Existence of positive solutions of Kirchhoff hyperbolic systems with multiple parameters

In this paper, by using sub-super solutions method, we study the existence of weak positive solution of Kirrchoff hyperbolic systems in bounded domains with multiple parameters. These results extend and improve many results in the literature

Symmetry and Nonexistence of Positive Solutions for a Fractional Laplacion System with Coupled Terms

In this paper, we study the problem for a nonlinear elliptic system involving fractional Laplacion: (equation 1.1) where 0 < α, β < 2, p, q > 0 and max{p, q} ≥ 1, α + γ > 0, β + τ > 0, n ≥ 2. First of all, while in the subcritical case, i.e. n + α + γ − p(n − α) − (q + 1)(n − β) > 0, n + β + τ − (p + 1)(n − α) − q(n − β) > 0, we prove the nonexistence of positive solution for the above system in R n . Moreover, though Doubling Lemma to obtain the singularity estimates of the positive solution on bounded domain Ω. In addition, while in the critical case, i.e. n+α+γ −p(n−α)−(q + 1)(n−β) = 0, n+β +τ −(p+ 1)(n−α)−q(n−β) = 0, we show that the positive solution of above system are radical symmetric and decreasing about some point by using the method of Moving planes in Rn Mathematics Subject Classification (2020): 35R11, 35A10, 35B06.

Metode Topsis Untuk Sistem Pendukung Keputusan Penilaian Mall Terbaik Di Kota Depok

Abstrak: Mall merupakan tempat berbelanja dan refreshing yang sangat disukai banyak kalangan. Selain untuk tempat berbelanja, di mall juga terdapat banyak kuliner, tempat bermain anak, bioskop, dan masih banyak fasilitas hiburan yang ada di mall. Ada banyak mall di Kota Depok seperti Detos, Margo City, CIPLAZ Depok (Ramayana), D’Mall, Pesona Square, ITC Depok, Cinere Bellevue Mall, dan Cimanggis Square. Banyaknya Mall yang ada kerap kali membuat pengunjung bingung untuk menentukan kemana harus pergi kemall yang membuat nyaman dan bisa semua dilakukan. Berangkat dari permasalahan ini maka tujuan penelitian ini dilakukan untuk menentukan Mall terbaik. Ada beberapa kriteria yang digunakan dalam menentukan Mall terbaik yaitu Kebersihan, Keamanan, Tempat parkir, Kelengkapan tenant, Lokasi strategis dan Fasilitas umum. Dalam penelitian ini penulis menggunakan Metode Topsis, metode ini pengambilan keputusan menggunakan multi kriteria dengan prinsip untuk alternaatif yang terpilih memiliki jarak terdekat untuk solusi terbaik positif. Hasil perhitungan didapat bahwa Mall Terbaik di Kota Depok yaitu Margo City dengan nilai 1, untuk rangking kedua yaitu Detos dengan nilai 0,9887, dan rangking ketiga yaitu ITC Depok dengan nilai 0,3246. Kata kunci: mall, metode Topsis, sistem pendukung keputusan Abstract: The mall is a place for shopping and refreshing that is very popular among many people. In addition to shopping, in the mall there are also many culinary delights, children's playgrounds, cinemas, and many entertainment facilities in the mall. There are many malls in Depok City such as Detos, Margo City, CIPLAZ Depok (Ramayana), D'Mall, Pesona Square, ITC Depok, Cinere Bellevue Mall, and Cimanggis Square. The number of existing malls often makes visitors confused about where to go to the mall which makes everything comfortable and can be done. Departing from this problem, the purpose of this study was to determine the best Mall. There are several criteria used in determining the best mall, namely Cleanliness, Security, Parking, Tenant Completeness, Strategic Location and Public Facilities. In this study, the author uses the Topsis method. This method uses multiple criteria for decision making with the principle that the chosen alternative has the closest distance to the best positive solution. The results of the calculation show that the Best Mall in Depok City is Margo City with a value of 1, for the second rank is Detos with a value of 0.9887, and the third rank is ITC Depok with a value of 0.3246. Keywords: decission support system, mall, Topsis methode

Positive solution of Hilfer fractional differential equations with integral boundary conditions

In this article, we have interested the study of the existence and uniqueness of positive solutions of the first-order nonlinear Hilfer fractional differential equation $$D_{0^{+}}^{\alpha ,\beta }y(t)=f(t,y(t)),\text{ }0<t\leq 1,$$ with the integral boundary condition $$I_{0^{+}}^{1-\gamma }y(0)=\lambda \int_{0}^{1}y(s)ds+d,$$ where $0<\alpha \leq 1,$ $0\leq \beta \leq 1,$ $\lambda \geq 0,$ $d\in \mathbb{R}^{+},$ and $D_{0^{+}}^{\alpha ,\beta }$, $I_{0^{+}}^{1-\gamma }$ are fractional ope\-rators in the Hilfer, Riemann-Liouville concepts, respectively. In this approach, we transform the given fractional differential equation into an equivalent integral equation. Then we establish sufficient conditions and employ the Schauder fixed point theorem and the method of upper and lower solutions to obtain the existence of a positive solution of a given problem. We also use the Banach contraction principle theorem to show the existence of a unique positive solution. The result of existence obtained by structure the upper and lower control functions of the nonlinear term is without any monotonous conditions. Finally, an example is presented to show the effectiveness of our main results.

Existence of multiple nontrivial solutions of the nonlinear Schrödinger-Korteweg-de Vries type system

In this paper we are concerned with the existence, nonexistence and bifurcation of nontrivial solution of the nonlinear Schrödinger-Korteweg-de Vries type system(NLS-NLS-KdV). First, we find some conditions to guarantee the existence and nonexistence of positive solution of the system. Second, we study the asymptotic behavior of the positive ground state solution. Finally, we use the classical Crandall-Rabinowitz local bifurcation theory to get the nontrivial positive solution. To get these results we encounter some new challenges. By combining the Nehari manifolds constraint method and the delicate energy estimates, we overcome the difficulties and find the two bifurcation branches from one semitrivial solution. This is an new interesting phenomenon but which have not previously been found.

Existence of Positive Solution for a Singular Fourth–Order Differential System

This paper investigates the existence of positive solutions for a fourth-order differential system using a fixed point theorem of cone expansion and compression type.

On a structure of the set of positive solutions to second-order equations with a super-linear non-linearity

We study the existence and multiplicity of positive solutions to the periodic problem u ′′ = p ⁢ ( t ) ⁢ u - q ⁢ ( t , u ) ⁢ u + f ⁢ ( t ) ; u ⁢ ( 0 ) = u ⁢ ( ω ) , u ′ ⁢ ( 0 ) = u ′ ⁢ ( ω ) , u^{\prime\prime}=p(t)u-q(t,u)u+f(t);\quad u(0)=u(\omega),\quad u^{\prime}(0)=u^{\prime}(\omega), where p , f ∈ L ⁢ ( [ 0 , ω ] ) p,f\in L([0,\omega]) and q : [ 0 , ω ] × R → R q\colon[0,\omega]\times\mathbb{R}\to\mathbb{R} is a Carathéodory function. By using the method of lower and upper functions, we show some properties of the solution set of the considered problem and, in particular, the existence of a minimal positive solution.

TWO-SIDED APPROXIMATIONS METHOD BASED ON THE GREEN’S FUNCTIONS USE FOR CONSTRUCTION OF A POSITIVE SOLUTION OF THE DIRICHLE PROBLEM FOR A SEMILINEAR ELLIPTIC EQUATION

Context. The question of constructing a method of two-sided approximations for finding a positive solution of the Dirichlet problem for a semilinear elliptic equation based on the use of the Green’s functions method is considered. The object of research is the first boundary value problem (the Dirichlet problem) for a second-order semilinear elliptic equation. Objective. The purpose of the research is to develop a method of two-sided approximations for solving the Dirichlet problem for second-order semilinear elliptic equations based on the use of the Green’s functions method and to study its work in solving test problems. Method. Using the Green’s functions method, the initial first boundary value problem for a semilinear elliptic equation is replaced by the equivalent Hammerstein integral equation. The integral equation is represented in the form of a nonlinear operator equation with a heterotone operator and is considered in the space of continuous functions, which is semi-ordered using the cone of nonnegative functions. As a solution (generalized) of the boundary value problem, it was taken the solution of the equivalent integral equation. For a heterotone operator, a strongly invariant cone segment is found, the ends of which are the initial approximations for two iteration sequences. The first of these iterative sequences is monotonically increasing and approximates the desired solution to the boundary value problem from below, and the second is monotonically decreasing and approximates it from above. Conditions for the existence of a unique positive solution of the considered Dirichlet problem and two-sided convergence of successive approximations to it are given. General guidelines for constructing a strongly invariant cone segment are also given. The method developed has a simple computational implementation and a posteriori error estimate that is convenient for use in practice. Results. The method developed was programmed and studied when solving test problems. The results of the computational experiment are illustrated with graphical and tabular informations. Conclusions. The experiments carried out have confirmed the efficiency and effectiveness of the developed method and make it possible to recommend it for practical use in solving problems of mathematical modeling of nonlinear processes. Prospects for further research may consist the development of two-sided methods for solving problems for systems of partial differential equations, partial differential equations of higher orders and nonstationary multidimensional problems, using semi-discrete methods (for example, the Rothe’s method of lines).