We study an anisotropic problem involving variable exponent growth conditions on a bounded domain $\Omega \subset \mathbb{R}^N$. We prove the existence of at least two nontrivial weak solutions using as main tool a result due to Ricceri.

In this paper, some existence results for optimal control problems governed by abstract quasi-variational inequalities are proved in reflexive Banach spaces. As an application, an existence of the optimal control for the bilateral obstacle optimal control problem is also given under some suitable conditions, in which the state satisfies a quasilinear elliptic variational inequality with a source term.

In this paper, we consider a problem of characterizing distribution through the constructive property of random sum $p S_N$, where $0 \lt p \lt 1$ and $N \geq 0$ is an integer-valued random variable. This problem will be solved under someregular conditions. We extend the characterization of exponential distribution to ageneral case. For example, the gamma distribution, the positive Linnik distributionand the scale mixture of stable distribution are characterized. Two new results inthe vein are obtained. Finally, the problem of characterizing distribution by theproperty of the first order statistics is also investigated.

We study the bifurcation diagrams of classical positive solutions $u$ with $\lVert u \rVert_{\infty} \in (0,\infty)$ of the $p$-Laplacian Dirichlet problem $$\left\{\begin{array}{ll} \varphi_{p}(u'(x)))' + \lambda f_{q,r}(u(x)) = 0,\ \ \ -1 \lt x \lt 1, \\ u(-1) = 0 = u(1), \end{array}\right.$$ where $p \gt 1$, $\varphi_{p}(y) = |y|_{p-2}$, $(\varphi(u'))'$ is the one dimensional $p$-Laplacian, $\lambda \gt 0$ is a bifurcation parameter, and $$f_{q,r}(u) = \left\{\begin{array}{ll} \mid 1-u \mid^q,\ \ \ \textrm{if } 0 \lt u \leq 1, \\ \mid 1-u \mid^r,\ \ \ \textrm{if } u \gt 0, \end{array}\right.$$ with positive constants $q$ and $r$. We give explicit formulas of bifurcation curves of classical positive solutions on the $\left( \lambda ,\left\Vert u\right\Vert _{\infty }\right) $-plane. More importantly, for different $(p,q,r)$, we give a complete classification of all bifurcation diagrams. Hence we are able to determine the (exact) multiplicity of classical positive solutions for each $(p,q,r,\lambda )$. Our results generalize the results of Lee et,al [J. Math. Anal. Appl., 330 (2007), 276-290] with nonlinearity $f_{q,r}$ generalized from $q = r \gt 0$ to $q,r \gt 0$.

In this paper, we initiate the study of ${\mathcal{P}} R$-warped products in para-Kähler manifolds and prove some fundamental results on such submanifolds. In particular, we establish a general optimal inequality for $\mathcal{P} R$-warped products in para-Kähler manifolds involving only the warping function and the second fundamental form. Moreover, we completely classify $\mathcal{P} R$-warped products in the flat para-Kähler manifold with least codimension which satisfy the equality case of the inequality. Our results provide an answer to the Open Problem (3) proposed in [19, Section 5].

In this paper, we suggest and analyze a new method which couple extra-gradient methods with KM's methods for solving some variational inequality problem and fixed points problem. It is shown that the proposed method has strong convergence in a general Hilbert space.

We study a nonlinear periodic problem driven by the scalar $p$-Laplacian. The reaction term is a Carathéodory function $f(t,x)$ which satisfies only a unilateral growth condition in the $x$-variable. Assuming strict monotonicity for the quotient $f(t,x)\big/x^{p-1}$ and using variational methods coupled with suitable truncation techniques, we produce necessary and sufficient conditions for the existence and uniqueness of positive solutions.

In this paper, a certain Herz-type Hardy spaces with variable exponent are introduced, and characterizations of these spaces are established in terms of atomic and molecular decompositions. Using these decompositions, the authors obtain the boundedness of some operators on the Herz-type Hardy spaces with variable exponent.

Consider the nonlinear matrix equation $X+A^{*}X^{-q}A = Q$ where $0 \lt q \leq 1$. A new sufficient condition for this equation to have positive definite solution is provided and two iterative methods for the maximal positive definite solution are proposed. Applying the theory of condition number developed by Rice, an explicit expression of the condition number of the maximal positive definite solution is obtained. The theoretical results are illustrated by numerical examples.

Let $\mu$ be a non-negative Borel measure on $\mathbb{R}^d$ which only satisfies some growth condition, we study two-weight norm inequalities for fractional maximal functions associated to such $\mu$. A necessary and sufficient condition for the maximal operator to be bounded from $L^p(v)$ into weak $L^{q}(u)$ $(1 \leq p \leq q \lt \infty)$ is given. Furthermore, by using certain Orlicz norm, a strong type inequality is obtained.

An interval-type computational procedure by combining the parametric method and discretization approach is proposed in this paper to solve a class of continuous-time linear fractional programming problems (CLFP). Using the different step sizes of discretization problems, we construct a sequence of convex, piecewise linear and strictly decreasing upper and lower bound functions. The zeros of upper and lower bound functions then determine a sequence of intervals shrinking to the optimal value of (CLFP) as the size of discretization getting larger. By using the intervals we can find corresponding approximate solutions to (CLFP). We also establish upper bounds of lengths of these intervals, and thereby we can determine the size of discretization in advance such that the accuracy of the corresponding approximate solution can be controlled within the predefined error tolerance. Moreover, we prove that the searched sequence of approximate solution functions weakly-star converges to an optimal solution of (CLFP). Finally, we provide some numerical examples to implement our proposed method.

In this paper, by using the generalization of Ljusternik theorem, the open mapping theorem of convex process, and the convex sets separation theorem, we give the necessary conditions for the efficient solution to the constrained vector equilibrium problems without requiring that the ordering cone in the objective space has a nonempty interior and without requiring that the the convexity conditions. By the assumption of the convexity, we give the sufficient conditions for the efficient solution. As applications, we give the necessary conditions and the sufficient conditions for efficient solution to the constrained vector variational inequalities and constrained vector optimization problems.

Ruled submanifolds of finite type in Lorentz-Minkowski space are studied. We construct a new example of ruled submanifolds with degenerate rulings called a $BS$-kind ruled submanifold, which is of finite type. Also, it is determined by the restricted minimal polynomial of the shape operator associated with the mean curvature vector field.

We introduce implicit and explicit iterative algorithms for the construction of fixed points of pseudocontractions $T$ in Hilbert spaces. We prove that the proposed iterative algorithms converge strongly to the minimum-norm fixed point of $T$. Moreover we show that some of the existing iterative algorithms for nonexpansive mappings fail to converge when applied to pseudocontractions.

In this article, we study the boundedness of matrix operators acting on weighted sequence Besov spaces $\dot{b}_{p,w}^{\alpha,q}$. First we obtain the necessary and sufficient condition for the boundedness of diagonal matrices acting on weighted sequence Besov space $\dot{b}_{p,w}^{\alpha,q}$, and investigate the duals of $\dot{b}_{p,w}^{\alpha,q}$, where the weight is non-negative and locally integrable. In particular, when $0 \lt p \lt 1$, we find a type of new sequence sapces which characterize the dual space of $\dot{b}^{\alpha,q}_{p,w}$.

We also use the duals of $\dot{b}_{p,w}^{\alpha,q}$ to characterize an algebra of matrix operators acting on weighted sequence Besov spaces $\dot{b}_{p,w}^{\alpha,q}$ and find the necessary and sufficient conditions to such a characterization. Note that we do not require that the given weight satisfies the doubling condition in this situation.

Using these results, we give some applications to characterize the boundedness of Fourier-Haar multipliers and paraproduct operators. In this situation, we need to require that the weight $w$ is an $A_p$ weight.

Let $C$ be a nonempty closed convex subset of a real Hilbert space $H$. Assume that $F: C \to H$ is a $\kappa$-Lipschitzian and $\eta$-strongly monotone operator with constants $\kappa,\eta \gt 0$, $f: C \to H$ is $L$-Lipschitzian with constant $L \geq 0$ and $T,V: C \to C$ are nonexpansive mappings with ${\rm Fix}(T) \neq \emptyset$. Let $0 \lt \mu \lt 2 \eta/\kappa^2$ and $0 \leq \gamma L \lt \tau$, where $\tau = 1 - \sqrt{1-\mu(2\eta-\mu\kappa^2)}$. Consider the hierarchical monotone variational inequality problem (in short, HMVIP):

VI (a): finding $z^* \in {\rm Fix}(T)$ such that $\langle(I-V)z^*, z-z^*\rangle \geq 0$, $\forall z \in {\rm Fix}(T)$;

VI (b): finding $x^* \in S$ such that $\langle(\mu F - \gamma f) x^*, x-x^*\rangle \geq 0$, $\forall z \in S$.

Here $S$ denotes the nonempty solution set of the VI (a). This paper combines hybrid steepest-descent method, viscosity method and projection method to design an explicit algorithm, that can be used to find the unique solution of the HMVIP. Strong convergence of the algorithm is proved under very mild conditions. Applications in hierarchical minimization problems are also included.

We investigate the $L^p$ boundedness for a class of singular integral operators associated to submanifolds, including surfaces of revolution, under the $L(\log L)(S^{n-1})$ or Block space condition on the kernel functions. Our results improve known results.