To the best **of** our knowledge, there are only very few results regarding multiple solu- tions to the **p**-**biharmonic** **equation**. In this paper, the **existence** **of** at least **three** **solutions** **of** problem (**P**) will be proved. The technical approach is based on the **three** critical points theorem **of** B. Ricceri [14]. Our theorem, under the new assumptions, ensures the **existence** **of** an open interval Λ ⊆ [0, +∞) and a positive real number ρ such that, for each λ ∈ Λ, problem (**P**) admits at least **three** weak **solutions** whose norms in X are less than ρ.

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Theorem 4. ([5], **p**-61). Let S be a convex subset **of** a normed linear space E and assume 0 ∈ S. Let F : S → S be a completely continuous operator, and let ε(F ) = {x ∈ S : x = λF x for some0 < λ < 1}. Then either ε(F ) is unbounded or F has a fixed point.

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[1] V. A. Ilin and E. I. Moiseev, Nonlocal boundary-value problem **of** the first kind for a Sturm-Liouville operator in its differential and finite difference aspects, J. Differential Equations 23(1987), 803-810. [2] C. **P**. Gupta, Solvability **of** a **three**-point nonlinear boundary value

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and multiplicity **of** positive **solutions** **of** a boundary value problem for second-order **three**-point nonlinear impulsive integrodifferential **equation** **of** mixed type in a real Banach space. In [32], Liu and Yu with the help **of** the coincidence degree continuation theorem, achieved a general result concerning the **existence** **of** **solutions** **of** m-point boundary value problems for second-order differential systems with impulses. They also give a definition **of** autonomous curvature bound set relative to this m-point boundary value problems, and by using this definition and the above **existence** theo- rem, obtained some simple **existence** conditions for **solutions** **of** these boundary value problems. Meiqiang and Dongxiu in [14], based on fixed point theory in a cone, dis- cussed the **existence** **of** **solutions** for the m-point BVPs for second-order impulsive differential systems, and Thaiprayoon et al. in [42], introduced a new definition **of** impulsive conditions for boundary value problems **of** first-order impulsive differential equations with multi-point boundary conditions was introduced, and used the method **of** lower and upper **solutions** in reversed order coupled with the monotone iterative technique to obtain the extremal **solutions** **of** the boundary value problem.

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free massless theory and is called the Gaussian FP (GFP). For all known formulations the ERG **equation** is stiff and most **of** its **solutions** singular at some finite value **of** the field, making them not acceptable from the physical point **of** view. Moreover, it turns out that only finitely many **solutions** **of** Eq. (2) do not end up in a singularity, thus giving massless continuum limits with the prescribed field content [22]. They correspond to particular values γ = γ ∗ (n) , n = 1, 2, . . ..

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Now consider those **solutions** **of** the equations **of** motion (5) and (7), which admit constant scalar curvature R. For these **solutions**, the **equation** (7) reduces to the **equation** **of** motion **of** the string coordinates, extracted from the Polyakov action with the curved background. However, for general **solutions** the scalar curvature R depends on the worldsheet coordinates σ and τ, and hence this coincidence does not occur.

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ratio **of** two polynomials. In addition, a disadvantage not referred can come through: the rational approximation may create inaccurate **solutions** near its poles when the real solution is not achieved. This drawback advises the search for the optimal order **of** the Pad´ e approximant to be used, which can be **of** lower order.

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V = − + + − has been presented. The energy eigenvalues and the corresponding eigenfunctions are calculated analytically by the use **of** Nikiforov-Uvarov (NU) method which is related to the **solutions** in terms **of** Jacobi polynomials. The bounded state eigenvalues are calculated numerically for the 1s state **of** N 2 CO and NO

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)n our opinion the main challenge for the Romanian health care systems is to improve the mix **of** health care resources doctors, hospital beds, and pharmaceuticals in order to offer greater access to medical services. Romania is still in the process **of** developing a reform strategy for the health sector. The aim **of** this reform should be an increasing role **of** private sector in health care. The reform measures should include:

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expansion method is quiet efficient and well suited for finding exact **solutions**. The reliability **of** the method and reduction in the size **of** computational domain give this method a wider applicability. With the aid **of** Maple and by putting them back into the original **equation**, we have assured the correctness **of** the obtained **solutions**. Finally, it is worthwhile to mention that the method is straightforward and concise and it can be applied to other nonlinear evolution equations in engineering and the physical sciences.

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Fig 2. Microbial yield on PD catheters external (A) and intraperitoneal (B) segments and subcutaneous (C) and deep (D) cuffs in patients with and without infection. Data is presented as colony-forming units (CFU)/segment with the corresponding microorganisms represented by specific symbols. dot, coagulase negative Staphylococci (including Staphylococcus epidermidis, S. haemolyticus, S. caprae/ capitis, S. hominis, S. auricularis); diamond filled, Staphylococcus aureus; square filled, Corynebacterium spp.; down-pointing triangle filled, Micrococcus luteus; up- pointing triangle filled, Enterococcus faecalis; hexagon filled, Streptococcus spp.; right-pointing triangle filled, Bacillus spp.; circle, Pseudomonas aeruginosa; square, Sphingomonas spp.; diamond, Alcaligenes faecalis; down-pointing triangle, Serratia marcescens; up-pointing triangle, Burkholderia sp.; left-pointing triangle, Stenotrophomonas maltophilia; right-pointing triangle, Escherichia coli; crossed out circle, Enterobacter aerogenes; star, Candida parapsilosis; asterisk, Candida glabrata; circle quartered, non-identified microorganism. Absence **of** microorganisms is represented by, vertical line. Polymicrobial cultures are identified with a half tick-up line. In the group “infection” the cause **of** removal is indicated as alpha, for refractory peritonitis; beta, for relapsing peritonitis; gamma, fungal peritonitis; delta, for catheter-related peritonitis and epsilon, for chronic catheter infections. The dashed line indicates the culture method detection limit.

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Existence of Solutions for Optimal Control Problems on Time Scales whose States are Absolutely Continuous.. I.L.D.[r]

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This **equation** can be solved by various analytical methods, such as the variational iteration method [2], the homotopy perturbation method [3-5], and the exp-function method [6, 7]. A complete review on various analytical method is available in [8, 9]. In this paper the double exp-function method [10] is adopted to elucidate the different velocities and different frequencies in the travelling wave.

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We study the initial value problem **of** BGK model [1] coupled with Poisson’s **equation**, which is a simple relax- ation model introduced by Bhatnagar, Gross & Krook to mimic Boltzmann flows, where f (x, v, t) is the density **of** plasma particles at time t in the space **of** position x and velocity v, and φ(x, t) is the potential **of** electric field **of** the plasma.

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for some positive constant M . Hence (Aϕ) is equicontinuous. Then by the Ascoli-Arzelà theorem we obtain that A is a compact map. Due to the continuity **of** all the terms in (2.16), we have that A is continuous. This completes the proof. Theorem 3.4. Let α, β and γ be given by (2.10). Suppose that conditions (2.1)–(2.4), (2.11)–(2.15) hold, then **equation** (1.1) has a positive periodic solution z satisfying K ≤ z ≤ L.

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The simple pendulum is a prototype for the study **of** nonlinear systems and their stability [1]. In fact, pendu- lar oscillations arise in distinct fields **of** physical science and technology, e.g. acoustic vibrations [2], molecular oscillations [3], optically torqued nanorods [4], Joseph- son superconducting junctions [3, 5], elliptic filters in electronics [6], gravitational lensing [7], smetic C liquid crystals [8], advanced field theory [9], oscillations **of** build- ings in earthquakes [10], etc. In the small-angle regime, the approximation sin θ ≈ θ works and the **equation** **of** motion can be linearized, becoming solvable in terms **of** a sinusoidal function **of** time. Beyond this regime, the nonlinear nature **of** the pendulum motion becomes apparent: (i) the period increases with the amplitude; and (ii) the function θ(t) departs more and more from a harmonic behavior. In the nonlinear regime, the sim- plicity **of** the sinusoidal solution is lost and the **equation** **of** motion becomes unsolvable in terms **of** elementary functions, so either a direct numerical solution or some analytical approximation is demanded. However, numeri- cal **solutions** for the pendulum **equation** loose accuracy

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In this work, we prove the **existence** and multiplicity **of** positive weak **solutions** for some classes **of** elliptic problems in plane involving exponential growth **of** the Trudinger-Moser type with Neumann boundary condition. To do this, we use the method **of** sub and supersolution in combination with variational methods and the maximum principle.

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Conditions for the **existence** and uniqueness **of** weak **solutions** for a class **of** nonlinear nonlocal degenerate parabolic equations are established. The asymptotic behaviour **of** the **solutions** as time tends to infinity are also studied. In particular, the finite time extinction and polynomial decay properties are proved.

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This is an alternative version **of** biquaternionic Schr¨o- dinger representation **of** Navier-Stokes equations. Numerical solution **of** the new Navier-Stokes-Schr¨odinger **equation** (14) can be performed in the same way with [12] using Maxima software package [7], therefore it will not be discussed here.

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