Some Contributions to the Theory of Abstract Degenerate Volterra Integro-Differential Equations

Email: marco.s@verat.net Abstract: In this study, we contribute to the existing theory of abstract degenerate Volterra integro-differential equations in sequentially complete locally convex spaces. We investigate a class of abstract degenerate Volterra inclusions by using the multivalued linear operator approach, as well as a class of abstract degenerate multi-term fractional differential equations with Caputo derivatives by using the pure Laplace transform techniques.


Introduction and Preliminaries
As mentioned in the abstract, the main aim of this paper is to contribute to the existing theory of abstract degenerate Volterra integro-differential equations in sequentially complete locally convex spaces (for the basic source of information on the abstract degenerate differential equations with integer order derivatives, we refer the reader to the monographs (Carroll and Showalter, 1976;Demidenko and Uspenskii, 2003;Favini and Yagi, 1998;Melnikova and Filinkov, 2001;Sviridyuk and Fedorov, 2003). The study of abstract degenerate Volterra equations starts presumably with the paper by Favini and Tanabe (2001), while the study of abstract degenerate fractional differential equations starts presumably with the paper by Fedorov and Debbouche (2013). In this study, we continue our research studies (Kostić, 2015b;2016;0000c) by investigating the abstract degenerate multi-term Volterra inclusions and the abstract degenerate multi-term fractional differential equations with Caputo derivatives (cf. (Kim, 2002;Knuckles and Neubrander, 1994;Obukhovskii and Zecca, 2003), the forthcoming monograph (Kostić,0000a) and references cited therein for further information on the above subjects).
The organization of paper can be briey described as follows. In section 2, we repeat some known facts and definitions about multivalued linear operators in locally convex spaces. The main purpose of section 3 is to prove some results on the well-posedness of the following abstract degenerate multi-term Volterra inclusion: [ (Kostić, 2015b), (3.7)] and [ (Kostić,0000b), (2.1)]). In Definition 3.1, we introduce the notion of a V-(pre)-solution of problem (1.1), while in Theorem 3.4 and Theorem 3.5 we analyze subordination principles for V-solutions of problem (1.1). The classes of (pre)-(k, C 1 , V)-existence families for (1.1) and k-regularized C 2 -uniqueness families for (1.1) are introduced in Definition 3.6 and Definition 3.7 (here, 0 n V N ⊆ ). The main objective of Proposition 3.8 and Theorem 3.9-Theorem 3.12 is to extend some known theoretical results from [ (Kostić, 2015b), section 3] to the introduced classes of existence and uniqueness families (because of a great similarity with our previous research study (Kostić, 2015b), the proof is given only for Theorem 3.11). Without going into exceptional details, we will demonstrate how our results can be used to provide certain applications at the end of section 3.
In section 4, we continue our previous research of entire solutions of higher order degenerate Cauchy problems with integer order derivatives (Kostić, 2016) by investigating (cf. Theorem 4.2) the existence and uniqueness of analytical solutions of abstract degenerate multi-term Cauchy problem: where, n∈N\{1}, A, B and A 1 ,⋅⋅⋅A n-1 are closed linear operators on a sequentially complete locally convex space X, 0≤α 1 <⋅⋅⋅< α n , 0≤α<α n , f(t) is a continuous Xvalued function and t D α denotes the Caputo fractional derivative of order α (Kim, 2002;Kostić, 2015a)). We apply Theorem 4.2 in the analysis of fractional analogons of abstract Boussinesq-Love equation, for which it is known that plays an important role in the modeling the longitudinal waves in an elastic bar with the transverse inertia and the fractional analogons of abstract Barenblatt-Zheltov-Kochina equation, for which it is known that plays an important role in the study of fluid filtration in fissured rocks (cf. Example 4.4). We use the standard notation throughout the paper. Unless specified otherwise, we assume that X is a Hausdorff sequentially complete locally convex space over the field of complex numbers, SCLCS for short. If Y is also an SCLCS over the field of complex numbers, then we denote by L(Y, X) the space consisting of all continuous linear mappings from Y into X; L(X) ≡ L(X, X). By ⊗ X (⊗, if there is no risk for confusion), we denote the fundamental system of seminorms which defines the topology of X. The fundamental system of seminorms which defines the topology on Y is denoted by ⊗ Y . The symbol I denotes the identity operator on X. Assume 0<τ≤∞. Then a strongly continuous operator family (W(t)) t∈[0,τ) ⊆ L(Y, X) is said to be locally equicontinuous iff, for every T ∈ (0, τ) and for every p∈⊗ X , there exist q p ∈⊗ Y and c p > 0 such that p(W(t)y)≤c p q p (y), y∈Y, t∈ [0, T]; the notions of equicontinuity of (W(t)) t∈ [0,τ) and the exponential equicontinuity of (W(t))t≥0 are defined similarly. Suppose that A is a closed linear operator acting on X. Then we denote the domain, kernel space and range of A by D(A), N(A) and R(A), respectively. Since no confusion seems likely, we will identify A with its graph. Set p A (x): = p(x) + p(Ax), x∈D(A), p ∈⊗. Then the calibration (p A ) p∈⊗ induces the Hausdorff sequentially complete locally convex topology on D(A); we denote this space simply by [D(A)].
If V is a general topological vector space, then a function f: Ω→V, where is an open non-empty subset of C, is said to be analytic if it is locally expressible in a neighborhood of any point z∈Ω by a uniformly convergent power series with coefficients in V. We refer the reader to [(Arendt et al., 2001;Kostić, 2015a), section 1.1] and references cited there for the basic information about vector-valued analytic functions. In our approach the space X is sequentially complete, so that the analyticity of a mapping f: Ω→X is equivalent with its weak analyticity. It is said that a function f: [0, ∞)→E is locally Holder continuous with the exponent r∈(0,1] iff for each p∈⊗ and T>0 there exists M≥1 such that p(f(t)-f(s)) ≤ M|t-s| r , provided 0≤t, s≤T. The reader may consult [ (Prüss, 1993), Definition 4.5, p. 96] for the notion of a completely positive function on [0, ∞) (cf. also [(Lizama, 2000), Remark 3.6, (3.3)]). Concerning the integration of functions with values in sequentially complete locally convex spaces, we will follow the approach of cf. [ (Martinez and Sanz, 2001), pp. 99-102]).
Let 0 <τ≤∞ and F: We denote the set consisting of all continuous sections of F by sec c (F).

Multivalued Linear Operators in Locally Convex Spaces
In this section, we will present some necessary definitions from the theory of multivalued linear operators. For more details about this topic, we refer the reader to the monographs by Cross (1998) and Favini and Yagi (1998).
Let X and Y be two sequentially complete locally convex spaces over the field of complex numbers. A multivalued map A: X→P(Y) is said to be a Multivalued Linear Operator (MLO) iff the following holds: If X = Y, then we say that A is an MLO in X. As an almost immediate consequence of definition, we have that the equality λAx + ηAy = A(λx + ηy) holds for every x, y ∈ D(A) and for every λ, η∈C with |λ|+ |η| ≠ 0. If A is an MLO, then A0 is a linear manifold in Y and Ax = f + y∈Ax}. It can be easily seen that A −1 is an MLO in X, as well as that N( We say that an MLO A: X→P(Y) is closed if for any nets (x τ ) in D(A) and (y τ ) in Y such that y τ ∈ Ax τ for all τ∈I we have that the suppositions lim τ→∞ x τ = x and lim τ→∞ y τ = y imply x∈D(A) and y∈Ax. As mentioned below, some of our results hold even for relatively closed MLOs (Knuckles and Neubrander, 1994).
We need the following auxiliary lemma from (Kostić, 0000c).

Lemma 2.1
Let Ω be a locally compact, separable metric space and let µ be a locally finite Borel measure defined on Ω.

Suppose that A: X→P(Y) is a closed MLO. Let f: Ω→X
and g: Ω→Y be µ-integrable and let g(x) ∈ Af(x), x ∈ Ω.
In the remaining part of paper, Ω will always be an appropriate subspace of R and µ will always be the Lebesgue measure defined on Ω.
We need also the following lemma from (Kostić,0000c), which can be proved by using the Phragmen-Doetsch inversion formula for the Laplace transform.
for ℜλ> max(abs(f), abs(l)). Then l(t)∈Af(t) for any t≥0 which is a point of continuity of both functions f(t) and l(t).
In the following definition, we introduce the notion of a V-(pre)-solution of problem (1.1).
t∈[0, τ), j∈V and: , then we can simply prove with the help of Lemma 2.1 that any V 1 -(pre)-solution of problem (1.1) is a V 2 -(pre)-solution of (1.1), provided that the operators A j are closed for j∈V 2 \V 1 . If V = φ then the notion of a pre-solution of (1.1) coincides with that of solution of (1.1); if this is the case, then any solution of (1.1) is also said to be a strong solution of (1.1). Here it is worth noting that the assumption u∈C ([0, τ): X) in Definition 3.1 (i) is slightly redundant in the case that V = φ becuse then we do not have a problem with defining the convolutions (a j *u)(t) for j∈V. Assuming V = φ and all the remaining assumptions from Definition 3.1(i) hold, u(t) will be called a p-strong solution of problem (1.1). In the case that n = 1, the most important examples of p-strong solutions of problem (1.1) with a 0 (t) = g α (t), where α > 0, are obtained by integrating α-times p-

Remark 3.2
Suppose that A 0 ,⋅⋅⋅,A n-1 , B are single-valued linear operators between the spaces X and Y and the mapping f: [0, τ)→D(B) is given in advance. Consider the following degenerate Volterra integral equation: . Then the following holds:
In the remaining part of this section, it will be assumed that Z is a sequentially complete locally convex space. We would like to propose the following general definition.
It is said that (E(t)) t∈[0,τ) is a pre-(k, C 1 , V)-existence family for (1.1) iff for each z∈Z there exist continuous sections A j , z ∈sec c (A(a j *E(⋅)z)) such that: One can simply verify that the notion of a pre-(k,C 1 , V)-existence family for (1.1) generalizes the notion of a k-regularized C 1 -existence family introduced in [( Kostić, 2015b), Definition 3.8(i)], as well as the notion of a mild (a, k)-regularized C 1 -existence family sub generated by a multivalued linear operator, introduced in [(Kostić, 0000c), Definition 3.8(i)]. It is clear that any kregularized C 1 -existence family, whose construction has been verified by applying [ (Kostić, 2015b), Theorem 3.9(i)/(b)], can serve as a particular example of a (k, C 1 , V)-existence family for (1.1), with the operators A 0 ,⋅⋅⋅,A n-1 , B being singe-valued and the set V appropriately chosen. Furthermore, if card (V)≤1, then any pre-(k, C 1 , V)-existence family for (1.1) is automatically a (k, C 1 , V)-existence family for (1.1).
If (E(t)) t∈[0,τ) is a (pre-)(k, C 1 , V)-existence family for (1.1), then it is clear that, for every z∈Z, the mapping In our analyses of uniqueness of solutions of problem (1.1), it will be crucial for us to assume that X = Y. We will use the following definition, in which we generalize the notions introduced in [( Kostić, 2015b), Definition 3.8(ii)] and [ (Kostić,0000c), Definition 5.1(iii)].
Observe that, in the formulation of Proposition 3.8, we do not require the closedness of multivalued linear operators A 0 ,⋅⋅⋅,A n-1 , B. Using Proposition 3.8, we can simply prove that there exists a unique V-solution of problem (1.1), provided the closedness of all operators A 0 ,⋅⋅⋅,A n-1 , B and the validity of conditions from Proposition 3.8.
The class of exponentially equicontinuous kregularized C 2 -uniqueness families for (1.1) can be simply characterized by using the vector-valued Laplace transform.
On the other hand, the class of exponentially equicontinuous pre-(k, C 1 , V)-existence families for (1.1) cannot be so simply characterized with the help of vector-valued Laplace transform. As the next theorem shows, this is not the case with the class of exponentially equicontinuous (k, C 1 , V)-existence families for (1.1); cf. also [(Kostić, 2015b), Theorem 3.9(i)/(a)].

Proof
Performing the Laplace transform, we get: Since the operators A j are closed for 0≤j≤n, we can apply Lemma 2.1 in order to see that: Substituting this in (3.6) and using the fact that ≠ , the result immediately follows.
The assertion of [ (Kostić, 2015b), Theorem 3.9(i)/(b)] can be extended to multivalued linear operators by assuming some very restrictive additional conditions. Contrary to this, the assertion of [ (Kostić, 2015b), Theorem 3.4(i)] admits a very simple reformulation in our context.

Theorem 3.12
Suppose that 0≤α 0 <⋅⋅⋅< α n and : , where, v i,j ∈Z satisfy A j u i ∩C 1 v i,j ≠φ for 0≤j≤n-1. Then the Caputo fractional derivative Before we briey describe how we can provide some illustrative examples of results obtained so far, it is worth noting that the assertion of [ (Kostić, 2015b), Theorem 3.6], concerning the inhomogeneous problem (1.1) with single-valued linear operators A j = A j (0≤j≤n), cannot be extended to the multivalued linear operator case.

On analytical Solutions of Abstract Degenerate Multi-Term Fractional Problems with Caputo Derivatives
In this section, we investigate the existence and uniqueness of analytical solutions of abstract degenerate Cauchy problem (1.2), provided that there exists an index 0 n i N ∈ such that α i ∉ N 0 , for the sake of brevity, we consider only the homogeneous case. Let us recall that n∈N\{1}, A, B and A 1 ,⋅⋅⋅⋅,A n-1 are closed linear operators on a sequentially complete locally convex space X, 0≤α 1 <⋅⋅⋅<α n , 0≤α<α n and t D α denotes the Caputo fractional derivative of order α. Define  Before stating the following theorem, it should be observed that P z,γ need not be equal to P ze-iγ for some particular values of γ and z: Theorem 4.2 Suppose k(t) satisfies (P1), C 1 ∈L(Y, X), C 2 ∈L(X) is injective and there exists a locally equicontinuous kregularized C 2 -uniqueness family for problem (1.2), with f(t)≡0 (in the sense of [ (Kostić, 2015b), Definition 3.11]). Suppose that for each number γ∈(-π, π] there exists a sufficiently large number R > 0 such that the operator P z,γ is injective for |z|≥R, as well as that the operator families

Proof
The proof of theorem is very similar to that of [ (Kostić, 2016), Theorem 2.2] and, because of that, we will only outline the most relevant details.