In this article, we prove the existence of mild and strong solutions for nonlinear impulsive integro-differential equations of Sobolev type with non- local initial conditions

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ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

EXISTENCE OF SOLUTIONS FOR NONLINEAR IMPULSIVE NEUTRAL INTEGRO-DIFFERENTIAL EQUATIONS OF

SOBOLEV TYPE WITH NONLOCAL CONDITIONS IN BANACH SPACES

BHEEMAN RADHAKRISHNAN, ARUCHAMY MOHANRAJ, VELU VINOBA

Abstract. In this article, we prove the existence of mild and strong solutions for nonlinear impulsive integro-differential equations of Sobolev type with nonlocal initial conditions. The results are obtained by using semigroup theory and the Schauder fixed point theorem. An example is provided to illustrate the theory.

1. Introduction

Differential equations arise in many areas of science and technology, specifically whenever a deterministic relation involving some continuously varying quantities (modeled by functions) and their rates of change in space and/or time are known or postulated. This is illustrated in classical mechanics where the motion of a body is described by its position and velocity as the time varies. It is well known that the systems described by partial differential equations can be expressed as abstract differential equations [20]. These equations occur in various fields of study and each system can be represented by different forms of differential or integro-differential equations in Banach spaces. Using the method of semigroups, various solutions of nonlinear and semilinear evolution equations have been discussed by Pazy [20].

The study of abstract nonlocal semilinear initial value problems was initiated by Byszewski [9, 10, 11]. Because it is demonstrated that the nonlocal problems have better effects in applications than the classical Cauchy problems. Such problems with nonlocal conditions have been extensively studied in the literature [1, 2, 4, 5, 6, 23]. Showalter [22] established the existence of solutions of semilinear evolution equations of Sobolev type in Banach spaces. This type of equations arise in various applications such as in the flow of fluid through fissured rocks, thermodynamics, and shear in second-order fluids. For more details, we refer the reader to [8, 16, 17].

Neutral differential equations arise in many areas of applied mathematics and for this reason these equations have received much attention during the last few decades. There are also a number of applications in which the delayed argument

2000Mathematics Subject Classification. 34A37, 47D06, 47H10, 74H20, 34K40.

Key words and phrases. Existence; neutral differential equation; fixed point theorem;

impulsive differential equation.

c

2013 Texas State University - San Marcos.

Submitted September 6, 2012. Published January 21, 2013.

1

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occurs in the derivative of the state variable as well as in the independent variable, as in the so called neutral differential difference equations. A neutral functional differential equation is one in which the derivatives of the past history or derivatives of functionals of the past history are involved as well as the present state of the system. A good guide to the literature for neutral functional differential equations is the book by Hale and Verduyn Lunel [13] and the references therein. Hernandez [14]

established the existence results for partial neutral functional differential equations with nonlocal conditions modeled as

d dt

u(t) +F(t, u(t))

=Au(t) +G(t, u(t)) uσ =ϕ+q(xt₁, xt₂, . . . , xt_n) in Ω,

(1.1) where Ais the infinitesimal generator of an analytic semigroup T(t) on a Banach space. He made use of fixed point theorems and the results mentioned in Pazy [20]. For results on neutral partial differential equations with nonlocal and classical conditions, we refer to the papers of Hernandez and Henryquez [15], Fu and Ezzinbi [12], and references therein. Controllability of functional differential systems of Sobolev type in Banach spaces has been first studied by Balachandran and Dauer [3].

Differential equations arise in many real world problems such as physics, popu- lation dynamics, ecology,biological systems, biotechnology, optimal control and so forth. Much has been done the assumption that the state variables and systems parameters change continuously. However, one may easily visualize that abrupt changes such as shock, harvesting and disasters may occur in nature. These phe- nomena are short time perturbations whose duration is negligible in comparison with the duration of the whole evolution process. Consequently, it is natural to assume, in modeling these problems, that these perturbations act instantaneously, that is in the form of impulses. The theory of impulsive differential equation [18, 21]

is much richer than the corresponding theory of differential equations without impulsive effects. The impulsive condition

∆u(t_i) =u(t⁺_i )−u(t⁻_i ) =I_i(u(t⁻_i )), i= 1, 2, . . . , m,

is a combination of traditional initial value problems and short-term perturbations whose duration is negligible in comparison with the duration of the process. Lin and Liu [19] discussed the iterative methods for the solution of impulsive functional differential systems.

Motivated by the above approach, the goal of this paper is to use the fixed point theorem to obtain the mild solution of the nonlinear impulsive neutral integrodifferential equation of Sobolev type with nonlocal conditions.

2. Preliminaries

Consider the nonlinear impulsive neutral integrodifferential equation of Sobolev type with nonlocal conditions of the form

d dt

Bu(t) +e(t, u(t))

+Au(t) =f(t, u(t)) + Z t

0

k(t, s, u(s))ds, t∈(0, a], t6=t_k,

(2.1)

u(0) +

n

X

i=1

c_iu(t_i) =u₀ (2.2)

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∆u(tk) =Ik(ut_k), k= 1,2, . . . , m, (2.3) where 0 ≤ t1 < t2 < · · · < tp ≤ a, B and A are linear operators with domains contained in a Banach space X and ranges contained in a Banach space Y and the nonlinear operators f : I×X → Y, k : I²×X → Y, e : I×X → Y and Ik :X →Y are appropriate functions and the symbol ∆u(tk) represent the jump of the functionuatt, which is defined by ∆u(tk) =u(t⁺)−u(t⁻). HereI= [0, a].

In this paper, we establish the existence of a nonlinear impulsive neutral integrodifferential equation of Sobolev type with nonlocal conditions using Schauder fixed point theorem.

To prove our main theorem we assume certain conditions on the operators A andB. Let X andY be Banach spaces with norm| · | andk · krespectively. The operators A : D(A) ⊂ X → Y and B : D(B) ⊂ X → Y satisfy the following hypothesis:

(M1) AandB are closed linear operators, (M2) D(B)⊂ D(A) andB is bijective, (M3) B⁻¹:Y → D(B) is continuous.

The hypothesis (M1)–(M3) and the closed graph theorem imply the boundedness of the linear operator AB⁻¹ : Y → Y and −AB⁻¹ generates a uniformly continuous semigroup S(t), t ≥ 0, of bounded linear operators from Y into Y and so maxt∈IkS(t)k is finite. We denote M = maxt∈IkS(t)k, R = kB⁻¹k. Let Br={x∈X:|x| ≤r}andc=Pp

i=1|ci|.

In this article, we assume that there exists an operatorE onD(E) =X given by the formula

E=h I+

n

X

i=1

ciB⁻¹S(ti)Bi−1

and Eu0∈ D(B),

with

En

B⁻¹e(t, u(t))−B⁻¹S(ti)e(0, u(0)) + Z t_i

0

AS(ti−s)B⁻¹e(s, u(s))ds +

Z ti

0

B⁻¹S(t_i−s)[f(s, u(s)) + Z s

0

k(s, τ, u(τ))dτ]ds

− X

0<tk<ti

B⁻¹S(ti−tk)Iku(tk)o

∈ D(B),

fori= 1,2, . . . , p.

The existence ofEcan be observed from the following fact (see [9]). Suppose that {S(t)}is aC₀ semigroup of operators onX such thatkB⁻¹S(t_i)Bk ≤Ce^−δtⁱ(i= 1,2, . . . , n) whereδis a positive constant andC≤1. IfPp

i=1|ci|e^−δtⁱ<1/C then kPp

i=1c_iB⁻¹S(t_i)Bk<1. So such an operatorE exists onX.

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Definition 2.1. A continuous solutionuof the integral equation u(t) =B⁻¹S(t)BEu0+

n

X

i=1

ciB⁻¹S(t)BE

×n

B⁻¹e(t, u(t))−B⁻¹S(t_i)e(0, u(0))− X

0<ti<t

B⁻¹S(t_i−t_k)I_ku(t_k)o

+

n

X

i=1

c_iB⁻¹S(t)BEnZ ti

0

B⁻¹S(t_i−s)h

Ae(s, u(s)) +f(s, u(s)) +

Z s

0

k(s, τ, u(τ))dτi dso

+B⁻¹S(t)e(0, u(0))−B⁻¹e(t, u(t)) +

Z t

0

S(t−s)B⁻¹h

Ae(s, u(s)) +f(s, u(s)) + Z s

0

k(s, τ, u(τ))dτi ds

+ X

0<t_i<t

B⁻¹S(t−tk)Iku(tk)

(2.4) is said to be a mild solution of problem (2.1)-(2.3) onI.

Definition 2.2. A functionuis said to be a strong solution of (2.1)-(2.3) onI if uis differentiable almost everywhere onI,u⁰ ∈L¹(I, X),u(0) +Pn

i=1ciu(ti) =u0

and d dt

(Bu(t) +e(t, u(t))

+Au(t) =f(t, u(t)) + Z t

0

k(t, s, u(s))ds, t∈(0, a], t6=t_k

∆u(tk) =Ik(ut_k), k= 1,2, , . . . , m almost everywhere onI.

Remark 2.3. A mild solution of the neutral integro-differential (2.1)-(2.3) satisfies the condition (2.2), for (2.4)

u(0) =Eu₀+

n

X

i=1

c_iEn

B⁻¹e(t, u(t))−B⁻¹S(t_i)e(0, u(0))

− X

0<t_i<t

+

n

X

i=1

ciEnZ t_i 0

S(ti−s)B⁻¹h

Ae(s, u(s)) +f(s, u(s)) +

Z s

0

k(s, τ, u(τ))dτi dso and

u(tj) =B⁻¹S(tj)BEu0+

n

X

i=1

ciB⁻¹S(tj)BEn

B⁻¹e(t, u(t))−B⁻¹S(ti)e(0, u(0))

− X

0<ti<t

+

n

X

i=1

c_iB⁻¹S(t_j)BEnZ ti

0

S(t_i−s)B⁻¹[Ae(s, u(s)) +f(s, u(s))

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+ Z s

0

k(s, τ, u(τ))dτ]dso

+B⁻¹S(tj)e(0, u(0))−B⁻¹e(tj, u(tj)) +

Z t_j

0

S(tj−s)B⁻¹h

Ae(s, u(s)) +f(s, u(s)) + Z s

0

+ X

0<t_i<t

B⁻¹S(tj−tk)Iku(tk).

Therefore, u(0) +

n

X

j=1

c_ju(t_j)

=h I+

n

X

j=1

cjB⁻¹S(tj)B

Eu0+h I+

n

X

j=1

cjB⁻¹S(tj)BiXⁿ

i=1

ciEn

B⁻¹e(t, u(t))

−B⁻¹S(t_i)e(0, u(0))− X

0<t_i<t

+h I+

n

X

j=1

c_jB⁻¹S(t_j)BiXⁿ

i=1

c_iEnZ t_i 0

S(t_i−s)B⁻¹[Ae(s, u(s)) +f(s, u(s))

+ Z s

0

k(s, τ, u(τ))dτ]dso +

n

X

j=1

c_jh

B⁻¹S(t_j)e(0, u(0))−B⁻¹e(t_j, u(t_j))

+ Z t_j

0

S(tj−s)B⁻¹h

Ae(s, u(s)) +f(s, u(s)) + Z s

0

+ X

0<ti<t

B⁻¹S(t_j−t_k)I_ku(t_k)i

=u0

To prove the existence result, we use the following hypotheses:

(M4) The functionf :I×X →Y is continuous in tand there exists a constant Lf >0 such that

kf(t, u)k ≤Lf, fort∈I andu∈X.

(M5) The functionk:I²×X →Y is continuous intand there exists a constant L_k >0 such that

kk(t, s, u)k ≤Lk, fors, t∈I andu∈X.

(M6) The function e :I×X →Y is continuous in t and there exist constants L_e>0, L₀>0 andL₁>0 such that

ke(t, u(t))k ≤Le, fort∈I andu∈X ke(0, u(0))k ≤L0, fort∈I andu∈X kAe(t, u(t))k ≤L₁, fort∈I andu∈X.

(M7) The maps Ik : X → Y are continuous and there exists a constantI > 0 such that

kIk(u)k ≤ I, fork∈Nandy∈X.

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(M8)

RkBEuokM+cR²kBEkM[L_e+MI+M L₀+aM(L₁+L_f+L_ka)]

+RM[L0+L1+Lf+Lka+a(L1+G1) +I+RK1]≤r.

3. Main Results

Theorem 3.1. If assumptions (M1)-(M7) hold, then Problem (2.1)-(2.3) has a mild solution on I.

Proof. Let E = C(I, Y) and Y0 = {u∈ Y : u(t) ∈ Br, t ∈ I}. Clearly, Y0 is a bounded closed convex subset ofY. We define a mappingF:Y₀→ Y₀by

(F u)(t) =B⁻¹S(t)BEu0+

n

X

i=1

ciB⁻¹S(t)BEn

B⁻¹e(t, u(t))

−B⁻¹S(t_i)e(0, u(0))− X

0<ti<t

+

n

X

i=1

c_iB⁻¹S(t)BEnZ ti

0

S(t_i−s)B⁻¹[Ae(s, u(s)) +f(s, u(s)) +

Z s

0

k(s, τ, u(τ))dτ]dso

+B⁻¹S(t)e(0, u(0))−B⁻¹e(t, u(t)) +

Z t

0

S(t−s)B⁻¹h

Ae(s, u(s)) +f(s, u(s)) + Z s

0

+ X

0<t_i<t

B⁻¹S(t−tk)Iku(tk)

Now we shown thatF :Y0→ Y0 is continuous. Let{un}^∞₀ ⊂ Y0 withu_n →uin Y0. Then there is an integerrsuch thatkun(t)k ≤r, for allnandt∈I, sou_n∈Br

andu∈Br. From the assumptions (M₁)−(M₇), we have (a) Ik,k= 1,2, . . . , pis continuous.

(b) e(t, un(t))→e(t, u(t)), fort∈I and since

ke(t, u_n(t))−e(t, u(t))k<2[L_e+L₀].

(c) Ae(t, un(t))→Ae(t, u(t)), fort∈I and since

kAe(t, un(t))−Ae(t, u(t))k<2[L₁+L₃].

(d) f(t, un(t))→f(t, u(t)), fort∈I and since

kf(t, un(t))−f(t, u(t))k<2[Lf+F0].

(e) k(t, s, u_n(s))→k(t, s, u(s)), fort, s∈I and since kk(t, s, un(s))−k(t, s, u(s))k<2[Lk+K0].

By the dominated convergence theorem, we have

kF u_n−F uk ≤R²M ckBEk{ke(t, u_n(t))−e(t, u(t))k}

+R²M ckBEk Z t_i

0

S(t_i−s)h

{kAe(s, un(s))−Ae(s, u(s))k}

+{kf(s, u_n(s))−f(s, u(s))k}

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+ Z s

0

{kk(s, τ, un(τ))−k(s, τ, u(τ))k}dτi ds +R²M ckBEk X

0<ti<t

S(ti−tk){kIk(un(tk))−Ik(u(tk))k}

+R{ke(t, un(t))−e(t, u(t))k}

+RM Z t

0

h{kAe(s, un(s))−Ae(s, u(s))k}

+{kf(s, un(s))−f(s, u(s))k}

+ Z s

0

{kk(s, τ, un(τ))−k(s, τ, u(τ))k}dτi ds

+RM X

0<ti<t

{kI_k(u_n(t_k))−I_k(u(t_k))k} →0 asn→ ∞.

ThusF is continuous. Moreover,F maps Y₀ into a precompact subset ofY₀. We prove that the setY0(t) ={(F u)(t) : u∈ Y0} is precompact in X for every fixed t∈I. We shall show thatF(Y0) =Z={F u:u∈ Y0}is an equicontinuous family of functions.

For 0< s < t, we have k(F u)(t)−(F u)(s)k

≤ kB⁻¹(S(t)−S(s))BEu₀k +

n

X

i=1

c_ikB⁻¹(S(t)−S(s))BEkn

kB⁻¹e(t, u(t))−B⁻¹S(t_i)e(0, u(0))

− X

0<ti<t

B⁻¹S(ti−tk)Iku(tk)ko +

n

X

i=1

cikB⁻¹(S(t)−S(s))BEk

× { Z t_i

0

kS(ti−s)B⁻¹h

Ae(s, u(s)) +f(s, u(s)) + Z s

0

k(s, τ, u(τ))dτi kds}

+kB⁻¹(S(t)−S(s))e(0, u(0))k+kB⁻¹(e(t, u(t))−e(s, u(s)))k +

Z t

0

k(S(t−θ)−S(s−θ)B⁻¹

Ae(θ, u(θ)) +f(θ, u(θ)) + Z θ

0

k(s, τ, u(τ))dτ kdθ +

Z t

θ

kS(t−θ)B⁻¹

Ae(θ, u(θ)) +f(θ, u(θ)) + Z θ

0

k(θ, τ, u(τ))dτ kdθ

+ X

0<t_i<t

kB⁻¹(S(t−s))Iku(tk)k

≤n

RkBEu0k+R²kBEk[Le+MI+M L₀]c +R²M akBEk[L₁+L_f+L_ka]c+RL₀o

ks(t)−S(s)k +{RL₀+RM[L₁+L_f+L_ka]}|t−s|

+R(L_e+L_f+L_ka) Z t

0

kS(t−θ)−S(s−θ)kdθ.

The right hand side of the above inequality is independent of u∈ Y0 and tends to zero as s→ t as a consequence of the continuity of S(t) in the uniform operator

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topology for t > 0 which follows from the compactness of S(t), t > 0. It is also clear that Z is bounded inY. Thus by Arzela-Ascoli’s theorem,Z is precompact.

Hence by the Schauder fixed point theorem,F has a fixed point inY0and any fixed point ofF is a mild solution of (2.1)-(2.3) onI such thatu(t)∈X, fort∈I.

Next we prove that the problem (2.1)-(2.3) has a strong solution.

Theorem 3.2. Assume that (i) Conditions(M1)–(M8)hold.

(ii) Y is a reflexive Banach space with normk · k.

(iii) f :I×X →Y is continuous int on I and there exists a constant G1>0 such that

kf(t, u)−f(s, v)k ≤G1[|t−s|+ku−vk], fort, s∈I andu, v∈X.

(iv) k:I²×X→Y is continuous intand there exists a constantK1>0such that

kk(t, τ, u)−k(s, τ, u)k ≤K1[|t−s|], forτ, s, t∈I,u∈X,

(v) e:I×X →Y is continuous and there exist constantsK >0 andK1>0 such that

kAe(t, u(t)−Ae(s, u(s))k ≤L2[|t−s|], fors, t∈I, u∈X, ke(t, u(t)−e(s, u(s))k ≤L[|t−s|], fors, t∈I, u∈X.

(vi) Eu₀∈ D(B), En

B⁻¹e(t, u(t))−B⁻¹S(ti)e(0, u(0)) +

Z t_i

0

B⁻¹S(ti−s)h

Ae(s, u(s)) +f(s, u(s)) + Z s

0

− X

0<t_i<t

∈D(B), fori= 1,2, . . . , p.

Thenuis a strong solution of problem (2.1)–(2.3)on I.

Proof. Since all the assumptions of Theorem 3.1 are satisfied, then (2.1)-(2.3) has a mild solution belonging toC(I, X). Now we shall show thatuis a strong solution of (2.1)-(2.3) onI. For anyt∈I, we have

ku(t+h)−u(t)k

≤ kB⁻¹[T(t+h)−T(t)]BEu0

+

n

X

i=1

c_ikB⁻¹(S(t+h)−S(t))BEkn

kB⁻¹e(t, u(t))−B⁻¹S(t_i)e(0, u(0))

− X

0<ti<t

B⁻¹S(t_i−t_k)I_ku(t_k)ko +

n

X

i=1

c_ikB⁻¹(S(t+h)−S(t))BEk

×nZ t_i 0

kS(ti−s)B⁻¹[Ae(s, u(s)) +f(s, u(s)) + Z s

0

k(s, τ, u(τ))dτ]kdso +kB⁻¹(S(t+h)−S(t))e(0, u(0))k+kB⁻¹(e(t+h, u)−e(t, u))k

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+ Z h

0

kS(t+h−s)B⁻¹

Ae(s, u(s)) +f(s, u(s) + Z s

0

k(s, τ, u(τ))dτ kds +

Z t+h

h

k(S(t+h−s)B⁻¹

Ae(s, u(s)) +f(s, u(s)) + Z s

0

k(s, τ, u(τ))dτ kds +

Z t

0

kS(t−s)B⁻¹

Ae(s, u(s)) +f(s, u(s)) + Z s

0

k(s, τ, u(τ))dτ kds

+ X

0<ti<t

kB⁻¹(S(t+h−tk)−S(t−tk))Iku(tk)k

≤ kB⁻¹S(t)[S(h)−I]BEu₀k +

n

X

i=1

c_ikB⁻¹S(t)(S(h)−I)BEkn

kB⁻¹e(t, u(t))k+kB⁻¹S(t_i)e(0, u(0))k

+ X

0<ti<t

kB⁻¹S(ti−tk)Iku(tk)ko +

n

X

i=1

cikB⁻¹S(t)(S(h)−I)BEk

×nZ t_i 0

kS(ti−s)B⁻¹kh

kAe(s, u(s))k+kf(s, u(s))k +

Z s

0

kk(s, τ, u(τ))kdτi dso

+kB⁻¹S(t)(S(h)−I)e(0, u(0))k +kB⁻¹(e(t+h, u)−e(t, u))k+

Z h

0

k(S(t+h−s)B⁻¹kh

kAe(s, u(s))k +kf(s, u(s))k+

Z s

0

kk(s, τ, u(τ))kdτi ds +

Z t+h

h

k(S(t+h−s)B⁻¹kh

Z s

0

kk(s, τ, u(τ))kdτi ds+

Z t

0

kS(t−s)B⁻¹kh

Z s

0

kk(s, τ, u(τ))kdτi

ds+ X

0<t_i<t

kB⁻¹S(t−t_k)(S(h)−I)I_ku(t_k)k

≤ kB⁻¹S(t)[S(h)−I]BEu0

+

n

X

i=1

c_ikB⁻¹S(t)(S(h)−I)BEk{kB⁻¹e(t, u(t))k+kB⁻¹S(t_i)e(0, u(0))k

+ X

0<ti<t

kB⁻¹S(t_i−t_k)I_ku(t_k)k}+

n

X

i=1

c_ikB⁻¹S(t)(S(h)−I)BEk

×nZ t_i 0

kS(ti−s)B⁻¹kh

Z s

0

kk(s, τ, u(τ))kdτi dso

+kB⁻¹S(t)(S(h)−I)e(0, u(0))k +kB⁻¹(e(t+h, u)−e(t, u))k

+ Z h

0

kS(t+h−s)B⁻¹kh

kAe(s, u(s))k+kf(s, u(s))k+ Z s

0

kk(s, τ, u(τ))kdτi ds

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+ Z t

0

kS(t−s)B⁻¹kh

kAe(s+h, u(s+h))−Ae(s, u(s))k +kf(s+h, u(s+h))−f(s, u(s))k+

Z s

0

kk(s+h, τ, u(τ))−k(s, τ, u(τ))kdτi ds

+ X

0<t_i<t

kB⁻¹S(t−tk)(S(h)−I)Iku(tk)k using our assumptions we observe that

ku(t+h)−u(t)k

≤RkBEu0kM hkAB⁻¹k+R²M hckBEk[Le+MI+M L0]kAB⁻¹k +R²M hckBEk[M a(L1+Le+Lka)]kAB⁻¹k

+RM hL0+RLh+RM h(L1+Le+K1a) +RM hI +RM

Z t

0

{L₂[h+ku(s+h)−u(s)] +G₁[h+ku(s+h)−u(s)]}ds +RM

Z t

0

nZ s

0

kk(s+h, τ, u(τ))−k(s, τ, u(τ))kdτ +

Z s+h

s

kk(s+h, τ, u(τ))kdτo ds

≤h{RkBEu0kMkAB⁻¹k

+R²M ckBEk[Le+MI+M L0+M a(L1+Le+Lka)]kAB⁻¹k +RM[L₀+L₁+L_e+L_ka+a(L₂+G₁+K₁+K₁a) +I] +RL_k} +RM(L₂+G₁)

Z t

0

ku(s+h)−u(s)kds

≤P h+Q Z t

0

ku(s+h)−u(s)kds where

P =RkBEu0kMkAB⁻¹+R²M ckBEkh

Le+MI+M L0

+M a(L1+Le+Lka)i

kAB⁻¹k+RM

L0+L1

+L_e+L_ka+a(L₂+G₁+K₁+K₁a) +I

+RL_k, Q=RM(L2+G1).

Hence by Gronwall’s inequality

ku(t+h)−u(t)k ≤ Phe^Q, fort∈I.

Therefore, u is Lipschitz continuous on I. The Lipschitz continuity of u on I combined with (iii)–(v) implies that

t→f(t, u(t)), t→e(t, u(t)), t→ Z t

0

k(t, s, u(s))ds.

Hence,uis strong solution of the problem (2.1)-(2.3) on (0, a].

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4. Example

Consider the partial integro-differential equation of neutral type

∂

∂t

hz(t, x)−z_xx(t, x) + Z t

−∞

a₁(s−t)z_t(s, x)dsi

−z_xx(t, x)

=ρ(t, z(t, x)) + Z t

0

a(t, s)z(s, x)ds, x∈[0, π], t∈I, z(t,0) =z(t, π) = 0, t∈I,

z(0, x) +

p

X

k=1

z(tk, x) =z0(x) 0< t1< t2<· · ·< tp< b; x∈[0, a]

∆z|t=t_i =Ii(z(x)) = (γi(z(x)) +ti)⁻¹, z∈X, 1≤i≤p,

(4.1)

wherea(t, s) is continuous such thatka(t, s)k ≤L₁ and the constantγ_i is small.

Let us take X =Y =L²[0, π] to be endowed with the usual normk · k_L₂. and let

e(t, z) = Z t

−∞

a1(s−t)zt(s, x)ds f(t, z) =ρ(t, z(t, x)) Z t

0

k(t, s, z)ds= Z t

0

a(t, s)z(s, x)ds Ii(z(x)) = (γi(z(x)) +ti)⁻¹.

Define the operatorA:D(A)⊂X→Y andB :D(B)⊂X→Y by Az=−z_xx, Bz=z−z_xx,

where each domainD(A) andD(B) is given by

{z∈X :z, zx are absolutely continuous,zxx∈X, z(0) =z(π) = 0}.

Then the above problem can be formulated abstractly as d

dt

Bu(t) +e(t, u(t))

+Au(t) =f(t, u(t)) + Z t

0

k(t, s, u(s))ds, t∈(0, a], t6=tk,

u(0) +

n

X

i=1

ciu(ti) =u0

∆u(tk) =Ik(ut_k), k= 1,2, . . . , m, ThenAandB can be written, respectively, as

Az=

∞

X

n=1

n²hz, znizn, z∈ D(A)

Bz=

∞

X

n=1

(1 +n²)hz, znizn, z∈ D(B),

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where zn(x) = p

2/πsin(nx), n= 1,2, . . ., is the orthogonal set of vectors of A.

Furthermore forz∈X, we have B⁻¹z=

∞

X

n=1

1

1 +n²hz, znizn,

−AB⁻¹z=

∞

X

n=1

−n²

1 +n²hz, znizn, S(t)z=

∞

X

n=1

exp −n²t 1 +n²

hz, z_niz_n.

It is easy to see thatAB⁻¹ generates a strongly continuous semigroupS(t) on Y and S(t) is compact such that |S(t)| ≤ e^−t for each t >0. For thisS(t), B, B⁻¹ we assume that the operatorEexists. So all the conditions of the Theorem 3.1 are satisfied. Hence the equation (4.1) has a mild solution.

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Bheeman Radhakrishnan

Department of Applied Mathematics & Computational Sciences, PSG College of Tech- nology, Coimbatore - 641 004, TamilNadu, India

E-mail address:[email protected]

Aruchamy Mohanraj

Department of Mathematics and Computer Sciences, SVS College of Engineering, Coim- batore - 642 109, TamilNadu, India

Velu Vinoba

Department of Mathematics, K. N. Govt. Arts College for Women, Thanjavur, Tamil- Nadu, India