Browsing by Subject "Green's function"

Now showing 1 - 5 of 5

Access status: Open Access ,
Existence of solutions for a four-point boundary value problem of a nonlinear fractional differential equation
(2011) Dou, Xiaoyan; Li, Yongkun; Liu, Ping
In this paper, we discuss a four-point boundary value problem for a nonlinear differential equation of fractional order. The differential operator is the Riemann-Liouville derivative and the inhomogeneous term depends on the fractional derivative of lower order. We obtain the existence of at least one solution for the problem by using the Schauder fixed-point theorem. Our analysis relies on the reduction of the problem considered to the equivalent Fredholm integral equation.
Access status: Open Access ,
Green's functions and existence of solutions of nonlinear fractional implicit difference equations with Dirichlet boundary conditions
(Wydawnictwa AGH, 2024) Cabada, Alberto; Dimitrov, Nikolay D.; Jonnalagadda, Jagan Mohan
This article is devoted to deduce the expression of the Green's function related to a general constant coefficients fractional difference equation coupled to Dirichlet conditions. In this case, due to the points where some of the fractional operators are applied, we are in presence of an implicit fractional difference equation. So, due to such a property, it is more complicated to calculate and manage the expression of the Green's function than in the explicit case studied in a previous work of the authors. Contrary to the explicit case, where it is shown that the Green's function is constructed as finite sums, the Green's function constructed here is an infinite series. This fact makes necessary to impose more restrictive assumptions on the parameters that appear in the equation. The expression of the Green's function will be deduced from the Laplace transform on the time scales of the integers. We point out that, despite the implicit character of the considered equation, we can have an explicit expression of the solution by means of the expression of the Green's function. These two facts are not incompatible. Even more, this method allows us to have an explicit expression of the solution of an implicit problem. Finally, we prove two existence results for nonlinear problems, via suitable fixed point theorems.
Access status: Open Access ,
On the blowing up solutions of the 4-d general q-Kuramoto-Sivashinsky equation with exponentially »dominated« nonlinearity and singular weight
(Wydawnictwa AGH, 2023) Baraket, Sami; Mahdaoui, Safia; Ouni, Taieb
Let $\Omega$ be a bounded domain in $\mathbb{R}^{4}$ with smooth boundary and let $x^{1},x^{2},\dots,x^{m}$ be $m$-points in $\Omega$. We are concerned with the problem $\Delta^{2} u - H(x,u,D^{k}u) = \rho^{4}\prod_{i=1}^{n}|x-p_{i}|^{4\alpha_{i}}f(x)g(u),$ where the principal term is the bi-Laplacian operator, $H(x,u,D^{k}u)$ is a functional which grows with respect to $Du$ at most like $|Du|^{q}$, $1 \leq q \leq 4$, $f:\Omega\to [0,+\infty[$ is a smooth function satisfying $f(p_{i}) \gt 0$ for any $i = 1,\ldots, n$, $\alpha_{i}$ are positives numbers and $g:\mathbb{R} \to [0,+\infty[$ satisfy $|g(u)|\leq ce^{u}$. In this paper, we give sufficient conditions for existence of a family of positive weak solutions $(u_\rho)_{\rho\gt 0}$ in $\Omega$ under Navier boundary conditions $u=\Delta u =0$ on $\partial \Omega$. The solutions we constructed are singular as the parameters $ho$ tends to 0, when the set of concentration $S=\{x^{1},\ldots,x^{m}\}\subset\Omega$ and the set $\Lambda :=\{p_{1},\ldots, p_{n}\}\subset\Omega$ are not necessarily disjoint. The proof is mainly based on nonlinear domain decomposition method.
Access status: Open Access ,
Region of existence of multiple solutions for a class of Robin type four-point BVPs
(Wydawnictwa AGH, 2021) Verma, Amit K.; Urus, Nazia; Agarwal, Ravi P.
This article aims to prove the existence of a solution and compute the region of existence for a class of four-point nonlinear boundary value problems (NLBVPs) defined as $\begin{gathered} -u''(x)=\psi(x,u,u'), \quad x\in (0,1),\\ u'(0)=\lambda_{1}u(\xi), \quad u'(1)=\lambda_{2} u(\eta),\end{gathered}$ where $I=[0,1]$, $0\lt\xi\leq\eta\lt 1$ and $\lambda_{1},\lambda_{2} \gt 0$. The nonlinear source term $\psi\in C(I\times\mathbb{R}^2,\mathbb{R})$ is one sided Lipschitz in $u$ with Lipschitz constant $L_1$ and Lipschitz in $u'$ such that $|\psi(x,u,u')-\psi(x,u,v')|\leq L_2(x)|u'-v'|$. We develop monotone iterative technique (MI-technique) in both well ordered and reverse ordered cases. We prove maximum, anti-maximum principle under certain assumptions and use it to show the monotonic behaviour of the sequences of upper-lower solutions. The sufficient conditions are derived for the existence of solution and verified for two examples. The above NLBVPs is linearised using Newton's quasilinearization method which involves a parameter $k$ equivalent to $\max_u\frac{\partial \psi}{\partial u}$. We compute the range of $k$ for which iterative sequences are convergent.
Access status: Open Access ,
Uniqueness and parameter dependence of positive doubly periodic solutions of nonlinear telegraph equations
(2014) Graef, John R.; Kong, Lingju; Wang, Min
The authors study a type of second order nonlinear telegraph equation. The existence and uniqueness of positive doubly periodic solutions are discussed. The parametric dependence of the solutions is also investigated. Two examples are given as applications of the results.