Schauder-Tychonoff Fixed-Point Theorem in Theory of Superconductivity

We study the existence of mild solutions to the time-dependent Ginzburg-Landau ((TDGL), for short) equations on an unbounded interval. The rapidity of the growth of those solutions is characterized. We investigate the local and global attractivity of solutions of TDGL equations and we describe their asymptotic behaviour.The TDGL equations model the state of a superconducting sample in a magnetic field near critical temperature. This paper is based on the theory of Banach space, Fréchet space, and Sobolew space.


Introduction
The objective of the paper is to investigate the existence and asymptotic behaviour of mild solutions on an unbounded interval of time-dependent Ginzburg-Landau equations (TDGL, for short) in superconductivity.
In the Ginzburg-Landau theory of phase transitions [1], the state of a superconducting material near the critical temperature is described by a complex-valued order parameter , a real vector-valued vector potential A, and, when the system changes with time, a real-valued scalar potential .The latter is a diagnostic variable;  and A are prognostic variables, whose evolution is governed by a system of coupled differential equations: The supercurrent density J  is a nonlinear function of  and A, The system of (1)-( 3) must be satisfied everywhere in Ω, the region occupied by the superconducting material, and at all times  > 0. The boundary conditions associated with the differential equations have the form n × (∇ × A − H) = 0 (5) on Ω, where Ω is the boundary of Ω and n is the local outer unit normal to Ω.They must be satisfied at all times  > 0.
We prove that the systems of ( 1)-( 5) can be reduced to a semilinear equation; to use the appropriate theorem, we investigate the local and global attractivity of solutions of equations in question and describe their asymptotic behaviour.
In this paper, we consider the existence and asymptotic behaviour of mild solutions on an unbounded interval of the semilinear evolution equation of the following form: where the operator A : (A) ⊂ X → X generates a  0semigroup { −A(−) } ≥0 and X is a real Banach space.
In this paper, we present conditions guaranteeing the existence of mild solutions on an unbounded interval of problem ( 6)- (7).We dispense with assumptions on the compactness of F(, ) or { −A(−) } ≥0 .
Moreover, we formulate theorems about asymptotic properties and both local and global attractivity of solutions of problem ( 6)- (7).The existence theorems concerning that problem will be proved with the help of the technique of a family of measures of noncompactness in the Fréchet space (R + , X) and Schauder-Tychonoff fixed-point principle.
The paper is organized as follows.In Section 2, there are given notation and auxiliary facts that are needed further on.In Section 3, we formulate and prove a theorem on the existence of mild solution of (6) with condition (7).Moreover, the rate of the growth of those solutions is characterized.
Section 4 contains a theorem on local and global attractivity of solutions of problem ( 6)- (7).In Section 5, we give a theorem describing the asymptotic behaviour of solutions of ( 6)- (7).
Finally, in Section 6, we formulate the gauged TDGL equation as an abstract evolution equation in a Hilbert space.Moreover, this section is devoted to present examples of application of previously obtained theorems for TDGL equations.

Preliminaries
Let (X, ‖ ⋅ ‖) be a real Banach space with the zero element .Denote by (, ) the closed ball in X centered at  and with radius .If  is a subset of a linear topological space, then the symbols  and Conv stand for the closure and the convex closure of , respectively.
Let  denote the Hausdorff measure of noncompactness in X, defined on bounded subsets  of X in the following way (see [27]):  () := inf { > 0 :  can be covered with finite numbers of balls of radius } .
Further, denote by (R + , X) the Fréchet space consisting of all functions defined and continuous on R + with values in a Banach space X.The space (R + , X) is furnished with the family of seminorms; Let us recall two facts: only if the restrictions to [0, ] of all functions from  form an equicontinuous set for each  > 0 and () is relatively compact in X for each  ∈ R + , where () = {() :  ∈ }.
Moreover, we recall that a nonempty subset Further, the family of all nonempty and bounded subsets of (R + , X) will be denoted by M  , while the family of all nonempty and relatively compact subsets of (R + , X) is denoted by N  .Obviously N  ⊂ M  .We will use a family of measures of noncompactness {  } ≥0 in the Fréchet space (R + , X) which was introduced in [20,35].In order to define these measures, recall some quantities [25][26][27].Let us fix a nonempty bounded subset  of the space (R + , X).For  ∈ ,  ≥ 0, and  ≥ 0 denote by   (, ) the modulus of continuity of the function  on the interval [0, ], that is, To prove the existence results in this paper, we need the following lemmas.

Main Result
In this section, we give an existence result for the semilinear equation of evolution ( 6)-( 7), and we describe the asymptotic behaviour of those solutions.First, we will assume that the functions involved in (6) satisfy the following conditions: Carathéodory condition; that is, F(⋅, ) is measurable for  ∈ X and F(, ⋅) is continuous for a.e. ∈ R + ; ( 3 )  : R + → R + and  : R + → R + are locally integrable functions such that for any  ∈ X and a.e. ∈ R + .
( 4 ) There exists a locally integrable function  : R + → R + such that for any bounded  ⊂ X, for a.e. ∈ R + .
Theorem 10.Under assumptions ( 1 )-( 4 ), (6) with initial condition (7) has, for every  0 ∈ X, at least one mild solution  = () which satisfied the following estimate: where Proof.Consider the operator  : (R + , X) → (R + , X) defined by the following formula: Now, let us observe that for any continuous function  : R + → X, in view of condition ( 3 ), we get the following estimate: which yields Next, consider the following integral equation: Solving this equation by standard methods, we get The function P() is continuous, nonnegative, and nondecreasing.Observe that the following implication is true: Indeed, linking ( 29) and ( 32 In the space (R + , X), let us consider the following set: Obviously, the set Γ is convex and closed.Moreover, in view of (32), we have that  is a self-mapping of Γ.
Using the criterion of convergence ( * ) in (R + , X) and standard techniques (see [31][32][33]35]), we can show that the operator  is continuous on (R + , X).Now, we consider the sequence of sets {Γ  } defined by induction as follows: This sequence is decreasing, that is, Γ  ⊃ Γ +1 for  = 0, 1, 2 . ... Further, let us fix  > 0 and for  ∈ [0, ], let us put The sequences {  } and {  ()} are nonincreasing for all  ∈ [0, ], so they have limits Moreover, each function   is nondecreasing; therefore   , and  ∞ are measurable on [0, ] for  = 1, 2, . ... Now, we apply the family of measures of noncompactness {  } ≥0 defined in (R + , X) by formula (13).In view of the above notation, we have We show that To fix a number  ∈ [0, ],  ∈ N and take an arbitrary number  ∈ [0, ].We know from Lemma 5 that for any  > 0, there is a sequence This implies that there is a sequence {  } ∞ =1 ⊂ Γ  , such that Hence, in view of Lemma 6, ( 4 ), and (36), we obtain Since  > 0 is arbitrary, it follows from the above inequalities that Using (36), we have Letting  → ∞, we derive the following inequality: This inequality, together with Gronwall's Lemma 8, implies that Next let us notice that in view of ( 4 ) and ( 36), we have for a.e. ∈ [0, ], and the function is measurable.Then, in virtue of equality   0 (Γ  ) =   0 (Γ  ) and Lemma 9, we get Hence, we derive Letting  → ∞, we get Keeping in mind (47), we deduce that This together with ( 39) and ( 47) yields lim Finally, using Remark 2 for the measure   , we deduce that the set Γ ∞ := ⋂ ∞ =0 Γ  is nonempty, convex, and compact.Then, by the Schauder-Tychonoff theorem, we conclude that operator  : Γ ∞ → Γ ∞ has at least one fixed-point  = ().Obviously, the function  = () is a solution of problem ( 6)- (7), and, in view of the definition of the set Γ ∞ , the estimate ‖()‖ ≤ P() holds to be true.This completes the proof.

Local and Global Attractivity
Following the concepts introduced in [36], we introduce first a few definitions of various kinds of the concept of attractivity of mild solution of ( 6).
Definition 11.The mild solution  = () of ( 6) with initial condition ( 7) is said to be globally attractive if for each mild solution V = V() of ( 6) with initial condition V(0) = V 0 we have that lim In other words, we may say that solutions of (6) are globally attractive if for arbitrary solutions () and V() of this equation condition (55) are satisfied.
In the case when the limit ( 55) is uniform with respect to all solutions V(), that is, when for each  > 0 there exist  > 0 such that for all V() being solutions of (6) with initial-value V(0) ∈ ((0), ), and for  ≥ , we will say that solution  = () is uniformly locally attractive on R + .Now, we formulate the main result of this section.We will consider (6) under the following conditions: A is the infinitesimal generator of an exponentially stable  0 -semigroup { −A } ≥0 ; that is, there exist  > 0,  > 0 such that ‖ −A ‖ ≤  − for all  ≥ 0; ( 5 ) there exist locally integrable functions  : R + → R + , such that for  ≥ 0 and , V ∈ X.Moreover, we assume that lim Remark 13.The property (  1 ) is generally satisfied in diffusion problem.A necessary and sufficient condition for (  1 ) is presented in [72].
The main result of this section is shown in the given theorem below.Theorem 14.Under assumptions (  1 ) and ( 2 )-( 5 ), problem (6)-( 7) has a mild solution  = () for each  0 ∈ X, which is globally attractive and locally uniformly attractive.
Let V = V() denote a mild solution of ( 6) with the initial condition V(0) = V 0 .Using (  1 ) and ( 3 ), we get Now, let us put Taking into account Lemma 5, we obtain Further, let  be the set of all mild solutions V() of ( 6) with the initial-value The estimate (61) implies that the function () is well defined.Applying ( 5 ) and (  1 ), we get or, equivalently, Using again Lemma 5 for the above estimate (where ℎ() =   ()), we obtain Elementary calculations lead to the following equality: Hence, Applying assumption ( 5 ), we derive lim and this proves that () is locally attractive.Finally, this equality together with definition of the function () implies that () is globally attractive.The proof is complete.
Remark 15.In the case when () ≡  is constant, the following condition lim means that  < .Observe that this condition cannot be weakened.This observation is illustrated by the following exampled.
Example 16.Let X = R, () ≡ , F(, ) = sin  + ,  = 1, and  −A =  − .Then the equation (for any fixed  0 ∈ R) has the solution () expressed by the following formula: Notice that for  ≥ , the solution () is neither globally attractive nor locally uniformly attractive, because for each other solution V() with initial condition V(0) = V 0 , obviously described by similar formula as (), we would have a contradiction:

Asymptotic Behaviour
In this section, we will give a theorem describing asymptotic behaviour of mild solutions of ( 6) with condition (7).This theorem generalizes the result included in [72,Theorem 4.4].First, we formulate the assumptions.
(  3 ) This condition is almost identical with ( 3 ) and the only difference is that we assume the functions  and  are locally essentially bounded on R + .
(  5 ) There exists  ∞ ∈ X such that there exists the limit lim  → ∞ F(,  ∞ ) and lim Moreover, there exists a number  < / such that for  ≥ 0 and  ∈ X.
Proof.The existence of a mild solution () is guaranteed by Theorem 10.Let us put We show that lim Recall that if assumption (  1 ) is fulfilled, then for each  ∈ X we have (see [72]) Using the above fact and (  5 ), we get Linking the above equality with (  1 ), we obtain Next, putting and applying (  5 ), we derive the following inequality: Hence, The above inequality in conjunction with Lemma 5 gives Journal of Function Spaces and Applications Hence, Before proving (76), we first show that lim To prove this equality, it is sufficient to show that the last component in the formula expressing (), that is, To this end, let us fix  > 0. Assumption (  5 ) implies that there exists  0 such that This inequality together with (  3 ) implies Now, fix Then using ( 86) and (88), we conclude that for  ≥  1 we have This fact proves (85).Further, using (  5 ) and (85) and employing de L'Hospital's rule for the fraction on the right-hand side of inequality (84), we obtain that condition (76) is satisfied.This fact completes the proof.

An Application to the Ginzburg-Landau Equations of Superconductivity
In this section, we formulate the gauged time-dependent Ginzburg-Landau (TDGL) equations as an abstract evolution equation in a Hilbert space.Moreover, we show applications of the above theorems to TDGL equations.We assume that Ω is a bounded domain in R  with boundary Ω of class  1,1 .That is, Ω is an open and connected set whose boundary Ω is a compact ( − 1)-manifold described by Lipschitz continuous differentiable charts.We consider two-and three-dimensional problems ( = 2 and  = 3, resp.).Assume that the vector potential A takes its values in R  .The vector H will represent the (externally) applied magnetic field, which is a function of space and time; similarly to A, it takes its values in R  .The function  is defined and satisfies Lipschitz condition on Ω, and () ≥ 0 for  ∈ Ω.The parameters in the TDGL equations are , a (dimensionless) friction coefficient, and , the (dimensionless) Ginzburg-Landau parameter.
The order parameter should be thought of as the wave function of the center-of-mass motion of the "superelectrons" (Cooper pairs), whose density is   = || 2 and whose flux is J  .The vector potential A determines the electromagnetic field; E = −  A − ∇ is the electric field and B = ∇ × A is the magnetic induction, where J, the total current, is the sum of a "normal" current J  = E, the supercurrent J  , and the transport current J  = ∇ × H.The normal current obeys Ohm's law J  =   E; the "normal conductivity" coefficient   is equal to one in the adopted system of units.The difference M = B−H is known as the magnetization.The trivial solution ( = 0, B = H, E = 0) represents the normal state, where all superconducting properties have been lost.Now we accept the following notion: all Banach spaces are real; the (real) dual of a Banach space  is denoted by   .The symbol   (Ω), for 1 ≤  ≤ ∞, denotes the usual Lebesgue space, with norm ‖ ⋅ ‖   ; (⋅, ⋅) is the inner product in  2 (Ω). ,2 (Ω), for nonnegative integer , is the usual Sobolev space, with norm ‖ ⋅ ‖  ,2 ;  ,2 (Ω) is a Hilbert space for the inner product (⋅, ⋅) ,2 given by (, V) ,2 = ∑ ||≤ (  ,   V) for , V ∈  ,2 (Ω).Fractional Sobolev space  ,2 (Ω), with a fractional , is defined by interpolation ([40, Chap.VII], and [41,49,50]). ] (Ω), for ] ≥ 0, ] =  +  with 0 ≤  < 1, is the space of  times continuous differentiable functions on Ω; those th-order derivatives satisfy the Hölder condition with exponent  if ] is a proper fraction; the norm ‖ ⋅ ‖  ] is defined in the usual way.
The definitions extend to the space of vector-valued functions in the standard way, with the caveat that the inner product in [ 2

(Ω)]
is defined by (, V) = ∫ Ω  ⋅ V, where the symbol (, V) indicates the scalar product in R  .Complexvalued functions are interpreted as vector-valued functions with two real components.Functions that vary in space and time, like the order parameter and the vector potential, are considered as mappings from the time domain, which is a subinterval of [0, ∞), into spaces of complex-or vector-valued functions defined in Ω.Let  = (, ‖ ⋅ ‖  ) be a Banach space of functions defined in Ω.Then, the functions are defined in Ω.Then, the functions of space and time defined on Ω × (0, ), for  > 0, may be considered as elements of   (0, ; ), for 1 ≤  ≤ ∞, or  2, (0, ; ), for nonnegative , or  ] (0, ; ), for ] ≥ 0, ] =  +  with 0 ≤  < 1. Detailed definitions can be found, for example, in [43].
Obviously, function spaces of ordered pairs (, A), where  : Ω → R 2 and A : Ω → R  ( = 2, 3), play an important role in the study of the gauged TDGL equations.We therefore adapt the following special notation: H = [(Ω)] 2 ×[(Ω)]  , for any Banach space for the order parameter  and the vector potential A, respectively.A suitable framework for the functional analysis of the gauged TDGL equations is the on the domain We now introduce the reduced vector potential A  : In terms of  and A  the gauged TDGL equations have the following form: Here,  and F are nonlinear functions of  and A  : The equations are supplemented by initial data, which is in the followimg form: where  0 and A 0 are given, and by (92), we have In the next part we connect the evolution of the solution (, A  ) of the system of (93) with the initial data ( 0 , A 0 − A H (0)) with the dynamics of a vector  in a Hilbert, space The following analysis is restricted to the case  > 0 and the case  = 0 (see [73]).
Namely, let us assume that X = L 2 and  0 ∈ L 2 , and  : R + × R  → R  for  ∈ N is an unknown function,  = (, ).In order to apply Theorems 10 and 14, we are not going to consider  as a function of  and  together, but rather as a mapping  of variable  into the space X = L 2 of functions , that is,  : R + → L 2 , ()() = (, ),  ∈ R  ,  ∈ R + .
Remark 19.Since A is the linear self-adjoint operator in L 2 associated with the quadratic form (100), then A generates a semigroup { −A } ≥0 (see [74]).
Below, we formulate the principal theorem of this paper.This theorem is a simple consequence of Theorem 10.
We observe that in the case when { −A } ≥0 is the semigroup of contractions and () ≡ , () ≡  for  ∈ R + are constant, after simple calculations based on estimate (25), we get that the solution  = () has the asymptotic characterization: ‖ ()‖ =  ( 3 ) , as  → ∞. (103) For further purposes, let us formulate the following assumption: (A) A is the infinitesimal generator of an exponentially stable  0 -semigroup { −A } ≥0 .
The next result of this chapter is shown in the given theorem below.