existence and uniqueness of the solution

... in the proof of the uniqueness of strong solutions Global existence of strong solutions In this section, we prove the global existence of strong solutions to the problem 1.2 – 1.7 by applying the ... This completes the proof of Theorem 1.1 except the uniqueness assertion because of the presence of vacuum , which will be proved in the next section Uniqueness and stability of strong solutions In ... 1.7 mean that the boundary is nonslip and impermeable The purpose of the present paper is to study the global existence and uniqueness of strong solutions of problem 1.2 – 1.7 The important...

Ngày tải lên: 22/06/2014, 06:20

... in the proof of the uniqueness of strong solutions Global existence of strong solutions In this section, we prove the global existence of strong solutions to the problem 1.2 – 1.7 by applying the ... This completes the proof of Theorem 1.1 except the uniqueness assertion because of the presence of vacuum , which will be proved in the next section Uniqueness and stability of strong solutions In ... 1.7 mean that the boundary is nonslip and impermeable The purpose of the present paper is to study the global existence and uniqueness of strong solutions of problem 1.2 – 1.7 The important...

Ngày tải lên: 22/06/2014, 19:20

... http://www.boundaryvalueproblems.com/content/2011/1/25 Page of and they proved the existence of positive solutions by means of the Krasnosel’skii fixed-point theorem and Legget-Williams fixed-point theorem Recently, in the paper [18] to ... comparable to x and y, (5) we have the following result Theorem Adding condition (5) to the hypotheses of Theorem 1, we obtain uniqueness of the fixed point Remark In Theorems and 2, the condition ... Page of In the sequel, we present a sufficient condition for the existence and uniqueness of positive solutions for Problem (2) (positive solution means x(t) > for t Î (0, 1)) The proof of this...

Ngày tải lên: 20/06/2014, 22:20

... http://www.boundaryvalueproblems.com/content/2011/1/25 Page of and they proved the existence of positive solutions by means of the Krasnosel’skii fixed-point theorem and Legget-Williams fixed-point theorem Recently, in the paper [18] to ... comparable to x and y, (5) we have the following result Theorem Adding condition (5) to the hypotheses of Theorem 1, we obtain uniqueness of the fixed point Remark In Theorems and 2, the condition ... Page of In the sequel, we present a sufficient condition for the existence and uniqueness of positive solutions for Problem (2) (positive solution means x(t) > for t Î (0, 1)) The proof of this...

Ngày tải lên: 21/06/2014, 00:20

... theorem, we have the conclusion of the theorem Remark 3.3 In Theorem 3.2, if we furthermore suppose that the hypothesis H4 − f t, v t f t, u t ≤L u−v , L > 0, 3.28 holds, then we can obtain the ... especially the estimating approaches given in 4, 6, 10, 24, 27 , we investigate the existence and uniqueness of mild solution of 1.1 in a Banach space X: −A generates a compact semigroup S · of uniformly ... the uniqueness of the mild solution in Theorem 3.2 Actually, from what we have just proved, 1.1 has a mild solution u t and t u t Q t u0 R t − s f s, u s 3.29 K u s ds Let v t be another mild solution...

Ngày tải lên: 21/06/2014, 16:20

... suggestions and comments, which improved the quality of this paper The work was supported by the NSF of China, the NSF of Jiangxi Province of China 2008GQS0057 , the Youth Foundation of Jiangxi ... special case and its variants, where ϕp s of 1.1 However, there are seldom results about the existence of periodic solutions to 1.1 The main difficulty lies in the p-Laplacian-like operator ϕ of 1.1 ... ϕp s , Liu 7, Theorem proved that 1.1 has a unique T -periodic solution ϕp s , Theorem 2.1 under the assumptions A1 and A2 Thus, even for the case of ϕ s is a generalization of 7, Theorem In...

Ngày tải lên: 21/06/2014, 18:20

... solution of the BVP 1.1 and 1.2 Next, we show the uniqueness of solutions of the BVP 1.1 and 1.2 Assume, to the contrary, that there exist two nontrivial solutions y1 and y2 of the BVP 1.1 and 1.2 ... relax the assumptions A1 and B1 on the nonlinear term, without demanding the existence of upper and lower solutions, we present conditions for the BVP 1.1 and 1.2 to have a unique solution and then ... 1–7 and references therein Especially, there was much attention focused on the existence and multiplicity of positive solutions of fourth-order problem, for example, 8–10 , and in particular the...

Ngày tải lên: 21/06/2014, 18:20

... completes the proof of the theorem Boundary Value Problems 13 Theorem 3.2 Assume that conditions C0 , C1 , and C2 are satisfied Then SBVP 1.1 , 1.2 has exactly one positive solution Proof The uniqueness ... completes the proof of the lemma Lemma 2.7 Assume that conditions C0 , C1 , and C2 are satisfied Then, the unique solution u t; h of BVP 2.1 h is nondecreasing in h Proof Let < h2 < h1 , and let ... s ds Take t ∈ 0, and Δt such that t Δt ∈ 0, , then from the definition of derivative, the mean value theorem of Boundary Value Problems integral, and the absolute continuity of integral, we have...

Ngày tải lên: 21/06/2014, 18:20

... operator by the Arzela-Ascoli theorem Therefore, Theorem 1.8 is applicable to T and T must have a fixed point Hence the BVP has a solution This concludes the proof The following theorem gives the existence ... Proofs The proofs follow the modification technique of Theorems 3.6 and 3.7 and so are omitted for brevity On nonincreasing solutions Some results about the qualitative nature of solutions for the ... satisfy y < R and are solutions to the BVP (1.12), (1.3) This concludes the proof Theorem 3.8 Let the conditions of Theorem 3.3 hold with (2.1) replaced by (3.23) and C /α < R Then the BVP (1.12),...

Ngày tải lên: 23/06/2014, 00:20

... QT , u1 , u2 ∈ R Existence and uniqueness In this section, we show the local existence and uniqueness of weak solutions of (1.1)(1.3) First, we show the local existence results Theorem 2.1 Assume ... u2) is a weak solution of (1.1)(1.3) □ The following is the uniqueness result to the solution of the system Theorem 2.2 Assume that f = (f1, f2) is Lipschitz continuous in (u1, u2), then (1.1)(1.3) ... carried out the proof of existence, BW conceived of the study, and participated in its design and coordination All authors read and approved the final manuscript Competing interests The authors...

Ngày tải lên: 21/06/2014, 03:20

... 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 proof of Theorem 1.1 a In the following, R and T ... > 0, and p > max{1, m} Then every nontrivial solution u x, t of 1.1 blows up in finite time Remark 1.3 Theorems 1.1 and 1.2 are the extension of the results of If we put σ these theorems, the ... 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 σ In this paper, we investigate the...

Ngày tải lên: 21/06/2014, 11:20

... study the existence and uniqueness of smooth positive solutions to the second-order singular m-point boundary value problem 1.1 and 1.2 A necessary and sufficient condition for the existence of smooth ... solutions is given by constructing lower and upper solutions and with the maximal principle Also, the uniqueness of the smooth positive solutions is studied A function α t is called a lower solution ... subintervals of 0, 10 Boundary Value Problems Proof of Theorem 1.5 Suppose that u1 t and u2 t are C 0, positive solutions to 1.1 and 1.2 , and at least one of ≡ them is a smooth positive solution...

Ngày tải lên: 21/06/2014, 20:20

... , 1.2 has a unique solution x t satisfying 3.3 Proof The existence of a solution for BVP 1.1 , 1.2 satisfying 3.3 follows from Theorem 3.1 Now, we prove the uniqueness of solution for BVP 1.1 ... only the trivial solution Main Results We may now formulate and prove our main results on the existence and uniqueness of solutions for nth-order three point boundary value problem 1.1 , 1.2 Theorem ... a solution of BVP 1.1 , 1.2 and satisfies 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...

Ngày tải lên: 21/06/2014, 20:20

... is comparable to x and y, 2.10 then we have the following theorem Theorem 2.9 Adding condition 2.10 to the hypotheses of Theorem 2.8 one obtains uniqueness of the fixed point of f In our considerations, ... uniqueness of the solution is not treated In this paper we will prove the existence and uniqueness of a positive and nondecreasing solution for the problem 1.1 by using a fixed point theorem in partially ... positive solution for 1.1 by using Krasnoselskii’s fixed point theorem and nonlinear alternative of Leray-Schauder type in a cone and assuming certain hypotheses on the function f In the uniqueness of...

Ngày tải lên: 21/06/2014, 20:20

... ∞ is continuous and nondecreasing; h : 0, ∞ → 0, ∞ is continuous and nonincreasing, and h may be singular at y Next, we consider the existence and uniqueness of solutions for the following singular ... 3.4 Then we have x2 t λf t, T x t x 0, x < t < 1, λ > 0, 3.5 Then from 3.4 , we have the next lemma Lemma 3.1 If x t is a solution of 3.5 , then y t is a solution of 1.1 Further, if y t is a solution ... Agarwal and F.-H Wong, Existence of positive solutions for higher order difference equations,” Applied Mathematics Letters, vol 10, no 5, pp 67–74, 1997 14 X Lin, D Jiang, and X Li, Existence and uniqueness...

Ngày tải lên: 22/06/2014, 00:20

... ∞ is continuous and nondecreasing; h : 0, ∞ → 0, ∞ is continuous and nonincreasing, and h may be singular at y Next, we consider the existence and uniqueness of solutions for the following singular ... 3.4 Then we have x2 t λf t, T x t x 0, x < t < 1, λ > 0, 3.5 Then from 3.4 , we have the next lemma Lemma 3.1 If x t is a solution of 3.5 , then y t is a solution of 1.1 Further, if y t is a solution ... Agarwal and F.-H Wong, Existence of positive solutions for higher order difference equations,” Applied Mathematics Letters, vol 10, no 5, pp 67–74, 1997 14 X Lin, D Jiang, and X Li, Existence and uniqueness...

Ngày tải lên: 22/06/2014, 06:20

... show the existence of approximate solutions of the auxiliary problems for the equilibrium problem, and establish the strong convergence of the approximate solutions to an exact solution of the ... Lemmas 2.12 and 2.13, and conditions (i) and (iii) (see [2]) Thus, the conclusions of the theorem follows from the argument similar to that given for Theorem 3.2 This completes the proof Theorem ... = ν Then, the assumption (iv) of Theorem 3.2 holds Therefore, our results extend, improve, and unify the corresponding results obtained by Chen and Wu in [19] Theorem 3.4 Let K and X be the same...

Ngày tải lên: 22/06/2014, 18:20

... for the existence and uniqueness of solutions of the dynamic equation (Ln ) For this, we take as reference the results obtained in [3, 4], where the existence and uniqueness of solutions of problem ... obtain the following result that assures the existence and uniqueness of solutions of the dynamic equation (L1 ) in the sector [α,β], with α and β a pair of coupled lower and upper solutions of (L1 ... Using the results proved in Sections and 3, we will obtain in Sections and the expression of Green’s function and a sufficient condition for the existence and uniqueness of solutions of the dynamic...

Ngày tải lên: 23/06/2014, 00:20

... a contraction, and show the existence and the uniqueness of the solution in this space Finally, Section recapitulates the overall procedure of the article, and explains some of the intuitions ... us to use the usual argument of the Banach fixed point theorem, and to prove the existence and the uniqueness of the fixed point of Ψ, which is the solution of the differential equation (2), at ... of the intuitions behind our formulation for the reader Definition of the problem We deal with the question of existence and uniqueness of solutions of nonlinear deflections for an infinitely long...

Ngày tải lên: 20/06/2014, 20:20

... http://www.boundaryvalueproblems.com/content/2011/1/5 Page of 11 were devoted to deal with the existence of positive solutions of nonlocal BVPs (see [3-9] and references therein) Inspired by the above references, our aim in the present paper ... studied the existence and multiplicity of positive solutions for an interesting nonlocal differential equation using the Cone-Compression and Cone-Expansion Theorem due to M Krasnosel’skii for the existence ... http://www.boundaryvalueproblems.com/content/2011/1/5 Page of 11 Then, Equation 12 holds Therefore, by Equations 10 and 12 and the first part of Lemma 2.3, T has a fixed point in P ∩ ( ¯ \ ), which is a positive solution of Equation □ Example...

Ngày tải lên: 21/06/2014, 02:20