34 results
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2. Maillet type theorem for singular first order nonlinear partial differential equations of totally characteristic type. Part II
- Author
-
Akira Shirai
- Subjects
formal solution ,Partial differential equation ,Formal power series ,Maillet type theorem ,General Mathematics ,lcsh:T57-57.97 ,Mathematical analysis ,First-order partial differential equation ,Mathematics::Analysis of PDEs ,Order (ring theory) ,Type (model theory) ,Nonlinear system ,Convergence (routing) ,lcsh:Applied mathematics. Quantitative methods ,singular partial differential equations ,Divergence (statistics) ,totally characteristic type ,Mathematics ,nilpotent vector field ,Gevrey order - Abstract
In this paper, we study the following nonlinear first order partial differential equation: \[f(t,x,u,\partial_t u,\partial_x u)=0\quad\text{with}\quad u(0,x)\equiv 0.\] The purpose of this paper is to determine the estimate of Gevrey order under the condition that the equation is singular of a totally characteristic type. The Gevrey order is indicated by the rate of divergence of a formal power series. This paper is a continuation of the previous papers [Convergence of formal solutions of singular first order nonlinear partial differential equations of totally characteristic type, Funkcial. Ekvac. 45 (2002), 187-208] and [Maillet type theorem for singular first order nonlinear partial differential equations of totally characteristic type, Surikaiseki Kenkyujo Kokyuroku, Kyoto University 1431 (2005), 94-106]. Especially the last-mentioned paper is regarded as part I of this paper.
- Published
- 2015
3. Existence of critical elliptic systems with boundary singularities
- Author
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Yimin Zhou and Jianfu Yang
- Subjects
Mean curvature ,Elliptic systems ,critical Hardy-Sobolev exponent ,General Mathematics ,lcsh:T57-57.97 ,Mathematical analysis ,existence ,Boundary (topology) ,nonlinear system ,Omega ,Delta-v (physics) ,Combinatorics ,Domain (ring theory) ,lcsh:Applied mathematics. Quantitative methods ,Exponent ,Gravitational singularity ,compactness ,Mathematics - Abstract
In this paper, we are concerned with the existence of positive solutions of the following nonlinear elliptic system involving critical Hardy-Sobolev exponent \begin{equation*}\label{eq:1}(*) \left\{ \begin{array}{lll} -\Delta u= \frac{2\alpha}{\alpha+\beta}\frac{u^{\alpha-1}v^\beta}{|x|^s}-\lambda u^p, & \quad {\rm in}\quad \Omega,\\[2mm] -\Delta v= \frac{2\beta}{\alpha+\beta}\frac{u^\alpha v^{\beta-1}}{|x|^s}-\lambda v^p, & \quad {\rm in}\quad \Omega,\\[2mm] u\gt 0, v\gt 0, &\quad {\rm in}\quad \Omega,\\[2mm] u=v=0, &\quad {\rm on}\quad \partial\Omega, \end{array} \right. \end{equation*} where \(N\geq 4\) and \(\Omega\) is a \(C^1\) bounded domain in \(\mathbb{R}^N\) with \(0\in\partial\Omega\). \(0\lt s \lt 2\), \(\alpha+\beta=2^*(s)=\frac{2(N-s)}{N-2}\), \(\alpha,\beta\gt 1\), \(\lambda\gt 0\) and \(1 \lt p\lt \frac{N+2}{N-2}\). The case when 0 belongs to the boundary of \(\Omega\) is closely related to the mean curvature at the origin on the boundary. We show in this paper that problem \((*)\) possesses at least a positive solution.
- Published
- 2013
4. Estimates of solutions for parabolic differential and difference functional equations and applications
- Author
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Lucjan Sapa
- Subjects
estimate of solution ,General Mathematics ,lcsh:T57-57.97 ,Mathematical analysis ,Finite difference method ,Finite difference ,Lipschitz continuity ,Dirichlet distribution ,Nonlinear system ,symbols.namesake ,Cover (topology) ,lcsh:Applied mathematics. Quantitative methods ,symbols ,Partial derivative ,Applied mathematics ,parabolic differential and discrete functional equations ,implicit difference method ,Differential (mathematics) ,Mathematics - Abstract
The theorems on the estimates of solutions for nonlinear second-order partial differential functional equations of parabolic type with Dirichlet's condition and for suitable implicit finite difference functional schemes are proved. The proofs are based on the com- parison technique. The convergent and stable difference method is considered without the assumption of the global generalized Perron condition posed on the functional variable but with the local one only. It is a consequence of our estimates theorems. In particular, these results cover quasi-linear equations. However, such equations are also treated separately. The functional dependence is of the Volterra type. The aim of the paper is to prove theorems on the estimates of solutions for non- linear second-order partial differential functional equations of parabolic type with Dirichlet's condition and for generated by them implicit finite difference functional schemes. We also give the applications of the results. More precisely, we prove the theorem on the convergence of a difference method to a classical solution for the differential functional problem, which by the given estimates, may be treated in the subspace C (,R) ⊂ C (,R), where R ⊂ R is an interval. It is a new idea in area of nonlinear implicit difference methods which was studied for explicit methods by K. Kropielnicka and L. Sapa (14). This considerably extends the class of problems which are solvable by the described method. Therefore, the Lipschitz, Perron or generalized Perron conditions posed onf with respect toz need not be global, inC (,R), as in the papers due to M. Malec, Cz. Mączka, W. Voigt, M. Rosati and L. Sapa (15-19,24,25)
- Published
- 2012
5. Quasilinearization method for finite systems of nonlinear RL fractional differential equations
- Author
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Zachary Denton and Juan Diego Ramírez
- Subjects
Nonlinear system ,lower and upper solutions ,quasilinearization method ,General Mathematics ,lcsh:T57-57.97 ,lcsh:Applied mathematics. Quantitative methods ,Finite system ,fractional differential systems ,Applied mathematics ,Fractional differential ,Mathematics - Abstract
In this paper the quasilinearization method is extended to finite systems of Riemann-Liouville fractional differential equations of order \(0\lt q\lt 1\). Existence and comparison results of the linear Riemann-Liouville fractional differential systems are recalled and modified where necessary. Using upper and lower solutions, sequences are constructed that are monotonic such that the weighted sequences converge uniformly and quadratically to the unique solution of the system. A numerical example illustrating the main result is given.
- Published
- 2020
6. A unique weak solution for a kind of coupled system of fractional Schrödinger equations
- Author
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Fatemeh Abdolrazaghi and Abdolrahman Razani
- Subjects
General Mathematics ,Weak solution ,lcsh:T57-57.97 ,fractional laplacian ,uniqueness ,weak solution ,Schrödinger equation ,Nonlinear system ,symbols.namesake ,lcsh:Applied mathematics. Quantitative methods ,symbols ,Uniqueness ,Fractional Laplacian ,nonlinear systems ,Mathematics ,Mathematical physics - Abstract
In this paper, we prove the existence of a unique weak solution for a class of fractional systems of Schrödinger equations by using the Minty-Browder theorem in the Cartesian space. To this aim, we need to impose some growth conditions to control the source functions with respect to dependent variables.
- Published
- 2020
7. Global existence and blow up of solution for semi-linear hyperbolic equation with the product of logarithmic and power-type nonlinearity
- Author
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Wei Lian, Runzhang Xu, and Salik Ahmed
- Subjects
global existence ,Logarithm ,General Mathematics ,lcsh:T57-57.97 ,Mathematical analysis ,logarithmic and polynomial nonlinearity ,Mathematics::Analysis of PDEs ,Type (model theory) ,potential well ,Power (physics) ,Nonlinear system ,Product (mathematics) ,lcsh:Applied mathematics. Quantitative methods ,Hyperbolic partial differential equation ,blow-up ,Mathematics - Abstract
In this paper we consider the semilinear wave equation with the multiplication of logarithmic and polynomial nonlinearities. We establish the global existence and finite time blow up of solutions at three different energy levels (\(E(0)\lt d\), \(E(0)=d\) and \(E(0)\gt 0\)) using potential well method. The results in this article shed some light on using potential wells to classify the solutions of the semilinear wave equation with the product of polynomial and logarithmic nonlinearity.
- Published
- 2020
8. Properties of solutions to some weighted p-Laplacian equation
- Author
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Garain, Prashanta, Pucci, Patrizia, Department of Mathematics and Systems Analysis, Aalto-yliopisto, and Aalto University
- Subjects
Pure mathematics ,weighted sobolev space ,35J70, 35J62, 35A01 ,General Mathematics ,Mathematics::Analysis of PDEs ,Degenerate elliptic equations ,01 natural sciences ,Mathematics - Analysis of PDEs ,FOS: Mathematics ,degenerate elliptic equations ,Nabla symbol ,0101 mathematics ,Weighted Sobolev space ,Mathematics ,lcsh:T57-57.97 ,p-Laplacian ,010102 general mathematics ,Degenerate energy levels ,010101 applied mathematics ,Nonlinear system ,Elliptic curve ,lcsh:Applied mathematics. Quantitative methods ,Domain (ring theory) ,Analysis of PDEs (math.AP) ,\(p\)-laplacian - Abstract
In this paper, we prove some qualitative properties for the positive solutions to some degenerate elliptic equation given by \[-\operatorname{div}(w|\nabla u|^{p-2}\nabla u)=f(x,u);\;\;w\in \mathcal{A}_p\] on smooth domain and for varying nonlinearity $f$., Opuscula Mathematica, 2020
- Published
- 2020
9. Existence of periodic positive solutions to nonlinear Lotka-Volterra competition systems
- Author
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Mimia Benhadri, Tomás Caraballo, and Halim Zeghdoudi
- Subjects
Competition (economics) ,variable delays ,Nonlinear system ,krasnoselskii's fixed point theorem ,lcsh:T57-57.97 ,General Mathematics ,lcsh:Applied mathematics. Quantitative methods ,Applied mathematics ,positive periodic solutions ,lotka-volterra competition systems ,Mathematics - Abstract
We investigate the existence of positive periodic solutions of a nonlinear Lotka-Volterra competition system with deviating arguments. The main tool we use to obtain our result is the Krasnoselskii fixed point theorem. In particular, this paper improves important and interesting work [X.H. Tang, X. Zhou, On positive periodic solution of Lotka-Volterra competition systems with deviating arguments, Proc. Amer. Math. Soc. 134 (2006), 2967-2974]. Moreover, as an application, we also exhibit some special cases of the system, which have been studied extensively in the literature.
- Published
- 2020
10. The existence of consensus of a leader-following problem with Caputo fractional derivative
- Author
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Ewa Schmeidel
- Subjects
leader-following problem ,General Mathematics ,02 engineering and technology ,caputo fractional differential equation ,nonlinear system ,01 natural sciences ,Leader following ,Schauder fixed point theorem ,Stability theory ,0202 electrical engineering, electronic engineering, information engineering ,Applied mathematics ,0101 mathematics ,schauder fixed point theorem ,Trajectory (fluid mechanics) ,Resolvent ,Mathematics ,lcsh:T57-57.97 ,010102 general mathematics ,Fractional calculus ,ComputingMilieux_GENERAL ,Computer Science::Multiagent Systems ,Nonlinear system ,Kernel (image processing) ,consensus ,lcsh:Applied mathematics. Quantitative methods ,020201 artificial intelligence & image processing - Abstract
In this paper, consensus of a leader-following problem is investigated. The leader-following problem describes a dynamics of the leader and a number of agents. The trajectory of the leader is given. The dynamics of each agent depends on the leader trajectory and others agents' inputs. Here, the leader-following problem is modeled by a system of nonlinear equations with Caputo fractional derivative, which can be rewritten as a system of Volterra equations. The main tools in the investigation are the properties of the resolvent kernel corresponding to the Volterra equations, and Schauder fixed point theorem. By study of the existence of an asymptotically stable solution of a suitable Volterra system, the sufficient conditions for consensus of the leader-following problem are obtained.
- Published
- 2019
11. Isotropic and anisotropic double-phase problems: old and new
- Author
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Vicenţiu D. Rădulescu
- Subjects
variable exponent ,Variable exponent ,lcsh:T57-57.97 ,General Mathematics ,010102 general mathematics ,Isotropy ,double-phase energy ,Differential operator ,01 natural sciences ,010101 applied mathematics ,Nonlinear system ,Double phase ,lcsh:Applied mathematics. Quantitative methods ,Content (measure theory) ,differential operator with unbalanced growth ,0101 mathematics ,Anisotropy ,Energy (signal processing) ,Mathematics ,Mathematical physics - Abstract
We are concerned with the study of two classes of nonlinear problems driven by differential operators with unbalanced growth, which generalize the \((p,q)\)- and \((p(x),q(x))\)-Laplace operators. The associated energy is a double-phase functional, either isotropic or anisotropic. The content of this paper is in relationship with pioneering contributions due to P. Marcellini and G. Mingione.
- Published
- 2019
12. Infinitely many solutions for some nonlinear supercritical problems with break of symmetry
- Author
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Anna Maria Candela and Addolorata Salvatore
- Subjects
perturbation method ,lcsh:T57-57.97 ,General Mathematics ,010102 general mathematics ,Mathematics::Analysis of PDEs ,weak cerami-palais-smale condition ,supercritical growth ,01 natural sciences ,Omega ,010101 applied mathematics ,Combinatorics ,Nonlinear system ,quasilinear elliptic equation ,Bounded function ,lcsh:Applied mathematics. Quantitative methods ,Domain (ring theory) ,break of symmetry ,Nabla symbol ,0101 mathematics ,Symmetry (geometry) ,Perturbation method ,ambrosetti-rabinowitz condition ,Mathematics - Abstract
In this paper, we prove the existence of infinitely many weak bounded solutions of the nonlinear elliptic problem \[\begin{cases}-\operatorname{div}(a(x,u,\nabla u))+A_t(x,u,\nabla u) = g(x,u)+h(x) &\text{in }\Omega,\\ u=0 &\text{on }\partial\Omega,\end{cases}\] where \(\Omega \subset \mathbb{R}^N\) is an open bounded domain, \(N\geq 3\), and \(A(x,t,\xi)\), \(g(x,t)\), \(h(x)\) are given functions, with \(A_t = \frac{\partial A}{\partial t}\), \(a = \nabla_{\xi} A\), such that \(A(x,\cdot,\cdot)\) is even and \(g(x,\cdot)\) is odd. To this aim, we use variational arguments and the Rabinowitz's perturbation method which is adapted to our setting and exploits a weak version of the Cerami-Palais-Smale condition. Furthermore, if \(A(x,t,\xi)\) grows fast enough with respect to \(t\), then the nonlinear term related to \(g(x,t)\) may have also a supercritical growth.
- Published
- 2019
13. Stochastic differential equations for random matrices processes in the nonlinear framework
- Author
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Hacène Boutabia, Sara Stihi, and Selma Meradji
- Subjects
Particle system ,Pure mathematics ,General Mathematics ,\(G\)-Brownian motion matrix ,lcsh:T57-57.97 ,010102 general mathematics ,eigenvalues ,Motion (geometry) ,eigenvectors ,01 natural sciences ,random matrices ,010104 statistics & probability ,Stochastic differential equation ,Nonlinear system ,Matrix (mathematics) ,Mathematics::Probability ,\(G\)-stochastic differential equations ,lcsh:Applied mathematics. Quantitative methods ,Symmetric matrix ,0101 mathematics ,Random matrix ,Eigenvalues and eigenvectors ,Mathematics - Abstract
In this paper, we investigate the processes of eigenvalues and eigenvectors of a symmetric matrix valued process \(X_{t}\), where \(X_{t}\) is the solution of a general SDE driven by a \(G\)-Brownian motion matrix. Stochastic differential equations of these processes are given. This extends results obtained by P. Graczyk and J. Malecki in [Multidimensional Yamada-Watanabe theorem and its applications to particle systems, J. Math. Phys. 54 (2013), 021503].
- Published
- 2018
14. On nonexistence of global in time solution for a mixed problem for a nonlinear evolution equation with memory generalizing the Voigt-Kelvin rheological model
- Author
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Volodymyr Il'kiv, Myroslava Vovk, Petro Pukach, and Zinovii Nytrebych
- Subjects
General Mathematics ,lcsh:T57-57.97 ,010102 general mathematics ,Mathematical analysis ,beam vibrations ,Voigt-Kelvin model ,01 natural sciences ,nonlinear evolution equation ,blowup ,010101 applied mathematics ,Vibration ,memory ,Nonlinear system ,Rheology ,boundary value problem ,lcsh:Applied mathematics. Quantitative methods ,Boundary value problem ,0101 mathematics ,Nonlinear evolution ,Time variable ,Mathematics - Abstract
The paper deals with investigating of the first mixed problem for a fifth-order nonlinear evolutional equation which generalizes well known equation of the vibrations theory. We obtain sufficient conditions of nonexistence of a global solution in time variable.
- Published
- 2017
15. Multiplicity results for perturbed fourth-order Kirchhoff-type problems
- Author
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Shapour Heidarkhani, Mohamad Reza Heidari Tavani, and Ghasem A. Afrouzi
- Subjects
multiple solutions ,Kirchhoff type ,General Mathematics ,Multiplicity results ,lcsh:T57-57.97 ,010102 general mathematics ,Mathematical analysis ,multiplicity results ,variational methods ,01 natural sciences ,Term (time) ,010101 applied mathematics ,Nonlinear system ,Fourth order ,critical point theory ,fourth-order Kirchhoff-type equation ,lcsh:Applied mathematics. Quantitative methods ,0101 mathematics ,Mathematics - Abstract
In this paper, we investigate the existence of three generalized solutions for fourth-order Kirchhoff-type problems with a perturbed nonlinear term depending on two real parameters. Our approach is based on variational methods.
- Published
- 2017
16. Controllability of semilinear systems with fixed delay in control
- Author
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Nagarajan Sukavanam and Surendra Kumar
- Subjects
General Mathematics ,lcsh:T57-57.97 ,Mathematical analysis ,first order delay system ,exact controllability ,Fixed point ,Space (mathematics) ,Lipschitz continuity ,Controllability ,Nonlinear system ,fixed point ,lcsh:Applied mathematics. Quantitative methods ,mild solution ,Point (geometry) ,Uniqueness ,Constant (mathematics) ,Mathematics - Abstract
In this paper, different sufficient conditions for exact controllability of semilinear systems with a single constant point delay in control are established in infinite dimensional space. The existence and uniqueness of mild solution is also proved under suitable assumptions. In particular, local Lipschitz continuity of a nonlinear function is used. To illustrate the developed theory some examples are given.
- Published
- 2015
17. Difference functional inequalities and applications
- Author
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Anna Szafrańska
- Subjects
difference functional inequalities ,Inequality ,General Mathematics ,media_common.quotation_subject ,interpolating operators ,lcsh:T57-57.97 ,Mathematical analysis ,initial boundary value problems ,Stability (learning theory) ,MathematicsofComputing_NUMERICALANALYSIS ,Finite difference coefficient ,Functional system ,Nonlinear system ,error estimates ,Convergence (routing) ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,lcsh:Applied mathematics. Quantitative methods ,Applied mathematics ,Boundary value problem ,difference methods ,stability and convergence ,Differential (infinitesimal) ,media_common ,Mathematics - Abstract
The paper deals with the difference inequalities generated by initial boundary value problems for hyperbolic nonlinear differential functional systems. We apply this result to investigate the stability of constructed difference schemes. The proof of the convergence of the difference method is based on the comparison technique, and the result for difference functional inequalities is used. Numerical examples are presented.
- Published
- 2014
18. Existence result for hemivariational inequality involving p(x)-Laplacian
- Author
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Sylwia Barnaś
- Subjects
Pure mathematics ,General Mathematics ,lcsh:T57-57.97 ,Mathematical analysis ,Palais-Smale condition ,Lipschitz continuity ,\(p(x)\)-Laplacian ,mountain pass theorem ,Nonlinear system ,Critical point (thermodynamics) ,Mountain pass theorem ,Variational inequality ,lcsh:Applied mathematics. Quantitative methods ,variable exponent Sobolev space ,Hemivariational inequality ,Laplace operator ,Mathematics - Abstract
In this paper we study the nonlinear elliptic problem with p(x)-Laplacian (hemi- variational inequality). We prove the existence of a nontrivial solution. Our approach is based on critical point theory for locally Lipschitz functionals due to Chang (J. Math. Anal. Appl. 80 (1981), 102-129). p(x)-Laplacian ( p(x)u ju(x)j p(x) 2 u(x)2 @j(x;u(x)) a.e. in
- Published
- 2012
19. Existence and asymptotic behavior of positive continuous solutions for a nonlinear elliptic system in the half space
- Author
-
Sameh Turki
- Subjects
Discrete mathematics ,elliptic system ,Class (set theory) ,Nonlinear system ,General Mathematics ,lcsh:T57-57.97 ,lcsh:Applied mathematics. Quantitative methods ,regular equation ,System U ,asymptotic behavior ,Half-space ,Potential theory ,Mathematics - Abstract
This paper deals with the existence and the asymptotic behavior of positive continuous solutions of the nonlinear elliptic system \(\Delta u=p(x)u^{\alpha}v^r\), \(\Delta v = q(x)u^s v^{\beta}\), in the half space \(\mathbb{R}^n_+ :=\{x=(x_1,..., x_n)\in \mathbb{R}^n : x_n \gt 0\}\), \(n \geq 2\), where \(\alpha, \beta \gt 1\) and \(r, s \geq 0\). The functions \(p\) and \(q\) are required to satisfy some appropriate conditions related to the Kato class \(K^{\infty}(\mathbb{R}^n_+)\). Our approach is based on potential theory tools and the use of Schauder's fixed point theorem.
- Published
- 2012
20. Weak solutions for nonlinear fractional differential equations with integral boundary conditions in Banach spaces
- Author
-
Mouffak Benchohra and Fatima-Zohra Mostefai
- Subjects
Class (set theory) ,Caputo fractional derivative ,General Mathematics ,Weak solution ,lcsh:T57-57.97 ,Mathematical analysis ,Banach space ,weak solution ,Pettis integrals ,Fractional calculus ,Nonlinear fractional differential equations ,Nonlinear system ,boundary value problem ,measure of weak noncompactness ,lcsh:Applied mathematics. Quantitative methods ,Boundary value problem ,Fractional differential ,Mathematics - Abstract
The aim of this paper is to investigate a class of boundary value problems for fractional differential equations involving nonlinear integral conditions. The main tool used in our considerations is the technique associated with measures of weak noncompactness.
- Published
- 2012
21. Boundary value problems for n-th order differential inclusions with four-point integral boundary conditions
- Author
-
Bashir Ahmad and Sotiris K. Ntouyas
- Subjects
four-point integral boundary conditions ,General Mathematics ,lcsh:T57-57.97 ,Mathematical analysis ,Regular polygon ,existence ,Fixed-point theorem ,Type (model theory) ,fixed point theorems ,Nonlinear system ,Schauder fixed point theorem ,Differential inclusion ,differential inclusions ,nonlinear alternative of Leray Schauder type ,lcsh:Applied mathematics. Quantitative methods ,Point (geometry) ,Boundary value problem ,Mathematics - Abstract
In this paper, we discuss the existence of solutions for a four-point integral boundary value problem of \(n\)-th order differential inclusions involving convex and non-convex multivalued maps. The existence results are obtained by applying the nonlinear alternative of Leray Schauder type and some suitable theorems of fixed point theory.
- Published
- 2012
22. Oscillation theorems concerning non-linear differential equations of the second order
- Author
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E. M. Elabbasy and Sh. R. Elzeiny
- Subjects
Complement (group theory) ,Pure mathematics ,second order ,Differential equation ,Oscillation ,General Mathematics ,lcsh:T57-57.97 ,Mathematical analysis ,differential equations ,oscillation ,Nonlinear system ,lcsh:Applied mathematics. Quantitative methods ,Order (group theory) ,nonlinear ,Sign (mathematics) ,Mathematics - Abstract
This paper concerns the oscillation of solutions of the differential eq. \[ \left[ r\left( t\right) \psi \left(x\left( t\right) \right) f\text{ }( \overset{\cdot }{x}(t))\right]^{\cdot }+q\left( t\right) \varphi (g\left( x\left( t\right) \right), r\left( t\right) \psi \left( x\left( t\right) \right) f(\overset{\cdot }{x}(t)))=0,\] where \(u\varphi \left( u,v\right) \gt 0\) for all \(u\neq 0\), \(xg\left( x\right) \gt 0\), \(xf\left( x\right)\gt 0\) for all \(x\neq 0\), \(\psi \left( x\right) \gt 0\) for all \(x\in \mathbb{R}\), \(r\left( t\right) \gt 0\) for \(t\geq t_{0}\gt 0\) and \(q\) is of arbitrary sign. Our results complement the results in [A.G. Kartsatos, On oscillation of nonlinear equations of second order, J. Math. Anal. Appl. 24 (1968), 665–668], and improve a number of existing oscillation criteria. Our main results are illustrated with examples.
- Published
- 2011
23. Initial value problem for the time-dependent linear Schrödinger equation with a point singular potential by the unified transform method
- Author
-
Yan Rybalko
- Subjects
Integral representation ,Integrable system ,lcsh:T57-57.97 ,General Mathematics ,010102 general mathematics ,interface problems ,Schrödinger equation ,the Fokas unified transform method ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Nonlinear system ,Mathematics - Analysis of PDEs ,lcsh:Applied mathematics. Quantitative methods ,symbols ,Initial value problem ,Applied mathematics ,Point (geometry) ,0101 mathematics ,Value (mathematics) ,Mathematics - Abstract
We study an initial value problem for the one-dimensional non-stationary linear Schr\"odinger equation with a point singular potential. In our approach, the problem is considered as a system of coupled initial-boundary value (IBV) problems on two half-lines, to which we apply the unified approach to IBV problems for linear and integrable nonlinear equations, also known as the Fokas unified transform method. Following the ideas of this method, we obtain the integral representation of the solution of the initial value problem. Since the unified approach is known as providing efficient solutions to both linear and nonlinear problems, the present paper can be viewed as a step in solving the initial value problem for the non-stationary {\em nonlinear} Schr\"odinger equation with a point singular potential., Comment: 12 pages, 1 figure
- Published
- 2018
24. On some impulsive fractional differential equations in Banach spaces
- Author
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YanLong Yang, JinRong Wang, and W. Wei
- Subjects
Pure mathematics ,General Mathematics ,fractional differential equations with impulses ,lcsh:T57-57.97 ,Mathematical analysis ,Banach space ,existence ,Fixed-point theorem ,\(PC\)-mild solutions ,Nonlinear system ,Gronwall's inequality ,lcsh:Applied mathematics. Quantitative methods ,Fractional differential ,C0-semigroup ,nonlinear impulsive singular version of the Gronwall inequality ,Mathematics - Abstract
This paper deals with some impulsive fractional differential equations in Banach spaces. Utilizing the Leray-Schauder fixed point theorem and the impulsive nonlinear singular version of the Gronwall inequality, the existence of \(PC\)-mild solutions for some fractional differential equations with impulses are obtained under some easily checked conditions. At last, an example is given for demonstration.
- Published
- 2010
25. Uniqueness of solutions of a generalized Cauchy problem for a system of first order partial functional differential equations
- Author
-
Milena Netka
- Subjects
Cauchy problem ,Differential equation ,General Mathematics ,lcsh:T57-57.97 ,Mathematical analysis ,estimates of the Perron type ,Type (model theory) ,Nonlinear system ,functional differential equations ,lcsh:Applied mathematics. Quantitative methods ,Uniqueness ,Hyperbolic partial differential equation ,Differential (mathematics) ,Mathematics ,Numerical partial differential equations ,comparison methods - Abstract
The paper is concerned with weak solutions of a generalized Cauchy problem for a nonlinear system of first order differential functional equations. A theorem on the uniqueness of a solution is proved. Nonlinear estimates of the Perron type are assumed. A method of integral functional inequalities is used.
- Published
- 2009
26. Oscillatory and asymptotic behavior of a third-order nonlinear neutral differential equation
- Author
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John R. Graef, Ercan Tunҫ, and Said R. Grace
- Subjects
Class (set theory) ,Third order nonlinear ,nonoscillatory ,Differential equation ,lcsh:T57-57.97 ,General Mathematics ,Mathematical analysis ,Zero (complex analysis) ,First-order partial differential equation ,010103 numerical & computational mathematics ,third order ,01 natural sciences ,010101 applied mathematics ,Nonlinear system ,Third order ,oscillatory solution ,lcsh:Applied mathematics. Quantitative methods ,asymptotic behavior ,0101 mathematics ,Neutral differential equations ,neutral differential equations ,Mathematics - Abstract
WOS:000416717000005 This paper discusses oscillatory and asymptotic properties of solutions of a class of third-order nonlinear neutral differential equations. Some new sufficient conditions for a solution of the equation to be either oscillatory or to converges to zero are presented. The results obtained can easily be extended to more general neutral differential equations as well as to neutral dynamic equations on time scales. Two examples are provided to illustrate the results.
- Published
- 2017
27. Monotone method for Riemann-Liouville multi-order fractional differential systems
- Author
-
Zachary Denton
- Subjects
Monotone method ,lower and upper solutions ,lcsh:T57-57.97 ,General Mathematics ,Mathematical analysis ,Linear system ,fractional differential systems ,Order (ring theory) ,Function (mathematics) ,01 natural sciences ,010305 fluids & plasmas ,010101 applied mathematics ,Nonlinear system ,multi-order systems ,Hybrid system ,lcsh:Applied mathematics. Quantitative methods ,0103 physical sciences ,Applied mathematics ,Development (differential geometry) ,0101 mathematics ,Fractional differential ,monotone method ,Mathematics - Abstract
In this paper we develop the monotone method for nonlinear multi-order \(N\)-systems of Riemann-Liouville fractional differential equations. That is, a hybrid system of nonlinear equations of orders \(q_i\) where \(0 \lt q_i \lt 1\). In the development of this method we recall any needed existence results along with any necessary changes. Through the method's development we construct a generalized multi-order Mittag-Leffler function that fulfills exponential-like properties for multi-order systems. Further we prove a comparison result paramount for the discussion of fractional multi-order inequalities that utilizes lower and upper solutions of the system. The monotone method is then developed via the construction of sequences of linear systems based on the upper and lower solutions, and are used to approximate the solution of the original nonlinear multi-order system.
- Published
- 2016
28. Analytic continuation of solutions of some nonlinear convolution partial differential equations
- Author
-
Hidetoshi Tahara
- Subjects
Partial differential equation ,convolution equations ,lcsh:T57-57.97 ,General Mathematics ,media_common.quotation_subject ,Analytic continuation ,Mathematical analysis ,summability ,Holomorphic function ,Infinity ,Convolution ,analytic continuation ,Nonlinear system ,lcsh:Applied mathematics. Quantitative methods ,partial differential equations ,sector ,Mathematics ,media_common ,Variable (mathematics) ,Numerical partial differential equations - Abstract
The paper considers a problem of analytic continuation of solutions of some nonlinear convolution partial differential equations which naturally appear in the summability theory of formal solutions of nonlinear partial differential equations. Under a suitable assumption it is proved that any local holomorphic solution has an analytic extension to a certain sector and its extension has exponential growth when the variable goes to infinity in the sector.
- Published
- 2015
29. Existence and uniqueness of the solutions of some degenerate nonlinear elliptic equations
- Author
-
Albo Carlos Cavalheiro
- Subjects
Dirichlet problem ,Pure mathematics ,degenerate nonlinear elliptic equations ,lcsh:T57-57.97 ,General Mathematics ,Degenerate energy levels ,Mathematical analysis ,Mathematics::Analysis of PDEs ,weighted Sobolev spaces ,Omega ,Sobolev space ,Nonlinear system ,lcsh:Applied mathematics. Quantitative methods ,Uniqueness ,Nabla symbol ,Mathematics - Abstract
In this paper we are interested in the existence of solutions for the Dirichlet problem associated with degenerate nonlinear elliptic equations \[\begin{split}&-\sum_{j=1}^n D_j{\bigl[}{\omega}(x) {\cal A}_j(x, u, {\nabla}u){\bigr]} + b(x, u, {\nabla}u)\,{\omega}(x) + g(x)\,u(x)=\\&= f_0(x) - \sum_{j=1}^nD_jf_j(x) \quad{\rm on}\quad {\Omega}\end{split}\] in the setting of the weighted Sobolev spaces \({\rm W}_0^{1,p}(\Omega, \omega)\).
- Published
- 2014
30. On the dependence on parameters for second order discrete boundary value problems with the p(k)-Laplacian
- Author
-
Joanna Smejda and Renata Wieteska
- Subjects
Class (set theory) ,geography ,geography.geographical_feature_category ,discrete boundary value problems ,lcsh:T57-57.97 ,General Mathematics ,Mathematical analysis ,variational methods ,Mathematics::Analysis of PDEs ,mountain pass theorem ,Discrete equation ,Nonlinear system ,lcsh:Applied mathematics. Quantitative methods ,Mountain pass theorem ,Order (group theory) ,Mountain pass ,Boundary value problem ,Laplace operator ,Mathematics - Abstract
In this paper we study the existence and the nonexistence of solutions for the boundary value problems of a class of nonlinear second-order discrete equations depending on a parameter. Variational (the mountain pass technique) and non-variational methods are applied.
- Published
- 2014
31. Global existence and asymptotic behavior for a nonlinear degenerate SIS model
- Author
-
Tarik Ali Ziane
- Subjects
global existence ,Degenerate diffusion ,reaction diffusion systems ,lcsh:T57-57.97 ,General Mathematics ,Degenerate energy levels ,Mathematical analysis ,Nonlinear system ,lcsh:Applied mathematics. Quantitative methods ,Reaction–diffusion system ,population dynamics ,Epidemic disease ,asymptotic behavior ,Statistical physics ,degenerate diffusion ,Mathematics - Abstract
In this paper we investigate the global existence and asymptotic behavior of a reaction diffusion system with degenerate diffusion arising in the modeling and the spatial spread of an epidemic disease.
- Published
- 2013
32. Stability by Krasnoselskii’s theorem in totally nonlinear neutral differential equations
- Author
-
Ahcene Djoudi, Ishak Derrardjia, and Abdelouaheb Ardjouni
- Subjects
Pure mathematics ,Work (thermodynamics) ,lcsh:T57-57.97 ,General Mathematics ,Direct method ,Mathematical analysis ,Zero (complex analysis) ,Fixed-point theorem ,Krasnoselskii-Burton theorem ,stability ,Fixed point ,Stability (probability) ,Nonlinear system ,nonlinear neutral equation ,fixed point ,Exponential stability ,lcsh:Applied mathematics. Quantitative methods ,Mathematics - Abstract
In this paper we use fixed point methods to prove asymptotic stability results of the zero solution of a class of totally nonlinear neutral differential equations with functional delay. The study concerns \[x'(t)=a(t)x^3(t)+c(t)x'(t-r(t))+b(t)x^3(t-r(t)).\] The equation has proved very challenging in the theory of Liapunov's direct method. The stability results are obtained by means of Krasnoselskii-Burton's theorem and they improve on the work of T.A. Burton (see Theorem 4 in [Liapunov functionals, fixed points, and stability by Krasnoselskii's theorem, Nonlinear Studies 9 (2001), 181-190]) in which he takes \(c=0\) in the above equation.
- Published
- 2013
33. On a nonlinear integrodifferential evolution inclusion with nonlocal initial conditions in Banach spaces
- Author
-
Zuomao Yan
- Subjects
Class (set theory) ,lcsh:T57-57.97 ,General Mathematics ,Mathematical analysis ,Banach space ,Fixed-point theorem ,nonlinear integrodifferential evolution inclusions ,Resolvent formalism ,resolvent operator ,Fixed point ,Nonlinear system ,fixed point ,lcsh:Applied mathematics. Quantitative methods ,Resolvent operator ,nonlocal initial condition ,Resolvent ,Mathematics - Abstract
In this paper, we discuss the existence results for a class of nonlinear integro- dierential evolution inclusions with nonlocal initial conditions in Banach spaces. Our results are based on a fixed point theorem for condensing maps due to Martelli and the resolvent operators combined with approximation techniques.
- Published
- 2012
34. Differential difference inequalities related to parabolic functional differential equations
- Author
-
Milena Netka
- Subjects
lcsh:T57-57.97 ,General Mathematics ,Mathematical analysis ,method of lines ,Delay differential equation ,Integrating factor ,Examples of differential equations ,Stochastic partial differential equation ,parabolic functional differential equations ,Nonlinear system ,lcsh:Applied mathematics. Quantitative methods ,stability and convergence ,Differential algebraic equation ,Numerical partial differential equations ,Mathematics ,Separable partial differential equation - Abstract
Initial boundary value problems for nonlinear parabolic functional differential equations are transformed by discretization in space variables into systems of ordinary functional differential equations. A comparison theorem for differential difference inequalities is proved. Sufficient conditions for the convergence of the method of lines is given. Nonlinear estimates of the Perron type for given operators with respect to functional variables are used. Results obtained in the paper can be applied to differential integral problems and to equations with deviated variables.
- Published
- 2010
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