Numerical extremal solutions for a mixed problem with singular {\phi} -Laplacian

We are concerned with extremal solutions for the mixed boundary value problem $$-\left(r^{N-1}\phi(u')\right)' = r^{N-1} g(r, u), \quad u'(0) = 0 = u(R),$$ where ${g : [0, R] \times \mathbb{R} \to \mathbb{R}}$ is a continuous function and ${\phi : (-\eta, \eta) \to \mathbb{R}}$ is an increasing homeomorphism with ${\phi(0) = 0.}$ We prove the existence of minimal and maximal solutions in presence of well-ordered lower and upper solutions and develop a numerical algorithm for theirs approximation. Also, we provide numerical experiments confirming the theoretical aspects.     

On some new nonlocal solvability theorems for various classes of nonlinear differential equations

We study nonlinear boundary value problems of the form $$ [\Psi u']' + F(x;u',u) = g, u(0) = u(1) = 0 $$ , where Φ is a coercive continuous operator from L _p to L _q , and $$ F(x;u'',u',u) = g, u(0) = u(1) = 0 $$ ; first- and second-order partial differential equations $$ \Phi (x_1 ,x_2 ;u'_1 ,u'_2 ,u) = 0, \sum\limits_{i = 1}^\infty {[\Psi _i (u'_{x_i } )]'_{x_i } + F(x; \ldots ,u'_{x_i } , \ldots ,u) = g_i } $$ ; and general equations F(x; ..., u″ _ii , ...., ...., u′ _i , ...; u) = g(x) of elliptic type. We consider the corresponding boundary value problems of parabolic and hyperbolic type. The proof is based on various a priori estimates obtained in the paper and a nonlocal implicit function theorem.     

The Robin and Wentzell-Robin Laplacians on Lipschitz Domains

Let $\Omega\subset{\Bbb R}^N$ be a bounded domain with Lipschitz boundary. We prove in the first part that a realization of the Laplacian with Robin boundary conditions $\frac{\partial u}{\partial \nu}+\beta u=0$ on the boundary $\partial \Omega$ generates a holomorphic $C_0$ -semigroup of angle $\pi/2$ on $C(\overline{\Omega})$ if $0<\beta_0\le \beta\in L^{\infty}(\partial \Omega)$ . With the same assumption on $\Omega$ and assuming that $0\le\beta\in L^{\infty}(\partial \Omega)$ , we show in the second part that one can define a realization of the Laplacian on $C(\overline{\Omega})$ with Wentzell-Robin boundary conditions $\Delta u+\frac{\partial u}{\partial \nu}+\beta u=0$ on the boundary $\partial \Omega$ and this operator generates a $C_0$ -semigroup.     

Energy decay rates for solutions of the wave equation with boundary damping and source term

In this paper, we consider the wave equation $$u'' - \Delta u = |u|^\rho u$$ with the following nonlinear boundary condition $$\frac{\partial u}{\partial \nu} + \int\limits^t_0 k(t-s,x)u'(s){\rm d}s + a(x)g(u') = 0.$$ We show energy decay rates for solutions of the wave equation in bounded domain with nonlinear boundary damping and source term.     

Strong convergence theorems for a Mann-type iterative scheme for a family of Lipschitzian mappings

In the paper, we obtain the existence of triple positive solutions for the following second order three-point boundary value problem, $$\left\{\begin{array}{l}(\phi_p(u'))'(t)+q(t)f(u(t),u'(t),(Tu)(t),(Su)(t))=0,\quad 0\leq t\leq1,\\[4pt]u'(0)=\beta u'(\eta),\qquad u(1)=g(u'(1)),\end{array}\right.$$ where $\phi_{p}(s)=|s|^{p-2}s,p>1,\beta\in[0,1),\eta\in(0,\frac{1}{2}]$ , T and S are all linear operators, g(t) is continuous.     

Ground state periodic solutions of second order Hamiltonian systems without spectrum 0

In this paper, we consider the second order Hamiltonian system $\left\{ \begin{gathered} u''(t) + A(t)u(t) + \nabla H(t,u(t)) = 0,t \in R, \hfill \\ u(0) = u(T),u'(0) = u'(T),T > 0. \hfill \\ \end{gathered} \right.$ Here, we assume 0 lies in a gap of σ(B) (the spectrum of B:= ?d ²/dt ² ?A(t)). We find nontrivial and ground state T-periodic solutions for the second order Hamiltonian system under conditions weaker than those previously assumed; also, our proof is much more direct.     

On the asymptotic behavior of certain solutions of the Dirichlet problem for the equation -\Delta _p u=\lambda |u|^{q-2}u

Let $p>1$ . We study the behavior of certain positive and nodal solutions of the problem $$\begin{aligned} \left\{ \,\, \begin{array}{lll} -\Delta _p u=\lambda |u|^{q-2}u \ \ &{}\mathrm{in} \ \ &{}{\varOmega } \\ u=0 &{}\mathrm{in} \ \ &{}\partial {\varOmega } \end{array}\right. \end{aligned}$$ on varying of the parameters $\lambda >0$ and $q>1$ .     

A singular boundary value problem for nonlinear differential equations of fractional order

We are concerned with the nonlinear differential equation of fractional order $$\mathcal{D}^{\alpha}_{0+}u(t)=f(t,u(t),u'(t)),\quad \mbox{a.\,e.}\ t\in (0,1),$$ where $\mathcal{D}^{\alpha}_{0+}$ is the Riemann-Liouville fractional order derivative, subject to the boundary conditions $$u(0)=u(1)=0.$$ We obtain the existence of at least one solution using the Leray-Schauder Continuation Principle.     

Existence, nonexistence and multiplicity of positive solutions for nonlinear fractional differential equations

In this paper, we consider the existence, nonexistence and multiplicity of positive solutions for nonlinear fractional differential equation boundary-value problem $$\left\{ \begin{array}{@{}l}-D^{\alpha}_{0+}u(t)=f(t,u(t)), \quad t\in[0,1]\\[3pt]u(0)=u(1)=u''(0)=0\end{array} \right.$$ where 2<????3 is a real number, and $D^{\alpha}_{0+}$ is the Caputo??s fractional derivative, and f:[0,1]×[0,+??)??[0,+??) is continuous. By means of a fixed-point theorem on cones, some existence, nonexistence and multiplicity of positive solutions are obtained.     

Numerical solutions to singular ϕ-Laplacianwith Dirichlet boundary conditions

In this note we are concerned with numerical solutions to Dirichlet problem $$[\phi(u')]' =f(x) \quad \mbox{in} [\alpha, \beta]; \quad u(\alpha)=A, \; u(\beta)=B, $$ where \(\phi :(-\eta , \eta ) \to \mathbb {R}\) \((\eta <+ \infty )\) is an increasing diffeomorphism with \(\phi '(y)\geq d >0\) for all \(y\in (-\eta , \eta )\) . The obtained algorithm combines the shooting method with Euler’s method and it is convergent whenever the problem is solvable. We provide numerical experiments confirming the theoretical aspects.     

Existence of triple symmetric positive solutions for four-point boundary-value problem with one-dimensional p-Laplacian

In this paper, we consider the four-point boundary value problem for one-dimensional p-Laplacian $$\bigl(\phi_{p}(u'(t))\bigr)'+q(t)f\bigl(t,u(t),u'(t)\bigr)=0,\quad t\in(0,1),$$ subject to the boundary conditions $$u(0)-\beta u'(\xi)=0,\qquad u(1)+\beta u'(\eta)=0,$$ where φ _p (s)=|s| ^p?2 s. Using a fixed point theorem due to Avery and Peterson, we study the existence of at least three symmetric positive solutions to the above boundary value problem. The interesting point is the nonlinear term f is involved with the first-order derivative explicitly.     

Fractional eigenvalues

We study the non-local eigenvalue problem $$\begin{aligned} 2\, \int \limits _{\mathbb{R }^n}\frac{|u(y)-u(x)|^{p-2}\bigl (u(y)-u(x)\bigr )}{|y-x|^{\alpha p}}\,dy +\lambda |u(x)|^{p-2}u(x)=0 \end{aligned}$$ for large values of $p$ and derive the limit equation as $p\rightarrow \infty $ . Its viscosity solutions have many interesting properties and the eigenvalues exhibit a strange behaviour.     

Convergence to steady states of solutions to nonlinear integral evolution equations

We prove that any global bounded solution of the nonlinear evolutionary integral equation $$\dot{u}(t) + \int\limits_0^t a(t-s)\mathcal{E}'(u(s))ds =f(t), \quad t >0 $$ tends to a single equilibrium state for long time (i.e., ${\mathcal{E}'(\vartheta)=0}$ where ${\vartheta= \lim_{t \rightarrow \infty} u(t)}$ on a real Hilbert space), where ${\mathcal{E}'}$ is the Fréchet derivative of a functional ${\mathcal{E}}$ , which satisfies the ?ojasiewicz?CSimon inequality near ${\vartheta}$ . The vector-valued function f and the scalar kernel a satisfy suitable conditions.     

Necessary and Sufficient Conditions for Associated Differential Operators in Quaternionic Analysis and Applications to Initial Value Problems

This paper deals with the initial value problem of type $$\begin{array}{ll} \qquad \frac{\partial u}{\partial t} = \mathcal{L} u := \sum \limits^3_{i=0} A^{(i)} (t, x) \frac{\partial u}{\partial x_{i}} + B(t, x)u + C(t, x)\\ u (0, x) = u_{0}(x)\end{array}$$ in the space of generalized regular functions in the sense of Quaternionic Analysis satisfying the differential equation $$\mathcal{D}_{\lambda}u := \mathcal{D} u + \lambda u = 0,$$ where ${t \in [0, T]}$ is the time variable, x runs in a bounded and simply connected domain in ${\mathbb{R}^{4}, \lambda}$ is a real number, and ${\mathcal{D}}$ is the Cauchy-Fueter operator. We prove necessary and sufficient conditions on the coefficients of the operator ${\mathcal{L}}$ under which ${\mathcal{L}}$ is associated with the operator ${\mathcal{D}_{\lambda}}$ , i.e. ${\mathcal{L}}$ transforms the set of all solutions of the differential equation ${\mathcal{D}_{\lambda}u = 0}$ into solutions of the same equation for fixedly chosen t. This criterion makes it possible to construct operators ${\mathcal{L}}$ for which the initial value problem is uniquely soluble for an arbitrary initial generalized regular function u ₀ by the method of associated spaces constructed by W. Tutschke (Teubner Leipzig and Springer Verlag, 1989) and the solution is also generalized regular for each t.     

Existence and extremal solutions of parabolic variational–hemivariational inequalities

We consider quasilinear parabolic variational–hemivariational inequalities in a cylindrical domain $Q=\Omega \times (0,\tau )$ of the form $$\begin{aligned} u\in K:\ \langle u_t+Au, v-u\rangle +\int _Q j^o(x,t, u;v-u)\,dxdt\ge 0,\ \ \forall \ v\in K, \end{aligned}$$ where $K\subset X_0=L^p(0,\tau ;W_0^{1,p}(\Omega ))$ is some closed and convex subset, $A$ is a time-dependent quasilinear elliptic operator, and $s\mapsto j(\cdot ,\cdot ,s)$ is assumed to be locally Lipschitz with $(s,r)\mapsto j^o(x,t, s;r)$ denoting its generalized directional derivative at $s$ in the direction $r$ . The main goal of this paper is threefold: first, an existence and comparison principle is proved; second, the existence of extremal solutions within some sector of appropriately defined sub-supersolutions is shown; third, the equivalence of the above parabolic variational–hemivariational inequality with an associated multi-valued parabolic variational inequality of the form $$\begin{aligned} u\in K:\ \langle u_t+Au, v-u\rangle +\int _Q \eta \, (v-u)\,dxdt\ge 0,\ \ \forall \ v\in K \end{aligned}$$ with $\eta (x,t)\in \partial j(x,t, u(x,t))$ is established, where $s\mapsto \partial j(x,t, s)$ denotes Clarke’s generalized gradient of the locally Lipschitz function $s\mapsto j(\cdot ,\cdot ,s)$ .     

On the exact solitary wave solutions of a special class of Benjamin-Bona-Mahony equation

The general form of Benjamin-Bona-Mahony equation (BBM) is $u_t + au_x + bu_{xxt} + (g(u))_x = 0,a,b \in \mathbb{R},$ where ab ≠ 0 and g(u) is a C ₂-smooth nonlinear function, has been proposed by Benjamin et al. in [1] and describes approximately the unidirectional propagation of long wave in certain nonlinear dispersive systems. In this payer, we consider a new class of Benjamin-Bona-Mahony equation (BBM) $u_t + au_x + bu_{xxt} + (pe^u + qe^{ - u} )_x = 0,a,b,p,q \in \mathbb{R},$ where ab ≠ 0, and qp ≠ 0, and we obtain new exact solutions for it by using the well-known (G′/G)-expansion method. New periodic and solitary wave solutions for these nonlinear equation are formally derived.     

Multiple positive solutions for fractional differential systems

In this paper, we study the existence of positive solution to boundary value problem for fractional differential system $$\left\{\begin{array}{ll}D_{0^+}^\alpha u (t) + a_1 (t) f_1 (t, u (t), v (t)) = 0,\;\;\;\;\;\;\;\quad t \in (0, 1),\\D_{0^+}^\alpha v (t) + a_2 (t) f_2 (t, u (t), v (t)) = 0,\;\;\;\;\;\;\;\quad t \in (0, 1), \;\; 2 < \alpha < 3,\\u (0)= u' (0) = 0, \;\;\;\; u' (1) - \mu_1 u' (\eta_1) = 0,\\v (0)= v' (0) = 0, \;\;\;\; v' (1) - \mu_2 v' (\eta_2) = 0,\end{array}\right.$$ where ${D_{0^+}^\alpha}$ is the Riemann-Liouville fractional derivative of order ??. By using the Leggett-Williams fixed point theorem in a cone, the existence of three positive solutions for nonlinear singular boundary value problems is obtained.     

Strongly nonlinear multivalued elliptic equations on a bounded domain

In this work we study the existence of nontrivial solution for the following class of multivalued quasilinear problems $$\begin{aligned} \displaystyle -\text{ div } ( \phi (|\nabla u|) \nabla u) - b(u)u \in \lambda \partial F(x,u)\;\text{ in }\;\Omega , \quad u=0\; \text{ on }\;\partial \Omega \end{aligned}$$ where $\Omega \subset \mathbb{R }^N$ is a bounded domain, $N\ge 2$ and $\partial F(x,u)$ is a generalized gradient of $F(x,t)$ with respect to $t$ . The main tools utilized are Variational Methods for Locally Lipschitz Functional and a Concentration Compactness Theorem for Orlicz space.     

A subelliptic analogue of Aronson–Serrin’s Harnack inequality

We study the Harnack inequality for weak solutions of a class of degenerate parabolic quasilinear PDE $$\begin{aligned} \partial _t u={-}X_i^* A_i(x,t,u,Xu)+ B(x,t,u,Xu), \end{aligned}$$ in cylinders $\Omega \times (0,T)$ where $\Omega \subset M$ is an open subset of a manifold $M$ endowed with control metric $d$ corresponding to a system of Lipschitz continuous vector fields $X=(X_1,\ldots ,X_m)$ and a measure $d\sigma $ . We show that the Harnack inequality follows from the basic hypothesis of doubling condition and a weak Poincaré inequality in the metric measure space $(M,d,d\sigma )$ . We also show that such hypothesis hold for a class of Riemannian metrics $g_\epsilon $ collapsing to a sub-Riemannian metric $\lim _{\epsilon \rightarrow 0} g_\epsilon =g_0$ uniformly in the parameter $\epsilon \ge 0$ .     

Semi-linear Wave Equations with Effective Damping

The authors study the Cauchy problem for the semi-linear damped wave equation $$u_{tt} - \Delta u + b\left( t \right)u_t = f\left( u \right), u\left( {0,x} \right) = u_0 \left( x \right), u_t \left( {0,x} \right) = u_1 \left( x \right)$$ in any space dimension n ≥ 1. It is assumed that the time-dependent damping term b(t) > 0 is effective, and in particular tb(t) → ∞ as t → ∞. The global existence of small energy data solutions for |f(u)| ≈ |u| ^p in the supercritical case of $p > \tfrac{2} {n}$ and $p \leqslant \tfrac{n} {{n - 2}}$ for n ≥ 3 is proved.