On well-posedness of vector-valued fractional differential-difference equations

We develop an operator-theoretical method for the analysis on well posedness of partial differential equations that can be modeled in the form \begin{equation*} \left\{ \begin{array}{rll} \Delta^{\alpha} u(n)&= Au(n+2) + f(n,u(n)), \quad n \in \mathbb{N}_0, \,\, 1<\alpha \leq 2; u(0)&= u_0; u(1)&= u_1, \end{array} \right. \end{equation*} where $A$ is an closed linear operator defined on a Banach space $X$. Our ideas are inspired on the Poisson distribution as a tool to sampling fractional differential operators into fractional differences. Using our abstract approach, we are able to show existence and uniqueness of solutions for the problem (*) on a distinguished class of weighted Lebesgue spaces of sequences, under mild conditions on strongly continuous sequences of bounded operators generated by $A,$ and natural restrictions on the nonlinearity $f$. Finally we present some original examples to illustrate our results.


Introduction
In an interesting paper published in 1943, H. Bateman [13] studied the differentialdifference equation where M, K, S, a, b, c are positive constants.This general formulation includes previous work of B. Taylor, J. Bernoulli and D. Bernoulli.In particular, it includes the temporal discretization of the diffusion equation and the wave equation Defining v(n)(x) := u(n, x) and A = ∂ 2 ∂x 2 the diffusion and wave equations can be rewritten as ∆v(n) = Av(n), and ∆ 2 v(n) = Av(n + 1), n ∈ N, respectively, where A is the generator of a C 0 -semigroup of operators given by f (s)ds.
Note that equation (1.2) is likewise used in the so called car-following problems, see for example [16].
Note that A is the generator of the translation semigroup T (t)f (x) = f (x − t).
The difference-differential equation with appropriate initial values define Chebyshev polynomials of the first and second kind.Indeed, take T (0, x) = 1 and T (1, x) = x in the first case and T (0, x) = 1and T (1, x) = 2x in the second case.Defining u(n)(x) := T (n, x), equation (1.3) can be rewritten in abstract form as where Af (x) = 2(x − 1)f (x).We observe that A is the generator of the multiplication semigroup: T (t)f (x) = e 2(x−1) f (x).These abstract models, with unbounded operators A defined on Banach spaces, are closely connected with numerical methods for partial differential equations, integro-differential equations [17,28] and evolution equations [24], [26].See also the recent monograph [3].Recently, it has been shown that the extension of some of this models to fractional difference equations is a promising tool for several biological and physical applications where a memory effect appears [8,12].
In spite of the significant increase of research in this area, there are still many significative questions regarding fractional difference equations.In particular, the study of fractional difference equations with closed linear operators and their qualitative behaviour remains an open problem.
In this paper, we propose a novel method to deal with this classes of abstract fractional difference problems.This method is inspired on sampling by means of the Poisson distribution.We will use it to develop a theory on well-posedness for the abstract fractional difference problem (1.4) where A : D(A) ⊂ X → X is a closed linear operator defined on a Banach space X and f a suitable function.
We notice that first studies on the model (1.4) when A is a complex or real valued matrix, have only recently appeared [9,19].However, the study of this equation when A is a closed linear operator, not necessarily bounded, has not been considered in the literature.
The approach followed here is purely operator-theoretic and has as main ingredient the use of the Poisson distribution p n (t) = e −t t n n! , n ∈ N 0 , t ≥ 0.
Our method relies in to take advantage of the properties of this distribution when it is applied to discrete phenomena.More precisely, given a continuous evolution (u(t)) t∈[0,∞) we can discretize it by means of that we will call the Poisson transformation In this paper, we will show that when this procedure is applied to continuous fractional processes, these transformations are well behaved and fit perfectly in the discrete fractional concepts.A remarkable feature of this work will be to show that by this method of sampling we recover the concept of fractional nabla sum and difference operator in [7], which has been used recently and independently of the method used here by other authors in order to obtain qualitative properties of fractional difference equations, notably concerning stability properties [14,15].The outline of this paper is as follows: In Section 2, we give some background on the definitions to be used.The remarkable fact here is that we highlight a particular choice of the definition introduced in [10] for the nabla operator; see the recent paper [23].This choice, that has been implicitly used by other authors [14,15], is proved to be the right notion in the sense that where, See Definition 2.2 and Theorem 5.5 below.Then, we can show an interesting connection between the Delta operator of order α (i.e. the Riemann-Liouville-like fractional difference) in the right hand side of (1.6) and the Caputo-like fractional difference by means of the identity (Theorem 2.5): In Section 3, we use successfully the preceding definitions and properties to solve the problem (1.4), firstly, in the homogeneous linear case.In order to do that, we construct a distinguished sequence of bounded and linear operators {S α (n)} n∈N 0 that solves the homogeneous linear initial value problem See Theorem 3.5.In particular, when the operator A is bounded, we derive an explicit representation of the solution (Proposition 3.2).From a different point of view, this representation can be considered as the discrete counterpart of the Mittag-Leffler function t α−1 E α,α (At α ) (when A is a complex number) which interpolates between the exponential and hyperbolic sine function for 1 < α < 2.
In Section 4 we study the fully nonlinear problem (1.4).After to introduce the notion of solution, which is motivated by the representation of the solution in the non-homogeneous linear case (Corollary 3.6), we consider a distinguished class of vector-valued spaces of weighted sequences, that behaves like This vector-valued Banach spaces of sequences will play a central role in the development of this section.The main ingredient for the success of our analysis is the observation that the special weight w(n) = nn!, that represents the factorial representation of a positive integer, proves to be suitable to find existence of solutions for (1.4) in the above defined space l ∞ w (N 0 ; X) under the hypothesis of only boundedness of the sequence of operators S α (n).We give two positive results in this direction.See Theorem 4.4 and Theorem 4.6.In Section 5, we prove several relations between the continuous and discrete setting, including the notable identity (1.6).See Theorem 5.5 below.This relations are obtained in the context of the Poisson transformation (1.5) whose main properties are established in Theorem 5.2.Note that the idea of discretization of the fractional derivative in time was employed in the paper [18] (see also [17] and references therein).Section 6 is devoted to the construction of sequences of operators {S α (n)} n∈N 0 via subordination by the Poisson transformation of α-resolvent families generated by A (Theorem 6.3).A remarkable consequence is Theorem 6.7, which proves existence of solution for the nonlinear problem (1.4) in the space l ∞ w (N 0 ; X) under the hypothesis that A is the generator of a bounded sine family such that the resolvent operator (λ − A) −1 is a compact operator for some λ large enough.
Finally, Section 7 provide us with several examples and applications of our general theorems, notably concerning the cases where either A is a multiplication operator or the second order partial differential operator ∂ 2 ∂x 2 .We also pay special attention to the case α = 2 and to some related problems formatted in a slighty different way than (1.4).
Notation We denote by N 0 := {0, 1, 2, ...}, the set of non-negative integer numbers and X a complex Banach space.We denote by s(N 0 ; X) the vectorial space consisting of all vectorvalued sequences u : N 0 → X.We recall that the Z-transform of a vector-valued sequence f ∈ s(N 0 ; X), is defined by where z is a complex number.Note that convergence of the series is given for |z| > R with R sufficiently large.
Recall that the finite convolution * of two sequences u ∈ s(N 0 ; C) and v ∈ s(N 0 ; X) is defined by where R 1 and R 2 are the radius of convergence of the Z-transforms of u and v respectively.The Banach space ℓ 1 (X) is the subset of s(N 0 ; X) such that u 1 := ∞ n=0 u(n) < ∞; and the Lebesgue space L 1 (R + ; X) is formed by measurable functions f : R + → X such that The usual Laplace transform is given by In the case X = C, the Banach space L 1 (R + ) is, in fact, a Banach algebra with the usual convolution product * given by The same holds in the case of (ℓ 1 , * ).The Banach space C (m) (R + ; X) is formed for continuous functions which have m-continuous derivatives defined on R + with m ∈ N 0 .Let S : R + → B(X) be strongly continuous, that is, for all x ∈ X the map t → S(t)x is continuous on R + .We say that a family of bounded and linear operators {S(t)} t≥0 is exponentially bounded if there exist real numbers M > 0 and ω ∈ R such that We say that {S(t)} t≥0 is bounded if ω = 0. Note that if {S(t)} t≥0 is exponentially bounded then the Laplace transform Ŝ(λ)x exists for all ℜ(λ) > ω.

Fractional difference operators
The forward Euler operator ∆ : s(N 0 ; X) → s(N 0 ; X) is defined by For m ∈ N, we define recursively ∆ m : s(N 0 ; X) → s(N 0 ; X) by ∆ 1 = ∆ and The operator ∆ m is called the m-th order forward difference operator and for u ∈ s(N 0 ; X).We also denote by ∆ 0 = I, where I is the identity operator.We define , n ∈ N 0 , α > 0.
The following definition of fractional sum (also called Cesàro sum in [32]) has appeared recently in some papers, see for example [1,23].It has proven to be useful in the treatment of fractional difference equations.Note that this definition is implicitly included in e.g.[4,7,25].Definition 2.1.[23, Definition 2.5] Let α > 0. The α-th fractional sum of a sequence u : N 0 → X is defined as follows One of the reasons to choose this operator in this paper is because their flexibility to be handled by means of Z-transform methods.Moreover, it has a better behavior for mathematical analysis when we ask, for example, for definitions of fractional sums and differences on subspaces of s(N 0 ; X) like e.g.l p spaces.We notice that, recently, this approach by means of the Z-transform has been followed by other authors, see [14,15].
The next concept is analogous to the definition of a fractional derivative in the sense of Riemann-Liouville, see [6,25].In other words, to a given vector-valued sequence, first fractional summation and then integer difference are applied.Definition 2.2.[23, Definition 2.7] Let α ∈ R + \N 0 .The fractional difference operator of order α in the sense of Riemann-Liouville, ∆ α : s(N 0 ; X) → s(N 0 ; X), is defined by We iterate m-times with m ∈ N to get for β > m that Let 0 < α < β and m − 1 < α < m for m ∈ N. By Definition 2.2 and (2.1), we get that for n ∈ N 0 .This equality extends [22,Corollary 3.6] given for 0 < α < 1.
Interchanging the order of the operators in the definition of fractional difference in the sense of Riemann-Liouville, and in analogous way as above, we can introduce the notion of fractional difference in the sense of Caputo as follows.
For further use, we note the following relation between the Caputo and Riemann-Liouville fractional differences of order 1 < α < 2. The connection between the Caputo and Riemann-Liouville fractional differences of order 0 < α < 1 is given in [22,Theorem 2.4].
Theorem 2.5.For each 1 < α < 2 and u ∈ s(N 0 ; X), we have Proof.By definition and (2.4) we have and so we obtain the desired result.
We also have the following property for the Riemann-Liouville fractional difference of the convolution.
proving the claim.
We notice that for 0 < α ≤ 1 the above property reads It has been proved only recently in [23,Lemma 3.6].

Linear fractional difference equations on Banach spaces
Let A be a closed linear operator defined on a Banach space X.In this section we study the problem We say that a vector valued sequence u ∈ s(N 0 ; X) is a solution of (3.1) if u(n) ∈ D(A) for all n ∈ N 0 and u satisfies (3.1).
We will use the notion of discrete α-resolvent family introduced in [1, Definition 3.1] to obtain the solution of the problem (3.1).Note that the knowledge of the abstract properties of this family of bounded operators provide insights on the qualitative behavior of the solutions of fractional difference equations.Definition 3.1.Let α > 0 and A be a closed linear operator with domain D(A) defined on a Banach space X.An operator-valued sequence An explicit representation of discrete α-resolvent family generated by bounded operators A with A| < 1 is given in the following proposition.Proposition 3.2.Let α > 0 and A ∈ B(X), with A < 1.Then the operator A generates a discrete α-resolvent family {S α (n)} n∈N 0 given by , for n ∈ N, (see for example [32, Vol.I, (1.18)]), then the series is convergent for A < 1.Take x ∈ X and n ∈ N 0 .Then we get that where we have applied the semigroup property of the kernel k α .Then we obtain and we conclude the proof.
For α > 0 fixed and each n ∈ N the sequence {β α,n (j)} j=1,...,n was introduced in [1, Section 3.1] as follows: For n = 1, For n = 3, For n ≥ 4, In case that A is closed, but not necessarily bounded, the authors in [1, Theorem 3.2] proved that given {S α (n)} n∈N 0 ⊂ B(X) a discrete α-resolvent family generated by A, then 1 ∈ ρ(A) and S α (0) = (I − A) −1 ; S α (0)x ∈ D(A) and S α (n)x ∈ D(A 2 ) for all n ∈ N, and x ∈ X; and The last equality provides an explicit representation of discrete α-resolvent families in terms of a bounded linear operators which is, in fact, a characterization of this family of operators as the next theorem shows.Theorem 3.3.Let λ, α > 0, (A, D(A)) be a closed operator on the Banach space X and {S α (n)} n∈N 0 ⊂ B(X) be a sequence of bounded operators.Then the following conditions are equivalent.
(i) The family If there exists λ 0 > 0 such that sup n∈N 0 λ −n 0 S α (n) < ∞, both equations are equivalent to Proof.The condition (i) implies the condition (ii) is given in [1, Theorem 3.2].Now we suppose that the condition (ii) holds.Then S α (n)x ∈ D(A) for any x ∈ X and n ∈ N 0 .For n ∈ N and x ∈ X we have that Applying that ( and clearly it follows that S α (n Finally we prove that if there exists λ 0 > 0 such that sup n∈N 0 λ −n 0 S α (n) < ∞, (iii) is equivalent to (i).Assume that {S α (n)} n∈N 0 ⊂ B(X) is a discrete α-resolvent family generated by A, then applying Z-transform we get for |λ| > max{λ 0 , 1} that where we have used that (1.7) and (2.3).Thus the operator λ−1 λ α − A is invertible, and we get (3.2).Conversely, let |λ|, |µ| > max{λ 0 , 1} and x ∈ D(A), then there exists y ∈ X such are bounded operators and commute, and A is closed we have that The uniqueness of Z-transform proves that Then we have S α (n)x ∈ D(A), and therefore AS α (n)x = S α (n)Ax for all x ∈ X. Finally note that for |λ| > max{λ 0 , 1} and x ∈ D(A) we have using (2.3) that and by the uniqueness of Z-transform we get the result.
A beautiful consequence of Theorem 3.3 is the following result about sums of combinatorial numbers which seems to be new.Corollary 3.4.Take α > 0, n ∈ N and {β α,n (j)} j=1,...,n defined as above.Then , for l ∈ N.
Proof.(i) We take |λ| < 1, then using Proposition 3.2 and Theorem 3.3 in the scalar case we have that where we have applied that . Then we apply (i) to get the result.
Our main result in this section is the following theorem.
Theorem 3.5.Suppose that A is the generator of a discrete α-resolvent family {S α (n)} n∈N 0 on a Banach space X.Then the fractional difference equation with initial conditions u(0) = u 0 ∈ D(A) and u(1) = u 1 ∈ D(A) admits the unique solution Proof.Convolving the identity given in Definition 3.1(ii) by k 2−α , we obtain Using the semigroup property for the kernels k α we have This is equivalent, by definition of fractional sum and convolution, to the following identity Therefore, we get using ∆ 2 k 2 (j) = 0 for j ∈ N 0 that for all n ∈ N 0 .We note that the left hand side in the above identity corresponds to the fractional difference of order α ∈ (0, 2) in the sense of Riemann-Liouville.Therefore, we obtain for all n ∈ N 0 and all x ∈ X. Define u(n) as It then follows from (3.4) that u solves (3.3).Finally, from the identities In the non homogeneous case, we derive the following result.
Corollary 3.6.Suppose that A is the generator of a discrete α-resolvent family {S α (n)} n∈N 0 on a Banach space X and f be a vector-valued sequence.The fractional difference equation with initial conditions u(0) = u 0 ∈ D(A) and u(1) = u 1 ∈ D(A), admits the unique solution for all n ≥ 2.
Proof.Indeed, by Theorem 3.5 and Theorem 2.6 we have u(n) ∈ D(A) for all n ≥ 2 and , and hence we obtain where we have used that S α (0) = (I − A) −1 and (I + A)S α (1) = αS α (0).For n = 0 and n = 1 it is a simple check, using the same above arguments, that u is solution of (3.5).

Non-linear fractional difference equations on Banach spaces
Let A be a closed linear operator defined on a Banach space X.In this section we study the non linear problem (4.1) The following definition is motivated by Corollary 3.6.In particular, it shows their consistence with the problem (4.1).
Definition 4.1.Under the assumption that the operator A is the generator of a discrete α-resolvent family {S α (n)} n∈N 0 on a Banach space X, we say that u : N 0 → X is a solution of the non-linear problem (4.1) if u satisfies The next concept of admissibility is one of the keys ingredients for the estimates that we will use in the proofs of our main results on existence of solutions to (4.1).Definition 4.2.We say that a sequence h : For each admissible weight sequence h, we consider the vector-valued weighted space where the norm h is defined by ξ h := sup The following is our first positive result on existence of solutions for the problem (4.1).It uses a Lipschitz type condition.Theorem 4.4.Let h be an admissible weight and define Let A be the generator of a bounded discrete α-resolvent family {S α (n)} n∈N 0 on a Banach space X for some 1 < α ≤ 2, and let f : N 0 × X → X be such that f (k, 0) = 0 for all k ∈ N 0 , verifying the following hypothesis: (L) The function f satisfies a Lipschitz condition in x ∈ X uniformly in k ∈ N 0 , that is, there exists a constant First, we show that G is well defined: Let u ∈ l ∞ h (N 0 ; X) be given.By using the assumption (L) for y = 0 and the boundedness of {S α (n)} n∈N 0 we get that, where we have used the assumption (L).Therefore

and consequently
Gu − Gv h ≤ S α ∞ HL u − v h , with S α ∞ HL < 1.Then, G has a unique fixed point in l ∞ h (N 0 ; X), by the Banach fixed point theorem.
The next Lemma provide a necessary tool for the use of the Schauder's fixed point theorem, needed in the second main result on existence and uniqueness of solutions to (4.1).Lemma 4.5.Let h be an admissible weight and U ⊂ l ∞ h (N 0 ; X) such that: Then U is relatively compact in l ∞ h (N 0 ; X).Proof.Let {u m } m be a sequence in U, then by (a) for n ∈ N 0 there is a convergent subse- and also Consequently, The next theorem is the second main result for this section.It gives one useful criteria for the existence of solutions without use of Lipchitz type conditions.Theorem 4.6.Let h be an admissible weight function.Let A be the generator of a bounded discrete α-resolvent family {S α (n)} n∈N 0 on a Banach space X for some 1 < α ≤ 2, and f : N 0 × X → X. Suppose that the following conditions are satisfied: (i) There exist a sequence M ∈ l ∞ (N 0 ) and a function W : R Then the problem (4.1) has an unique solution in l ∞ h (N; X).

Proof. Let us define the operator
To prove that G has a fixed point in l ∞ h (N 0 ), we will use Leray-Schauder alternative theorem.We verify that the conditions of the theorem are satisfied: , then by the assumption (i) Therefore, for each n ∈ N 0 , we have Therefore

Hence, by the assumption (ii) and admissibility of h we obtain Gu
) is relatively compact, we will use Lemma 4.5.We check that the conditions in such Lemma are satisfied: and then, , where co(K n ) denotes the convex hull of K n for the set Note that each set K n is relatively compact by the assumption (iii).From the inclusions ) such that u = γGu, γ ∈ (0, 1).Again by (i), Then for each n ∈ N 0 we have We deduce that U is a bounded set in l ∞ h (N 0 ; X).Finally, by using the Leray-Schauder alternative theorem, we conclude that G has a fixed point u ∈ l ∞ h (N 0 ).

The Poisson transformation of fractional difference operators
For each n ∈ N 0 , the Poisson distribution is defined by The Poisson distribution arises in connection with classical Poisson processes and semigroups of functions; note that it is also called fractional integral semigroup in [29,Theorem 2.6].
In this section we study in detail this sequence of functions (Proposition 5.1), the Poisson transformation (considered deeply in Theorem 5.2) and give their connection with fractional difference and differential operators in Theorem 5.5.
(iii) Given t ≥ 0, then (iv) For m, n ∈ N 0 , we have n+m .(v) The Z-transform and the Laplace transform are given by Proof.The proof of (i) and (ii) is straightforward, and also may be found in [29,Theorem 2.6].To show (iii), note that for n ∈ N 0 and t ≥ 0. Now we get that and we iterate to obtain the equality ∆ m p n = (−1) m p (m) n+m for m, n ∈ N 0 .Finally the Z-transform and the Laplace transform of (p n ) n≥0 are easily obtained.Now we introduce an integral transform using the Poisson distribution as integral kernel.Some of their properties are inspired in results included in [22,Section 3] in particular a remarkable connection between the vector-valued Z-transform and the vector-valued Laplace transform, Theorem 5.2 (ii).Theorem 5.2.Let ψ ∈ L 1 (R + ; X) and we define (Pψ) ∈ s(N 0 ; X) by Then the following properties hold.
(i) The map P defines a bounded linear operator from L 1 (R + ; X) to ℓ 1 (N 0 ; X) and P = 1.(ii) For ψ ∈ L 1 (R + ; X), we have that In particular the map P is inyective.
Also the identity given in Theorem 5.2 (iii) holds for the Dirac distribution δ t for t > 0, see Proposition 5.1 (v).
By the uniqueness theorem for the Laplace transform, a 1-resolvent family is the same as a C 0 -semigroup, while a 2-resolvent family corresponds to a strongly continuous sine family.See for example [5] and the references therein for an overview on these concepts.Some properties of (S α (t)) t>0 are included in the following Lemma.For a proof, see for example [21].Lemma 6.2.Let α > 0. The following properties hold: = x for all x ∈ X in case 0 < α < 1).
(iv) For all x ∈ X : (g α * S α )(t)x ∈ D(A) and The next theorem is the main result of this section.
Theorem 6.3.Suppose that A is the generator of an α-resolvent family (S α (t)) t>0 on a Banach space X, of exponential bound less than 1.Then A is the generator of a discrete α-resolvent family (S α (n)) n∈N 0 defined by Proof.Take x ∈ D(A).Since (A, D(A)) is a closed operator and the condition in Lemma 6.2(ii) we have that From the identity S α (t)x = g α (t)x + A t 0 g α (t − s)S α (s)xds, t ≥ 0, valid for all x ∈ X, we obtain where we have applied Example 5.4(iii) and Theorem 5.2 (iv) and the second condition in Definition 3.1.The theorem is proved.
Example 6.4.Consider the Mittag-Leffler function E α,β studied in Example 5.4 (iv).Suppose that A is a bounded operator on the Banach space X.It then follows from Definition 6.1 that for x ∈ X. Compare with Proposition 3.2.
Example 6.5.Let us recall the definition of ω-sectorial operator.A closed and densely defined operator A is said to be ω-sectorial of angle θ if there exist 0 < θ < π/2, M > 0 and ω ∈ R such that its resolvent exists outside the sector ω+Σ Suppose that A is a ω-sectorial operator of angle θ < απ/2 and ω < 0. Then A is the generator of a bounded α-resolvent family (S α (t)) t>0 on X for 1 < α < 2 given by where Γ is a suitable path where the resolvent operator is well defined.By Theorem 3.5 and 5.2, its Poisson transformation S α (n) := P(S α )(n) defines a bounded discrete α-resolvent family {S α (n)} n∈N 0 ⊂ B(X).
Example 6.6.Suppose that A is the generator of a bounded sine family (S(t)) t>0 on X.
Our next corollary imposes a natural and useful condition of compactness on a given family of operators in order to obtain existence and uniqueness of solutions.Theorem 6.7.Suppose that A is the generator of a bounded sine family (S(t)) t>0 on X such that (λ − A) −1 is a compact operator for some λ large enough.Let h be an admissible weight and f : N 0 × X → X satisfying the following conditions: (i) There exist a function M ∈ l ∞ (N 0 ) and a function W : R + → R + , with W (x) ≤ Cx for x ∈ R + , such that f (k, x) ≤ M(k)W ( x ) for all k ∈ N 0 and x ∈ X. (ii) The Nemytskii operator N f is continuous in l ∞ h (N 0 ; X).Then, for each 1 < α ≤ 2, the problem (4.1) has an unique solution in l ∞ h (N 0 ; X).Proof.To prove this result we only have to check that the assumption (iii) in Theorem 4.6 is satisfied.Indeed, by hypothesis we have that (λ α − A) −1 is compact for all λ α ∈ ρ(A) and all 1 < α ≤ 2. By Example 6.6 we obtain that A is the generator of a bounded α-resolvent family (S α (t)) t>0 , which is moreover compact by [31,Corollary 2.3].From Theorem 5.2 (vii) it follows that {S α (n)} n∈N 0 is compact.Also, for all a ∈ N 0 and σ > 0, the set {f

Examples, applications and final comments
In this section, we provide several concrete examples and applications of the abstract results developed in the previous sections.Finally we present some related problems with problem (1.4) for α = 2.
Since 0 < m(x) < 1 we obtain by subordination Observe that the operator A can be written as where z n (s) = 2/π sin ns, n = 1, 2, . . ., is an orthonormal set of eigenvectors of A. Note that A is the infinitesimal generator of a sine family S(t), t ∈ R, in L 2 [0, π], given by The resolvent of A is given by The compactness of R(λ; A) follows from the fact that eigenvalues of R(λ; A) are λ n = 1 λ+n 2 , n = 1, 2, . . ., and thus lim n→∞ λ n = 0, see for example [30].
Let us consider the weighted space (ii) For u 1 , u 2 ∈ l ∞ h (N 0 ; L 2 [0, π]) and each n ∈ N 0 , we have Consequently, by Corollary 6.7, we conclude that the problem (7.1) has an unique solution u ∈ l ∞ h (N 0 ), that is, u satisfies where B is a linear operator defined on a Banach space X.In such cases, and under mild conditions, we can still handle this problem with our theory.That is the content of the following two results.