The main solutions in the sense of Kantorovich of nonlinear Volterra operator-integral equations are constructed. Convergence of the successive approximation method is established through studies of the majorant integral equations and the majorant algebraic equations. Estimates are derived for the solutions and for the intervals on the right margin of which the solution of nonlinear Volterra operator-integral equation has blow-up or solution start branching.

In this paper, we introduce a new hybrid projection algorithm based on the shrinking projection methods for two hemi-relatively nonexpansive mappings. Using the new algorithm, we prove some strong convergence theorems for finding a common element in the fixed points set of two hemi-relatively nonexpansive mappings, the solutions set of a variational inequality and the solutions set of an equilibrium problem in a uniformly convex and uniformly smooth Banach space. Furthermore, we apply our results to finding zeros of maximal monotone operators. Our results extend and improve the recent ones announced by Li [J. Math. Anal. Appl. 295 (2004) 115--126], Fan [J. Math. Anal. Appl. 337 (2008) 1041--1047], Liu [J. Glob. Optim. 46 (2010) 319--329], Kamraksa and Wangkeeree [J. Appl. Math. Comput. DOI: 10.1007/s12190-010-0427-2] and many others.

In the paper, necessary and sufficient conditions are presented for a function involving a ratio of gamma functions to be logarithmically completely monotonic. This extends and generalizes the main result of Guo and Qi [Taiwanese J. Math. 7 (2003), no. 2, 239--247] and others. As applications, several inequalities involving the volume of the unit ball in $\mathbb{R}^n$ are derived, which refine, generalize and extend some known inequalities.

We discuss properties of pseudospectrum and condition spectrum of an element in a complex unital Banach algebra and its $\epsilon$-perturbation. Several results are proved about linear maps preserving pseudospectrum/ condition spectrum. These include the following: (1) Let $A, B$ be complex unital Banach algebras and $\epsilon$ is positive. Let $\Phi: A\rightarrow B$ be an $\epsilon$-pseudospectrum preserving linear onto map. Then $\Phi$ preserves spectrum. If $A$ and $B$ are uniform algebras, then, $\Phi$ is an isometric isomorphism. (2) Let $A, B$ be uniform algebras and $\epsilon \in (0,1)$. Let $\Phi:A\rightarrow B$ be an $\epsilon$-condition spectrum preserving linear map. Then $\Phi$ is an $\epsilon^{'}$-almost multiplicative map, where $\epsilon, \epsilon^{'}$ tend to zero simultaneously.

Consider an operator $T:E \to X(\mu)$ from a Banach space $E$ to a Banach function space $X(\mu)$ over a finite measure $\mu$ such that its dual map is $p$-th power factorable. We compute the optimal range of $T$ that is defined to be the smallest Banach function space such that the range of $T$ lies in it and the restricted operator has $p$-th power factorable adjoint. For the case $p=1$, the requirement on $T$ is just continuity, so our results give in this case the optimal range for a continuous operator. We give examples from classical and harmonic analysis, as convolution operators, Hardy type operators and the Volterra operator.

In this paper we obtain existence and uniqueness results for quadruple fixed points of operators $F: X^4 \rightarrow X$. We also give some examples to support our results.

In this work we deals with almost automorphic behavior of solutions of a class of semilinear evolution equations. To achieve our goal we use interpolation theory and fixed point theory. As application, we examine sufficient conditions for existence of almost automorphic solutions of equations of the heat conduction theory.

We prove the existence of the unique common fixed point theorems of a pair of weakly compatible mappings satisfying $\Phi$-maps in $G$-metric spaces. These results generalize the well-known results in the literature.

Let $X$ and $Y$ be complete metric spaces of analytic functions over the unit disk in the complex plane. A linear operator $T: X \to Y$ is a bounded operator with respect to metric balls if $T$ takes every metric ball in $X$ into a metric ball in $Y$. We also say that $T$ is metrically compact if it takes every metric ball in $X$ into a relatively compact subset in $Y$. In this paper we will consider these properties for composition operators from Nevanlinna type spaces to Bloch type spaces.

Spectral synthesis deals with the description of translation invariant function spaces. It turns out that the basic building blocks of this description are the exponential monomials, which are built up from exponential functions and polynomial functions. The author collaborated with Laczkovich [Math. Proc. Cambridge Philos. Soc. 143 (2007), no. 1, 103--120] proved that spectral synthesis holds on an Abelian group if and only if the torsion free rank of the group is finite. The author [Aequationes Math. 70 (2005), no. 1-2, 122--130] showed that the torsion free rank of an Abelian group is strongly related to the properties of polynomial functions on the group. Here we show that spectral synthesis holds on an Abelian group if and only if the ring of polynomial functions on the group is Noetherian.

Given an inclusion $N\subset M$ of $\rm{II}_{1}$ factors with trivial relative commutant, this paper lists all operators $x,y\in M$ such that the left $N$-module generated by $x$ is equal to or contained in the right $N$-module generated by $y$.

We prove that for a separable Hilbert space $\mathcal{H}$ with an orthonormal basis $\{e_i\}_{i=1}^\infty$, the equality $\|\cdot\| =\|\sum_{i=1}^{\infty}s_i(\cdot)e_i\otimes e_i \|$ holds for all unitarily invariant norms on $\mathbb{B}(\mathcal{H})$ and Ky Fan's dominance theorem remains valid on $\mathbb{B}(\mathcal{H})$.

The study of operators satisfying Bishop's property $(\beta)$ is of significant interest and is currently being done by a number of mathematicians around the world. Recently Uchiyama and Tanahashi [Oper. Matrices 4 (2009), 517--524] showed that a paranormal operator has Bishop's property $(\beta)$. In this paper we introduce a new class of operators which we call the class of $k$-quasi-paranormal operators. An operator $T$ is said to be a $k$-quasi-paranormal operator if it satisfies $||T^{k+1}x||^{2}\leq||T^{k+2}x|||T^{k}x||$ for all $x\in H$ where k is a natural number. This class of operators contains the class of paranormal operators and the class of quasi-class $A$ operators. We prove basic properties and give a structure theorem of $k$-quasi-paranormal operators. We also show that Bishop's property $(\beta)$ holds for this class of operators. Finally, we prove that if $E$ is the Riesz idempotent for a nonzero isolated point $\lambda_{0}$ of the spectrum of a $k$-quasi-paranormal operator $T$, then $E$ is self-adjoint if and only if the null space of $T-\lambda_{0},\, \ker(T-\lambda_{0})\subseteq \ker(T^{*}-\overline{\lambda_{0}})$.

We provide a sufficient condition for strong (Arens) irregularity of certain bounded bilinear maps, which applies in particular to the adjoint of Banach module actions. We then apply our result to improve several known results concerning to the relation between Arens regularity of certain Banach module actions and reflexivity.

In this paper we consider the generalized Walsh system and a problem $L^1$- convergence of greedy algorithm of functions after changing the values on small set.