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Volterra-stieltjes integral operators
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eC,,'NCm~ o,REolr.ELSEVIER
MATHEMATICAL AND COMPUTER MODELLING
Mathematical and Computer Modelling 41 (2005) 335-344 www.elsevier.com/locate/mcm
Volterra-Stieltjes Integral Operators J . BANAS Department of Mathematics Rzesz6w University of Technology 35-959 Rzesz6w, W. Pola 2, Poland D. O'REGAN Department of Mathematics National University of Ireland Galway, Ireland (Received and accepted February 2003)
A b s t r a c t - - W e will study some properties of the integral operator of Volterra-Stieltjes type which is defined with help of the Stieltjes integral with the kernel depending on two real variables. The obtained results are applied to prove an existence result concerning nonlinear integral equation of Volterra-Stieltjes type. (~) 2005 Elsevier Ltd. All rights reserved. Keywords--Function of bounded variation, Monotonic function, Stieltjes integral, Integral operator, Compact operator.
1. I N T R O D U C T I O N The theory of integral operators of various types creates an important and significant branch of modern nonlinear functional analysis (cf. [1-7], for example). Let us mention that both the theory of linear integral operators (Fredholm, Volterra, etc.) and the theory of various types of nonlinear integral operators (Hammerstein, Urysohn, etc.) find numerous applications in mathematical physics, engineering, biology and economics, among others [2,3,6,8-10]. Several integral operators mentioned above can be treated as special cases of integral operators of Stieltjes type with the kernel depending on two variables (see e.g., [11-14]). Integral operators of such a form will be also studied in this paper. More precisely, this paper is devoted to the study of some properties of the integral operators of Volterra-Stieltjes type. These operators are defined with help of the Riemann-Stieltjes integral with the kernel depending on two real variables. We prove a few results concerning the continuity, bounded variation, monotonicity and compactness of these operators in the space of continuous functions. Results of such a kind were obtained also in the paper [13]. Here, we generalize and improve these results. Moreover, we show that the mentioned results can be formulated under weaker assumptions. For example, we do not assume in this paper that the kernel of our integral operator has a bounded variation in the sense of Arz~la (cf. [13], where this assumption was used).
0895-7177/05/$ - see front matter ~) 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j .mcm.2003.02.014
Typeset by ,42M,.q-TEX
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J. BANA§ AND D. O'REGAN
Using the obtained results, we will also study the solvability of a nonlinear Volterra-Stieltjes integral equation in the space of functions continuous on a bounded interval.
2. N O T A T I O N ,
DEFINITIONS,
AND
AUXILIARY
FACTS
In this section, we collect a few auxiliary facts which will be needed in the sequel. At the beginning, we recall some basic concepts and results concerning functions of bounded variation and Stieltjes integral. b For a given real function x defined on the interval [a, b] the symbol Va x denotes the variation of x on [a, b]. We say that x is of bounded variation whenever vb~ x is finite. If u(t, s) = u : [a, b] x [c, aq --* R, then, we denote by Vqt=pu(t, s), the variation of the function t --* u(t, s) on the interval ~p, q] C [a, b], where s is arbitrarily fixed in [c, a~. Similarly, we define the quantity Vq=p u(t, s). We refer to [1, 15] for the properties of functions of bounded variation. If x and ~ are two real bounded functions defined on the interval [a, b], then, under some additional conditions [15], we can define the Stieltjes integral (in the Riemann-Stieltjes sense),
~
b x (t) d~v (t),
of the function x with respect to the function ~. In such a case, we say that x is Stieltjes integrable on the interval [a, b] with respect to ~v. There are known several conditions guaranteeing the Stieltjes integrability [15,16]. One of the most frequently used requires that x is continuous and ~v is of bounded variation on the interval [a, hi. The properties of the Stieltjes integral used further on are formulated in below-given lemmas (cf. [15-17]). LEMMA 1. • x is Stieltjes integrable on In, b] with respect to a function ~ of bounded variation,
then, x(t) d~(t) <
]x(t)[d
~
.
Moreover, the following inequality holds,
Iz
I
z(t) d~(t) < sup Iz(t) l a
(?) ~
•
Let xl, x2 be Stidtjes integrable /'unctions on the interval [a,b] with respect t o a nondecreasing function ~v and, such that xl (t) < x2(t), for t E [a, hi. Then, LEMMA 2.
fa Xl (t) d~v(t) <
b
f[ X2 (t) d~(t).
In what follows, we will also consider the Stieltjes integral of the form,
~
bx(s)d~g(t,s),
where g : [a, b] x [a, b] --* R and the symbol ds indicates the integration with respect to s. The details concerning the integral of this type will be given later. Now, let us assume that f : In, b] x R -* R is a given function. Then, to every real function x defined on [a,b], we may assign the function (Fx)(t) = f(t,x(t)), t e [a,b]. The operator F defined in this way is called the superposition operator generated by the function f(t, x). The properties of the superposition operator may be found in [18]. For our further purposes, we shall need the following result concerning the behaviour of the superposition operator F on the space C[a, b] consisting of all continuous functions acting from the interval In, b] into R and furnished with the standard maximum norm I]xll = max{Ix(t)] : t e In, b]} (cf. [18]).
Volterra-Stieltjes Integral Operators
337
LEMMA 3. Let F be the superposition operator generated by the function f : [a, b] x R --* R. Then, F transforms the space C[a, b] into itself and is continuous if and only if the function f is continuous on the set [a, b] x ~,. Further, let us assume that x is a real function defined on [a,b]. Then, by w(x,E), we denote the modulus of continuity of the function x, i.e., w (x,¢) = sup [Ix (t) - x (s)l: t, s e [a,b], It - s I < 6]. If p(t, s) -- p : [a, b] x [e, a~ --+ R, then, the modulus of continuity of the function s ~ p(t, s) on the interval [c, d], for a fixed t e [a, b], is defined as
w ( p ( t , . ) , E ) = sup[Ip(t,u ) - p ( t , v ) [
: u, v E [c,~, l u - v l
<_ z].
In the similar way, we define the modulus w(p(., s), 6). 3. P R O P E R T I E S VOLTERRA-STIELTJES
OF NONLINEAR INTEGRAL OPERATOR
Let I be a bounded and closed interval in R. For convenience, we assume that I = [0, 1]. In this section, we will investigate the nonlinear integral operator of Volterra-Stieltjes type having the form,
(Vx) (t) --
/o'
v (s, x (s)) dsg (t, s),
t e I.
(1)
In our further considerations, we shall always assume t h a t the following conditions are satisfied. (i) g : I x I --* R and, for every tl, t2 E I, such that tl < t2, the function s --* g(t2, s) - g ( t l , s) is nondecreasing on the interval I. (ii) g(0, s) = 0, for any s E I. (iii) v : I x R -~ R is a continuous function such that there exist a continuous function a : I ~ I and a continuous and nondecreasing function ~ : ]~+ --+ R+, for which the following inequality holds,
for t E I and x E R. REMARK 1. Observe that Assumptions (i) and (ii) imply that the function s ~ g(t, s) is nondecreasing on the interval I, for any fixed t E I (cf. Remark 3 in [13]). Indeed, putting in (i), t2 -- t, tl = 0 and keeping in mind (ii), we obtain the desired conclusion. From this observation, it follows immediately that, for every t E I, the function s ~ g(t, s) is of bounded variation on I. Such a condition was assumed in [13], but we showed above that it is superfluous. REMARK 2. In paper [13], the following condition was assumed in place of (iii). (iii') v : I × ]~ --, R is continuous and, such that there exist a function a = a(t) continuous on I and a constant b > 0 with the property,
Iv (t,x)l < a(t)+ blxl, for t E ~ and x E I. Obviously, Assumption (iii) is less restrictive than (iii'). Moreover, we show later that (iii) admits several natural realizations. Now, we prove a few results about the properties of the integral operator V defined by (1). We start with the following theorem.
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J. BANA$ANDD. O'REGAN
THEOREM 1. Assume that Assumptions (i)-(iii) are satisfied. Then, for every function x E C(I), the function V z is of bounded variation on I. PROOF. Observe first that taking into account our assumptions, in view of Lemma 3 and Remark 1, we infer t h a t the Volterra-Stieltjes operator V defined by (1) is well-defined on the space C(I). Next, fix a partition 0 = to < tl < ... < t= = 1 of the interval I = [0, 1]. Then, in view of Lemma I and Remark 1, we get 7%
I(Vx) (t~) - ( V x ) (t~-l)l i=l
1/o'
<
L
v ( s , x ( s ) ) d s g ( t i , s) -
i=l
+
~li[ '-1v ( s , x ( s ) ) d s g ( t i ,
s) -
i=l
v ( s , x ( s ) ) d s g ( t i , s)
S[*-'v ( s , x ( s ) ) d s g ( t i _ l , S ) I
v (s, x (s)) dsg (t~, s) +
= i=l
v (s, x (s)) d, [g (ti, s) - g (ti-1, s)] i=1
-i
i=1 Jti--I
+
I
tt~$1_l
I.(s,x(s))ld~ i=l
[g(t.u) -g(ti_~,u)] . u=O
Consequently, keeping in mind the above estimate, Lemma 2, and Remark 1, we obtain,
7%
I(v~) (t~) - (v~) (t,-~)l i----1
<
±f i=1
Iv(s,x(s))ldsg(t.s)+
tl-i
Iv(s,z(s))lds[g(t.s)-g(t~-l,s)] ~=i
5; a(s)~(lx(s)[)d~g(t~,s)+ D/oa(s)~v(lx(s)l)d.[g(t.s)-g(t~-~,s)]
<_
i=l
t~--I
_.
=
Ilall ~ (11~11){ ~i=1[g (ti'
ti) -- g (tl,
ti--1)]
+ ~ b (t~, t~_~) - 9 (t~_i, t~_~) - g (t~, o) + g (t~_~, o)1 i=1
= Ilall ~o (llxll)
{ ~=1[9(ti,ti)
)
- g ( t i - l , t i - 1 ) - g(t~, O) + g(ti-1,0)] }
= Ilall ~ (llxll) [g (1,1) - g (1, 0)] < c¢. This completes the proof. For our further purposes, we will need the following Lemma.
Volterra-Stieltjes Integral Operators
339
LEMMA 4. Assume that the function g(t, s) = g : I x I --* R satisfies Assumption (i). Then, for every Sl, s2 E I, such that Sl < s2, the function t --* g(t, s2) - g ( t , sl) is nondecreasing on the interval I. We omit easy proof requiring only a few simple calculations. Now, we prove a generalization of the result contained in Theorem 1. THEOREM 2. Suppose that there are satisfied the assumptions of Theorem i. Moreover, assume that the values of the function v are nonnegative i.e., v : I x R --* R+. Then, for every function x E C(I), the function V x is nondecreasing on I. PROOF. Fix arbitrarily tl, t2 E I, tl < t2. Observe that, for every arbitrarily fixed function x E C(I), the following inequality holds to be true,
(2)
~ i 2 v(s, x(s))dsg(t2, s) >_ O.
Indeed, take a partition tl --- so < Sl < ... < sn -- t2 of the interval [tl,t2]. Further, let us choose arbitrarily points ci E [si-l,si] (i = 1, 2 , . . . , n). Consider the Pdemann-Stieltjes sum corresponding to the integral from the inequality (2), i.e., n
on = ~
i=l
~(c,, x(c,))[g(t~, s,) - g(t2, s,_l)].
Since v(ci, x(ci)) >_ 0 and g(t2, si) >_ g(t2, si-1), for i = 1, 2 , . . . , n (see Remark 1), we infer that an _> 0, for n E N. This shows that the inequality (2) is true. In what follows, let us take an arbitrary partition 0 = s0 < Sl < -.- < s,~ = tl of the interval [0, tl] and choose arbitrarily points ci E [S~-l, s~l (i = 1, 2 , . . . , n). Then, keeping in mind Lemma 4 and (2), we deduce the following inequalities, }2
v (e~. x (~)) [, (tl. s.) - g (tl. S~_l)] 4=1 n
< ~v
(c~, x (c,)) [g (t2, s,) - g (t2, s,_ll]
i=l °
_< y~'v (e. x (~)) [g (t2. s,) - g (t~, s,_l)] +
v (s,~ (s)) e~g (t2. s).
i=l
Hence, we derive the following inequality,
~0tl v (s, x (s)) dsg (tl, s) < ~0tl v (s, x (s)) d,g (t2, s) + ~i 2v (s, x (s)) dsg (ts, s) 0t2 v (s, x (s) ) dsg (t2, s) Thus, the proof is complete.
|
THEOREM 3. Suppose there are satisfied the assumptions of Theorem 1. Moreover, assume that the function s --* g(1, s) is continuous on I and the function t --* g(t, s) is continuous on I, for any fixed s E I. Then, the operator V acts continuously from the space C ( I ) into itself. PROOF. Observe that the operator V can be written as the composition V = G F of the superposition operator, (Fx) (t) = v (t, x (t)),
340
J. BANA§ANDD. O'REGAN
and the linear Volterra-Stieltjes integral operator G defined by the formula,
(Gx) (t) =
x (8) dsg (t, s).
Further, let us notice t h a t in view of Lemma 3 and Assumption (iii) the superposition operator F maps continuously the space C(I) into itself. Thus, it is sufficient to show that the operator G acts continuously from C(I) into C(I). To do this, fix arbitrarily x E C(I). Next, fix ~ > 0 and take tl,t2 E I, tl < t2, such that It2 - tl[ _< z. Then, we obtain
/?
I(Gz) (t2) - (Gz) (tl)l <_
<_
x (s) dsg (t2, s) -
/?
x (s) dsg (t2, s)
x (s) d,g (t2, s) +
I
x (s) d8 [g (t2, s) - g (tl,
•
Hence, applying Lemmas 1 and 2, we get I(Gx) (t2) - (ax)(tl)l
<_
[x(s)[ds
1g(t2'u)
+
u=$1
= Ilxll
[x(s)lds
~=o~/[g(t2'u)-g(tl'u)]
~t~---0
d~g (t2, 8) + Llxll
d~ [g(t~,s) - g (t~,s)]
= Ilxll {[g (t2, t2) - g (t2, tl)] + [g (t~, tl) - g (tl, t~) - g (t2, O) + g (tl, 0)]}
_< IIx 11{ [g (t2, t2) - g (t2, t~)] + Ig (t2, t~) - g (t l, t l)l + Ig (t2, 0) - g (t~, 0) 1}. Hence, in view of Remark 1 and Lemma 4, we arrive to the following estimate,
I(G:c) (t2) - (az) (t~)l _< Ilxll {[g (1, t2) - g (1, t~)] + Ig (t~, t~) - g (t,, t~)l + Ig (t2, 0) - g (t~, 0)1} <-- II~ll {~ (g (1, "/, ~) + ~ (g (', t~), s) + ~ (g (., 0), e)}, where the symbols w(g(1, .),e), co(g(-, tl), s), w(g(., 0), e) were introduced in Section 2. From the above estimate, we conclude that Gx C C(I). In order to show that G is continuous, fix x E C(I). Then, in view of Lemmas 1 and 2, we have
I(Gx)(t)[ <_
Ix(s)ld~
g(t,u)
<_ llxH
d~ - - g(t,u) u=O
= ]lxll
d,g (t, s) < Hxl[
d~g (t, s) = I]xH [g (t, 1) - g (t, 0)].
Hence, in virtue of Lemma 4, we get
I(ax) (t)l <_ llxll [g (1, 1) - g (1, 0)1, and consequently,
IlOxll <_ Ilxll [g (1, 1) - g (1, 0)]. The above inequality shows that G transforms continuously the space C(I) into itself. The proof is complete. | REMARK 3. Theorem 3 remains true if we assume that the hypotheses of Theorem 1 are satisfied, and additionally, the following one as well. (iv) The functions t ~ g(t, t) and t ~ g(t, 0) are continuous on the interval I.
Volterra-Stieltjes Integral Operators
341
Indeed, evaluating analogously as in the proof of Theorem 3, we get,
I(Oz) (t=) - (Oz) (tl)l < Ilzll {[g (t=, t=) - g (t=, t,)] + [g (t=, t~) - g (t~, t,) - g (t=, 0) + (tl, 0)]} =
Ilzll {[g
(t=,
t=) -
g (t,, tl) ] - [g (t2, 0) - g (tl, 0)]}
_< Ilzll {Ig (t2, t=) - g (tl, t~)l + Ig (t2, o) - g (t,, o)1}. From the above estimate we obtain our assertion. Our next result shows that the Volterra-Stieltjes integrM operator V is Mso compact under suitable assumptions. THEOREM 4. Under the assumptions of Theorem 3, the operator V is compact on the space
C(I). PROOF. Let us take a bounded subset X of the space C(I) and choose an arbitrary element x E X. Next, fix e > 0 and t l , t l E I, tl < t2, such that It2 --tll < e. Then, arguing similarly as in the proof of Theorem 3, we obtain the following estimate,
I(Vz) (t=) - (vz) (t~)l ~ I1~11~ (Hzll) {~o (g (1, .), e) + ¢o (g (., tl), ~) + w (g (-, 0), ~)}. Hence, we infer that the functions of the set V X are equicontinuous on the interval I. On the other hand, for an arbitrary function x c X , we have
I(vm)(x)l <
/:
[v(s,x(s))ld,
-< I1~11~(llxll)
i
(20) g(t,u)
d,g(t,s) < Ilall~o(llxll)
jo1d , g ( t , s )
= I1~11~o (llxll) [g (t, 1) - g (t, o)]. In view of Lemma 4, the above inequality yields
](Vx) (t)l < [laH ~o(HxH)[g (1,1) - g (1, 0)]. Consequently, we obtain
Ilvzll ~ I1~11~ (llXrl) [g (1,1) - g (1,0)], where we have denoted
IIXll = s,,p [llzll : x e x]. In view of the above estimate, we infer that the set V X is bounded in the space C(I). Now, applying Arzdla criterion for compactness in the space U(I), we deduce that the set V X is relatively compact in this pace. Thus, the proof is complete. | REMARK 4. The conclusion of Theorem 4 remains true if we assume the hypotheses formulated in Remark 3. In what follows, we provide a few examples illustrating the above obtained results. EXAMPLE 1. Let a, b : I --+ I be nondecreasing functions on I, such that a(0) = 0. Assume additionally that b is differentiable and a is continuous on the interval I. P u t g(t, s) = a(t). b(s). Then it is easy to check that the function g(t, s) satisfies Assumptions (i), (ii), and the additional assumption formulated in Theorem 3 (cf. also Remark 3). In this case, the Volterra-Stieltjes operator V defined by (1) has the form,
(Vx) (t) =
/0
a (t) v (s, x (s)) b' (s) ds = a (t)
/0
v (s, x (s)) b' (s) ds.
342
J. BANAS AND D. O'REGAN
EXAMPLE 2. Consider the function g : I x I --~ l~ defined by the formula,
g(t,s)
~tln t+s -~-
(
t
0,
for t e [0,1],
seI,
'
fort=0,
sEI.
It can be easily seen that the function g(t, s) satisfies Assumptions (i), (ii), and the assumption given in Theorem 3 (see also Remark 3). In this case, the Volterra-Stieltjes integral operator V has the form,
( v ~ ) (t) =
t + s v (s' ~ (s)) d~
end creates the Volterra counterpart of the well-known Chandrasekhar integral operator defined by the formula,
(Hx) (t) = ~ l t ~ o
(s) x (s) ds,
(cf. [8,9,12,141). EXAMPLE 3. Let us take a function a E C(I) and the function T : R+ ~ R+ defined by the formula ~(r) = l + r . Denote b = I]all = max[la(t)l : t E I]. Then, the estimate from Assumption (iii) has the form, [v (t, x)L < ~ (t) (1 + Ixl) < a (t) + b IxL. Notice that such an assumption was accepted in paper [13]. Thus, we see that this assumption is a special case of Assumption (iii) accepted in this paper. EXAMPLE 4. Similarly as above, take a function a(t) E C(I). Assume that the function T : N[+ --~ R+ has the form ~(r) = 1 + r a, where a > 0 is a fixed number. Then, the inequality from Assumption (iii) has the form,
Iv (t, ~)l < a (t) (1 + IxF) < a (t) + b I x F , where b = II"lt. EXAMPLE 5. Now, let us take a(t) E C(I) and consider the function to : ]~+ --* ]~+ defined by the formula ~o(r) = exp r. Then, the estimate from (iii) has the form, Iv (t, x)l < a (t) exp Ix[. The above given examples show that the assumptions accepted in Theorems 1-4 admit several natural realizations.
4. SOLVABILITY OF VOLTERRA-STIELTJES INTEGRAL EQUATION In this section, we show that the results obtained in the previous section can be applied in the study of the solvability of the following nonlinear integral equation of Volterra-Stieltjes type,
x (t) ----p (t) +
I'
v (s, x (s)) dsg (t, s),
(3)
where t e I = [0,1]. Obviously, the above equation may be written in the form x = p+ Vx, where V is the VolterraStieltjes operator defined by (1). We formulate below two existence results concerning the equation (3).
Volterra-Stieltjes Integral Operators
343
THEOREM 5. S u p p o s e p = p(t) E C ( I ) and the functions v and g satisfy Assumptions (i)-(iii) and the assumption formulated in Theorem 3 or Remark 3. Moreover, assume that there exists a positive solution ro of the inequality
Ilpll + Ilall (9 (1,1) - 9 (1, o))
(r) < r.
Then, the equation (3) has at least one solution in the space C(I). PROOF. Consider the operator S defined on the space C ( I ) by the formula, S x = p + Vx, where V is defined by (1). In view of Theorem 3, we see that S transforms continuously the space C ( I ) into itself. Further, consider the ball B~o = B(0, r0) in the space C(I), where r0 is a number appearing in our assumptions. For an arbitrarily fixed function, x E B~ o and, for t E I, in virtue of Lemmas 1, 2, and 4, we get I(Sx)(t)r < Ip(t)l + <
[[ptl +
v(s,z(s))d~g(t,s) [v(s,x(s))Ids
-- g(t,u) u~O
< Ilpl[+
a(s)~@(s)l)ds U-~0
< [IPl[ +[lal[ ~ (llxll)
/o
dsg (t, s) < IIPl[ + [lal] ~ (to)
< IlP[[ + [[all ~ (to) [g (t, 1) - g (t, 0)]
/0
dsg (t, s)
--< [[PII + [tail (g (1, 1) -- g (1, 0)) ~ (r0) _< ro. The above inequality shows that S transforms the ball B~o into itself. Obviously, in view of Theorem 4 the operator S is completely continuous. Thus, applying the Schauder fixed-point principle, we complete the proof. | Our next existence result is based mainly on Theorem 2. THEOREM 6. Assume that there are satisfied the hypotheses of Theorem 5 and, in addition, the function v is nonnegative (i.e., v : I x R ~ R+ ) and p(t) is a nondecreasing function on I. Then, the equation (3) has at least one solution in the space C ( I ) being a nondecreasing function on the interval I. The proof can be carried over in the same way as the proof of Theorem 5. Moreover, we use the fact that the image SB~ o of the ball B~o under the operator S contains only nondecreasing function on I, which is a simple consequence of Theorem 2. To complete the proof, it is sufficient to notice that the solutions of the equation (3) belong to the set SB~ o. REFERENCES N. Dunford and J. Schwartz, Linear Operators I, Int. Publ., Leyden, (1963). K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, (1985). C. Corduneanu, Integral Equations and Applications, Cambridge Univ. Press, Cambridge, (1991). D. O'Regan and M. Meehan, Existence Theory for Nonlinear Integral and Integrodifferential Equations, Kluwer Academic, Dordrecht, (1998). 5. R.P. Agarwal, D. O'Regan and P.J.Y. Wong, Positive Solutions of Differential, Difference and Integral Equations, Kluwer Academic Publishers, Dordrecht, (1999). 6. P.P. Zabrejko, A.I. Koshelev, M.A. Krasnosel'skii, S.G. Mikhlin, L.S. Rakovschik and V.J. Stecsenko, Integral Equations, Noordhoff, Leyden, (1976). 1. 2. 3. 4.
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7. M. V~ith~ Volterra and Integral Equations of Vector Functions Pure and Applied Math, Marcel Dekker, New York, (2000). 8. B. Cahlon and M. Eskin, Existence theorems for an integral equation of the Chandrasekhar H-equation with perturbation, J. Math. Anal. Appl. 83, 159-171, (1981). 9. S. Chandrasekhar, Radiative Transfer, Oxford Univ. Press, London, (1950). 10. S. Hu, M. Khavanin and W. Zhuang, Integral equations arising in the kinetic theory of gases, Appl. Analysis 34, 201-266, (1989). 11. J. Banal, Some properties of Urysohn-Stieltjes integral operators, Intern. J. Math. and Math. Sci. 21, 78-88, (1998). 12. J. Banal, J.R. Rodriguez and K. Sadarangani, On a class of Urysohn-Stieltjes quadratic integral equations and their applications, J. Comput. Appl. Math. I13, 35-50, (2000). 13. J. Banag and J. Dronka, Integral operators of Volterra-Stieltjes type, their properties and applications, Mathl. Comput. Modelling 32 (11-13), 1321-1331, (2000). 14. J. Banag and K. Sadarangani, Solvability of Volterra-Stieltjes operator-integral equations and their applications, Computers Math. Applic. 41 (12), 1535-1544, (2001). 15. I.P. Natansoa, Theory of Functions of a Real Variable, Ungar, New York, (1960). 16. R. Sikorski, Real Functions, (in Polish), PWN, Warszawa, (1958). 17. A.D. Mygkis, Linear Differential Equations with Retarded Argumsnt~ (in Russian), Nauka, Moscow, (1972). 18. J. Appell and P.P. Zabrejko, Nonlinear Superposition Operators, Cambridge Tracts in Mathematics, Camb. Univ. Press, Cambridge, (1990).
eC,,'NCm~ o,REolr.ELSEVIER
MATHEMATICAL AND COMPUTER MODELLING
Mathematical and Computer Modelling 41 (2005) 335-344 www.elsevier.com/locate/mcm
Volterra-Stieltjes Integral Operators J . BANAS Department of Mathematics Rzesz6w University of Technology 35-959 Rzesz6w, W. Pola 2, Poland D. O'REGAN Department of Mathematics National University of Ireland Galway, Ireland (Received and accepted February 2003)
A b s t r a c t - - W e will study some properties of the integral operator of Volterra-Stieltjes type which is defined with help of the Stieltjes integral with the kernel depending on two real variables. The obtained results are applied to prove an existence result concerning nonlinear integral equation of Volterra-Stieltjes type. (~) 2005 Elsevier Ltd. All rights reserved. Keywords--Function of bounded variation, Monotonic function, Stieltjes integral, Integral operator, Compact operator.
1. I N T R O D U C T I O N The theory of integral operators of various types creates an important and significant branch of modern nonlinear functional analysis (cf. [1-7], for example). Let us mention that both the theory of linear integral operators (Fredholm, Volterra, etc.) and the theory of various types of nonlinear integral operators (Hammerstein, Urysohn, etc.) find numerous applications in mathematical physics, engineering, biology and economics, among others [2,3,6,8-10]. Several integral operators mentioned above can be treated as special cases of integral operators of Stieltjes type with the kernel depending on two variables (see e.g., [11-14]). Integral operators of such a form will be also studied in this paper. More precisely, this paper is devoted to the study of some properties of the integral operators of Volterra-Stieltjes type. These operators are defined with help of the Riemann-Stieltjes integral with the kernel depending on two real variables. We prove a few results concerning the continuity, bounded variation, monotonicity and compactness of these operators in the space of continuous functions. Results of such a kind were obtained also in the paper [13]. Here, we generalize and improve these results. Moreover, we show that the mentioned results can be formulated under weaker assumptions. For example, we do not assume in this paper that the kernel of our integral operator has a bounded variation in the sense of Arz~la (cf. [13], where this assumption was used).
0895-7177/05/$ - see front matter ~) 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j .mcm.2003.02.014
Typeset by ,42M,.q-TEX
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J. BANA§ AND D. O'REGAN
Using the obtained results, we will also study the solvability of a nonlinear Volterra-Stieltjes integral equation in the space of functions continuous on a bounded interval.
2. N O T A T I O N ,
DEFINITIONS,
AND
AUXILIARY
FACTS
In this section, we collect a few auxiliary facts which will be needed in the sequel. At the beginning, we recall some basic concepts and results concerning functions of bounded variation and Stieltjes integral. b For a given real function x defined on the interval [a, b] the symbol Va x denotes the variation of x on [a, b]. We say that x is of bounded variation whenever vb~ x is finite. If u(t, s) = u : [a, b] x [c, aq --* R, then, we denote by Vqt=pu(t, s), the variation of the function t --* u(t, s) on the interval ~p, q] C [a, b], where s is arbitrarily fixed in [c, a~. Similarly, we define the quantity Vq=p u(t, s). We refer to [1, 15] for the properties of functions of bounded variation. If x and ~ are two real bounded functions defined on the interval [a, b], then, under some additional conditions [15], we can define the Stieltjes integral (in the Riemann-Stieltjes sense),
~
b x (t) d~v (t),
of the function x with respect to the function ~. In such a case, we say that x is Stieltjes integrable on the interval [a, b] with respect to ~v. There are known several conditions guaranteeing the Stieltjes integrability [15,16]. One of the most frequently used requires that x is continuous and ~v is of bounded variation on the interval [a, hi. The properties of the Stieltjes integral used further on are formulated in below-given lemmas (cf. [15-17]). LEMMA 1. • x is Stieltjes integrable on In, b] with respect to a function ~ of bounded variation,
then, x(t) d~(t) <
]x(t)[d
~
.
Moreover, the following inequality holds,
Iz
I
z(t) d~(t) < sup Iz(t) l a
(?) ~
•
Let xl, x2 be Stidtjes integrable /'unctions on the interval [a,b] with respect t o a nondecreasing function ~v and, such that xl (t) < x2(t), for t E [a, hi. Then, LEMMA 2.
fa Xl (t) d~v(t) <
b
f[ X2 (t) d~(t).
In what follows, we will also consider the Stieltjes integral of the form,
~
bx(s)d~g(t,s),
where g : [a, b] x [a, b] --* R and the symbol ds indicates the integration with respect to s. The details concerning the integral of this type will be given later. Now, let us assume that f : In, b] x R -* R is a given function. Then, to every real function x defined on [a,b], we may assign the function (Fx)(t) = f(t,x(t)), t e [a,b]. The operator F defined in this way is called the superposition operator generated by the function f(t, x). The properties of the superposition operator may be found in [18]. For our further purposes, we shall need the following result concerning the behaviour of the superposition operator F on the space C[a, b] consisting of all continuous functions acting from the interval In, b] into R and furnished with the standard maximum norm I]xll = max{Ix(t)] : t e In, b]} (cf. [18]).
Volterra-Stieltjes Integral Operators
337
LEMMA 3. Let F be the superposition operator generated by the function f : [a, b] x R --* R. Then, F transforms the space C[a, b] into itself and is continuous if and only if the function f is continuous on the set [a, b] x ~,. Further, let us assume that x is a real function defined on [a,b]. Then, by w(x,E), we denote the modulus of continuity of the function x, i.e., w (x,¢) = sup [Ix (t) - x (s)l: t, s e [a,b], It - s I < 6]. If p(t, s) -- p : [a, b] x [e, a~ --+ R, then, the modulus of continuity of the function s ~ p(t, s) on the interval [c, d], for a fixed t e [a, b], is defined as
w ( p ( t , . ) , E ) = sup[Ip(t,u ) - p ( t , v ) [
: u, v E [c,~, l u - v l
<_ z].
In the similar way, we define the modulus w(p(., s), 6). 3. P R O P E R T I E S VOLTERRA-STIELTJES
OF NONLINEAR INTEGRAL OPERATOR
Let I be a bounded and closed interval in R. For convenience, we assume that I = [0, 1]. In this section, we will investigate the nonlinear integral operator of Volterra-Stieltjes type having the form,
(Vx) (t) --
/o'
v (s, x (s)) dsg (t, s),
t e I.
(1)
In our further considerations, we shall always assume t h a t the following conditions are satisfied. (i) g : I x I --* R and, for every tl, t2 E I, such that tl < t2, the function s --* g(t2, s) - g ( t l , s) is nondecreasing on the interval I. (ii) g(0, s) = 0, for any s E I. (iii) v : I x R -~ R is a continuous function such that there exist a continuous function a : I ~ I and a continuous and nondecreasing function ~ : ]~+ --+ R+, for which the following inequality holds,
for t E I and x E R. REMARK 1. Observe that Assumptions (i) and (ii) imply that the function s ~ g(t, s) is nondecreasing on the interval I, for any fixed t E I (cf. Remark 3 in [13]). Indeed, putting in (i), t2 -- t, tl = 0 and keeping in mind (ii), we obtain the desired conclusion. From this observation, it follows immediately that, for every t E I, the function s ~ g(t, s) is of bounded variation on I. Such a condition was assumed in [13], but we showed above that it is superfluous. REMARK 2. In paper [13], the following condition was assumed in place of (iii). (iii') v : I × ]~ --, R is continuous and, such that there exist a function a = a(t) continuous on I and a constant b > 0 with the property,
Iv (t,x)l < a(t)+ blxl, for t E ~ and x E I. Obviously, Assumption (iii) is less restrictive than (iii'). Moreover, we show later that (iii) admits several natural realizations. Now, we prove a few results about the properties of the integral operator V defined by (1). We start with the following theorem.
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J. BANA$ANDD. O'REGAN
THEOREM 1. Assume that Assumptions (i)-(iii) are satisfied. Then, for every function x E C(I), the function V z is of bounded variation on I. PROOF. Observe first that taking into account our assumptions, in view of Lemma 3 and Remark 1, we infer t h a t the Volterra-Stieltjes operator V defined by (1) is well-defined on the space C(I). Next, fix a partition 0 = to < tl < ... < t= = 1 of the interval I = [0, 1]. Then, in view of Lemma I and Remark 1, we get 7%
I(Vx) (t~) - ( V x ) (t~-l)l i=l
1/o'
<
L
v ( s , x ( s ) ) d s g ( t i , s) -
i=l
+
~li[ '-1v ( s , x ( s ) ) d s g ( t i ,
s) -
i=l
v ( s , x ( s ) ) d s g ( t i , s)
S[*-'v ( s , x ( s ) ) d s g ( t i _ l , S ) I
v (s, x (s)) dsg (t~, s) +
= i=l
v (s, x (s)) d, [g (ti, s) - g (ti-1, s)] i=1
-i
i=1 Jti--I
+
I
tt~$1_l
I.(s,x(s))ld~ i=l
[g(t.u) -g(ti_~,u)] . u=O
Consequently, keeping in mind the above estimate, Lemma 2, and Remark 1, we obtain,
7%
I(v~) (t~) - (v~) (t,-~)l i----1
<
±f i=1
Iv(s,x(s))ldsg(t.s)+
tl-i
Iv(s,z(s))lds[g(t.s)-g(t~-l,s)] ~=i
5; a(s)~(lx(s)[)d~g(t~,s)+ D/oa(s)~v(lx(s)l)d.[g(t.s)-g(t~-~,s)]
<_
i=l
t~--I
_.
=
Ilall ~ (11~11){ ~i=1[g (ti'
ti) -- g (tl,
ti--1)]
+ ~ b (t~, t~_~) - 9 (t~_i, t~_~) - g (t~, o) + g (t~_~, o)1 i=1
= Ilall ~o (llxll)
{ ~=1[9(ti,ti)
)
- g ( t i - l , t i - 1 ) - g(t~, O) + g(ti-1,0)] }
= Ilall ~ (llxll) [g (1,1) - g (1, 0)] < c¢. This completes the proof. For our further purposes, we will need the following Lemma.
Volterra-Stieltjes Integral Operators
339
LEMMA 4. Assume that the function g(t, s) = g : I x I --* R satisfies Assumption (i). Then, for every Sl, s2 E I, such that Sl < s2, the function t --* g(t, s2) - g ( t , sl) is nondecreasing on the interval I. We omit easy proof requiring only a few simple calculations. Now, we prove a generalization of the result contained in Theorem 1. THEOREM 2. Suppose that there are satisfied the assumptions of Theorem i. Moreover, assume that the values of the function v are nonnegative i.e., v : I x R --* R+. Then, for every function x E C(I), the function V x is nondecreasing on I. PROOF. Fix arbitrarily tl, t2 E I, tl < t2. Observe that, for every arbitrarily fixed function x E C(I), the following inequality holds to be true,
(2)
~ i 2 v(s, x(s))dsg(t2, s) >_ O.
Indeed, take a partition tl --- so < Sl < ... < sn -- t2 of the interval [tl,t2]. Further, let us choose arbitrarily points ci E [si-l,si] (i = 1, 2 , . . . , n). Consider the Pdemann-Stieltjes sum corresponding to the integral from the inequality (2), i.e., n
on = ~
i=l
~(c,, x(c,))[g(t~, s,) - g(t2, s,_l)].
Since v(ci, x(ci)) >_ 0 and g(t2, si) >_ g(t2, si-1), for i = 1, 2 , . . . , n (see Remark 1), we infer that an _> 0, for n E N. This shows that the inequality (2) is true. In what follows, let us take an arbitrary partition 0 = s0 < Sl < -.- < s,~ = tl of the interval [0, tl] and choose arbitrarily points ci E [S~-l, s~l (i = 1, 2 , . . . , n). Then, keeping in mind Lemma 4 and (2), we deduce the following inequalities, }2
v (e~. x (~)) [, (tl. s.) - g (tl. S~_l)] 4=1 n
< ~v
(c~, x (c,)) [g (t2, s,) - g (t2, s,_ll]
i=l °
_< y~'v (e. x (~)) [g (t2. s,) - g (t~, s,_l)] +
v (s,~ (s)) e~g (t2. s).
i=l
Hence, we derive the following inequality,
~0tl v (s, x (s)) dsg (tl, s) < ~0tl v (s, x (s)) d,g (t2, s) + ~i 2v (s, x (s)) dsg (ts, s) 0t2 v (s, x (s) ) dsg (t2, s) Thus, the proof is complete.
|
THEOREM 3. Suppose there are satisfied the assumptions of Theorem 1. Moreover, assume that the function s --* g(1, s) is continuous on I and the function t --* g(t, s) is continuous on I, for any fixed s E I. Then, the operator V acts continuously from the space C ( I ) into itself. PROOF. Observe that the operator V can be written as the composition V = G F of the superposition operator, (Fx) (t) = v (t, x (t)),
340
J. BANA§ANDD. O'REGAN
and the linear Volterra-Stieltjes integral operator G defined by the formula,
(Gx) (t) =
x (8) dsg (t, s).
Further, let us notice t h a t in view of Lemma 3 and Assumption (iii) the superposition operator F maps continuously the space C(I) into itself. Thus, it is sufficient to show that the operator G acts continuously from C(I) into C(I). To do this, fix arbitrarily x E C(I). Next, fix ~ > 0 and take tl,t2 E I, tl < t2, such that It2 - tl[ _< z. Then, we obtain
/?
I(Gz) (t2) - (Gz) (tl)l <_
<_
x (s) dsg (t2, s) -
/?
x (s) dsg (t2, s)
x (s) d,g (t2, s) +
I
x (s) d8 [g (t2, s) - g (tl,
•
Hence, applying Lemmas 1 and 2, we get I(Gx) (t2) - (ax)(tl)l
<_
[x(s)[ds
1g(t2'u)
+
u=$1
= Ilxll
[x(s)lds
~=o~/[g(t2'u)-g(tl'u)]
~t~---0
d~g (t2, 8) + Llxll
d~ [g(t~,s) - g (t~,s)]
= Ilxll {[g (t2, t2) - g (t2, tl)] + [g (t~, tl) - g (tl, t~) - g (t2, O) + g (tl, 0)]}
_< IIx 11{ [g (t2, t2) - g (t2, t~)] + Ig (t2, t~) - g (t l, t l)l + Ig (t2, 0) - g (t~, 0) 1}. Hence, in view of Remark 1 and Lemma 4, we arrive to the following estimate,
I(G:c) (t2) - (az) (t~)l _< Ilxll {[g (1, t2) - g (1, t~)] + Ig (t~, t~) - g (t,, t~)l + Ig (t2, 0) - g (t~, 0)1} <-- II~ll {~ (g (1, "/, ~) + ~ (g (', t~), s) + ~ (g (., 0), e)}, where the symbols w(g(1, .),e), co(g(-, tl), s), w(g(., 0), e) were introduced in Section 2. From the above estimate, we conclude that Gx C C(I). In order to show that G is continuous, fix x E C(I). Then, in view of Lemmas 1 and 2, we have
I(Gx)(t)[ <_
Ix(s)ld~
g(t,u)
<_ llxH
d~ - - g(t,u) u=O
= ]lxll
d,g (t, s) < Hxl[
d~g (t, s) = I]xH [g (t, 1) - g (t, 0)].
Hence, in virtue of Lemma 4, we get
I(ax) (t)l <_ llxll [g (1, 1) - g (1, 0)1, and consequently,
IlOxll <_ Ilxll [g (1, 1) - g (1, 0)]. The above inequality shows that G transforms continuously the space C(I) into itself. The proof is complete. | REMARK 3. Theorem 3 remains true if we assume that the hypotheses of Theorem 1 are satisfied, and additionally, the following one as well. (iv) The functions t ~ g(t, t) and t ~ g(t, 0) are continuous on the interval I.
Volterra-Stieltjes Integral Operators
341
Indeed, evaluating analogously as in the proof of Theorem 3, we get,
I(Oz) (t=) - (Oz) (tl)l < Ilzll {[g (t=, t=) - g (t=, t,)] + [g (t=, t~) - g (t~, t,) - g (t=, 0) + (tl, 0)]} =
Ilzll {[g
(t=,
t=) -
g (t,, tl) ] - [g (t2, 0) - g (tl, 0)]}
_< Ilzll {Ig (t2, t=) - g (tl, t~)l + Ig (t2, o) - g (t,, o)1}. From the above estimate we obtain our assertion. Our next result shows that the Volterra-Stieltjes integrM operator V is Mso compact under suitable assumptions. THEOREM 4. Under the assumptions of Theorem 3, the operator V is compact on the space
C(I). PROOF. Let us take a bounded subset X of the space C(I) and choose an arbitrary element x E X. Next, fix e > 0 and t l , t l E I, tl < t2, such that It2 --tll < e. Then, arguing similarly as in the proof of Theorem 3, we obtain the following estimate,
I(Vz) (t=) - (vz) (t~)l ~ I1~11~ (Hzll) {~o (g (1, .), e) + ¢o (g (., tl), ~) + w (g (-, 0), ~)}. Hence, we infer that the functions of the set V X are equicontinuous on the interval I. On the other hand, for an arbitrary function x c X , we have
I(vm)(x)l <
/:
[v(s,x(s))ld,
-< I1~11~(llxll)
i
(20) g(t,u)
d,g(t,s) < Ilall~o(llxll)
jo1d , g ( t , s )
= I1~11~o (llxll) [g (t, 1) - g (t, o)]. In view of Lemma 4, the above inequality yields
](Vx) (t)l < [laH ~o(HxH)[g (1,1) - g (1, 0)]. Consequently, we obtain
Ilvzll ~ I1~11~ (llXrl) [g (1,1) - g (1,0)], where we have denoted
IIXll = s,,p [llzll : x e x]. In view of the above estimate, we infer that the set V X is bounded in the space C(I). Now, applying Arzdla criterion for compactness in the space U(I), we deduce that the set V X is relatively compact in this pace. Thus, the proof is complete. | REMARK 4. The conclusion of Theorem 4 remains true if we assume the hypotheses formulated in Remark 3. In what follows, we provide a few examples illustrating the above obtained results. EXAMPLE 1. Let a, b : I --+ I be nondecreasing functions on I, such that a(0) = 0. Assume additionally that b is differentiable and a is continuous on the interval I. P u t g(t, s) = a(t). b(s). Then it is easy to check that the function g(t, s) satisfies Assumptions (i), (ii), and the additional assumption formulated in Theorem 3 (cf. also Remark 3). In this case, the Volterra-Stieltjes operator V defined by (1) has the form,
(Vx) (t) =
/0
a (t) v (s, x (s)) b' (s) ds = a (t)
/0
v (s, x (s)) b' (s) ds.
342
J. BANAS AND D. O'REGAN
EXAMPLE 2. Consider the function g : I x I --~ l~ defined by the formula,
g(t,s)
~tln t+s -~-
(
t
0,
for t e [0,1],
seI,
'
fort=0,
sEI.
It can be easily seen that the function g(t, s) satisfies Assumptions (i), (ii), and the assumption given in Theorem 3 (see also Remark 3). In this case, the Volterra-Stieltjes integral operator V has the form,
( v ~ ) (t) =
t + s v (s' ~ (s)) d~
end creates the Volterra counterpart of the well-known Chandrasekhar integral operator defined by the formula,
(Hx) (t) = ~ l t ~ o
(s) x (s) ds,
(cf. [8,9,12,141). EXAMPLE 3. Let us take a function a E C(I) and the function T : R+ ~ R+ defined by the formula ~(r) = l + r . Denote b = I]all = max[la(t)l : t E I]. Then, the estimate from Assumption (iii) has the form, [v (t, x)L < ~ (t) (1 + Ixl) < a (t) + b IxL. Notice that such an assumption was accepted in paper [13]. Thus, we see that this assumption is a special case of Assumption (iii) accepted in this paper. EXAMPLE 4. Similarly as above, take a function a(t) E C(I). Assume that the function T : N[+ --~ R+ has the form ~(r) = 1 + r a, where a > 0 is a fixed number. Then, the inequality from Assumption (iii) has the form,
Iv (t, ~)l < a (t) (1 + IxF) < a (t) + b I x F , where b = II"lt. EXAMPLE 5. Now, let us take a(t) E C(I) and consider the function to : ]~+ --* ]~+ defined by the formula ~o(r) = exp r. Then, the estimate from (iii) has the form, Iv (t, x)l < a (t) exp Ix[. The above given examples show that the assumptions accepted in Theorems 1-4 admit several natural realizations.
4. SOLVABILITY OF VOLTERRA-STIELTJES INTEGRAL EQUATION In this section, we show that the results obtained in the previous section can be applied in the study of the solvability of the following nonlinear integral equation of Volterra-Stieltjes type,
x (t) ----p (t) +
I'
v (s, x (s)) dsg (t, s),
(3)
where t e I = [0,1]. Obviously, the above equation may be written in the form x = p+ Vx, where V is the VolterraStieltjes operator defined by (1). We formulate below two existence results concerning the equation (3).
Volterra-Stieltjes Integral Operators
343
THEOREM 5. S u p p o s e p = p(t) E C ( I ) and the functions v and g satisfy Assumptions (i)-(iii) and the assumption formulated in Theorem 3 or Remark 3. Moreover, assume that there exists a positive solution ro of the inequality
Ilpll + Ilall (9 (1,1) - 9 (1, o))
(r) < r.
Then, the equation (3) has at least one solution in the space C(I). PROOF. Consider the operator S defined on the space C ( I ) by the formula, S x = p + Vx, where V is defined by (1). In view of Theorem 3, we see that S transforms continuously the space C ( I ) into itself. Further, consider the ball B~o = B(0, r0) in the space C(I), where r0 is a number appearing in our assumptions. For an arbitrarily fixed function, x E B~ o and, for t E I, in virtue of Lemmas 1, 2, and 4, we get I(Sx)(t)r < Ip(t)l + <
[[ptl +
v(s,z(s))d~g(t,s) [v(s,x(s))Ids
-- g(t,u) u~O
< Ilpl[+
a(s)~@(s)l)ds U-~0
< [IPl[ +[lal[ ~ (llxll)
/o
dsg (t, s) < IIPl[ + [lal] ~ (to)
< IlP[[ + [[all ~ (to) [g (t, 1) - g (t, 0)]
/0
dsg (t, s)
--< [[PII + [tail (g (1, 1) -- g (1, 0)) ~ (r0) _< ro. The above inequality shows that S transforms the ball B~o into itself. Obviously, in view of Theorem 4 the operator S is completely continuous. Thus, applying the Schauder fixed-point principle, we complete the proof. | Our next existence result is based mainly on Theorem 2. THEOREM 6. Assume that there are satisfied the hypotheses of Theorem 5 and, in addition, the function v is nonnegative (i.e., v : I x R ~ R+ ) and p(t) is a nondecreasing function on I. Then, the equation (3) has at least one solution in the space C ( I ) being a nondecreasing function on the interval I. The proof can be carried over in the same way as the proof of Theorem 5. Moreover, we use the fact that the image SB~ o of the ball B~o under the operator S contains only nondecreasing function on I, which is a simple consequence of Theorem 2. To complete the proof, it is sufficient to notice that the solutions of the equation (3) belong to the set SB~ o. REFERENCES N. Dunford and J. Schwartz, Linear Operators I, Int. Publ., Leyden, (1963). K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, (1985). C. Corduneanu, Integral Equations and Applications, Cambridge Univ. Press, Cambridge, (1991). D. O'Regan and M. Meehan, Existence Theory for Nonlinear Integral and Integrodifferential Equations, Kluwer Academic, Dordrecht, (1998). 5. R.P. Agarwal, D. O'Regan and P.J.Y. Wong, Positive Solutions of Differential, Difference and Integral Equations, Kluwer Academic Publishers, Dordrecht, (1999). 6. P.P. Zabrejko, A.I. Koshelev, M.A. Krasnosel'skii, S.G. Mikhlin, L.S. Rakovschik and V.J. Stecsenko, Integral Equations, Noordhoff, Leyden, (1976). 1. 2. 3. 4.
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