The theory of diﬀerence equations occupies now a central position in applicable analysis. We just refer to the recent results of Agarwal et al. 1, Yu and Guo 3, Kon´e and Ouaro 4, Guiro et al. 5, Cai and Yu 6, Zhang and Liu 7, Mih˘ailescu et al. 8, Candito and D Agui 9, Cabada et al. 10, Jiang and Zhou 11, and the references therein. In 7, the authors studied the following **problem**:

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by using an algorithmic technique we developed firstly for perturbed differ- ence equations (see [10], [11], [16], [17], [18], [19]). Recently, many researchers have studied **discrete** versions of **boundary** **value** problems (BVPs) (see [1], [4], [5], [8]), and applications of **two**-**point** BVP algorithms arise in pollu- tion control problems, nuclear reactor heat transfer and vibration. For an abbreviated writing, we denote the partial derivative ∂ k 1+ k 2+ ···+kp f (x 1 ,x 2 , ··· ,x p )

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The construction of the projection W is analagous to a projection constructed for the special case of a **two**-**point** **boundary** **value** **problem** in [9]. The projections P and E enable us to study the existence of solutions to the **boundary** **value** **problem** (1)-(2) using the classic Lyapunov Schmidt Reduction. Our approach to this procedure is entirely self-contained. We do provide references, [4, 5, 6, 7, 11, 16, 17, 19, 21, 22] for readers interested in a more general formulation and for those interested in applications to differential or difference equations.

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studied the question of existence and uniqueness of a solution for this type of differential equations, see e.g. [4, 5] and references cited therein. Fi- nite differences method [4], collocation methods [10, 15], Richardson extrapolation [6], shooting techniques [7, 11], projection method [16], suc- cessive and Pad approximations [5, 12] and suc- cessive interpolations [8] are some of the existing numerical methods for **boundary** **value** problems of differential equations with deviating argument. In this paper, based on reproducing kernel with polynomial form [13, 14], we propose an effec- tive numerical technique for solving the following second order **two**-**point** **boundary** **value** problems with a deviating argument:

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[3] C. D. ALIPRANTIS and K. C. BORDER, Infinite dimensional analysis, Springer-Verlag, 1994. [4] D. ALSPACH, A fixed **point** free nonexpansive map, Proc. Amer. Math. Soc., 82 (1981), 423–424. [5] M. ALTMAN, A fixed **point** theorem for completely continuous operators in Banach spaces, Bull. Acad. Polon. Sci., 3 (1955), 409–413.

By using the fixed-**point** index theory and Leggett-Williams fixed-**point** theorem, we study the exis- tence of multiple solutions to the three-**point** **boundary** **value** **problem** u t atft, ut, u t 0, 0 < t < 1; u0 u 0 0; u 1 − αu η λ, where η ∈ 0, 1/2, α ∈ 1/2η, 1/η are constants, λ ∈ 0, ∞ is a parameter, and a, f are given functions. New existence theorems are obtained, which extend and complement some existing results. Examples are also given to illustrate our results.

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Finally, we note that when at ≡ 1 in 1.2, the existence of odd solutions or even solutions was investigated by R. Ma and H. Ma 16 under some parity conditions on the nonlinearities. The existence of solutions of second-order **discrete** **problem** at resonance was studied by Rodriguez in 17 , in which the nonlinearity is required to be bounded. For other results on **discrete** **boundary** **value** problems, see Kelley and Peterson 18 , Agarwal and O’Regan 19 , Rachunkova and Tisdell 20 , Yu and Guo 21 , Atici and Cabada 22 , Bai and Xu 23 . However, these papers do not address the **problem** under “asymptotic nonuniform resonance” conditions.

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2. Exact discretization of the **problem**. In this section, we present the exact dis- cretization of (1.1). We ﬁrst introduce special sets of basis functions that we use for the discretization. Then we get a system of equations for the exact solution at the mesh points. Our working space is the space of continuous functions on the interval [0,1]. Let π = { 0 = x 0 < x 1 < ··· < x N = 1 } be a given partition of the interval [0,1]. We

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larities at the points a and b, are studied. Also the ordinary diﬀerential equations with strong singularities under **two**-**point** **boundary** conditions are studied in the articles [, ] by Kiguradze. In the papers [–] these results are generalized for a linear diﬀer- ential equation with deviating arguments, i.e., the Agarwal-Kiguradze type theorems are proved, which guarantee the Fredholm property for the linear diﬀerential equation with deviating arguments. In this paper, on the basis of articles [, ], we prove the a priori boundedness principle for **problem** (.), (.) from which several suﬃcient conditions of the solvability of this **problem** follow.

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reason for many authors to use these approximation for solving problems. Numerical solutions of **boundary** **value** problems by using Sinc functions have been studied first by Frank Stenger more than thirty years ago [12]. The efficiency of the method has been formally proved by many researchers. Bialecki [1] used Sinc-collocation method to solve a linear **two** **point** **boundary** **value** problems. Lund [6] applied symmetrization Sinc-Galerkin for **boundary** **value** problems. Dehghan and Saadatmandi [2] used Sinc-collocation method for solving **nonlinear** system of second order. El-Gamel [3] solving a class of linear and **nonlinear** **two** **point** **boundary** **value** **problem** by Sinc-Galerkin method. The books by Stenger [13], [14] and by Lund and Bowers [6] provide excellent over wive of existing methods based on Sinc function for solving integral equations, ordinary and partial differential equations. In our pervious work we applied Sinc collocation method for solution of linear and **nonlinear** integral equations [9] [10] .

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As is well known, the diﬀerential equations with right hand sides satisfying the Lipschitz conditions (Lipschitz equations for short) are important, and thus their solvability has at- tracted much attention from many researchers. Among a substantial number of works dealing with higher order Lipschitz equations with three-**point** **boundary** conditions, we mention [–] and references therein. Most of these results are obtained via applying control theory methods (Pontryagin maximum principle), matching methods, and topo- logical degree methods etc. To the best of our knowledge, most of the three-**point** bound- ary conditions in the above mentioned references are limited to simple **boundary** condi- tions.

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a constant to be determined. Assuming that the dimensionless pressure p is a qua- dratic function of y, we ﬁnd that the r -component of the acceleration equation in the Navier-Stokes energy system describing the ﬂow of ﬂuid and its temperature in the cylinder becomes (1.1a). The physical **boundary** conditions reduce to the condi- tions (1.1b) if we make the assumption that the free **boundary** is time-independent but not “ﬂat.”

tioned works, in this paper, we study the existence and uniqueness of solutions for the four-**point** fractional **boundary** **value** problems with the p-Laplacian operator. To the best knowledge of the authors, no work has been done to obtain the positive solution of the **problem** (1), (2). It is interesting to note that the fractional differential equation with the p- Laplacian operator.

A reference ellipsoid, an ellipsoid of rotation, is a suitable approximation of the shape of the Earth (Vanichek & krawinsky, 1986). Reference ellipsoid is a rotational one that formulated using **two** the size and shape parameters, i.e., semi-major axis and the eccentricity ( , ) a e of the ellipsoid. The geometry of ellipsoid is fully explained using the size and shape parameters. Therefore, all the geometrical computation (**point** positioning, area and volume calculation and etc.) is formulated using the size and shape parameters. The position vector of a **point** located on the surface of the reference ellipsoid in terms of the curvilinear coordinates (the geodetic latitude 𝜑 and geodetic longitude 𝜆 ) is expressed as (Jekeli, 2006):

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This paper aims to study a fractional diﬀerence equation of **two** **point** BVP type, which was realized as the **discrete** BVP. Certain classes are formulated in which the **discrete** **boundary** **value** problems will have a single solution. The novelty hither comprises a method selection of metric and employment of Hölder’s inequality. This investigation al- lows the related functions to be contractive, which were earlier non-contractive in classical regularities. This work, unlike those which appeared in the current designs, consequently grants the enhanced applications of Banach’s ﬁxed **point** theorem for classifying an exten- sive framework of issues.

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A New Fourth-Order Finite Difference Method for Computing Eigenvalues of Fourth-Order Two-Point Boundary Value Problem, IMA Journal of Numerical Analysis, 1983 291-293.. Discrete Approxi[r]

as well as on the interval [, b], where b < ∞ [] (further see []). We should note that there are several works in the ﬁeld of the existence and uniqueness theorems of the second order **nonlinear** **boundary** **value** problems. Some of them can be found in, for example, [–]. In particular, in [] Guseinov and Yaslan have investigated the existence and uniqueness of the solutions of the second order **nonlinear** **boundary** **value** **problem** on the semi-inﬁnite interval [, ∞)

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Recently, there have been many works related to the existence of solutions for multipoint **boundary** **value** problems (BVPs for short in the remaining) at nonresonance of FDEs (see [–]). Motivated by the above articles and recent studies on FDEs (see [–]), we con- sider the existence of solutions for a **nonlinear** fractional multipoint BVPs at resonance in this article.

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tegral equations are considered by using the Schauder ﬁxed **point** theorem and the con- traction mapping theorem, respectively. A new numerical method is further proposed to construct the approximate solutions of Hammerstein integral equations. Some numerical examples are computed to show the eﬀectiveness of the proposed methods.

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of perturbed convex variational problems in Sobolev spaces (see [7] for details.) We also prove that the solu- tion is bounded in certain Sobolev norms. In Section 3, we derive an error estimates for the semi-**discrete** error between the week solution and the Galerkin’s finite ele- ment solution of (1.3) for the standard conformal finite elements. The results of this section are based on the re- sults in Section 2. We believe that the results established in these sections are novel and preliminary.