces of Locally Compact Groups

Let be a locally compact group with a fixed left Haar measure and be a system of weights on . In this paper, we deal with locally convex space equipped with the locally convex topology generated by the family of norms . We study various algebraic and topological properties of the locally convex space . In particular, we characterize its dual space and show that it is a semireflexive space. Finally, we give some conditions under which with the convolution multiplication is a topological algebra and then characterize its closed ideals and its spectrum.

THE STRICT TOPOLOGY ON THE DISCRETE LEBESGUE SPACES

Let Σ be a set and σ be a positive function on Σ. We introduce and study a locally convex topology β1(Σ,σ) on the space ℓ1(Σ,σ) such that the strong dual of (ℓ1(Σ,σ),β1(Σ,σ)) can be identified with the Banach space $(c_0(\Sigma ,1/\sigma ),\|\cdot \|_{\infty ,\sigma })$. We also show that, except for the case where Σ is finite, there are infinitely many such locally convex topologies on ℓ1(Σ,σ). Finally, we investigate some other properties of the locally convex space (ℓ1(Σ,σ),β1(Σ,σ)) , and as an application, we answer partially a question raised by A. I. Singh [‘L∞0(G)* as the second dual of the group algebra L1 (G) with a locally convex topology’, Michigan Math. J.46 (1999), 143–150].