Surveys in Mathematics and its Applications
ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 5 (2010), 61 – 72
EXISTENCE OF SOLUTIONS FOR NONLINEAR
MIXED TYPE INTEGRODIFFERENTIAL
EQUATION OF SECOND ORDER
H. L. Tidke and M. B. Dhakne
Abstract. In this paper, we investigate the existence of solutions for nonlinear mixed VolterraFredholm integrodifferential equation of second order with nonlocal conditions in Banach spaces.
Our analysis is based on Leray-Schauder alternative, rely on a priori bounds of solutions and the
inequality established by B. G. Pachpatte.
1
Introduction
Let X be a Banach space with norm k · k. Let B = C([t0 , t0 + β], X) be the Banach
space of all continuous functions from [t0 , t0 + β] into X endowed with supremum
norm
kxkB = sup{kx(t)k : t ∈ [t0 , t0 + β]}.
Motivated by the work of [6], in this paper we consider the following nonlinear
mixed Volterra-Fredholm integrodifferential equation of the form:
Z t
00
x (t) + Ax(t) = f (t, x(t),
a(t, s)k(s, x(s))ds,
t0
t0 +β
Z
b(t, s)h(s, x(s))ds),
t ∈ [t0 , t0 + β]
(1.1)
t0
x(t0 ) + g(t1 , t2 , · · · , tp , x(·)) = x0 ,
x0 (0) = η,
(1.2)
where −A is an infinitesimal generator of a strongly continuous cosine family {C(t) :
t ∈ R} in Banach space X, 0 ≤ t0 < t1 < t2 < · · · < tp ≤ β, f : [t0 , t0 + β] × X ×
X × X → X, k, h : [t0 , t0 + β] × X → X, g(t1 , t2 , · · · , tp , ·) : C([t0 , t0 + β], X) → X,
a, b : [t0 , t0 + β] × [t0 , t0 + β] → R are functions and x0 is a given element of X.
The nonlocal condition, which is a generalization of the classical initial condition,
was motivated by physical problems. The problem of existence of solutions of
2010 Mathematics Subject Classification: Primary 45N05; Secondary 47B38; 47H10.
Keywords:
Volterra-Fredholm integrodifferential equation; Leray-Schauder alternative;
Semigroup theory; priori bounds of solutions; nonlocal condition and Pachpatte’s integral inequality.
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62
H. L. Tidke and M. B. Dhakne
evolution equation with nonlocal conditions in Banach space was first studied by
[2] and he investigated the existence and uniqueness of mild, strong and classical
solutions of the nonlocal Cauchy problem. As indicated in [2, 3] and references
therein, the nonlocal condition y(0) + g(y) = y0 can be applied in physics with
better effect than the classical condition y(0) = y0 . For example, in [3], the author
used
X
ci y(ti ),
(1.3)
g(y) =
i=1p
where ci , i = 1, 2, · · · , p and 0 < t1 < t2 < · · · ≤ b, to describe the diffusion
phenomenon of a small amount of gas in a transparent tube. In this case, the
equation (1.3) allows the additional measurements at ti , i = 1, 2, · · · , p. The study
of differential and integrodifferential equations in abstract spaces with nonlocal
condition has received much attention in recent years. We refer to the papers
[4, 6, 7, 8, 9, 12, 13, 14] and the references cited therein.
The objective of the present paper is to study the global existence of solutions
of the equations (1.1)–(1.2).
The main tool used in our analysis is based on an application of the topological
transversality theorem known as Leray-Schauder alternative, rely on a priori bounds
of solutions and the inequality established by B. G. Pachpatte. The interesting and
useful aspect of the method employed here is that it yields simultaneously the global
existence of solutions and the maximal interval of existence.
The paper is organized as follows. In section 2, we present the preliminaries and
the statement of our main result. Section 3 deals with proof of Theorem.
2
Preliminaries and Main Result
We give the following preliminaries and hypotheses used in our subsequent discussion.
In many cases it is advantageous to treat second order abstract differential
equations directly rather than to convert into first order systems. A useful technique
for the study of abstract second order equations is the theory of strongly continuous
cosine family. We only mention a few results and notations needed to establish our
results. A one parameter family {C(t) : t ∈ R} of bounded linear operators mapping
the Banach space X into itself is called a strongly continuous cosine family if and
only if
(a) C(0) = I (I is the identity operator);
(b) C(t)x is strongly continuous in t on R for each fixed x ∈ X;
(c) C(t + s) + C(t − s) = 2C(t)C(s) for all t, s ∈ R.
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Existence of solutions for nonlinear second order
If {C(t) : t ∈ R} is a strongly continuous cosine family in X, then {S(t) : t ∈ R},
associated to the given strongly continuous cosine family, is defined by
Z t
C(s)xds, x ∈ X, t ∈ R.
S(t)x =
0
The infinitesimal generator A : X → X of a cosine family {C(t) : t ∈ R} is defined
by
d2
Ax = 2 C(t)x|t=0 , x ∈ D(A),
dt
where D(A) = {x ∈ X : C(.)x ∈ C 2 (R, X)}.
Definition 1. Let f ∈ L1 (t0 , t0 + β; X). The function x ∈ B given by
Z t
S(t − s)f (s, x(s),
x(t) = C(t − t0 )(x0 − g(t1 , t2 , · · · , tp , x(·))) + S(t − t0 )η +
t0
Z
s
Z
t0 +β
a(s, τ )k(τ, x(τ ))dτ,
t0
b(s, τ )h(τ, x(τ ))dτ )ds,
t ∈ [t0 , t0 + β],
t0
(2.1)
is called mild solution of the initial value problem (1.1)–(1.2).
In the sequel we will use the following results:
Lemma 2. ([15]). Let C(t), (resp.S(t)), t ∈ R be a strongly continuous cosine
(resp. sine) family on X. Then there exist constants N ∗ ≥ 1and w ≥ 0 such that
kC(t)k ≤ N ∗ e|t| , for all t ∈ R,
Z t2
kS(t1 ) − S(t2 )k ≤ N ∗ |
ew|s| ds|, for all t1 , t2 ∈ R.
t1
Theorem 3. ([11], p-47). Let z(t), u(t), v(t), w(t) ∈ C([α, β], R+ ) and k ≥ 0 be
a real constant and
Z t
Z s
Z β
z(t) ≤ k +
u(s)[z(s) +
v(σ)z(σ)dσ +
w(σ)z(σ)dσ]ds, for t ∈ [α, β].
α
α
α
If
Z
β
r=
Z
σ
w(σ) exp(
α
[u(τ ) + v(τ )]dτ )dσ < 1,
α
then
Z t
k
z(t) ≤
exp( [u(s) + v(s)]ds),
1−r
α
f ort ∈ [α, β].
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64
H. L. Tidke and M. B. Dhakne
Theorem 4. ([5], p-61). Let S be a convex subset of a normed linear space E
and assume 0 ∈ S. Let F : S → S be a completely continuous operator, and let
ε(F ) = {x ∈ S : x = λF x for some0 < λ < 1}. Then either ε(F ) is unbounded or
F has a fixed point.
We list the following hypotheses for our convenience.
(H1 ) −A is the infinitesimal generator of a strongly continuous cosine family {C(t) :
t ∈ R} which is compact for t > 0, and there exists a constant K such that
K = sup{C(t) : t ∈ [t0 , t0 + β]}.
(H2 ) There exists a constant G such that
kg(t1 , t2 , · · · , tp , x(·))k ≤ G,
for all x ∈ C([t0 , t0 + β], X).
(H3 ) There exists a continuous function p : [t0 , t0 + β] → R+ such that
kk(t, , x(t))k ≤ p(t)kx(t)k,
for every t ∈ [t0 , t0 + β]
and x ∈ X.
(H4 ) There exists a continuous function q : [t0 , t0 + β] → R+ such that
kh(t, x(t))k ≤ q(t)kx(t)k,
for every t ∈ [t0 , t0 + β]
and x ∈ X.
(H5 ) There exists a continuous function l : [t0 , t0 + β] → R+ such that
kf (t, x, y, z)k ≤ l(t)(kxk + kyk + kzk),
for every t ∈ [t0 , t0 + β]
and x, y, z ∈ X.
(H6 ) There exists a constant M such that
|a(t, s)| ≤ M,
for t ≥ s ≥ t0 .
(H7 ) There exists a constant N such that
|b(t, s)| ≤ N,
for t, s ∈ [t0 , t0 + β].
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65
Existence of solutions for nonlinear second order
(H8 ) For each t ∈ [t0 , t0 + β] the function f (t, ·, ·, ·) : [t0 , t+ β] × X × X × X → X
is continuous and for each x, y, z ∈ X the function f (·, x, y, z) : [t0 , t0 + β] ×
X × X × X → X is strongly measurable.
(H9 ) For each t ∈ [t0 , t0 + β] the functions k(t, ·), h(t, ·) : [t0 , t0 + β] × X → X are
continuous and for each x ∈ X the functions k(·, x), h(·, x) : [t0 , t0 + β] × X →
X are strongly measurable.
(H10 ) For every positive integer m there exists αm ∈ L1 (t0 , t0 + β) such that
sup
kf (t, x, y, z)k ≤ αm (t),
for t ∈ [t0 , t0 + β]
a. e.
kxk≤m,kyk≤m,kzk≤m
3
Existence Result
Theorem 5. Suppose that the hypotheses (H1 ) − (H10 ) hold. If
Z t0 +β
Z σ
r∗ =
N q(σ) exp(
[Kβl(τ ) + M p(τ )]dτ )dσ < 1,
t0
(3.1)
t0
then the initial value problem (1.1)–(1.2) has a mild solution on [t0 , t0 + β].
Proof. To prove the existence of a solution of nonlinear mixed Volterra-Fredholm
integrodifferential equations (1.1)–(1.2), we apply topological transversality theorem
and Pachpatte’s inequality. First we establish the priori bounds for the initial value
problem
Z t
00
x (t) + Ax(t) = λf (t, x(t),
a(t, s)k(s, x(s))ds,
Z
t0
t0 +β
b(t, s)h(s, x(s))ds),
t ∈ [t0 , t0 + β].
(3.2)
t0
with condition (1.2). Let x(t) be a mild solution of the IVP (1.1)–(1.2). Then from
Z t
x(t) = [C(t − t0 )(x0 − g(t1 , t2 , · · · , tp , x(·))) + S(t − t0 )η] + λ
S(t − s)f (s, x(s),
t0
Z
s
Z
t0 +β
a(s, τ )k(τ, x(τ ))dτ,
t0
b(s, τ )h(τ, x(τ ))dτ )ds,
t ∈ [t0 , t0 + β]
t0
(3.3)
and using hypotheses (H1 ) − (H7 ) and the fact that λ ∈ (0, 1), we have
Z t
kx(t)k ≤ kC(t − t0 )(x0 − g(t1 , t2 , · · · , tp , x(·)))k + kS(t − t0 )ηk +
kS(t − s)k
t0
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66
H. L. Tidke and M. B. Dhakne
s
Z
× kf (s, x(s),
Z
t0 +β
b(s, τ )h(τ, x(τ ))dτ )kds
a(s, τ )k(τ, x(τ ))dτ,
t0
t0
t
Z
≤ K(kx0 k + G) + Kβkηk +
Kβl(s)
t0
Z
s
Z
t0 +β
kb(s, τ )h(τ, x(τ ))kdτ ]ds
ka(s, τ )k(τ, x(τ ))kdτ +
× [kx(s)k +
t0
t0
Z
t
≤ K[(kx0 k + G) + βkηk] +
Kβl(s)
t0
Z
s
× [kx(s)k +
Z
t0 +β
N q(τ )kx(τ )kdτ ]ds.
M p(τ )kx(τ )kdτ +
(3.4)
t0
t0
Using the condition (3.1) and Pachpatte’s inequality given in Theorem 3 with z(t) =
kx(t)k in (3.4) , we obtain
Z t
K[(kx0 k + G) + βkηk]
kx(t)k ≤
exp( [Kβl(s) + M p(s)]ds)
1 − r∗
t0
k∗
exp(β[KβL + M P ]) = γ,
≤
1 − r∗
where k ∗ = K[(kx0 k + G) + βkηk],
L = sup {l(t)},
(3.5)
t∈[t0 ,t0 +β]
and
P =
sup
{p(t)}.
t∈[t0 ,t0 +β]
Hence there exists a constant γ independent of λ ∈ (0, 1) such that kx(t)k ≤ γ and
consequently
kxkB = sup{kx(t)k : t ∈ [t0 , t0 + β]} ≤ γ.
Now, we rewrite the problem (1.1)–(1.2) as follows: If y ∈ B and x(t) = C(t −
t0 )(x0 − g(t1 , t2 , · · · , tp , x(·))) + y(t), t ∈ [t0 , t0 + β], where y(t) satisfies
Z t
y(t) = S(t − t0 )η +
S(t − s)f (s, y(s) + C(s − t0 )(x0 − g(t1 , t2 , · · · , tp , x(·))),
t0
Z s
a(s, τ )k(τ, y(τ ) + C(τ − t0 )(x0 − g(t1 , t2 , · · · , tp , x(·))))dτ,
t0
t0 +β
Z
b(s, τ )h(τ, y(τ ) + C(τ − t0 )(x0 − g(t1 , t2 , · · · , tp , x(·))))dτ )ds,
t0
t ∈ [t0 , t0 + β] if and only if x(t) satisfies
Z
t
x(t) = C(t − t0 )(x0 − g(t1 , t2 , · · · , tp , x(·))) + S(t − t0 )η +
S(t − s)
t0
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67
Existence of solutions for nonlinear second order
Z
s
× f (s, x(s),
Z
t0 +β
b(s, τ )h(τ, x(τ ))dτ )ds.
a(s, τ )k(τ, x(τ ))dτ,
t0
t0
Define F : B0 → B0 ,
B0 = {y ∈ B : y(t0 ) = 0} by
Z t
(F y)(t) = S(t − t0 )η +
S(t − s)f (s, y(s) + C(s − t0 )(x0 − g(t1 , t2 , · · · , tp , x(·))),
t0
Z s
a(s, τ )k(τ, y(τ ) + C(τ − t0 )(x0 − g(t1 , t2 , · · · , tp , x(·))))dτ,
t0
t0 +β
Z
b(s, τ )h(τ, y(τ ) + C(τ − t0 )(x0 − g(t1 , t2 , · · · , tp , x(·))))dτ )ds,
t0
(3.6)
t ∈ [t0 , t0 + β].
First, we prove that F : B0 → B0 is continuous. Let {un } be a sequence of
elements of B0 converging to u in B0 . Then
Z t
(F un )(t) = S(t)η +
S(t − s)f (s, un (s) + C(s)(x0 − g(t1 , t2 , · · · , tp , x(·))),
t0
Z s
a(s, τ )k(τ, un (τ ) + C(τ )(x0 − g(t1 , t2 , · · · , tp , x(·))))dτ,
t0
t0 +β
Z
b(s, τ )h(τ, un (τ ) + C(τ )(x0 − g(t1 , t2 , · · · , tp , x(·))))dτ )ds, (3.7)
t0
t ∈ [t0 , t0 + β]. Now, kF un − F ukB = supt∈[t0 ,t0 +β] k(F un )(t) − (F u)(t)k. Since
{un } be the sequence of elements of B0 converging to u in B0 and by hypotheses
(H8 ) − (H9 ), we have
f (t, un (t) + C(t)(x0 − g(t1 , t2 , · · · , tp , x(·))),
Z t
a(t, s)k(s, un (s) + C(s)(x0 − g(t1 , t2 , · · · , tp , x(·))))ds,
t0
t0 +β
Z
b(t, s)h(s, un (s) + C(s)(x0 − g(t1 , t2 , · · · , tp , x(·))))ds)
t0
→ f (t, u(t) + C(t)(x0 − g(t1 , t2 , · · · , tp , x(·))),
Z t
a(t, s)k(s, u(s) + C(s)(x0 − g(t1 , t2 , · · · , tp , x(·))))ds,
t0
t0 +β
Z
b(t, s)h(s, u(s) + C(s)(x0 − g(t1 , t2 , · · · , tp , x(·))))ds)
t0
for each t ∈ [t0 , t0 + β]. Then by dominated convergence theorem, we have
k(F un )(t) − (F u)(t)k
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68
H. L. Tidke and M. B. Dhakne
t
Z
kS(t − s)kkf (s, un (s) + C(s)(x0 − g(t1 , t2 , · · · , tp , x(·))),
≤
t0
Z
s
a(s, τ )k(τ, un (τ ) + C(τ )(x0 − g(t1 , t2 , · · · , tp , x(·))))dτ,
t0
Z t0 +β
b(s, τ )h(τ, un (τ ) + C(τ )(x0 − g(t1 , t2 , · · · , tp , x(·))))dτ )
t0
− f (s, u(s) + C(s)(x0 − g(t1 , t2 , · · · , tp , x(·))),
Z s
a(s, τ )k(τ, u(τ ) + C(τ )(x0 − g(t1 , t2 , · · · , tp , x(·))))dτ,
t0
t0 +β
Z
b(s, τ )h(τ, u(τ ) + C(τ )(x0 − g(t1 , t2 , · · · , tp , x(·))))dτ )kds
t0
→0
and consequently kF un −F ukB → 0 as n → ∞ i.e. F un → F u in B0 as un → u ∈ B0 .
Therefore, F is continuous.
Now, we prove that F maps a bounded set of B0 into a precompact set of B0 .
Let Bm = {y ∈ B0 : kykB ≤ m} for some m ≥ 1. We first show that F maps
Bm into an equicontinuous family of functions with values in X. Let y ∈ Bm and
t0 ≤ s < t ≤ t0 + β. Then we have,
k(F y)(s) − (F y)(t)k
≤ kS(s)η − S(t)ηk
Z s
+
kS(s − τ ) − S(t − τ )kl(τ )[ky(τ ) + C(τ )(x0 − g(t1 , t2 , · · · , tp , x(·)))k
t0
Z τ
+k
a(τ, σ)k(σ, y(σ) + C(σ)(x0 − g(t1 , t2 , · · · , tp , x(·))))dσk
t0
t0 +β
Z
+k
b(τ, σ)h(σ, y(σ) + C(σ)(x0 − g(t1 , t2 , · · · , tp , x(·))))dσk]dτ
t0
Z
+
t
Kβl(τ )[ky(τ ) + C(τ )(x0 − g(t1 , t2 , · · · , tp , x(·)))k
Z τ
+k
a(τ, σ)k(σ, y(σ) + C(σ)(x0 − g(t1 , t2 , · · · , tp , x(·))))dσk
s
t0
t0 +β
Z
+k
b(τ, σ)h(σ, y(σ) + C(σ)(x0 − g(t1 , t2 , · · · , tp , x(·))))dσk]dτ
Z s
≤ kS(s)η − S(t)ηk +
kS(s − τ ) − S(t − τ )kL[m + K(kx0 k + G)
t0
t0
Z
τ
Z
t0 +β
M P (m + K(kx0 k + G))dσ +
+
t0
N Q(m + K(kx0 k + G))dσ]dτ
t0
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Existence of solutions for nonlinear second order
Z
t
KβL[m + K(kx0 k + G)
Z β
Z τ
N Q(m + K(kx0 k + G))dσk]dτ
M P (m + K(kx0 k + G))dσk +
+
0
t0
Z s
≤ kS(s)η − S(t)ηk +
kS(s − τ ) − S(t − τ )k
+
s
t0
× L[(m + k1∗ ) +
Z
τ
M P (m + k1∗ )dσ +
N Q(m + k1∗ )dσ]dτ
t0
t0
Z
t0 +β
Z
t
Z
τ
M P (m + k1∗ )dσ +
KβL[(m + k1∗ ) +
s
Z s t0
≤ kS(s)η − S(t)ηk +
kS(s − τ ) − S(t − τ )k
Z
t0 +β
+
N Q(m + k1∗ )dσ]dτ
t0
t0
k1∗ )
× L[(m +
+ βM P (m + k1∗ ) + βN Q(m + k1∗ )]dτ
Z t
+
KβL[(m + k1∗ ) + βM P (m + k1∗ ) + βN Q(m + k1∗ )dσ]dτ,
(3.8)
s
where k1∗ = K(kx0 k + G) and Q = supt∈[0,β] {q(t)}. The right hand side of (3.8) is
independent of y ∈ Bm and tends to zero as s−t → 0, since C(t), S(t) are uniformly
continuous for t ∈ [t0 , t0 + β] and the compactness of C(t), S(t) for t > 0 imply the
continuity in the uniform operator topology. Thus FBm is an equicontinuous family
of functions with values in X.
We next show that FBm is uniformly bounded. From the equation (3.6) and
using hypotheses (H1 ) − (H7 ) and the fact that kykB ≤ m, we obtain
Z t
Kβl(s)[ky(s) + C(s)(x0 − g(t1 , t2 , · · · , tp , x(·)))k
k(F y)(t)k ≤ Kβkηk +
t0
Z s
+k
a(s, τ )k(τ, y(τ ) + C(τ )(x0 − g(t1 , t2 , · · · , tp , x(·))))dτ k
t0
t0 +β
Z
+k
b(s, τ )h(τ, y(τ ) + C(τ )(x0 − g(t1 , t2 , · · · , tp , x(·))))dτ k]ds
t0
Z
t
≤ Kβkηk +
KβL[m + K(kx0 k + G)
t0
Z s
+
M p(τ )ky(τ ) + C(τ )(x0 − g(t1 , t2 , · · · , tp , x(·)))kdτ
t0
t0 +β
Z
N q(τ )ky(τ ) + C(τ )(x0 − g(t1 , t2 , · · · , tp , x(·)))kdτ ]ds
+
t0
Z
t
≤ Kβkηk +
KβL[(m +
t0
k1∗ )
Z
s
+
M P (m + k1∗ )dτ
t0
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70
H. L. Tidke and M. B. Dhakne
Z
t0 +β
+
N Q(m + k1∗ )dτ ]ds
t0
Z
t
≤ Kβkηk +
KβL[(m + k1∗ ) + βM P (m + k1∗ ) + βN Q(m + k1∗ )]ds
t0
≤ Kβkηk + Kβ 2 L(m + k1∗ )[1 + βM P + βN Q].
This implies that the set {(F y)(t) : kykB ≤ m, t0 ≤ t ≤ t0 + β} is uniformly
bounded in X and hence {FBm } is uniformly bounded.
We have already shown that FBm is an equicontinuous and uniformly bounded
collection. To prove the set FBm is precompact in B, it is sufficient, by ArzelaAscoli’s argument, to show that the set {(F y)(t) : y ∈ Bm } is precompact in X
for each t ∈ [t0 , t0 + β]. Since (F y)(t0 ) = 0 for y ∈ Bm , it suffices to show this
for t0 < t ≤ t0 + β. Let t0 < t ≤ t0 + β be fixed and a real number satisfying
t0 < < t. For y ∈ Bm , we define
Z
t−
(F y)(t) = S(t)η +
S(t − s)f (s, y(s) + C(s)(x0 − g(t1 , t2 , · · · , tp , x(·))),
t0
Z s
a(s, τ )k(τ, y(τ ) + C(τ )(x0 − g(t1 , t2 , · · · , tp , x(·))))dτ,
t0
t0 +β
Z
b(s, τ )h(τ, y(τ ) + C(τ )(x0 − g(t1 , t2 , · · · , tp , x(·))))dτ )ds.
(3.9)
t0
Since C(t), S(t) are compact operators , the set Y (t) = {(F y)(t) : y ∈ Bm } is
precompact in X, for every , t0 < < t. Moreover for every y ∈ Bm , we have
(F y)(t) − (F y)(t)
Z t
=
S(t − s)f (s, y(s) + C(s)(x0 − g(t1 , t2 , · · · , tp , x(·))),
t−
Z s
a(s, τ )k(τ, y(τ ) + C(τ )(x0 − g(t1 , t2 , · · · , tp , x(·))))dτ,
t0
t0 +β
Z
b(s, τ )h(τ, y(τ ) + C(τ )(x0 − g(t1 , t2 , · · · , tp , x(·))))dτ )ds.
(3.10)
t0
By making use of hypotheses (H1 ) − (H7 ) and the fact that ky(s)k ≤ m, we have
k(F y)(t) − (F y)(t)k
Z t
≤
Kβl(s)[ky(s) + C(s)(x0 − g(t1 , t2 , · · · , tp , x(·)))k
t−
Z s
+k
a(s, τ )k(τ, y(τ ) + C(τ )(x0 − g(t1 , t2 , · · · , tp , x(·))))dτ k
t0
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71
Existence of solutions for nonlinear second order
Z
t0 +β
b(s, τ )h(τ, y(τ ) + C(τ )(x0 − g(t1 , t2 , · · · , tp , x(·))))dτ k]ds
+k
t0
Z
t
KβL[m + K(kx0 k + G)
≤
t−
Z
s
M p(τ )ky(τ ) + C(τ )(x0 − g(t1 , t2 , · · · , tp , x(·)))kdτ
+
t0
Z t0 +β
N q(τ )|ky(τ ) + C(τ )(x0 − g(t1 , t2 , · · · , tp , x(·)))kdτ ]ds
+
t0
Z
t
≤
KβL[(m +
t−
k1∗ )
Z
s
+
M P (m +
k1∗ )dτ
Z
t0
≤ KβL(m +
k1∗ )[1
t0 +β
+
N Q(m + k1∗ )dτ ]ds
t0
+ βM P + βN Q].
This shows that there exists precompact sets arbitrarily close to the set {(F y)(t) :
y ∈ Bm }. Hence the set {(F y)(t) : y ∈ Bm } is precompact in X. Thus we have
shown that F is completely continuous operator.
Moreover, the set
ε(F ) = {y ∈ B0 : y = λF y
for some
0 < λ < 1},
is bounded in B, since for every y in ε(F ), the function x(t) = y(t) + C(t)(x0 −
g(t1 , t2 , · · · , tp , x(·))) is a mild solution of (3.2)–(1.2) for which we have proved
kxkB ≤ γ and hence kykB ≤ γ + k1∗ . Now, by virtue of Theorem 4, the operator
F has a fixed point in B0 . Therefore, the initial value problem (1.1)–(1.2) has a
solution on [t0 , t0 + β].
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Haribhau Laxman Tidke
M. B. Dhakne
Department of Mathematics,
Department of Mathematics,
North Maharashtra University,
Dr. Babasaheb Ambedkar Marathwada University,
Jalgaon-425 001, India.
Aurangabad-431004, India.
e-mail: tharibhau@gmail.com
e-mail: mbdhakne@yahoo.com
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Surveys in Mathematics and its Applications 5 (2010), 61 – 72
http://www.utgjiu.ro/math/sma