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The paper deals with a class of discrete fractional boundary value problems. We construct the corresponding Green's function, analyse it in detail and establish several of its key properties. Then, by using the fixed point index theory, the existence of multiple positive solutions is obtained, and the uniqueness of the solution is proved by a new theorem on an ordered metric space established by M. Jleli, et al. (2012).

Multipoint boundary value problems for discrete equations

Pavel Drábek, Harold Bevan Thompson, Christopher Tisdell (2001)

Commentationes Mathematicae Universitatis Carolinae

In this work we establish existence results for solutions to multipoint boundary value problems for second order difference equations with fully nonlinear boundary conditions involving two, three and four points. Our results are also applied to systems.

Multisommabilité des séries entières solutions formelles d’une équation aux $q$ -différences linéaire analytique

Fabienne Marotte, Changgui Zhang (2000)

Annales de l'institut Fourier

Nous introduisons une version $q$ -analogue du procédé d’accélération élémentaire d’Écalle-Martinet-Ramis et définissons la notion de série entière $G q$ -multisommable. Nous montrons que toute série entière solution formelle d’une équation aux $q$ -différences linéaire analytique est $G q$ -multisommable.

Multisummability for some classes of difference equations

Boele L. J. Braaksma, Bernard F. Faber (1996)

Annales de l'institut Fourier

This paper concerns difference equations $y (x + 1) = G (x, y)$ where $G$ takes values in $C^{n}$ and $G$ is meromorphic in $x$ in a neighborhood of $\infty$ in $C$ and holomorphic in a neighborhood of 0 in $C^{n}$ . It is shown that under certain conditions on the linear part of $G$ , formal power series solutions in $x^{- 1 / p}, p \in N,$ are multisummable. Moreover, it is shown that formal solutions may always be lifted to holomorphic solutions in upper and lower halfplanes, but in general these solutions are not uniquely determined by the formal solutions.

Natural and artificially controlled connections among steady states of a climate model.

Jesús Ildefonso Díaz, Víctor García (2007)

RACSAM

Necessary and sufficient conditions for oscillations of delay partial difference equations

Bing Gen Zhang, Shu Tang Liu (1995)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

This paper is concerned with the delay partial difference equation (1) $A_{m + 1, n} + A_{m, n + 1} - A_{m, n} + Σ_{i = 1}^{u} p_{i} A_{m - k_{i}, n - l_{i}} = 0$ where $p_{i}$ are real numbers, $k_{i}$ and $l_{i}$ are nonnegative integers, u is a positive integer. Sufficient and necessary conditions for all solutions of (1) to be oscillatory are obtained.

Necessary and sufficient conditions for oscillations of delay partial difference equations.

Bing Liu, Aimin Zhao, Jurang Yan (1997)

Collectanea Mathematica

In this paper we study two classes of delay partial difference equations with constant coefficients. Explicit necessary and sufficient conditions for the oscillation of the solutions of these equations are obtained.

Necessary and sufficient conditions for stability of Volterra integro-dynamic equation on time scales

Youssef N. Raffoul (2016)

Archivum Mathematicum

In this research we establish necessary and sufficient conditions for the stability of the zero solution of scalar Volterra integro-dynamic equation on general time scales. Our approach is based on the construction of suitable Lyapunov functionals. We will compare our findings with known results and provides application to quantum calculus.

Necessary and sufficient conditions for the oscillation of delay differential equations with a piecewise constant argument.

Agwo, H.A. (1998)

International Journal of Mathematics and Mathematical Sciences

Necessary and sufficient conditions for the oscillation of forced nonlinear second order delay difference equation

Ethiraju Thandapani, L. Ramuppillai (1999)

Kybernetika

In this paper the authors give necessary and sufficient conditions for the oscillation of solutions of nonlinear delay difference equations of Emden– Fowler type in the form $Δ^{2} y_{n - 1} + q_{n} y_{σ (n)}^{γ} = g_{n}$ , where $γ$ is a quotient of odd positive integers, in the superlinear case $(γ > 1)$ and in the sublinear case $(γ < 1)$ .

Necessary and sufficient conditions for the oscillatory and asymptotic behaviour of solutions to neutral delay dynamic equations.

Karpuz, Basak, Ocalan, Ozkan, Rath, Radhanath (2009)

Electronic Journal of Differential Equations (EJDE) [electronic only]

Necessary conditions for linear noncooperative N-player delta differential games on time scales

Natália Martins, Delfim F.M. Torres (2011)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

We present necessary conditions for linear noncooperative N-player delta dynamic games on an arbitrary time scale. Necessary conditions for an open-loop Nash-equilibrium and for a memoryless perfect state Nash-equilibrium are proved.

Necessary conditions for the solutions of second order nonlinear neutral delay difference equations to be oscillatory or tend to zero.

Rath, R.N., Dix, J.G., Barik, B.L.S., Dihudi, B. (2007)

International Journal of Mathematics and Mathematical Sciences

Nevanlinna theory for the difference operator.

Halburd, R.G., Korhonen, R.J. (2006)

Annales Academiae Scientiarum Fennicae. Mathematica

Nonlinear delay discrete inequalities and their applications to Volterra type difference equations.

Wu, Yu, Li, Xiaopei, Deng, Shengfu (2010)

Advances in Difference Equations [electronic only]

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