In this article, we present various compactness and minimizations from very classical theorems such as Bolzano–Weierstrass theorem to modern theorems such as Palais–Smale conditions. Then apply them to assert the existence and multiplicity of solutions of partial differential equations.

In a Five-volume work published recently, K. Nishimoto [1] has presented a systematic account of the theory and applications of fractional calculus in a number of areas (such as ordinary and partial differential equations, special functions, and summation of series). In 2001, K. Nishimoto, D.-K. Chyan, S.-D. Lin and S.-T. Tu [11] derived the following interesting families of infinite series via fractional calculus, $$ \displaystyle\sum_{k=2}^{\infty}\ \frac{(-c)^k}{k(k-1)}\frac{(kz-c)}{(z-c)^{k-1}}=c^2\ \biggr(\biggr|\displaystyle\frac{-c}{z-c}\biggr|\lt 1\biggr).$$ The object of the present paper is to extend the above families of infinite series to more general closed form relations. Various numerical results are also provided.

In several recent works, many different families of infinite series were evaluated by applying certain operators of fractional calculus (that is, calculus of derivatives and integrals of any arbitrary real or complex order). In the present sequel to some of these recent investigations, it is observed that much more general classes of infinite sums can be derived without using fractional calculus. Some other related evaluations of finite and infinite sums are also considered.

In this paper we consider the relation ²* defined on an Hv-moduleM and we interprete the lower and upper approximations as subsets of themodule M=²* and we give some results in this connection.

We prove that a generic hypersurface of general type in ${\Bbb P}^n$ for $n\ge 3$ does not contain a reduced irreducible divisor which admits a desingularization having nef anticanonical bundle.

In this paper, we study $(d, 2)$-dominating numbers for an important class of parallel networks - butterfly networks $B(n)$. The main result of this paper is to determine their $(d, 2)$-dominating numbers for $2n-1\le d\le 2n+1$.

We prove the Hyers-Ulam-Rassias stability of the Jensen’s equation in Banach modules over a Banach algebra.

Let $s_{mn}(x,y)$ denote the rectangular partial sums of the double Laguerre series with the coefficients $\{c_{jk}\}$. We give sufficient conditions on $\{c_{jk}\}$ to obtain the regular convergence and weighted $L^r$-convergence of $s_{mn}(x,y)$.

A solvability theorem is obtained for a quasilinear elliptic boundary value problem. The linear part of the problem is an elliptic operator of order 2m which has a nontrivial kernel not necessarily symmetric. The nonlinear part may grow sublinearly and contain derivatives of order up to 2m. The proof is based on Borsuk’s Theorem and the Nussbaum–Sodovskii degree.

In this paper, the unique positive solution of the nonlinear elliptic equation $¢ _pu + |u|^{qi2}u=0$, where $p \neq q$, is described and its behaviors relative to certain limiting conditions on $p$ and $q$ are discussed.

We determine two types of generalized exponent sets for tournaments with given order. In the course of proving the main results we find the following result, which may be interesting in its own right: When $n$ is large enough, almost all tournaments on $n$ vertices have the property that there is a path of length $2$ from each vertex $u$ to each vertex $v \ne u$.

In this note we discuss the Karatsuba conjectures on lower bound estimates for the maximum modulus of the Riemann Zeta-Function over very short intervals of the critical line and in small domains of the critical strip.