on volterra integral equations in banach spaces proof. put

Funkcialaj Ekvacioj, 20 (1977) 247-258

On Volterra Integral Equations in Banach Spaces

By

Stanislaw SZUFLA

(A. Mickiewicz University, Poland)

Let $J=[0, a]$ be a compact interval in $R$ , and let $E$ be a Banach space withthe norm $||¥cdot||$ . Denote by $C=C(J, E)$ the Banach space of all continuous functions$u:J¥rightarrow E$ with the norm $||u||_{c}=¥sup$ $¥{||u(t)||:t¥in J¥}$ . For the case where $g$ is abounded continuous function of $(t, s, x)$ on $¥{0¥leq s¥leq t¥leq a¥}¥times R^{n}$, M. Hukuhara [2] hasshown that the $¥epsilon$ -approximate solutions set of the Volterra integral equation

(1) $x(t)=f(t)+¥int_{0}^{t}g(t, s, x(s))ds$

is connected in $C(J, R^{n})$ . In this paper we prove the connectedness of the set$¥{x ¥in C:||x-F(x)||_{c}<_{¥epsilon}¥}$ for a certain class of non-linear mappings $F:C¥rightarrow C$ . As aneasy application of our result, we show that Hukuhara’ $¥mathrm{s}$ theorem is also true when$g$ satisfies the Caratheodory conditions. Moreover, this note contains an existencetheorem for the equation (1) in Banach spaces.

1. First we shall show the following main theorem.

Theorem 1. Assume that $F:C¥rightarrow C$ is a continuous mapping such that1o $F(C)$ is an equiunijormly continuous set of mappings $J¥rightarrow E$ ;2o there exists $b¥in E$ such that $F(x)(0)=b$ for every $x¥in C$ ;3o $x|[0, d]=y|[0, d]¥Rightarrow F(x)|[0, d]=F(y)|[0, d]$

for every $x$ , $y¥in C$ and $0<d¥leq a$ .Then for any $¥eta>0$ the set

$S_{¥eta}=¥{u¥in C:u(0)=b, ||u-F(u)||_{c}<_{¥eta}¥}$

is non-empty and connected.

Proof. Put

$ w(h)=¥sup$ $¥{||F(x)(t)-F(x)(s)||:x¥in C, t, s¥in J, |t-s|¥leq h¥}$ .

From 1o it follows that $¥lim_{h-0+}w(h)=0$ .Let $x¥in S_{¥eta}$ . Choose $¥epsilon>0$ such that $||x-F(x)||_{c}+w(¥epsilon)<_{¥eta}$ . For any $p$ , $0¥leq p¥leq a$ ,

248 S. SZUFLA

we define a function $y(¥cdot, p)$ by the formula

(2) $y(t, p)=z_{k}(t, p)$ for $0¥leq t¥leq a$ ,

where $k$ is an integer $¥geq 1$ such that $ p+(k-1)¥epsilon¥leq a<p+k¥epsilon$, and $z_{i}(t, p)$ , $i=1$ , $¥cdots$ , $k$ ,are defined inductively by

$z_{1}(t, p)=¥left¥{¥begin{array}{l}x(t)¥mathrm{f}¥mathrm{o}¥mathrm{r}0¥leq t¥leq p¥¥x(p)¥mathrm{f}¥mathrm{o}¥mathrm{r}p¥leq t¥leq a¥end{array}¥right.$

and

$z_{i+1}(t, p)=¥left¥{¥begin{array}{l}z_{i}(t,p)¥mathrm{f}¥mathrm{o}¥mathrm{r}0¥leq t¥leq p+i¥epsilon¥¥ x(p)-F(x)(p)+F(z_{i}(¥cdot,p))(t-¥epsilon)¥mathrm{f}¥mathrm{o}¥mathrm{r}p+i¥epsilon¥leq t¥leq a.¥end{array}¥right.$

Then $y(¥cdot, p)¥in C$ and

$y(t, p)=¥{xxx(p)-F(x)(p)+F(y(¥cdot, p))(t-¥epsilon)(t)¥mathrm{f}¥mathrm{o}¥mathrm{r}0¥leq t¥leq p(p)¥mathrm{f}¥mathrm{o}¥mathrm{r}p¥leq t¥leq¥min(a,p+¥epsilon)$

for $¥min(a, p+¥epsilon)¥leq t¥leq a$ .

Moreover

$||y(t, p)-F(y(¥cdot, p))(t)||=||x(t)-F(x)(t)||<_{¥eta}$ for $0¥leq t¥leq p$ ;$||y(t, p)-F(y(¥cdot, p))(t)||=||x(p)-F(x)(p)+F(y(¥cdot, p))(p)$

$-F(y(¥cdot, p))(t)||¥leq||x-F(x)||_{c}+w(¥epsilon)<_{¥eta}$

for $p¥leq t¥leq¥min(a, p+¥epsilon)$ ;$||y(t, p)-F(y(¥cdot, p))(t)||=||x(p)-F(x)(p)+F(y(¥cdot, p))(t-¥epsilon)$

$-F(y(¥cdot, p))(t)||¥leq||x-F(x)||_{c}+w(¥epsilon)<_{¥eta}$

for $¥min(a, p+¥epsilon)¥leq t¥leq a$ .

This proves that $y(¥cdot, p)¥in S_{¥eta}$ . We shall now show that

(3) $¥lim$ $||y(¥cdot, p)-y(¥cdot, q)||_{c}=0$ .$p-q$

Let $0¥leq p¥leq q¥leq a$ and $ q-p¥leq¥epsilon$ . We see that

$||y(t, p)-y(t, q)||=0$ for $0¥leq t¥leq p$ ;

$||y(t, p)-y(t, q)||=||x(p)-x(t)||$ for $p¥leq t¥leq q$ ;

$||y(t, p)-y(t, q)||=||x(p)-x(q)||$ for $q¥leq t¥leq¥min(a, p+¥epsilon)$ .

Moreover,

On Volterra Integral Equations in Banach Spaces 249

a) if $ 0¥leq p¥leq q¥leq a-¥epsilon$ , then

$||y(t, p)-y(t, q)||=||x(p)-F(x)(p)+F(y(¥cdot, p))(t-¥epsilon)-x(q)||$

$=||x(p)-x(q)+F(y(¥cdot, p))(t-¥epsilon)-F(y(¥cdot, p))(p)||$

$¥leq||x(p)-x(q)||+w(q-p)$ for $ p+¥epsilon¥leq t¥leq q+¥epsilon$ ,

and

$||y(t, p)-y(t, q)||=||x(p)-F(x)(p)+F(y(¥cdot, p))(t-¥epsilon)$

$-x(q)+F(x)(q)-F(y(¥cdot, q))(t-¥epsilon)||¥leq||x(p)-x(q)||+w(q-p)$

$+||F(y(¥cdot, p))(t-¥epsilon)-F(y(¥cdot, q))(t-¥epsilon)||$ for $q+¥epsilon¥leq t¥leq a$ .

b) if $0¥leq p¥leq a-¥epsilon¥leq q¥leq a$ , then

$||y(t, p)-y(t, q)||=||x(p)-F(x)(p)+¥Gamma’(y(¥cdot, p))(t-¥epsilon)-x(q)||$

$¥leq||x(p)-x(q)||+w(q-p)$ for $p+¥epsilon¥leq t¥leq a$ .

c) if $a-¥epsilon¥leq p¥leq q¥leq a$ , then

$||y(t, p)-y(t, q)||=||x(p)-x(q)||$ for $q¥leq t¥leq a$ .

From this we deduce that if $0¥leq p$ , $q¥leq a$ and $|q-p|¥leq¥epsilon$ , then

(4) $||y(t, p)-y(t, q)||¥leq r(x, |p-q|)+w(|p-q|)$ for $ 0¥leq t¥leq¥epsilon$ ,

and

$||y(t, p)-y(t, q)||¥leq r(x, |p-q|)+w(|p-q|)$(5)

$+||F(y(¥cdot, p))(t-¥epsilon)-F(y(¥cdot, q))(t-¥epsilon)||$ for $¥epsilon¥leq t¥leq a$ ,

where $ r(x, h)=¥sup$ $¥{||x(t)-x(s)||:t, s¥in J, |t-s|¥leq h¥}$ . From (4) it is clear that

$¥lim$ $y(t, p)=y(t, q)$ uniformly on $[0, ¥epsilon]$ .$p-q$

By 3o and the continuity of $F$ from this it follows that

$¥lim$ $||F(y(¥cdot, p))(t-¥epsilon)-F(y(¥cdot, q))(t-¥epsilon)||=0$ uniformly on $[¥epsilon, 2¥epsilon]$ ,$p-q$

and hence, by (5),

$¥lim$ $y(t, p)=y(t, q)$ uniformly on $[0, 2¥epsilon]$ .$p-q$

By repeating this argument we find $¥lim_{p-q}y(t, p)=y(t, q)$ uniformly on $[0, n¥epsilon]$ , $n=$

$1,2$ , $¥cdots$ , $[a/¥epsilon]$ , which proves (3).Choose a positive number $¥beta$ such that $ w(¥beta)<¥eta$ and $¥beta<a$ . For any $¥epsilon$ , $0<_{6}$

250 S. SZUFLA

$<¥beta$ , we define a function $v(¥cdot, ¥epsilon)$ by the formula

$v(t, ¥epsilon)=w_{k}(t, ¥epsilon)$ for $0¥leq t¥leq a$ ,

where $k$ is an integer $¥geq 1$ such that $(k-1)¥epsilon¥leq a<k¥epsilon$ , and $w_{i}(t, ¥epsilon)$ , $i=1$ , $¥cdots$ , $k$ , aredefined inductively by

$w_{1}(t, ¥epsilon)=b$ for $0¥leq t¥leq a$

and

$w_{i+1}(t, ¥epsilon)=¥left¥{¥begin{array}{l}w_{i}(t,¥epsilon)¥mathrm{f}¥mathrm{o}¥mathrm{r}0¥leq t¥leq i¥epsilon¥¥ F(w_{i}(¥cdot,¥epsilon))(t-¥epsilon)¥mathrm{f}¥mathrm{o}¥mathrm{r}i¥epsilon¥leq t¥leq a.¥end{array}¥right.$

Obviously $v(¥cdot, ¥epsilon)$ is continuous on $J$ and

$v(t, ¥epsilon)=¥left¥{¥begin{array}{l}b¥¥F(v(¥cdot,¥epsilon))(t-¥epsilon)¥end{array}¥right.$ $¥mathrm{f}¥mathrm{f}¥mathrm{o}¥mathrm{r}¥epsilon¥leq t¥leq a¥mathrm{o}¥mathrm{r}0¥leq t¥leq¥epsilon$

.

Moreover $v(¥cdot, ¥epsilon)¥in S_{¥eta}$ , because

$||v(t, ¥epsilon)-F(v(¥cdot, ¥epsilon))(t)||=||b-F(v(¥cdot, ¥epsilon))(t)||$

$=||F(v(¥cdot, ¥epsilon))(0)-F(v(¥cdot, ¥epsilon))(t)||¥leq w(¥epsilon)<¥eta$ for $ 0¥leq t¥leq¥epsilon$ ,

and

$||v(t, ¥epsilon)-F(v(¥cdot, ¥epsilon))(t)||$

$=||F(v(¥cdot, ¥epsilon))(t-¥epsilon)-F(v(¥cdot, ¥epsilon))(t)||¥leq w(¥epsilon)<_{¥eta}$ for $¥epsilon¥leq t¥leq a$ .

Furthermore, if $ 0<_{¥epsilon}<¥delta<¥beta$ , then

$||v(t, ¥delta)-v(t, ¥epsilon)||=0$ for $ 0¥leq t¥leq¥epsilon$ ,$||v(t, ¥delta)-v(t, ¥epsilon)||=||b-F(v(¥cdot, ¥epsilon))(t-¥epsilon)||$

$=||F(v(¥cdot, ¥epsilon))(0)-F(v(¥cdot, ¥epsilon))(t-¥epsilon)||¥leq w(¥delta-¥epsilon)$ for $¥epsilon¥leq t¥leq¥delta$ ,

and

$||v(t, ¥delta)-v(t, ¥epsilon)||=||F(v(¥cdot, ¥delta))(t-¥delta)-F(v(¥cdot, ¥epsilon))(t-¥epsilon)||$

$¥leq||F(v(¥cdot, ¥delta))(t-¥delta)-F(v(¥cdot, ¥delta))(t-¥epsilon)||$

$+||F(v(¥cdot, ¥delta))(t-¥epsilon)-F(v(¥cdot, ¥epsilon))(t-¥epsilon)||$

$¥leq w(¥delta-¥epsilon)+||F(v(¥cdot, ¥delta))(t-¥epsilon)-F(v(¥cdot, ¥epsilon))(t-¥epsilon)||$ for $¥delta¥leq t¥leq a$ .

Thus for each $¥epsilon$ , $¥delta$ , $0<_{¥epsilon}$ , $¥delta<¥beta$ , we have

$||v(t, ¥delta)-v(t, ¥epsilon)||¥leq w(|¥delta-¥epsilon|)$ for $ 0¥leq t¥leq¥epsilon$


and

$||v(t, ¥delta)-v(t, ¥epsilon)||¥leq w(|¥delta-¥epsilon|)+||F(v(¥cdot, ¥delta))(t-¥epsilon)-F(v(¥cdot, ¥epsilon))(t-¥epsilon)||$ for $¥epsilon¥leq t¥leq a$ .

Hence, using the same argument as in the proof of (3), we can prove that

$¥lim_{¥delta-¥mathrm{e}}||v(¥cdot, ¥delta)-v(¥cdot, ¥epsilon)||_{c}=0$ .

From this we conclude that the set $V=¥{v(¥cdot, ¥epsilon):0<_{¥epsilon}<¥beta¥}$ is connected in C.Further, for any $x¥in S_{¥eta}$ we choose $¥epsilon=¥epsilon_{x}$ such that $ 0<_{¥epsilon}<¥beta$ and $||x-F(x)||_{c}+$

$w(¥epsilon)<_{¥eta}$ . Put $T_{x}=¥{y(¥cdot, p):0¥leq p¥leq a¥}$ , where $y(¥cdot, p)$ denotes the function definedby (2). From (3) it follows that $T_{x}$ is a connected subset of $C$ . As $y(¥cdot, 0)=$

$v(¥cdot, ¥epsilon)¥in V¥cap T_{x}$ , the set $V¥cup T_{x}$ is connected, and therefore the set $W=¥bigcup_{x¥in S¥eta}T_{x}$

$¥cup V$ is connected in C. Moreover $S_{¥eta}¥subset W$ , because $x=y(¥cdot, a)¥in T_{x}$ for every $x¥in S_{¥eta}$ .

On the other hand $W¥subset S_{¥eta}$ , since $T_{x}¥subset S_{¥eta}$ and $V¥subset S_{¥eta}$ . Consequently $S_{¥eta}=W$, whichends the proof.

2. Now we consider the Volterra integral equation.

(1) $x(t)=i(t)+¥int_{0}^{t}g(t, s, x(s))ds$ ,

where $t¥in J$ and $x$ , $f$ and $g$ are functions with values in E.Assume thatI. $f:J¥rightarrow E$ is a continuous function,

$¥mathrm{I}¥mathrm{I}$ . $(t, s, x)¥rightarrow g(t, s, x)$ is a function of the set $¥{0¥leq s¥leq t¥leq a, x¥in E¥}$ into $E$ , whichsatisfies the following conditions:

1o for each fixed $x¥in E$ and $t¥in J$ the function $s¥rightarrow g(t, s, x)$ is strongly L-measurable on $[0, t]$ ;

2o for each fixed $t$ , $s$ , $0¥leq s¥leq t¥leq a$ , the function $x¥rightarrow g(t, s, x)$ is continuous on$E$ ;

3o there exist real-valued functions $(¥tau, t, s)¥rightarrow r(¥tau, t, ¥mathrm{s})$ and $(t, s)¥rightarrow m(t, s)$

$(0¥leq s¥leq t¥leq¥tau¥leq a)$ such that(i) for each fixed $t$ , $¥tau$ the functions $s¥rightarrow r(¥tau, t, ¥mathrm{s})$ and $s¥rightarrow m(t, s)$ are $¥mathrm{L}$ -integrable

on $[0, t]$ ;(ii) $¥sup$ $¥{||g(¥tau, s, x)-g(t, s, x)||:x¥in E¥}¥leq r(¥tau, t, ¥mathrm{s})$ and $¥sup$ $¥{||g(t, s, x)||:x¥in E¥}$

$¥leq m(t, s)$ ;

(iii) $¥lim_{¥tau-t-0+}[¥int_{t}^{¥tau}m(¥tau, s)ds+¥int_{0}^{t}r(¥tau, t, s)ds]=0$ for fixed $t$ or $¥tau$ .

A function $x:J¥rightarrow E$ is called an $¥epsilon$-approximate solution of (1) if $x(0)=f(0)$ and

$||x(t)-f(t)-¥int_{0}^{t}g(t, s, x(s))ds||<_{¥epsilon}$ for every $t¥in J$ .

252 S. SZUFLA

Theorem 2. Under the above assumptions the set of all $¥epsilon-$approximate solu-tions of (1) is non-empty and connected in $C$ .

Proof. It is sufficient to show that the mapping $F$ , defined by the formula

$F(x)(t)=f(t)+¥int_{0}^{t}g(t, s, x(s))ds$ for $x¥in C$ and $t¥in J$ ,

satisfies the assumptions of Theorem 1.Let us fix $t_{0}¥in J$ . For any $x¥in C$ and $t¥in J$ we have

$||F(x)(t)-F(x)(t_{0})||¥leq||f(t)-f(t_{0})||$

$+¥int_{t¥mathrm{o}}^{t}||g(t, s, x(s))||ds+¥int_{0}^{t¥mathrm{o}}||g(t, s, x(s))-g(t_{0}, s, x(s))||ds$

$¥leq||f(t)-f(t_{0})||+¥int_{t¥mathrm{o}}^{t}m(t, s)ds+¥int_{0}^{t¥mathrm{o}}r(t, t_{0}, s)ds$ when $t_{0}¥geq t$ ,

and

$||F(x)(t)-F(x)(t_{0})||¥leq||f(t)-f(t_{0})||+¥int_{t}^{t¥mathrm{o}}m(t_{0}, s)ds+¥int_{0}^{t}r(t_{0}, ¥mathrm{t}, s)ds$ when $t¥leq t_{0}$ .

By 3o and the continuity of $f$ from this it follows that

$¥lim_{t}$ $F(x)(t)=F(x)(t_{0})$ uniformly in $x¥in C$ ,

which proves that $F(C)$ is an equicontinuous subset of $C$ . This implies that the set$F(C)$ is equiuniformly continuous, since $J$ is compact.

Assume that $x_{n}$ , $x¥in C$ and $¥lim_{n-¥infty}||x_{n}-x||_{c}=0$ . Since

$¥lim_{n¥rightarrow¥infty}g(t, s, x_{n}(s))=g(t, s, x(s))$ and $||g(t, s, x_{n}(s))-g(t, s, x(s))||¥leq 2m(t, s)$

for each $0¥leq s¥leq t¥leq a$ , the Lebesgue theorem proves that $¥lim_{n¥rightarrow¥infty}¥int_{0}^{t}||g(t, s, x_{n}(s))-$

$g(t, s, x(s))||ds=0$, i.e. $¥lim_{n-¥infty}F(x_{n})(t)=F(x)(t)$ for every $t¥in J$ , and hence, by theequicontinuity of $F(C)$ , $¥lim_{n-¥infty}||F(x_{n})-F(x)||_{c}=0$ . This shows that $F$ is a contin-uous mapping $C¥rightarrow C$ .

Remark. The use of the time delay $ t-¥epsilon$ for constructing the $¥eta$-approximatesolutions is due to Caratheodory in the case of ordinary differential equations (seealso [5] $)$ .

Now we shall show an existence theorem for the equation (1). Assume thata non-negative real-valued function $(¥mathrm{t}, s, z)¥rightarrow h(t, s, z)$ defined on $¥{0¥leq s¥leq t¥leq a¥}¥times R^{+}$

is a Kamke function, i.e. $h$ satisfies the Caratheodory conditions 1o-3o and the


function identically equal to zero is the unique continuous solution of the inequality$u(t)¥leq¥int_{0}^{t}h(t, s, u(s))ds$ for $t¥in J$ satisfying the condition $u(0)=0$ .

Furthermore, for any bounded subset $A$ of $E$ we denote by $¥alpha(A)$ the infimumof all $¥epsilon>0$ such that there exists a finite covering of $A$ by sets of diameter $¥leq¥epsilon$ (cf.[4] $)$ . The number $¥alpha(A)$ is called the measure of non-compactness of the set $A$ .For properties of $¥alpha$ see [1].

Theorem 3. Assume that for any $t¥in J$ the inequality

(7) $¥varliminf_{¥delta 0}¥alpha(g(t, I_{S,¥delta}, X))¥leq h(t, s, ¥alpha(X))$ ,

where $I_{S,¥delta}=(s-¥delta, s+¥delta)¥cap[0, t]$ , is satisfied for almost every $s¥in[0, t]$ and for everybounded subset $X$ of E. Then there exists at least one solution of (1) defined on$J$ .

The proof of Theorem 3 is suggested by a paper of Pianigiani [7] concerningdifferential equations.

Proof. Let $F$ denote the mapping defined in the proof of Theorem 2. Thereexists a sequence $(u_{n})$ such that $u_{n}¥in C$ and

(8) $n¥varliminf¥infty||u_{n}-F(u_{n})||_{c}=0$.

Put $V=¥{u_{n} : n=1,2, --¥}$ . Denote by I the identity mapping on $C$ . From (8) itfollows that $(I-F)(V)$ is an equiuniformly continuous subset of $C$ . Since

(9) $V¥subset(I-F)(V)+F(V)$

and the set $F(V)$ is equiuniformly continuous, we see that the set $V$ is also equiuni-formly continuous.

Put $ w(V, ¥delta)=¥sup$ $¥{||u(t)-u(s)||:u¥in V, t, s¥in J, |t-s|¥leq¥delta¥}$ and $V(t)=¥{u(t)$ :$u¥in V¥}$ . Since

$||u(t)-v(t)||¥leq||u(s)-v(s)||+2w(V, |t-s|)$ for $u$ , $v¥in V$ and $t$ , $s¥in J$ ,

we see that

$|¥alpha(V(t))-¥alpha(V(s))|¥leq 2w(V, |t-s|)$ for $t$ , $s¥in J$ ,

and therefore the function $t¥rightarrow¥alpha(V(t))$ is continuous on /. Moreover, let $b=$

$¥sup$ $¥{||u(t)||:u¥in V, t¥in J¥}$ .Fix $t¥in J$ and $¥epsilon>0$ . Scorza-Dragoni’ $¥mathrm{s}$ theorem [10] proves that there is an

open set $J_{¥epsilon}¥subset[0, t]$ such that

254 S. SZUFLA

$¥mu(J_{¥mathrm{e}})<_{¥epsilon}$ ($¥mu-$the Lebesgue measure); $¥int_{J_{¥mathrm{g}}}m(t, s)ds<_{6}$ ;

the function $(s, u)¥rightarrow h(t, s, u)$ is uniformly continuous on $([0, t]¥backslash J_{¥epsilon})¥times[0,2b]$ ;the inequality $¥lim_{¥delta-0}¥alpha(g(t, I_{S,¥delta}, X))¥leq h(t, s, ¥alpha(X))$ is satisfied for every $s¥in[0, t]¥backslash J_{¥epsilon}$

and each bounded subset $X$ of $E$ . Let $T=[0, t]¥backslash J_{¥mathrm{e}}$ . Putting $¥int_{Z}g(t, s, V(s))ds=$

$¥{¥int_{Z}g(t, s, u(s))ds:u¥in V¥}$ , we see that

$F(V)(t)¥subset f(t)+¥int_{T}g(t, s, V(s))ds+¥int_{J_{¥mathrm{e}}}g(t, s, V(s))ds$,

and, consequently,

(10) $¥alpha(F(V)(t))¥leq¥alpha(¥int_{T}g(t, s, V(s))ds)+¥alpha(¥int_{J_{¥xi}}g(t, s, V(s))ds)$ .

Choose $¥delta>0$ in such a way that $|s_{1}-s_{2}|<¥delta$. $|u_{1}-u_{2}|<¥delta$, $s_{1}$ , $s_{2}¥in T$ , $u_{1}$ , $u_{2}¥in[0,2b]$ ,implies $|h(t, s_{1}, u_{1})-h(t, s_{2}, u_{2})|<_{¥epsilon}$ , and choose $¥eta$ , $ 0<¥eta<¥delta$ , such that $|s_{1}-s_{2}|<¥eta$ ,$s_{1}$ , $s_{2}¥in T$ , implies $|¥alpha(V(s_{1}))-¥alpha(V(s_{2}))|<¥delta$ . We divide the interval $[0, t]$ into $n$ parts$0=t_{0}<t_{1}<¥cdots<t_{n}=t$ in such a way that $ t_{i}-t_{i-1}<¥eta$ for $i=1$ , $¥cdots$ , $n$ . Let $T_{i}=$

$[t_{i-1}, t_{i}]¥backslash J_{¥mathrm{e}}$ . Because

$¥int_{T}g(t, s, V(s))ds¥subset¥sum_{i=1}^{n}¥int_{¥tau_{i}}g(t, s, V(s))ds$,

we get

(11) $¥alpha(¥int_{T}g(t, s, V(s))ds)¥leq¥sum_{i=1}^{n}¥alpha(¥int_{¥tau_{i}}g(t, s, V(s))ds)$ .

Set $V_{i}=¥{v(s):v¥in V, s¥in T_{i}¥}$ . For any $s¥in T_{i}$ there exists $d_{s}>0$ such that

(12) $¥alpha(g(t, I_{s,a_{s}}, V_{i}))¥leq h(t, s, ¥alpha(V_{i}))+¥epsilon$.

Since the family $¥{I_{s,l_{S}}( : s¥in T_{i}¥}$ covers $T_{i}$ , there is a finite cover $I_{s_{1},d_{s_{1}}}$ , $¥cdots$ , $I_{s_{m},a_{s_{m}}}$ of$T_{i}$ . Choose sets $P_{1}$ , $¥cdots$ , $P_{m}$ such that $P_{k}¥subset I_{s_{k},a_{Sk}}$ , $¥bigcup_{k=1}^{m}P_{h}=T_{i}$ , $ P_{j}¥cap P_{k}=¥emptyset$ for$j¥neq k$ . Then

$¥int_{¥tau_{i}}g(t, s, V(s))ds¥subset¥sum_{k=1}^{m}¥int_{P_{h}}g(t, s, V(s))ds¥subset¥sum_{h=1}^{m}¥mu(P_{k})¥overline{¥mathrm{c}¥mathrm{o}¥mathrm{n}¥mathrm{v}}g(t, P_{k}, V_{i})$ .

By (12) and the corresponding properties of $¥alpha$ from this it follows that


$¥alpha(¥int_{¥tau_{i}}g(t, s, V(s))ds)¥leq¥sum_{h=1}^{m}¥mu(P_{k})¥alpha(g(t, P_{k}, V_{i}))$

(13)$¥leq¥sum_{k=1}^{m}¥mu(P_{k})h(t, s_{k}, ¥alpha¥iota^{¥prime}V_{i}))+¥epsilon¥mu(T_{i})$ .

By Ambrosetti’ $¥mathrm{s}$ lemma [1; Th. 2.2] and the continuity of $¥alpha(V(¥cdot))$ there is $q_{i}¥in T_{i}$

such that $¥alpha(V_{i})=¥sup$ $¥{¥alpha(V(s)):s¥in T_{i}¥}=¥alpha(V(q_{i}))$ . Therefore $|¥alpha(V_{i})-¥alpha(V(s_{k}))|=$

$|¥alpha(V(q_{i}))-¥alpha(V(s_{k}))|<¥delta$ , because $|q_{i}-s_{k}|<¥eta$ . Consequently,

$|h(t, s_{h}, ¥alpha(V_{i}))-h(t, s_{k}, ¥alpha(V(s_{k})))|<¥epsilon$ ,

so that

(14) $¥sum_{k=1}^{m}¥mu(P_{h})h(t, s_{k}, ¥alpha(V_{i}))¥leq¥sum_{k=1}^{m}¥mu(P_{k})h(t, s_{k}, ¥alpha(V(s_{h})))+¥epsilon¥mu(T_{i})$ .

Since $s^{¥prime}$ , $s^{¥prime¥prime}¥in T_{i}$ implies $|h(t, s^{¥prime}, ¥alpha(V(s^{¥prime})))-h(t, s^{¥prime¥prime}, ¥alpha(V(s^{¥prime¥prime})))|<_{¥epsilon}$ , we have

(15) $¥mu(P_{k})h(t, s_{lc}, ¥alpha(V(s_{k})))¥leq¥int_{P_{k}}h(t, s, ¥alpha(V(s)))ds+¥epsilon¥mu(P_{k})$ .

Now from (13)?(15) it follows that

$¥alpha(¥int_{¥tau_{i}}g(t, s, V(s))ds)¥leq¥int_{¥tau_{i}}h(t, s, ¥alpha(V(s)))ds+3¥epsilon¥mu(T_{i})$ for $i=1$ , $¥cdots$ , $n$ ,

and therefore, by (11),

$¥alpha(¥int_{T}g(t, s, V(s))ds)¥leq¥sum_{i=1}^{n}[¥int_{¥tau_{i}}h(t, s, ¥alpha(V(s)))ds+3¥epsilon¥mu(T_{i})]$

(16)$=¥int_{T}h(t, s, ¥alpha(V(s)))ds+3¥epsilon¥mu(T)$ .

On the other hand, for any $v¥in V$ we have

$||¥int_{J_{8}}g(t, s, v(s))ds||¥leq¥int_{J_{¥mathrm{e}}}m(t, s)ds<¥epsilon$ ,

which implies

$¥alpha(¥int_{J_{¥epsilon}}g(t, s, V(s))ds)¥leq 2¥epsilon$ .

Finally, by (10) and (16), we obtain

$¥alpha(F(V)(t))¥leq¥int_{T}h(t, s, ¥alpha(V(s)))ds+2¥epsilon+3¥epsilon¥mu(T)$

$¥leq¥int_{0}^{t}h(t, s, ¥alpha(V(s)))ds+2¥epsilon+3¥epsilon¥mu(T)$.

256 S. SZUFLA

Since the last inequality is satisfied for every $¥epsilon>0$ , we get

$¥alpha(F(V)(t))¥leq¥int_{0}^{t}h(t, s, ¥alpha(V(s)))ds$ .

Further, from (8) it follows that $¥alpha((I-F)(V))=0$ . Consequently, by (9) andAmbrosetti’ $¥mathrm{s}$ lemma [1; Th. 2. 3], we get

$¥alpha(V(t))¥leq¥alpha((I-F)(V)(t))+¥alpha(F(V)(t))=¥alpha(F(V)(t))$ ,

so that

$¥alpha(V(t))¥leq¥int_{0}^{t}h(t, s, ¥alpha(V(s)))ds$ for $t¥in J$ .

As $h$ is a Kamke function, this implies

$¥alpha(V(t))=0$ for $t¥in J$ .

Therefore by Ambrosetti’ $¥mathrm{s}$ lemma

$¥alpha(V)=¥sup$ $¥{¥alpha(V(t)):t¥in J¥}=0$ ,

$¥mathrm{i}.¥mathrm{e}.¥overline{V}$ is compact in $C$ . Hence we can find a subsequence $(u_{n_{k}})$ of $(u_{n})$ whichconverges in $C$ to a limit $u¥in C$ . From (8) it follows that

$||u-F(u)||_{c}=¥lim_{k-¥infty}||u_{n_{k}}-F(u_{n_{k}})||_{c}=0$ .

This shows that $u=F(u)$ , i.e. $u$ is a solution of (1).

3. In this section we present a simple theorem on the connectedness of theset $T^{-1}(y)$ , where $T$ is a continuous function from a metric space $X$ into a metricspace $¥mathrm{Y}$ , and $y$ is a fixed element of Y. Put $S_{¥mathrm{e}}=¥{x¥in X:d(y, T(x))<_{¥epsilon}¥}$ for any$¥epsilon>0$ , where $d$ denotes the metric of Y.

Theorem 4. Assume that $T:X¥rightarrow ¥mathrm{Y}$ is a continuous function such that1o for any $¥epsilon>0$ the set $S_{¥mathrm{e}}$ is connected;2o $y¥in¥overline{T(V)}¥Rightarrow y¥in T(V)$ for every closed subset $V$ of X. Then the set $T^{-1}(y)$ is

connected.

Proof. Suppose that the set $S=T^{-1}(y)$ is disconnected. Thus there are non-empty closed sets $W_{1}$ , $W_{2}$ such that $S=W_{1}¥cup W_{2}$ and $ W_{1}¥cap W_{2}=¥emptyset$ , and, consequent-ly, there are two disjoint open sets $U_{1}$ , $U_{2}$ of $X$ such that $W_{1}¥subset U_{1}$ , $W_{2}¥subset U_{2}$ . Sup-pose that for every $n¥in N=¥{1,2, --¥}$ there exists an $x_{n}¥in V_{n}¥backslash U$ , where $V_{n}=¥overline{S}_{1/n}$ and$U=U_{1}¥cup U_{2}$ . From 1o it follows that $V_{n}$ is connected for every $n¥in N$ . Put $V=$

$¥{¥overline{x_{n}..n¥in N}¥}$ . Since $d(y, T(x_{n}))¥leq 1/n$ , we see that $T(x_{n})¥rightarrow y$ , i.e. $y¥in¥overline{T(V)}$ . By 2o


this implies that there exists $x_{0}¥in V$ such that $T(x_{0})=y$ . Furthermore $V¥subset X¥backslash U$ ,since $U$ is open in $X$ , so that $x_{0}¥in S¥backslash U$ , a contradiction. Therefore there is $m¥in N$

such that $V_{m}¥subset U$ . Because $S¥subset V_{m}$ , we see that $U_{1}¥cap V_{m}¥neq¥emptyset¥neq U_{2}¥cap V_{m}$, and hence$V_{m}$ is disconnected, in contradiction with the connectedness of $V_{n}$ for every $n¥in N$ .This proves that $S$ is connected.

Remark. Let $B$ be a Banach space, and let $F$ be a continuous mapping $B¥rightarrow B$ .Denote by I the identity mapping on $B$ . Then the fixed points set of $F$ is connect-ed whenever the mapping $T=I-F$ satisfies the assumptions 1, $2^{¥mathrm{o}}$ of Theorem 4.In particular, the solutions set of (1) is connected whenever the mapping $T=I-F$satisfies the assumption 2o of Theorem 4, where $F$ denotes the mapping defined inthe proof of Theorem 2. In this way we obtain a simple proof of Kneser’ $¥mathrm{s}$ theoremfor a certain class of Volterra integral equations in Banach spaces.

Corollary. Under the assumptions of Theorem 3 the solutions set of (1) is $a$

continuum $¥dot{¥tau}nG$ .

Acknowledgement. The author expresses his appreciation to the referee forhis suggestions to amend the earlier version of the paper.

References

[1] Ambrosetti, A., Un teorema di esistenza per le equazioni differenziali negli spazi diBanach, Rend. Sem. Mat. Univ. Padova, 39 (1967), 349-360.

[2] Hukuhara, M., Sur l’application qui fait correspondre a un point un continu bicompact,Proc. Japan Acad., 31-1 (1955), 5-7.

[3] Kelley, W. G., A Kneser theorem for Volterra integral equations, Proc. Amer. Math.Soc., 40-1 (1973).

[4] Kuratowski, K., Topologie, Warszawa, 1958.[5] Murakami, H. and Nakagiri, S., Approximate solutions for some non-linear Volterra

integral equations, Proc. Japan Acad., 50-3 (1974), 212-217.[6] Murakami, H. and Nakagiri, S., Kneser’s property of solution families of non-linear

Volterra integral equations, ibid., 50-4 (1974), 296-300.[7] Pianigiani, G., Existence of solutions for ordinary differential equations in Banach

spaces, Bull. Acad. Polon. Sci., Ser. Sci. Math. Astronom. Phys., 23-8 (1975), 853-857.[8] Sato, T., Sur l’equation integrale non lineaire de Volterra, Compositio Math., 11-3

(1953), 271-290.[9] Sato, T., Sur l’application qui fait correspondre a une corbe une famille des courbes,

Proc. Japan Acad., 31-1 (1955), 1-4.[10] Scorza-Dragoni, G., Un teorema sulle funzioni continue rispetto a una e misurabili

rispetto all’altra variable, Rend. Sem. Mat. Univ. Padova, 17 (1948), 102-108.[11] Vidossich, G., On the structure of solutions set of nonlinear equations, J. Math. Anal.

Appl., 34 (1971), 602-617.[12] Vidossich, G., A fixed point theorem for function spaces, ibid., 36 (1971), 581-587.

S. SZUFLA

nuna adreso:Os. Poustan Narodowych 5961216 PoznanPoland

(Ricevita la 8-an de decembro, 1976)(Reviziita la 15-an de februaro, 1977)

on volterra integral equations in banach spaces proof. put

Documents