ON THE CORRECT FORMULATION OF A NONLINEAR DIFFERENTIAL EQUATION IN BANACH SPACES

We study the existence and uniqueness of the initial value problems in a Banach space E for the abstract nonlinear differential equation (dn−1/dtn−1)(du/dt+ Au)= B(t)u+f(t,W(t)), and consider the correct solution of this problem. We also give an application of the theory of partial differential equations. 2000 Mathematics Subject Classification. 34G20, 47N20.

Suppose that the range of these operators are in E, therefore consider the abstract nonlinear differential equation ) where all the elements g 0 ,g 1 ,g 2 ,...,g n−1 ∈ S, W = (B 1 (t)u, B 2 (t)u,...,B ν (t)u) and f is a given abstract nonlinear function defined on I × E ν with values in E. Without loss of generality, we assume that where θ is the zero element of the Banach space E. Let f be uniformly Hölder continuous for all t ∈ I, that means for all t and t * in I and all W in E ν , the constant K and β are positive and β < 1, where ..,w * ν ), and t ∈ I, the function f satisfies the Lipschitz condition Suppose that −A generates a semigroup {T (t), t ∈ I} strongly continuous for all t ≥ 0, this class of semigroup is called C 0 (see [11,12]).Furthermore, suppose that T (t)v ∈ S for all v ∈ E, t > 0 (see [6]).Assume that if there exist 0 < δ < 1 and a positive constant M, then where M is a positive constant and 0 < δ < 1 for all v ∈ E, t 2 ∈ I, t 1 ∈ (0,T 0 ] with i = 1, 2,...,ν.In this paper, we prove the existence and the uniqueness of the solution of the Cauchy problem (1.1) and (1.2).The correct formulation of the considered problem is also proved, finally we give an application of the theory of partial differential equations.

The solution of the problem.
In this section, we discuss the existence and uniqueness of the solution of the initial value problem (1.1) and (1.3).Define on C E (I), a distance function (metric) ρ by where u 1 ,u 2 ∈ C E (I), λ > 1 being a fixed number.It is clear that (C E (I), ρ) is a metric space, (see [2,3]).
Theorem 2.1.The abstract initial value problem (1.1) and (1.3) has a weak solution in the metric space (C E (I), ρ) for every t ∈ I.

Proof. From equation (1.1), let
The desired solution u of the above equation can be written in the form (see [1,11,12]) where v satisfies ) (2.6) Let Q be an operator defined on C E (I) by (2.7) We prove that Q is a contraction mapping.We notice that (2.9) For a sufficiently large λ we deduce that Q is a contraction operator therefore there exists a unique fixed point such that (see [2,3,1]) Qv = v ∈ C E (I), which proves the existence and uniqueness of a weak solution u in C E (I). We (2.10) From (1.8), we get the required result.Theorem 2.3 (see [1,4,11]).The function u(t) is an element of S for every t ∈ I and so u ∈ C S [0,T 0 ].

Proof. To prove this theorem, it is enough to show that v(t) satisfies the Lipschitz condition in
The theorem is proved by using the above equation and (1.6).
To complete the proof of the existence and uniqueness of the solution (strongly) we prove that each of the following derivative From (1.1), we can write formally (2.13) To get the required result, we must prove that Ψ 1 and Ψ 2 satisfies a uniform Hölder condition for t ∈ I. Suppose that t 2 > t 1 , therefore it is easy to show that where J is the identity operator on E, where K 1 and K 2 are positive constants.Similarly, as in [9], we can prove that Ψ 1 and Ψ 2 satisfies Hölder condition in t ∈ I, therefore (du/dt) ∈ C S (I) and (dv/dt) is continuous for all t ∈ I. Now, Au(t) can be written in the form (2.16) Thus differentiate (n − 1) times we get (2.17) Therefore, is continuous on I. Consequently, represent the unique solution of the considered Cauchy problem (compare with [8,10,11]).

Correct solution.
In this section, we prove the correct formulation of the considered initial value problem (1.1) and (1.2).In other words, we prove the continuous dependent of the solution of the problem on the initial conditions.Let {u m } be a sequence of solutions of the initial value problem where W m is the sequence W m = (B 1 (t)u m ,B 2 (t)u m ,...,B ν (t)u m ).

.,g n−1 are elements in G, then the sequence of solutions {u m (t)} of the problem (3.1) and (3.2) converges in the metric space C E (I) to the solution u(t) of the considered problem with the initial conditions,
substitute in equation (1.8), we get with the initial conditions where therefore, It is clear that we can easily deduce that P m (t) − P r (t) Multiply by e −λt and using the metric defined by equation (2.1), we get (3.12) According to all the conditions before, the sequence {P m } is fundamental and hence converges to P in C E (I).But Therefore, the sequence {u m (t)} uniformly converges with respect to t ∈ I in E to the required solution (compare with [6,5,4,9]).

Application.
Consider the Cauchy problem with the initial conditions where an open set in the n-dimensional Euclidean space R n , and let L 2 (M) be the space of all square integrable functions on M. We denote by C m (M) the set of all continuous real-valued functions in M together with all their m-partial derivatives, and we denote by C m 0 (M) the subset of C m (M) consisting of all functions having a compact support.Let H m (M) be the complete space of C m (M) with respect to the norm (see [2,3]) For any 0 < b < ∞ denote by Ω b the cylinder {(x, t) : x ∈ M, 0 < t < b}, and by Γ b the boundary {(x, t) : x ∈ ∂M, 0 < t < b}, where ∂M is the boundary of M. We say that L is uniformly parabolic in M, the closure of M if the coefficients b α are continuous on M and if for all x ∈ M and for all η ∈ R n , where  where i = 1, 2,...,ν.The domain of these operators can be taken H 2m−1 (M) ∩ H m 0 (M) which is dense in L 2 (M) (see [1,2,10]).Therefore, we can assume that which depends only on the continuity of the functions g 0 ,g 1 ,...,g n−1 .
Suppose also that the coefficients a α , C α,i are continuous on Ω b and satisfies a uniform Hölder condition in t ∈ [0,b].The Cauchy problem (4.1) and (4.2) can be written in the abstract form (1.1) and (1.2), where A is the operator with domain S = H 2m (M) ∩ H m 0 (M) given by Au = A(x, D)u = |α|≤2m b α (x)D α u. (4.6)Let E = L 2 (M).Then the domain S is dense in L 2 (M).
[2,6]ppose that g 0 (x), g 1 (x),...,g n−1 (x) are given functions in S. Since Lu is uniformly parabolic, it follows that −A = −A(x, D) generates a semi-group {T (t)} of class C 0 .It can be proved that T (t) satisfies the condition (1.7) and (1.8).Consequently,[2,6]can be applied to the Cauchy problem (4.1) and (4.2).This means that the considered problem can be solved in S without any restrictions on the characteristic forms of the operators