In studying diﬀerential **equations** in the complex plane C, it is always an interesting and quite diﬃcult problem to prove the existence or uniqueness of the entire or meromorphic solution of a given equation. There have been many studies and results obtained lately that relate to the existence or growth of entire or meromorphic solutions of various types of complex diﬀerence **equations** and diﬀerence-diﬀerential **equations**. By our methods, one can discuss the entire solutions for the algebraic diﬀerence equation and diﬀerence- diﬀerential equation of the form

In this paper we have built a rigorous algebraic setting for **difference** and rational (pseudo–**difference**) operators with coefficients in a **difference** field F and study their properties. In particular, we formulate a criteria for a rational operator to be weakly nonlocal. We have defined and studied preHamiltonian pairs, which is a generalization of the well known bi-Hamiltonian structures in the theory of integrable systems. By definition a preHamiltonian operator is an operator whose images form a Lie subalgebra in the Lie algebra of evolutionary derivations of F. The latter can be directly verified and it is a relatively simple problem comparing to the verification of the Jacobi identity for Hamiltonian operators. We have shown that a recursion Nijenhuis operator is a ratio of **difference** operators from a preHamiltonian pair. Thus for a given rational operator, to test whether it is Nijenhuis or not can be done systematically. We applied our theoretical results to integrable **differential** **difference** **equations** in two aspects:

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In El’sgol’ts [10], similar boundary value problems with solutions that show quick oscillations are considered. Kadalbajoo and Sharma [2],[3], Kadalbajoo and Ramesh [4] and Kadalbajoo and Kumar [5],[6] started a broad numerical work for solving singularly perturbed delay **differential** **equations** based on finite **difference** scheme, fitted mesh and B-spline method, piecewise uniform mesh. Lange and Miura [11],[12] gave asymptotic approaches in the study of class of boundary value problems for linear second order **differential** **difference** **equations** in which the highest order derivative is multiplied by small parameter. They have also additionally talked about the impact of small shifts on the oscillatory solution of the problem.Duressa et al. and Sirisha L et al. [13],[14] exhibited numerical methods to approximate the solution of boundary value problems for SPDDEs with delay as well as advance.

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solutions concerned. During the last few decades, much work has been done for developing various Gronwall-Bellman type inequalities. These inequalities involve Gronwall-Bellman type **differential** and integral inequalities [13-18], retarded inequalities [19-23], **difference** inequalities [24-28], Volterra- Fredholm type inequalities [24,27], and dynamic inequal- ities on time scales [29-33]. Recently, some authors have researched and established some Gronwall-Bellman type fractional **differential** and integral inequalities [34-37], which have contributed much in the research of qualitative and quantitative properties of solutions of certain fractional d- ifferential and integral **equations**. However, the earlier in- equalities established are inadequate in the research of the properties of solutions of some certain fractional **differential** **equations**, and it is necessary to establish new Gronwall- Bellman type inequalities so as to fulfill corresponding analysis.

. The **equations** of this form arise at the description of the phenomena of diffusion or heat conductivity with the sources, nonlocally dependent on the state . Physically the state u means the density distribution of the con- centration or the temperature. In particular, the **equations** of a kind (1) arise at research of problems of manage- ment by the phenomena of a heat transfer in which dy- namics of a state u is given by the **differential** equation

u t S t B T t W t As an application we consider some control systems governed by partial **differential** **equations**, integrodiffer- ential **equations** and **difference** **equations** that can be studied using these results. Particularly, we work in de- tails the following controlled damped wave equation

The theory of upper and lower solutions is well known to be an eﬀective method to deal with the existence of solutions for the boundary value problems of the fractional diﬀeren- tial **equations**. In [] the authors used the method of upper and lower solutions and in- vestigated the existence of solutions for initial value problems. By the same method some people got the solutions of boundary value problems for fractional diﬀerential **equations**, such as [, ]. To the best of our knowledge, only few papers considered the existence of solutions by using the method of upper and lower solutions for boundary value problems with fractional coupled systems.

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The rest of this paper is organized as follows. In Section 2, we give some necessary no- tions and state some preliminary results. In Section 3, we establish a necessary condition (maximum principle) and a suﬃcient condition (veriﬁcation theorem) for the Nash equi- librium point. In Section 4, we study a linear-quadratic game problem under partial infor- mation. We derive the ﬁltering **equations** and prove the existence and uniqueness for the Nash equilibrium point. In Section 5, we solve a pension fund management problem with nonlinear expectation and obtain the explicit solution.

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**Difference** **equations** usually occur due to certain phenomena over time, and it play an important role in describing discrete dynamical systems [1]. **Difference** equation and their associated operators not only play a role in their own right as direct mathematical models of physical phenomena but also provide the field of numerical analysis with powerful tools. **Difference** **equations** also occur in combined form with **differential** **equations**, commonly called **differential**-**difference** **equations** yielding rich models, particularly in control theory. **Difference** **equations** are widely used in the theory of probability, biology, engineering, social and behavioral sciences.

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In a recent study [24], the Hölder continuity assumption is relaxed to C p continuity of the functions involved in the Riemann-Liouville fractional **differential** equation. In the following we also prove a comparison result by not requiring the Hölder continuity with a different argument for Caputo fractional **differential** **equations**. It is obvious that this result is essential to extend the applicability of iterative techniques such as the monotone iterative technique and the method of quasilinearization.

Abstract. This paper presents a new approach to constructing Non-Standard Finite **Difference** Method (NSFDM) for the solution of autonomous Ordinary **Differential** **Equations** (ODEs). The need for this method came up due to the shortcomings of the standard methods; in which the qualitative properties of the exact solutions are not usually transferred to the numerical solutions. These shortcomings affect the stability of the standard approach. The new nonstandard finite **difference** method have the property that its solution do not exhibit numerical instabilities. Keywords: Autonomous function; Denominator function; Numerator function; NSFDM.

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concept of Hukuhara derivatives with diﬀerent diﬀerences [, ]. The diﬀerent deﬁnition of the Hukuhara derivative will lead to a diﬀerent level-cut system. The new generalization of the Hukuhara diﬀerence, due to Stefanini [], raising the fuzzy diﬀerential **equations** associated with the level-cut system involving the min-max terms, is as follows:

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The variational iteration method (VIM) [–] has been one of the often used non- linear methods in initial boundary value problems of diﬀerential **equations**. In this study, the extension of the method into FQC is undertaken and the Caputo q-fractional initial value problems are investigated. Our study is organized as follows. In Section , the basic idea of the VIM is illustrated. In Section , the VIM is extended to q-diﬀerence **equations**, and the Lagrange multipliers of the method are presented for the **equations** of high-order q-derivatives. In Section , recent development of the method in fractional calculus is introduced. Following Section , the application of the VIM in q-fractional calculus is considered. Then the method is applied to the Caputo q-fractional initial value problem.

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3. Beardon, AF: Entire solutions of f(kz) = kf (z)f (z). Comput. Methods Funct. Theory 12(1), 273-278 (2012) 4. Goldstein, R: Some results on factorization of meromorphic functions. J. Lond. Math. Soc. 4(2), 357-364 (1971) 5. Goldstein, R: On meromorphic solutions of certain functional **equations**. Aequ. Math. 18(1), 112-157 (1978) 6. Gundersen, G: Finite order solutions of second order linear diﬀerential **equations**. Trans. Am. Math. Soc. 305(1),

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The conventional subtraction arithmetic on interval numbers makes studies on interval systems diﬃcult because of irreversibility on addition, whereas the gH-diﬀerence as a popular concept can ensure interval analysis to be a valuable research branch like real analysis. However, many properties of interval numbers still remain open. This work focuses on developing a complete normed quasi-linear space composed of continuous interval-valued functions, in which some fundamental properties of continuity, diﬀerentiability, and integrability are discussed based on the gH-diﬀerence, the gH-derivative, and the Hausdorﬀ-Pompeiu metric. Such properties are adopted to investigate semi-linear interval diﬀerential **equations**. While the existence and uniqueness of the (i)- or (ii)-solution are studied, a necessary condition that the (i)- and the (ii)-solutions to be strong solutions is obtained. For such a kind of equation it is demonstrated that there exists at least a strong solution under certain assumptions.

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Sabbavarapu Nageswara Rao has received M.Sc. from Andhra University, M. Tech (CSE) from Jawaharlal Nehru Technological University, and Ph.D from Andhra University under the esteemed guidance of Prof. K. Rajendra Prasad. He served in the cadre of Assistant professor and Associate professor in Sriprakash College of Engineering and Professor in AITAM, Tekkali. Presently, Dr. Rao is working in the Department of Mathematics, Jazan University, Jazan, Kingdom of Saudi Arabia. His major research interest includes ordinary **differential** **equations**, **difference** **equations**, dynamic **equations** on time scales, p-Laplacian, fractional order **differential** **equations** and boundary value problems. He published several research papers on the above topics in various national and international journals.

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In this paper we review two concepts directly related to the Lax representations: Darboux trans- formations and Recursion operators for integrable systems. We then present an extensive list of integrable di ff erential-di ff erence **equations** together with their Hamiltonian structures, recursion oper- ators, nontrivial generalised symmetries and Darboux-Lax representations. The new results include multi-Hamiltonian structures and recursion operators for integrable Volterra type **equations**, integrable discretization of derivative nonlinear Schr ¨odinger **equations** such as the Kaup-Newell lattice, the Chen- Lee-Liu lattice and the Ablowitz-Ramani-Segur (Gerdjikov-Ivanov) lattice. We also compute the weakly nonlocal inverse recursion operators.

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[13] A. A. Opanuga, E. A. Owoloko, H. I. Okagbue and O. O. Agboola, “Finite **difference** method and Laplace transform for boundary value problems,”in Lecture Notes in Engineering and Computer Science: World Congress on Engineering and Computer Science 2017, pp. 65-69. [14] A. J. Jerri, Linear **difference** **equations** with discrete transform meth-

9. Liu, K., Liu, X., Cao, T.B. : Value distribution of the **difference** opera- tor, Advances in **Difference** **equations**, Vol.2011, Art ID 234215,12 pages. 10. Yang, C.C., Hua, X.H.: Uniqueness and value sharing of meromorphic functions, Ann. Acad.Sci. Fenn. Math. 22(1997), pp.395-406.

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of an Euler-Maruyuma (explicit and implicit) finite **difference** scheme is of order one in space and one-half in time. More recently, it was shown by I. Gy¨ongy and N.V. Krylov that under certain regularity conditions, the rate of convergence in space of a semi-discretized finite **difference** approximation of a linear second order SPDE driven by continuous martingale noise can be accelerated to any order by Richardson’s extrapolation method. For the non- degenerate case, we refer to [10] and [11], and for the degenerate case, we refer to [7]. In [14] and [15], E. Hall proved that the same method of acceleration can be applied to implicit time-discretized SPDEs driven by continuous martingale noise.

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