... but also the fact that they are themselves solutions to a similar system ofequations This allowed us to improve the exponent s, needed √ in the proof of well-posedness ofequationsof type (6),8 ... ofthe Main Theorem with the exception of those results which concern the asymptotic properties ofthe Ricci coefficients (the Asymptotics Theorem), and the straightforward modifications ofthe ... desired in the proof of (93) The remaining term on the left-hand side of (93) is easier to treat Proof ofthe L2 − L∞ decay estimate; Theorem 2.1 In this section we rely on the Boundedness Theorem...
... results of If we put σ these theorems, the same results as Theorem in are obtained in We will prove Theorem 1.1 a and b in Sections and 4, respectively The proof of Theorem 1.2 is included in the ... 373–390, 1993 R Suzuki, “Existence and nonexistence of global solutionsof quasilinear parabolic equations, ” Journal ofthe Mathematical Society of Japan, vol 54, no 4, pp 747–792, 2002 C Bandle, ... is called the critical exponent On the other hand, 6–9 and so on study the inhomogeneous equations i.e., f x / ≡ in 1.1 Bandle et al study the case m 1, σ 0, and Zeng and Zhang study the case...
... behavior ofsolutionsofthe system of difference equations xn+1 = a + xn c + yn e + zn , yn+1 = , zn+1 = b + yn d + zn f + xn In [8], Özban studied the positive solutionsofthe system of rational ... rational difference equations xn+1 = yn−k , yn+1 = yn xn−m yn−m−k In [9], Zhang et al investigated the behavior ofthe positive solutionsofthe system ofthe difference equations xn = A + yn−p ... behavior of a three-dimensional linear fractional system of difference equations J Math Anal Appl 310, 673–689 (2005) Özban, AY: On the positive solutionsofthe system of rational difference equations...
... mentioning that the conditions of our theorems are easily to verify, so they are applicable to a variety of problems, see Examples 4.1 and 4.2 The proof of our main results is based upon the following ... t∈[η,1] β(1 − η) , η} − βη Proof Note that u“ (t) ≤ 0, by applying the concavity of u, the proof is easy So we omit it where γ = min{βη, Proofs of main theorems Define the operator T : C[0, 1] ® ... aspects of these equations are yet to be explored For some recent work on equationsof fractional order with deviated arguments, see [46-48] and the references therein In this article, we consider the...
... differential equations, ” Bulletin ofthe Institute of Mathematics Academia Sinica, vol 13, no 4, pp 367– 384, 1985 20 N Parhi and S Parhi, “On the behaviour ofsolutionsofthe differential equations ... ¸ ofthe Greek Mathematical Society, vol 39, pp 131–137, 1997 26 C Tunc and E Tunc, “On the asymptotic behavior ofsolutionsof certain second-order differential ¸ ¸ equations, ” Journal ofthe ... nonpositive Journal of Inequalities and Applications Nonoscillation Behaviors ofSolutionsof 1.1 In this section, we obtain sufficient conditions for the nonoscillation ofsolutionsof 1.1 Theorem 2.1...
... if 1.16 holds true, then we have u Lq ≤C u r/q Lr u 1−r/q , ˙0 B∞,∞ ≤ r < q < ∞ 1.21 Proof of Theorem 1.2 This section is devoted to the proof of Theorem 1.2 First, we recall the following result ... Applied Mathematics, vol 41, no 4, pp 437–458, 1988 6 Journal of Inequalities and Applications H Sohr and W von Wahl, “On the regularity ofthe pressure of weak solutionsof Navier-Stokes equations, ” ... regularity of weak solutionsofthe Navier-Stokes system,” Journal of Differential Equations, vol 62, no 2, pp 186–212, 1986 M Struwe, “On partial regularity results for the Navier-Stokes equations, ”...
... i 1, The proof of a is complete b In virtue of Lemma 5.5 the proof ofthe theorem will be complete if we show that the sequences H and h satisfy the conditions H2 – H5 in Section Since the series ... ofsolutionsof difference Volterra equations, ” Applied Mathematics Letters, vol 16, no 6, pp 857–862, 2003 18 Z H Li, The asymptotic estimates ofsolutionsof difference equations, ” Journal of ... some lemmas Lemma 5.1 The hypotheses of Theorem 3.1 imply that the hypotheses of Theorem A are satisfied, and hence the solution z : z n n≥0 of 3.1 , 3.2 is bounded Proof Let > be such that ρ W is...
... solution of a fourth-order algebraic matrix equation is required during the derivation ofthe bounds, and the numerical aspect has not been discussed In the present paper, we derive the bounds ofsolutions ... defined in the statement ofthe theorem Finally, from (2.20) and (2.22), we get (2.17) The proof is completed Theorem 2.8 If there exist an SPD matrix X and a constant ε > satisfying the linear ... positive definite solutions P to (1.1) and P < X The proof is completed Remark 2.9 From the relations between the solution ofthe perturbed discrete Lyapunov equation and that of an appropriate...
... completes the proof ofthe proposition Behavior ofthe positive solutionsof fuzzy equation (1.1) In this section, we study the behavior ofthe positive solutionsof (1.1) Firstly, we study the periodicity ... uniqueness ofthe positive solutionsof fuzzy difference equations (1.1) and (1.2) In this section, we study the existence and the uniqueness ofthe positive solutionsofthe fuzzy difference equations ... use the following technique: we investigate the behavior ofthesolutionsof a related family of systems of two parametric ordinary difference equations and then, using these results and the fuzzy...
... By this the proofs ofthe part ofthe results (which require the rather long proofs) are given in Section In Section 4, the main results are formulated, in Sections 5–7 the proofs ofthe main ... respect to γk ¯ The definition ofthe L-condensing map is given in [3] The proof of theorem repeats the proof of Theorem 2.2 in [21] on the strength of ¯ ¯ Lemmas 5.6–5.8 and the inequality u ... [17]), and γn is the operator of taking ofthe normal component of a trace on the boundary of a function defined on Ω0 The operator γn 236 On weak solutionsoftheequationsof motion 1/2 is...
... 8 Numerical Solutionsofthe Black Scholes Equation The left-hand side ofthe heat equation, on the other hand, cannot be unambiguously approximated The following are some ofthe more common ... exponentials ofthe first column Note that the S values in this grid correspond to those ofthe Jarrow– Rudd scheme of Figure 7.4 If the reader is unsure ofthe reason for this, he will find the answer ... the initial values at t = and work across the grid towards the point (N, M) But because ofthe simple way in which the u n+1 only depend on the adjacent values to the m immediate left, only solutions...
... by the standard deviation ofthe actual P&L; and The variance ofthe unexplained P&L divided by the variance ofthe actual P&L The specific time period that would be used in the computation of ... design ofthe regime also contributed to, and amplified, the problems exposed during the crisis These include: The role ofthe regulatory boundary: The Committee believes that its definition ofthe ... address the shortcomings ofthe framework As a result, the Committee initiated a fundamental review ofthe trading book regime, beginning with an assessment of “what went wrong” The fundamental...
... the categories of unity ofthe form ofthe will (its universality), plurality ofthe matter (the objects, i.e., the ends), and totality ofthe system of these In forming our moral judgement of ... beings On the other hand, that which merely contains the ground of possibility ofthe action of which the effect is the end, this is called the means The subjective ground ofthe desire is the spring, ... the conception of an object within the reach of our faculties can exercise on the will ofthe subject, in consequence of its natural properties, depends on the nature ofthe subject, either the...
... under the Agreement II (“Verständigung II”) and are under the sole ownership ofthe Federal States On the basis ofthe public legal framework and under the ownership ofthe Federal States, they ... are owned either by the municipalities and/or savings banks associations, on the one hand, or by the Federal States on the other Therefore, the separation of responsibilities within the savings ... the weaknesses of one side with the strengths ofthe other The tie-up is aimed rather at securing – by means of balanced business and balance sheet structures – a structural contribution to the...
... vacuum solutions 29.1 Twisting vacuum solutions – the field equations 29.1.1 The structure ofthe field equations 29.1.2 The integration ofthe main equations 29.1.3 The remaining field equations ... Exact Solutionsof Einstein’s Field Equations A revised edition ofthe now classic text, Exact Solutionsof Einstein’s Field Equations gives a unique survey ofthe known solutionsof Einstein’s ... surprise for the reader may lie in the enormous number of known exact solutions Those who not work in the field often suppose that the 1.2 The development ofthe subject intractability ofthe full...
... vacuum solutions 29.1 Twisting vacuum solutions – the field equations 29.1.1 The structure ofthe field equations 29.1.2 The integration ofthe main equations 29.1.3 The remaining field equations ... Exact Solutionsof Einstein’s Field Equations A revised edition ofthe now classic text, Exact Solutionsof Einstein’s Field Equations gives a unique survey ofthe known solutionsof Einstein’s ... surprise for the reader may lie in the enormous number of known exact solutions Those who not work in the field often suppose that the 1.2 The development ofthe subject intractability ofthe full...
... carried out the mathematical calculations, participated in the interpretations ofthe results and drafted the manuscript JJAG conceived ofthe study, participated in the analysis ofthe results ... precedence ofthe heat flux vector to the temperature gradient at all In fact, whether the heat flux vector precedes the temperature gradient or not depends on the combined effects ofthe thermal ... the difference ofthe time delays t = tq-tT These studies have pointed out that the usefulness ofthe Cattaneo-Vernotte and DPL exact models is limited In this study, by means ofthe method of...
... and they give rise to many applications see Not only the new theory ofthe so-called “dynamic equations unifies the theories of differential equations and difference equations but also extends these ... G ut p τ Δτ ≤ G u T 2.53 P T The rest ofthe proof is similar to that of Theorem 2.8, and the details are omitted The proof is completed Theorem 2.10 Assume that the function f t, u0 , , un−1 ... discrete analysis The cases when a time scale is equal to the real numbers or to the integers represent the classical theories of differential and of difference equations Many other interesting...
... multiple of k On the other hand, in the remaining two cases, the study ofthe regular variation ofthesolutions gives the additional information that the positive solutions, even if they are ... M∞,0 The above theorem shows how the study ofthe regular variation ofthesolutions and the M-classification supplement each other to give an asymptotic description of nonoscillatory solutions ... SV \ NSV The main properties of regularly varying sequences, useful to the development ofthe theory in the subsequent sections, are listed in the following proposition The proofs ofthe statements...
... 2.7 The proof of Lemma 2.3 is completed Main Results In this section, we investigate the boundedness ofsolutionsof 1.1 Let q > p > 1, and let {xn }∞ −1 be a positive solution of 1.1 with the ... for any n ≥ Let S be the set of all initial values x−1 , x0 ∈ 0, ∞ × 0, ∞ such that the positive solutions {xn }∞ −1 of 1.1 are bounded Then we have the following theorem n Theorem 3.6 S ∞ n Qn ... follows 16 Theorem A If p q ≤ 1, then the zero equilibrium of 1.1 is globally asymptotically stable If q 1, then the equilibrium x p of 1.1 is globally asymptotically stable If < q < p, then every...