Sobolev-Type Volterra-Fredholm Functional Integrodifferential Equations In Banach Spaces

Kishor D. Kucche and M. B. Dhakne

Consider the Volterra-Fredholm functional integrodifferential equations of sobolev type in Banach spaces of the form (Bx(t)) ′ + Ax(t) = f t, x t , t 0 k(t, s)w(s, x s )ds, b 0 l(t, s)h(s, x s )ds , t ∈ [0, b], (1.1)  3)- (1.4) and their special forms arise in various physical phenomena such as in flow of fluid through fissured rocks, the propagation of long waves of small amplitudes, thermodynamics and shear in second order fluids, see for example, [4,5,9,14] and the references cited therein.Many authors have studied the existence, uniqueness, continuation and other properties of solutions of various special forms of the equations (1.1)-(1.2) and (1.3)-(1.4)by using different techniques, see for example, [7]- [11], [16]- [18], [21] and some of the references given therein.
Brill [6] and Showalter [23] established the existence of solutions of semilinear evolution equations of Sobolev type in Banach spaces.Lightbourne and Rankin [15] discussed the solution of partial functional differential equation of Sobolev type.Dhakne and Pachpatte [12] discussed mild and classical solutions of functional integrodifferential equations by using various approaches.Han [13] has studied controllability problem for the special form of (1.1)-(1.2) with w = 0, h = 0 and without functional arguments assuming the resolvent operator is compact.Balchandran and coauthors [1,2] established existence and qualitative properties of (1.1)-(1.2) when k(t, s)w(s, x s ) = k(t, s, x s ) and h = 0 with nonlocal condition.Also, Balachandran et al. [3] have studied existence of solution of special form of (1.3)-(1.4)when f = t 0 w(s, x s )ds using the Schaefer fixed point theorem.In the present paper, we prove the existence results of mild solutions for (1.1)-(1.2) and (1.3)-(1.4)using the Banach fixed point principle.Using the tools of Pachpatte's integral inequality and Gronwall-Bellman inequality we establish the results pertaining to continuous dependence, uniqueness, and boundedness of mild solution of (1.1)-(1.2) cosidering the different cases on the argument t of The paper is organized as follows.In section 2, we present the preliminaries and the hypothesis.Section 3 deals with existence results of (1.1)-(1.2) and (1.3)- (1.4).In section 4 we prove continuous dependence and boundedness of mild solution of (1.1)-(1.2).In section 5 we give applications of some of our results obtained in section 3. Finally in section 6 we give example to illustrate the results obtained in section 3.

Preliminaries and Hypotheses
In this section we shall set forth some preliminaries from [6] and the hypotheses that will be used in our subsequent discussion.Here and hereafter, we assume that (H 0 ) the operators A : D(A) ⊂ X → Y and B : D(B) ⊂ X → Y satisfy the following conditions (i) A and B are closed linear operators.
(ii) D(B) ⊂ D(A) and B is bijective.
The assumptions (i) − (ii) and the closed graph theorem imply the boundedness of the linear operator AB −1 : Y → Y .Further −AB −1 generates a uniformly continuous semigroup T (t), t ≥ 0 on Y and so max{ T (t ) and the integral equation is satisfied.
The following integral inequalities plays the crucial role in our analysis.20], p-11) Let u and f be continuous functions defined on R + and c be a nonnegative constant.If We list the following hypothesis.
(H 1 ) There exists constant F such that (3.1) Observe that using the definition of the operator Γ in (3.1), the equivalent integral equations of initial value problem (1.1)-(1.2) can be written as x = Γx, whose fixed point is the mild solution of (1.1)-(1.2).Now we will show that Γ is contraction on D. Consider, Using the condition (3.3), the hypothesis (H 1 ) and (H 2 ) in equation (3.2), we get

holds.
Proof: Define an operator Ω : D → D by Define the function µ as in the proof of Theorem 3.1, then from (3.9) we have This shows that Ω is a contraction on D .Therefore, in space D there is a unique fixed point for Ω and this point is the mild solution of the initial problem (1.3)- (1.4).✷

Continuous Dependence and Boundedness
Theorem 4.1.Suppose that the hypotheses (H 0 )-(H 2 ) are satisfied.Let x and x be mild solutions of (1.1) with initial functions φ and φ ∈ C respectively.Then the following inequality Proof: Let x and x be mild solutions of (1.1) with initial functions φ and φ ∈ C respectively.Then we have, Using the condition (3.3) and the hypotheses (H 1 ) and (H 2 ), from the equation (4.2) for t ∈ [0, b], we obtain Case 1 : Suppose t ≥ r.Then for every θ ∈ [−r, 0] we have t + θ ≥ 0. For such θ's Volterra-Fredholm Functional Integrodifferential Equations 245 from (4.4) we get, which yields, Case 2 : Suppose 0 ≤ t < r.Then for every θ ∈ [−r, −t) we have t + θ < 0. For such θ's we get, which yields, For θ ∈ [−t, 0], t + θ ≥ 0 then from (4.4) we get as in the case 1, Thus for every θ ∈ [−r, 0], (0 ≤ t < r), from (4.6) and (4.7), we obtain Therefore, for every t ∈ [0, b], from (4.5) and (4.8) we have Thus for every, t ∈ [0, b], we get By applying Grownwall-Bellman inequality given in Lemma 2.2 to the inequality (4.9), we obtain This gives that, Volterra-Fredholm Functional Integrodifferential Equations 247 and hence, the inequality (4.1) holds.Hence the proof is complete.✷ Remark 4.1.We note that estimates obtained in the Theorem 4.1 yields not only the continuous dependence of mild solution on initial functions φ and φ, but also gives the criterion to prove the uniqueness of mild solution of (1.1)-(1.2).It follows by putting φ = φ.
In the following theorem we give condition for the boundedness of mild solution of the initial value problem (1.1)-(1.2) Theorem 4.2.Assume that there exists p, q, r ∈ be the mild solution of (1.1)-(1.2).Using condition (3.3) and the assumption of theorem, in the equation (4.10), we get Thanks to pachpatte integral inequality given in Lemma 2.1 and applying it to the inequality (4.17) with z(t) = x t C , we get {p(t), q(t)}.This gives that, Therefore, Hence the proof is complete.✷ Remark 4.2.We observe that, by making similar argument and suitable modifications, one can study the continuous dependence and boundedness of solutions of initial value problem (1.3)-(1.4).

Applications
As an application of the Theorem 3.1, we shall consider the system (1.1)-(1.2) with the control parameter (Bx(t)) ′ + Ax(t) = Eu(t) + f t, x t , )where B and A are linear operators with the domains contained in a Banach space X and ranges contained in a Banach space Y .The nonlinear functions f :[0, b] × C × Y × Y → Y , g, w, h : [0, b] × C → Y are continuous.The kernel functions k, l : [0, b] × [0, b] → R are continuous and φ is a given element of C.Equations of the form (1.1)-(1.2) and (1.

3. Existence results Theorem 3 . 1 . 2 +
If the hypotheses (H 0 )-(H 2 ) are satisfied then initial value problem (1.1)-(1.2) has a unique mild solution on [−r, b], provided the condition RF b 1 + KW LH b < 1, holds, where K and L are constants as in condition (3.3).Proof: Define an operator Γ : D → D by ) for x, y ∈ D and t ∈ [0, b].We note that, the kernel functions k and l are continuous on compact set [0, b]×[0, b], therefore there exists constants K and L such that |k(t, s)| ≤ K, for t ≥ s ≥ 0 and |l(t, s)| ≤ L, for s, t ∈ [0, b].