For functions belonging to the generalization class $\mathcal{R}_{p}(k,\alpha ,\lambda )$ of analytic functions in the open unit disc $\mathbb{U}=\{z:\,\left| z\right| \lt 1\}$, which investigated in this paper, some applications of differential subordination are obtained which contain interesting property of Hurwitz-Lerch Zeta function.

The exact distributions of the product $XY$ and the ratio $X/Y$ are derived when $(X, Y)$ has the elliptically symmetric Pearson type II distribution, elliptically symmetric Pearson type VII distribution and the elliptically symmetric Kotz type distribution.

In this paper, the isomorphic path decompositions of $\lambda$-fold balanced complete tripartite graphs $\lambda K_{n,n,n}$ and $\lambda$-fold balanced complete tripartite digraphs $\lambda K^*_{n,n,n}$ are investigated for odd $n$. We prove that the obvious necessary conditions for such decompositions in the undirected case are also sufficient; we also provide sufficient conditions for the directed case.

In this paper, a class of $\eta$-PPM mappings $F$ where $F$ and $-F$ are both $\eta$-pseudomonotone are introduced, which are proper generalizations of the PPM mappings considered by Bianchi and Schaible. The solution sets of two kinds of vector variational-like inequality problems involving $\eta$-PPM mappings are characterized in Banach spaces. Furthermore, the solution sets of two classes of vector variational-like inequalities involving set-valued mappings are also characterized via the scalarization approach due to Konnov.

Let $G$ be a simple graph. If we place $p$ pebbles on the vertices of $G$, then a pebbling move is taking two pebbles off one vertex and then placing one on an adjacent vertex. The optimal pebbling number of $G$, $f'(G)$, is the least positive integer $p$ such that $p$ pebbles are placed suitably on vertices of $G$ and for any target vertex $v$ of $G$, we can move one pebble to $v$ by a sequence of pebbling moves. In this paper, we find the optimal pebbling number of the caterpillars.

In this paper, we present discrete fixed point theorems. They are based on monotonicity of the mapping. We apply them to a non-cooperative n-person game and give an existence theorem of a Nash equilibrium of pure strategies. As a special case, we consider bimatrix games.

The necessary and sufficient conditions for the existence of a subgroup of finite index in a Fuchsian group $\Gamma = \prod^{\ast n}_{j=1} {\bf Z}_{p_j}$, where each $p_j \geq 2$, are the Riemann-Hurwitz and diophantine conditions. For torsion free subgroups of index $d$ in $\Gamma$, the diophantine condition reduces to the one that $d$ is divisible by each $p_j$. The purpose of this paper is to study the realizability problem of torsion free subgroups of finite index with given signatures in $\Gamma$. In general, the Riemann-Hurwitz and diophantine conditions are not sufficient for our realizability problem if $n \geq 3$. An additional necessary end-condition for the existence of a subgroup $\Phi$ of index $d$ in $\Gamma$ is that the number $t$ of punctures of the Riemann surface ${\bf H^2} / \Phi$ is at most $d$. A major goal is to completely determine all possible $t \leq d$. Such signatures can always be realized under the Riemann-Hurwitz, diophantine and certain end conditions.

Let K be a weakly compact, convex subset of a Banach space X with normal structure. Brodskii and Milman proved that there exists a point p ∈ K which is fixed under all isometries of K onto K. Suppose now WCC(X) is the collection of all non-empty weakly compact convex subsets of X. We shall define a certain weak topology Tw on WCC(X) and have the above-mentioned result extended to the hyperspace (WCC(X), Tw)

We prove that suitable wavelets and scaling functions give characterizations and unconditional bases of the weighted Sobolev space $L^{p,s}(w)$ with $A_p$ or $A_p^{\mathop{\mathrm{loc}}}$ weights. In the case of $w \in A_p$, we use only wavelets with proper regularity. Meanwhile, if we assume $w \in A_p^{\mathop{\mathrm{loc}}}$, not only compactly supported $C^{s+1}$-wavelets but also compactly supported $C^{s+1}$-scaling functions come into play. We also establish that our bases are greedy for $L^{p,s}(w)$ after normalization.

We will say that the permutations $f_1,...,f_n$ are an $\epsilon$-solution of an equation if the normalized Hamming distance between its l.h.p. and r.h.p. is $\leq\epsilon$. We give a sufficient conditions when near to an $\epsilon$-solution exists an exact solution and some examples when there does not exist such a solution.

In this paper, we establish some coincidence theorems, generalized variational inequality theorems and minimax inequality theorems for the family $KKM^{∗}(X, Y)$ and the generalized Φ-mapping on a nonconvex set.

In this paper we derive by means of the duality theory necessary and sufficient optimality conditions for convex optimization problems having as objective function the composition of a convex function with a linear mapping defined on a finite-dimensional space with values in a Hausdorff locally convex space. We use the general results for deriving optimality conditions for two portfolio optimization problems having as objective functions different convex deviation measures.

Let $X$ be a Banach space which is continuously embedded in another Banach space $Y$ and is an invariant subspace for an $(a,k)$-regularized resolvent family $R(\cdot)$ of operators on $Y$. It is shown that the restriction of $R(\cdot)$ to $X$ is strongly continuous with respect to the norm of $X$ if and only if all its partial orbits are relatively weakly compact in $X$. This property is shared by many particular cases of $(a,k)$-regularized resolvent families, such as integrated solution families, integrated semigroups, and integrated cosine functions.

We consider the initial boundary value problem for a Petrovsky system with nonlinear damping \begin{equation*} u_{tt}+\Delta ^{2}u+a\left| u_{t}\right| ^{m-2}u_{t}=b\left| u\right| ^{p-2}u, \end{equation*} in a bounded domain. We showed that the solution is global in time under some conditions without the relation between $m$ and $p$. We also prove that the local solution blows-up in finite time if $p\gt m$ and the initial energy is nonngeative. The decay estimates of the energy function and the estimates of the lifespan of solutions are given. In this way, we can extend the result of ([6]).

In this article, we consider the semilinear elliptic equation $$(*)_\lambda \,\,\,\,\,\,\, -\Delta u+u=\lambda K(x)u^p+h(x)\mbox{ in }\Omega, u\gt 0\mbox{ in }\Omega\mbox{ }, u\in H_0^1(\Omega ), $$ where $\lambda \geq 0,$ $N\ge 2$, $1\lambda^*$. We also establish the asymptotic behavior and some further properties of positive solutions of equation $(*)_\lambda$.

A convergence criterion of the family of Euler-Halley type methods for the vector fields on Riemannian manifolds whose covariant derivatives satisfy the $\gamma$-condition is established. The corresponding results due to [12] are extended. An application to analytic vector fields is provided.

Let $\{S_i\}^N_{i=1}$ be $N$ strict pseudo-contractions defined on a nonempty closed convex subset $C$ of a real Hilbert space $H$. Consider the problem of finding a common element of the set of common fixed points of these mappings $\{S_i\}^N_{i=1}$ and the set of solutions of the variational inequality for a monotone Lipschitz continuous mapping of $C$ into $H$, and consider the parallel-extragradient and cyclic-extragradient algorithms for solving this problem. We will derive the weak convergence of these algorithms. Moreover, these weak convergence results will be applied to finding a common zero point of a finite family of maximal monotone mappings. Further we prove that these algorithms can be modified to have strong convergence by virtue of additional projections. Our results represent the improvement, generalization and development of the previously known results in the literature.