This paper deals with a nonlinear $p$-Laplacian equation with singular boundary conditions. Under proper conditions, the solution of this equation quenches in finite time and the only quenching point thatis $x=1$ are obtained. Moreover, the quenching rate of this equation is established. Finally, we give an example of an application of our results.

We discuss the existence of subharmonic solutions for nonautonomous second order differential equations with singular nonlinearities. Simple sufficient conditions are provided enable us to obtain infinitely many distinct subharmonic solutions. Our approach is based on a variational method, in particular the saddle point theorem.

In a general context, that of the locally convex spaces, we characterize the existence of a solution for certain variational equations with constraints. For the normed case and in the presence of some kind of compactness of the closed unit ball, more specifically, when we deal with reflexive spaces or, in a more general way, with dual spaces, we deduce results implying the existence of a unique weak solution for a wide class of linear elliptic boundary value problems that do not admit a classical treatment. Finally, we apply our statements to the study of linear impulsive differential equations, extending previously stated results.

Combining with the Crank-Nicolson/leapfrog scheme in time discretization, Chebyshev-Legendre spectral method is applied to space discretization for numerically solving the Benjamin-Bona-Mahony-Burgers (gBBM-B) equations. The proposed approach is based on Legendre Galerkin formulation while the Chebyshev-Gauss-Lobatto (CGL) nodes are used in the computation. By using the proposed method, the computational complexity is reduced and both accuracy and efficiency are achieved. The stability and convergence are rigorously set up. Optimal error estimate of the Chebyshev-Legendre method is proved for the problem with Dirichlet boundary condition. The convergence rate shows “spectral accuracy.” Numerical experiments are presented to demonstrate the effectiveness of the method and to confirm the theoretical results.

This paper is concerned with the existence of solutions for Sturm-Liouville boundary value problem of a class of second-order impulsive differential equations, under different assumptions on the nonlinearity and impulsive functions, existence criteria of single and multiple solutions are established. The main tools are variational method and critical point theorems. Some examples are also given to illustrate the main results.

We consider a nonlinear Dirichlet elliptic equation driven by a nonhomogeneous differential operator and with a Carathéodory reaction $f\left(z,\zeta \right)$, whose primitive $f\left(z,\zeta \right)$ is $p$-superlinear near $\pm \infty$, but need not satisfy the usual in such cases, the Ambrosetti-Rabinowitz condition. Using a combination of variational methods with the Morse theory (critical groups), we show that the problem has at least three nontrivial smooth solutions. Our result unifies the study of “superlinear” equations monitored by some differential operators of interest like the $p$-Laplacian, the $\left(p,q\right)$-Laplacian, and the $p$-generalized mean curvature operator.

Some new vectorial Ekeland variational principles in conequasi-uniform spaces are proved. Some new equivalent principles, vectorialquasivariational inclusion principle, vectorial quasi-optimizationprinciple, vectorial quasiequilibrium principle are obtained. Also, severalother important principles in nonlinear analysis are extended to conequasi-uniform spaces. The results of this paper extend, generalize, andimprove the corresponding results for Ekeland's variational principles ofthe directed vectorial perturbation type and other generalizations ofEkeland's variational principles in the setting of $F$-type topologicalspace and quasi-metric spaces in the literatures. Even in usual real metricspaces, some of our results are new.

We study the nonsteady Stokes flow in a thin tube structure composed by two thin rectangles with lateral elastic boundaries which are connected by a domain with rigid boundaries. After a variational approach of the problem which gives us existence, uniqueness, regularity results, and some a priori estimates, we construct an asymptotic solution. The existence of a junction region between the two rectangles imposes to consider, as part of the asymptotic solution, some boundary layer correctors that correspond to this region. We present and solve the problems for all the terms of the asymptotic expansion. For two different cases, we describe the order of steps of the algorithm of solving the problem and we construct the main term of the asymptotic expansion. By means of the a priori estimates, we justify our asymptotic construction, by obtaining a small error between the exact and the asymptotic solutions.

We consider a differential inclusion system involving the $\left(p\left(x\right),q\left(x\right)\right)$-Laplacian with Dirichlet boundary condition on a bounded domain and obtain two nontrivial solutions under appropriate hypotheses. Our approach is variational and it is based on the nonsmooth critical point theory for locally Lipschitz functions.

The paper is concerned with the variational formulation of the oligopolistic market equilibrium problem in presence of both production and demand excesses. In particular, we generalize a previous model in which the authors, instead, considered only the problem with production excesses, by allowing also the presence of demand excesses. First we examine the equilibrium conditions in terms of the well-known dynamic Cournot-Nash principle. Next, the equilibrium conditions will be expressed in terms of Lagrange multipliers by means of the infinite dimensional duality theory. Then, we show the equivalence between the two conditions that are both expressed by an appropriate evolutionary variational inequality. Moreover, thanks to the variational formulation, some existence and regularity results for equilibrium solutions are proved. At last, a numerical example, which illustrates the features of the problem, is provided.

We provide the existence of a solution for quasilinear elliptic equation $-\text{div}(a_{\infty }(x)|\nabla u{|}^{p-2}\nabla u+\tilde{a}(x,|\nabla u|)\nabla u)=\lambda m(x)|u{|}^{p-2}u+f(x,u)+h(x)$ in $\mathrm{\Omega }$ under the Neumann boundary condition. Here, we consider the condition that $\tilde{a}(x,t)=o(t^{p-2})$ as $t\to +\infty$ and $f(x,u)=o(|u{|}^{p-1})$ as $|u|\to \infty$. As a special case, our result implies that the following $p$-Laplace equation has at least one solution: $-{\mathrm{\Delta }}_{p}u=\lambda m(x)|u{|}^{p-2}u+\mu |u{|}^{r-2}u+h(x)$ in $\mathrm{\Omega },\partial u/\partial \nu =0$ on $\partial \mathrm{\Omega }$ for every , $\lambda \in ℝ$, $\mu \ne 0$ and $m,h\in L^{\infty }(\mathrm{\Omega })$ with ${\int }_{\mathrm{\Omega }}mdx\ne 0$. Moreover, in the nonresonant case, that is, $\lambda$ is not an eigenvalue of the $p$-Laplacian with weight $m$, we present the existence of a solution of the above $p$-Laplace equation for every , $\mu \in ℝ$ and $m,h\in L^{\infty }(\mathrm{\Omega })$.

We study the solvability of Dirichlet and Neumann problems for different classes of nonlinear elliptic systems depending on parameters and with nonmonotone operators, using existence theorems related to a general system of variational equations in a reflexive Banach space. We also point out some regularity properties and the sign of the found solutions components. We often prove the existence of at least two different solutions with positive components.

We consider subclasses of functions with bounded turning for normalized analytic functions in the unit disk. The geometric representation is introduced, some subordination relations are suggested, and the upper bound of the pre-Schwarzian norm for these functions is computed. Moreover, by employing Jack's lemma, we obtain a convex class in the class of functions of bounded turning and relations with other classes are posed.

We discuss nonlinear homogeneous eigenvalue problems and the variational characterization of their eigenvalues. We focus on the Ljusternik-Schnirelmann method, present one possible alternative to this method and compare it with the Courant-Fischer minimax principle in the linear case. At the end we present a special nonlinear eigenvalue problem possessing an eigenvalue which allows the variational characterization but is not of Ljusternik-Schnirelmann type.

Employing a recent three critical points theorem due to Bonanno and Marano (2010), the existence of at least three solutions for the following multipoint boundary value system $-\left(\right|u_i^{\prime }{|}^{p_i-2}{u}_i^{\prime })\prime =\lambda F_{u_i}(x,u_1,\dots ,u_n)$ in $\left(\mathrm{0,1}\right)$, $u_i\left(0\right)={\sum }_{j=1}^m{a}_j{u}_i\left(x_j\right)$, $u_i\left(1\right)={\sum }_{j=1}^m{b}_j{u}_i\left(x_j\right)$ for $1\le i\le n$, is established.

We estimate the dimension of the global attractor of an evolution equation by the study of the evolution of the n-dimensional volumes under the flow. We compare these results with the estimate of the dimension of the inertial manifold.

We introduce a general implicit iterative scheme base on viscosity approximation method with a ϕ-strongly pseudocontractive mapping for finding a common element of the set of solutions for a system of mixed equilibrium problems, the set of common fixed point for a nonexpansive semigroup, and the set of solutions of system of variational inclusions with set-valued maximal monotone mapping and Lipschitzian relaxed cocoercive mappings in Hilbert spaces. Furthermore, we prove that the proposed iterative algorithm converges strongly to a common element of the above three sets, which is a solution of the optimization problem related to a strongly positive bounded linear operator.

By establishing a new proper variational framework and using the critical pointtheory, we establish some new existence criteria to guarantee that the 2$n$th-order nonlinear difference equation containing both advance and retardation with $p$-Laplacian ${\Delta }^n\left(r\left(t-n\right){\phi }_p\left({\Delta }^{n}u\left(t-1\right)\right)\right)+q\left(t\right){\phi }_p\left(u\left(t\right)\right)=f\left(t,u\left(t+n\right),\dots ,u\left(t\right),\dots ,u\left(t-n\right)\right)$, $n\in \mathbb{Z}\left(3\right)$, $t\in \mathbb{Z}$, has infinitely many homoclinic orbits, where ${\phi }_p\left(s\right)$ is $p$-Laplacian operator; $r$, $q$, $f$ are nonperiodic in $t$. Our conditions on the potential are rather relaxed, and some existing results in the literature are improved.

We study the existence and multiplicity of solutions for the following fractional boundary value problem: $\mathrm{(d}/d\mathrm{t)}((1/\mathrm{2)}{}_0{D}_t^{-\beta }\left(u\text{'}\right(t\left)\right)+\mathrm{(1}/\mathrm{2)}{}_t{D}_T^{-\beta }\left(u\text{'}\right(t\left)\right))+\nabla F(t,u\left(t\right))=0,\text{a}.\text{e}.t\in [0,T],u(0)=u(T)=0$, where $F(t,\cdot )$ are superquadratic, asymptotically quadratic, and subquadratic, respectively. Several examples are presented to illustrate our results.