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ONE PARAMETER FAMILY OF LINEAR DIFFERENCE EQUATIONS AND THE STABILITY PROBLEM FOR THE NUMERICAL pdf
... k odd are known in literature as top order methods (TOMs) [2] Theorem 4.4 Let k = 2ν The k-step methods in the Top Order family are Aν,ν -stable Proof From (2.22) and Theorem 3.8 we obtain qk ... overcome this drawback, since only y0 is provided by the continuous problem, we can choose to fix some of the (k − 1) additional conditions at the beginning of the interval of integration and the ... equal to the number of roots of the characteristic polynomial inside the unit circle and, of course, the number of conditions at the end of the interval of integration should be equal to the number...
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báo cáo hóa học:" Research Article Existence of Positive Solutions of Nonlinear Second-Order Periodic Boundary Value Problems" ppt
... Krein-Rutman theorem, see 15, Theorem 19.3 , we may get the desired results Remark 3.4 Theorem 3.2 is a partial generalization of Lemma 3.1 It is enough to prove that the condition i on f in Theorem ... positive linear operator V on E is said to be a linear minorant for A if A λ, u ≥ λV x for λ, u ∈ 0, ∞ × K If B is a continuous linear operator on E, denote r B the spectrum radius of B Define ... global structure of the set of positive solutions of parameterized nonlinear operator equations, which is essentially a consequence of Dancer 12, Theorem Suppose that E is a real Banach space with...
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báo cáo hóa học:" Nodal solutions of second-order two-point boundary value problems" potx
... subinterval [α, β] of for all r ∈ [α, β] and s = In the celebrated study [4], Rabinowitz established Rabinowitz’s global bifurcation theory [4, Theorems 1.27 and 1.40] However, as pointed out by Dancer ... corrected o o version of unilateral bifurcation theorem in [7] By applying the bifurcation theorem of L´pez-G´mez [7, Theorem 6.4.3], we shall establish the following: o o Theorem 1.1 Suppose that f ... Similarly, Therefore, Theorem A is the corollary of Theorems 1.1 and 1.2 Using the similar proof with the proof Theorems 1.1 and 1.2, we can obtain the more general results as follows Theorem 1.3 Suppose...
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Báo cáo hóa học: " Research Article Multiplicity of Positive Periodic Solutions of Singular Semipositone Third-Order Boundary Value Problems" doc
... index theory Recently, instead of Schauder fixed-point theorem and fixed-point index theory, Chu and Zhou 10 employed a nonlinear alternative principle of Leray-Schauder and a fixed-point theorem ... Furthermore, an illustrating example will be given Preliminaries In this section, we present some preliminary results First, as in 13 , we transform the problem into an integral equation For any ... ω g r/ρ − ω 1 n0 3.8 which is a contradiction to the choice of n0 and the claim is proved From this claim, the nonlinear alternative of Leray-Schauder guarantees that 3.4 has a fixed point, denoted...
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CONSTRUCTION OF UPPER AND LOWER SOLUTIONS FOR SINGULAR DISCRETE INITIAL AND BOUNDARY VALUE PROBLEMS potx
... holds Theorem 2.3 guarantees that (2.47) has a solution u ∈ C(N + , R) with u(k) > for k ∈ N Acknowledgment The research is supported by National Natural Science Foundation (NNSF) of China Grant ... (2.27) j =k Thus (2.3) holds Theorem 2.1 guarantees that (2.23) has a solution u ∈ C(N + , R) with u(k) > for k ∈ N Next we present a result for initial value problems Theorem 2.3 Let n0 ∈ {1,2, ... O’Regan, A generalized upper and lower solution method for singular discrete initial value problems, Demonstratio Math 37 (2004), no 1, 115–122 R P Agarwal, H L¨ , and D O’Regan, Existence theorems...
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báo cáo hóa học:" Research Article Exponential Stability and Estimation of Solutions of Linear Differential Systems of Neutral Type with Constant Coefficients" pot
... Usually, it uses positive definite functionals of a quadratic form generated from terms of 1.1 and the integral over the interval of delay of a quadratic form A possible form of such a functional is ... achieved by reducing the initial neutral system 1.1 to a neutral system having the same solution on the intervals indicated in which the “neutrality” is concentrated only on the Boundary Value Problems ... hardly applicable in practice Acknowledgments J Baˇ tinec was supported by Grant 201/10/1032 of Czech Grant Agency, by the Council s of Czech Government MSM 0021630529, and by Grant FEKT-S-10-3 of...
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Báo cáo hóa học: "Research Article Existence and Uniqueness of Solutions for Higher-Order Three-Point Boundary Value Problems" ppt
... and μi ∈ R, i 0, 1, , n − are arbitrary given constants The tools we mainly used are the method of upper and lower solutions and Leray-Schauder degree theory Note that for the cases of a b ... 3.3 Now we give a uniqueness theorem by assuming additionally the differentiability for functions f, g and h, and a kind of estimating condition in Theorem 3.1 Theorem 3.2 Assume that i there ... 3.64 i Consequently, by Taylor’s theorem there exists t2 ∈ t1 , c such that M0 z n−2 t > γ n−2 ∀t ∈ t1 , t2 , t , 3.65 which is a contradiction A similar contradiction can be obtained if t1 ∈...
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Báo cáo hóa học: "Research Article Asymptotic Representation of the Solutions of Linear Volterra Difference Equations" pdf
... and hence the proof of Theorem 3.1 is complete 5.2 Proof of Theorem 3.3 Theorem 3.3 will be proved after some preparatory lemmas In the next lemma, we show that 3.7 can be transformed into an equation ... function Proofs of the main theorems 5.1 Proof of Theorem 3.1 To prove Theorem 3.1 we need the next result from 20 −2 − 2α−1 −1 ς α , 4.30 I Gy˝ ri and L Horv´ th o a 11 Theorem A Let us consider ... complete Now, we prove Theorem 3.1 Proof Let n > N ≥ be arbitrarily fixed Then, 3.1 can be written in the form N z n n H n, i z i H n, i z i i hn 5.34 n > N ≥ 5.35 i N Subtracting 5.29 from the...
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Báo cáo hóa học: "EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR SOME THREE-POINT NONLINEAR BOUNDARY VALUE PROBLEMS" pptx
... no solution for λ > λ∗ Remark 2.3 Our theorems generalize Theorems 1.1–1.4 and the main results in [9] In fact, Theorems 1.1–1.4 are corollaries of our theorems Moreover, the nonlinear term f ... = 0, Theorem 2.1 cannot be obtained by Theorems 1.1–1.4 and the abstract results in [12] Remark 2.4 The nonlinear term f was assumed to be nondecreasing in Theorems 1.2 and 1.4, but in Theorem ... value problem and the spectral radius of a related linear operator, Nonlinear Analysis Theory, Methods & Applications An International Multidisciplinary Journal Series A: Theory and Methods 34 (1998),...
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Báo cáo hóa học: "REPRESENTATION OF SOLUTIONS OF LINEAR DISCRETE SYSTEMS WITH CONSTANT COEFFICIENTS AND PURE DELAY" ppt
... Oberwolfach, Germany, during the RiP stay The first author has been supported by Grant 201/04/0580 of Czech Grant Agency (Prague) and by the Council of Czech Government MSM 00216 30503 References [1] ... of two discrete variables be given The operator Δk acting by the formula Δk F(k,n) := F(k + 1,n) − F(k,n) (3.17) is said to be a partial difference operator, provided that the right-hand side exists ... this function into (3.22) This ends the proof Collecting the results of Theorems 3.1 and 3.5 we get immediately the following Theorem 3.6 Solution x = x(k) of the problem (1.1), (1.2) can be on Z∞m...
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Báo cáo hóa học: "Research Article Periodic and Almost Periodic Solutions of Functional Difference Equations with Finite Delay" pdf
... − xnk +rk ≥ ε k→∞ (4.23) This is a contradiction This contradiction shows that Lemma 4.3 is true We are now in a position to prove the following result Theorem 4.4 Suppose that for each G ∈ H(F), ... Oscillation Theory of Delay Differential Equations, Oxford Mathematical o Monographs, The Clarendon Press, Oxford University Press, New York, NY, USA, 1991 [10] Y Song and C T H Baker, “Perturbation theory ... with G(n,φ) = F(n + j∗ ,φ) This completes the proof Using Theorem 3.1 and Lemma 3.2, we can show that (3.1) has an almost periodic solution Theorem 3.3 If the bounded solution {u(n)}n≥0 of (3.1)...
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Báo cáo hóa học: "SECOND-ORDER ESTIMATES FOR BOUNDARY BLOWUP SOLUTIONS OF SPECIAL ELLIPTIC EQUATIONS CLAUDIA ANEDDA, ANNA BUTTU, AND GIOVANNI " pot
... on {x ∈ Ω : δ(x) < δ0 } The theorem is proved 12 Second-order estimates References [1] L Andersson and P T Chru´ ciel, Solutions of the constraint equations in general relativity satiss fying ... and Integral Equations 11 (1998), no 1, 23–34 , Dependence of blowup rate of large solutions of semilinear elliptic equations, on the curva[8] ture of the boundary, Complex Variables Theory and ... of domain geometry in boundary blow-up elliptic problems, Nonlinear Analysis Theory, Methods & Applications Series A: Theory and Methods 48 (2002), no 6, 897–904 [12] D Gilbarg and N S Trudinger,...
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ON THE SOLVABILITY OF INITIAL-VALUE PROBLEMS FOR NONLINEAR IMPLICIT DIFFERENCE EQUATIONS PHAM KY ANH pdf
... the Hadamard theorem on homeomorphism, (3.1) has a unique solution Pn xn+1 and Qn−1 xn for all n This completes the proof of Theorem 3.3 In the next theorem, without loss of generality, we will ... projection Q ∈ C (J, Rm×m ) onto KerA(t) such that the matrix G(t) = A(t) + B(t)Q(t) is nonsingular for all t ∈ J It is proved that the index property (transferability) of linear DAEs does not ... LIDEs, which is quite similar to that of index (transferable) linear DAEs Definition 2.3 The LIDEs (2.2) are said to be of index if, for all n (i) rank An = r; (ii) Gn := An + Bn Qn−1,n is nonsingular...
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NONOSCILLATORY HALF-LINEAR DIFFERENCE EQUATIONS AND RECESSIVE SOLUTIONS ˇ ´ MARIELLA CECCHI, pdf
... the tools used in [7] for proving limit and integral characterization of principal solutions is based on certain properties of a suitable quadratic functional studied in [9] Since in the discrete ... for any recessive solution u of (4.9) Concluding remarks Theorems 4.2 and 4.3, and Example 4.4 illustrate some difficulties concerning the characterization of the recessive solution via summation ... n [1] lim xn = ∞, n (5.2) as follows from [13, Theorems and 10] or [16, Theorems 3.4 and 3.5] Hence it seems to be difficult to prove the limit characterization and the summation properties of recessive...
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Tài liệu Solution of Linear Algebraic Equations part 1 docx
... number can readily be extended to N as large as several hundred, after which point the limiting factor is generally machine time, not accuracy Even larger linear sets, N in the thousands or greater, ... already seen, in §1.2, that this C notation can in fact hide a rather subtle and versatile physical storage scheme, “pointer to array of pointers to rows.” You might wish to review that section ... but to use sophisticated black-box program packages Several good ones are available, though not always in C LINPACK was developed at Argonne National Laboratories and deserves particular mention...
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Tài liệu Solution of Linear Algebraic Equations part 2 ppt
... how to recognize a particularly desirable pivot when we see one The answer to this is not completely known theoretically It is known, both theoretically and in practice, that simply picking the ... does not have to exist as separate storage: The matrix inverse of A is gradually built up in A as the original A is destroyed Likewise, the solution vectors x can gradually replace the right-hand ... Sciences (New York: McGraw-Hill), Program B-2, p 298 Westlake, J.R 1968, A Handbook of Numerical Matrix Inversion and Solution of Linear Equations (New York: Wiley) Ralston, A., and Rabinowitz, P 1978,...
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Tài liệu Solution of Linear Algebraic Equations part 11 ppt
... its predecessor Instead of doing O(N ) operations each time to solve the equations from scratch, one can often update a matrix factorization in O(N ) operations and use the new factorization to ... (North America only),or send email to trade@cup.cam.ac.uk (outside North America) The standard algorithm for the QR decomposition involves successive Householder transformations (to be discussed later ... Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software Permission is granted for internet users to make one paper copy for their...
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Tài liệu Solution of Linear Algebraic Equations part 3 pdf
... elimination is simply that the former is faster in raw operations count: The innermost loops of Gauss-Jordan elimination, each containing one subtraction and one multiplication, are executed N and ... a routine CITED REFERENCES AND FURTHER READING: Ralston, A., and Rabinowitz, P 1978, A First Course in Numerical Analysis, 2nd ed (New York: McGraw-Hill), §9.3–1 Sample page from NUMERICAL RECIPES ... Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software Permission is granted for internet users to make one paper copy for their...
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Tài liệu Solution of Linear Algebraic Equations part 4 docx
... computer’s floating-point dynamic range In this case you can modify the loop of the above fragment and (e.g.) divide by powers of ten, to keep track of the scale separately, or (e.g.) accumulate the ... practically the same operations count Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) 1988-1992 by Cambridge University Press.Programs ... Numerical Analysis (New York: Springer-Verlag), §4.2 Ralston, A., and Rabinowitz, P 1978, A First Course in Numerical Analysis, 2nd ed (New York: McGraw-Hill), §9.11 Horn, R.A., and Johnson, C.R 1985,...
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