In the present paper, the concept of nonlinear $F$-contraction formultivalued mappings in metric spaces is introduced and considering the new proof technique, which was used for single valued maps by Wardowski [D. Wardowski, Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory Appl. 2012, 2012:94, 6 pp.], we demonstrate that multivalued nonlinear $F$-contractions of Ćirić type are weakly Picard operators on complete metric spaces. Finally, we give a nontrivial example to guarantee that our result is veritable generalization of recent result of Ćirić [Lj. B. Ćirić, Multi-valued nonlinear contraction mappings, Nonlinear Analysis, 71 (2009), 2716-2723]. Also, we show that many fixed point results in the literature can not be applied to this example.

In this paper, we consider the following first-order Hamiltonian system $$\dot{z} = \mathcal{J} H_{z}(t,z),$$ where $H\in C^{1}(\mathbb{R}\times\mathbb{R}^{2N},\mathbb{R})$ isthe form $H(t,z)=\frac{1}{2}L(t)z\cdot z + R(t,z)$. By applying the variant generalized weak linking theorem for strongly indefinite problem developed by Schechter and Zou, we establish nontrivial and ground state solutions for the above system under conditions weaker than those in [39].

In this paper we prove two sharp inequalities involving the normalized scalar curvature and the generalized normalized $\delta$-Casorati curvatures for slant submanifolds in quaternionic space forms. We also characterize those submanifolds for which the equality cases hold. These results are a generalization of some recent results concerning the Casorati curvature for a slant submanifold in a quaternionic space form obtained by Slesar et al., J. Inequal. Appl., 2014, 2014:123.

Let $(X,d,\mu)$ be a metric measure space satisfying both thegeometrically doubling and the upper doubling measure conditions,which is called non-homogeneous metric measure space. In thispaper, via a sharp maximal operator, the boundedness of commutatorsgenerated by multilinear singular integral with $RBMO(\mu)$ functionon non-homogeneous metric measure spaces in $m$-multiple Lebesgue spacesis obtained.

In the paper, we study the normality of families of meromorphicfunctions related a Hayman Conjecture. We consider whether afamily meromorphic functions $\mathcal{F}$ is normal in $D$, iffor each function $f$ in $\mathcal{F}$, $f' + af^n =b$ has at most one zero, where $n$ is a positive integer, $a$ and $b\neq 0$ are two finite complex numbers. Some examples show that the conditions in our results are best possible.

Based on a special concept of the difference of sets, a new notion ofthe directional derivative of a set-valued map is presented. This theoryis applied to set optimization problems with the known set less orderrelation, and it results in necessary and sufficient optimalityconditions.

In this article we obtain a classification of special Lagrangian submanifolds in complex space forms subject to a pointwise $SO(2)\rtimes S_3$-symmetry on the second fundamental form. The algebraic structure of this form has been obtained by Marianty Ionel in [8]. However, the classification of special Lagrangian submanifolds in $\mathbb{C}^4$ having this $SO(2)\rtimes S_3$ symmetry in [8] is incomplete. In this paper we give a complete classification of such submanifolds, and extend the classification to special Lagrangian submanifolds of arbitrary complex space forms with a pointwise $SO(2)\rtimes S_3$-symmetry in the second fundamental form.

In this paper, we investigate the growth of meromorphic solutions of the equation: $f(z+n)+\sum_{j=0}^{n-1}\{P_{j}(e^{A(z)})+Q_{j}(e^{-A(z)})\}f(z+j)=0$, where $A(z)$, $P_{j}(z)$ and $Q_{j}(z)$ are polynomials in $z$. This article extends earlier results by Li et al [7, 15].

Let $p$ be an odd prime and let $m\not\equiv 0\pmod p$ be a rational p-adic integer. In this paper we reveal the connection between quartic residues and the sum $\sum_{k=0}^{[p/4]}\binom{4k}{2k}\frac 1{m^k}$, where $[x]$ is the greatest integer not exceeding $x$. Let $q$ be a prime of the form $4k+1$ and so $q=a^2+b^2$ with $a,b\in\Bbb Z$. When $p\nmid ab(a^2-b^2)q$, we show that for $r=0,1,2,3$, $$ \begin{array}{ll} & \displaystyle p^{\frac{q-1}4} \equiv \Big(\frac ab\Big)^r\pmod q\\ & \displaystyle\Leftrightarrow \sum_{k=0}^{[p/4]}\binom{4k}{2k}\Big(\frac{a^2}{16q}\Big)^k\equiv(-1)^{\frac{p^2-1}8a+\frac{p-1}2\cdot \frac{q-1}4}\Big(\frac pq\Big) \Big(\frac ab\Big)^r\pmod p, \end{array} $$ where $(\frac pq)$ is the Legendre symbol. We also establish congruences for $\sum_{k=0}^{[p/4]}\binom{4k}{2k}\frac 1{m^k}\pmod p$ in the cases $m=17,18,20,32,52,80,272$.

We apply wavelets to study the generalized local Morrey-Campanato spaces $M_{\phi, p}(\mathbb{R}^{n})$ and their preduals. As applications, we characterize the multipliers on $M_{\phi, p}(\mathbb{R}^{n})$ and the stability of these spaces under the perturbation of Calderón-Zygmund operators. Our results indicate that there exist some $M_{\phi, p}(\mathbb{R}^{n})$ without unconditional basis. This fact shows that $M_{\phi, p}(\mathbb{R}^{n})$ have some different characteristics unlike the classical Morrey spaces.

Given a graph $G=(V,E)$, a function $f$ on $V$ is an $L(3,2,1)$-labeling if for each pair of vertices $u,v$ of $G$, it holds that $|f(u)-f(v)|\ge 4-{\text {dist}}(u,v)$. $L(3,2,1)$-labeling number for $G$, denoted by $\lambda_{3,2,1}(G)$, is the minimum span of all $L(3,2,1)$-labeling $f$ for $G$. In this paper, when $G=K_m\Box C_n$ is the Cartesian product of the complete graph $K_m$ and the cycle $C_n$, we show that the lower bound of $\lambda_{3,2,1}(G)$ is $5m-1$ for $m\ge 3$, and the equality holds if and only if $n$ is a multiple of 5. Moreover, we show that $\lambda_{3,2,1}(K_3\Box C_n)=15$ when $n\ge 28$ and $n\not \equiv 0 \pmod 5$.

Chen conjecture states that every Euclidean biharmonic submanifold is minimal. In this paper we consider the Chen conjecture for $ L_k $-operators. The new conjecture ($ L_k $-conjecture) is formulated as follows: If $ L_k^{2}x=0 $ then $ H_{k+1}=0$ where $ x:M^{n}\rightarrow\Bbb{R}^{n+1} $ is an isometric immersion of a Riemannian manifold $ M^n $ into the Euclidean space $ \Bbb{R}^{n+1} $, $ H_{k+1} $ is the $ (k+1) $-th mean curvature of $ M $, and $ L_k $ is the linearized operator of the $ (k+1) $-th mean curvature of the Euclidean hypersurface $ M $. We prove the $ L_k $-conjecture for the hypersurface $ M $ with at most two principal curvatures.

We study the structure of $0$-primitive near-rings and are able to answer an open question in the theory of minimal ideals in near-rings to the negative, namely if the heart of a zero symmetric subdirectly irreducible near-ring is subdirectly irreducible again. Also, we will be able to classify when a simple near-ring with an identity and containing a minimal left ideal is a Jacobson radical near-ring. Such near-rings are known to exist but have unusual properties. Along the way we prove results on minimal ideals and left ideals in near-rings which so far were known to hold or have been established in the DCCN case, only.

In this paper, the uniqueness problems of entire functions and their difference operators are investigated. It is shown that if a finite order entire function $f$ shares $0,\alpha$ CM with its difference operator $\Delta_\eta f(z)=f(z+\eta)-f(z)$, then $\Delta_\eta f\equiv f$, where $\alpha$ is an entire function with order less than $f$. The research results also include a difference analogue of Brück conjecture, and extend some results in Chen-Yi Results Math., 63 (2013), 557-565).

In this note we show that the problem of characterizing two weight norm inequalities for the Bergman projection under the assumption of doubling measures admits a surprisingly simple solution. Our principal discovery is that Sawyer-type testing can be avoided. This stands in sharp contrast with the current folklore in two weight theory and with the corresponding result for the Hilbert transform.

In this paper we are concerned with the inverse spectral problems for energy-dependent Sturm-Liouville problems (that is, differential pencils) defined on interval $[0,1]$ with two potentials known on a subinterval $ [a_{1},a_{2}]\subset [0,1]$. We prove that the potentials on the entire interval $[0,1]$ and the boundary condition at $x=1$ are uniquely determined in terms of partial knowledge of the spectrum in the situation of $a_{1}=0$ and $a_{2}\geq 1/2$. Furthermore, in the situation of $a_{1}\gt 0$ and $1/2\in [a_{1},a_{2}]$ we need additional information on the eigenfunctions at some interior point to obtain the uniqueness of the potentials on $[0,1]$ and two boundary conditions at $x=0,1$.

Let $R$ be a noncommutative primering with its Utumi ring of quotients $U$, $C=Z(U)$ the extendedcentroid of $R$, $F$ a generalized derivation of $R$ and $I$ anonzero ideal of $R$. Suppose that there exists $0\neq a\in R$ such that $a(F([x,y])^n-[x,y])=0$ for all $x,y \in I$, where $n\geq 1$ is a fixedinteger. Then either $n=1$ and $F(x)=bx$ for all $x\in R$ with$a(b-1)=0$ or $n\geq 2$ and one of the following holds:

1. char $(R)\neq 2$, $R\subseteq M_2(C)$, $F(x)=bx$ for all$x\in R$ with $a(b-1)=0$ (In this case $n$ is an odd integer);

2. char $(R)= 2$, $R\subseteq M_2(C)$ and $F(x)=bx+[c,x]$ forall $x\in R$ with $a(b^n-1)=0$.

Endomorphism rings of modules appear as the center of a ring, as the fix ring of a ring with group action or as the subring of constants of a derivation. This note discusses the question whether certain $*$-prime modules have a prime endomorphism ring. Several conditions are presented that guarantee the primeness of the endomorphism ring. The contours of a possible example of a $*$-prime module whose endomorphism ring is not prime are traced.

We study a differential game of optimal approach of finite or countable number of pursuers with one evader in the Hilbert space $l_{2}$. On control functions of the players integral constraints are imposed. Such constraints arise in modeling the constraint on energy. The duration of the game $\theta$ is fixed. The payoff functional is the greatest lower bound of distances between the pursuers and evader when the game is terminated. The pursuers try to minimize the payoff functional, and the evader tries to maximize it. In this paper, we find formula for the value of the game and construct explicitly optimal strategies of the players. Important point to note is that the energy resource of any pursuer needs not be greater than that of the evader.