EXISTENCE OF SOLUTIONS FOR
VOLTERRA
INTEGRO-DIFFERENTIAL
EQUATIONS WITH IMPLICIT
DERIVATIVE
Giuseppe Anichini,
Dipartimento di Matematica
e Informatica ”U.Dini”,
Universit`
a di Firenze,
Firenze, Italy
giuseppe.anichini@unifi.it
Giuseppe Conti,
Dipartimento di Matematica
e Informatica ”U.Dini”,
Universit`
a di Firenze,
Firenze, Italy
g.conti@unifi.it
March 14, 2018
Classification: AbstractThe purpose of this paper is to study the existence of continuous solutions of an integro-differential equation with implicit derivative, on bounded intervals. In our investigations we aimed to extend the use of fixed-point theorems for upper semicontinuous mappings with acyclic val-ues in a Banach space to get the result.
Keywords: Metric spaces, multivalued maps, AR (absolute retract),
cohomology, acyclic sets, integro-differential equations, Ascoli-Arzel`a the-orem.
1
Introduction
In this paper, we study, in an abstract setting, the solvability of a nonlinear integro-differential equation of Volterra type with implicit derivative like:
x′(t) = ∫ t
0
We will look for solutions of these equations in the Banach space of all real
C1 functions which are defined in the real interval J = [0, 1].
Equation (1) is a special case of integro-differential equation. It is well known that the theory of integro-differential equations has been emerging as an impor-tant area of investigation in recent years and has been developed very rapidly due to the fact that such equations find a wide range of applications modeling adequately many real processes observed in physics, chemistry, biology and engi-neering (see, e.g, [AC, BS, HL, MS, P-1998, P-2010] and the references therein). Particularly the integro-differential equations are involved through modeling in the framework of heat flow in material, kinetic theory, electrical engineering, vehicular traffic theory, biology, queuing theory, population dynamics, control theory, mathematical economics, mechanics. For example, Balachandran and Somasundaram ([BS]) proved an existence theorem for the optimal control of non-linear systems having an implicit derivative with quadratic performance in-volving an integro-differential term also by using the same method of functional analysis i.e. a fixed point theorem.
The integro-differential equations have been studied in various papers with the help of several tools of functional analysis, topology and fixed point theory. For instance, we can refer to [AC], [BS], [MS] [P-1998] [P-2010] [S]. So, the crucial key of our approach in order to find solutions of equation (1), consists in the use a very useful fixed point theorem for multivalued, compact uppersemi-continuous maps with acyclic values in a Banach space.
2
Preliminaries and Notations
Let B = C1(J, IRn) be a Banach space of all continuously differentiable functions
defined in J = [a, b] with the norm||x||1= max{||x||, ||x′||} where:
||x|| = max{|x(t)|, t ∈ [a, b]} and ||x′|| = max{|x′(t)|, t ∈ [a, b]}.
A subset A⊂ C1(J, IRn
) is a relatively compact set if and only if the functions of A are equicontinuous and uniformly bounded (together their derivatives on
J ).
Let M be a subset of the Banach space B and let T : M −→ B. Let {ϵn}
be an infinitesimal sequence of positive real numbers. A sequence{Tn} of maps
Tn : M −→ B is said to be an ϵn-approximation of T if||Tn(x)− T (x)||1 ≤ ϵn
for every x∈ M.
Let us denote by C(B) the family of all nonempty and compact subset of B and denote by B(0, r) the ball of B defined by: B(0, r) ={x ∈ B : ||x||1≤ r}
(and denote by B(0, r) its closure).
Let X be a subset of B; a multivalued mapping S : X −→ C(B) is said to be upper semi-continuous (u.s.c.) if the graph of S is closed in X× B, i.e. if for any sequence {xn}n∈IN⊂ X, xn → x0, yn → y0 as n → +∞, we have
y0∈ S(x0).
A mapping from B to C(B) is said to be compact if it sends bounded sets into relatively compact sets.
We say that A⊂ B is an Rδ –set in the space B if A is the intersection of
countable decreasing sequence of absolute retracts contained in B. It is known that Rδ –set is an acyclic set, i.e. it is acyclic with respect to any continuous
theory of cohomology (see, for istance, [G]).
Let M be a closed and nonempty subset of B and let T : M −→ B be a compact mapping. Let Tn : M −→ B be an ϵn-approximation where Tn are
compact mappings. It is know that if the equation x− Tn(x) = y has at most
one solution belonging to B(0, ϵn) for every n∈ IN, then the set of fixed points
of T is a compact Rδ-set ([LR]).
The well known Gronwall Lemma, from the standard theory of Ordinary Differential Equations, will be used.
Proposition 1 Let U (t), h(t) : J −→ J be continuous (nonnegative)
func-tions such that U (t)≤ K(t) +∫ath(s)U (s)ds, t∈ J, where K is a nonnegative
continuous and nondecreasing function on J . Then we have
U (t)≤ K(t) exp(∫ath(s)ds), t∈ J.
The following proposition can be deduced from Theorem 1 of ([FP]) and it will be useful in the sequel.
Proposition 2 (fixed point theorem): Let B be a Banach space, let M
be a closed and convex subset of B. Assume that T : M −→ C(B) be an uppersemicontinuous compact multivalued mapping with acyclic values. Then, if T (M )⊂ M, T has a fixed point.
3
Main result
We want to deal with the existence of solutions of the following integro-differential equation
x′(t) = ∫ t
0
k(t, s)f (s, x(s), x′(s))ds, x(0) = 0, t∈ J = [0, 1] (1)
The following theorem holds:
Theorem 1 : Let assume the following conditions:
• i) f : [0, 1] × IRn× IRn−→ IRn is a continuous function such that for every
(s, x, y)∈ [0, 1] × IRn× IRn we have: |f(s, x, y)| ≤ a|x| + b|y|, a, b ≥ 0. • ii) k : J × J −→ IRn is a C1function such that there a continuous positive
function h : J −→ IR such that |k(t, s)| ≤ h(s) and |∂k
• iii) putting h = sup{|h(s)|, s ∈ [0, 1]}, the following inequality holds:
2ha + hb < 1.
Then, if the conditions i), ii), iii) hold, the equation (1) has at least one solution.
Proof: Let q be a function belonging to C1(J, IRn) and consider the following integro-differential equation y(t) = ∫ t 0 k(t, s)f (s, ∫ s 0 q(τ )dτ, y(s))ds, t∈ J = [0, 1] (1q) Let Σ : C1(J, IRn ) −→ C1(J, IRn
) be the map which associates to every q ∈
C1(J, IRn
) the set of solutions of the (1q) equation. Clearly, putting x(t) =
∫t
0y(s)ds ( hence x′(t) = y(t) and x(0) = 0) we have that the fixed points of
the map Σ are the solutions of equation (1).
To that aim, the following steps in the proof have to be established:
• a) there exists a closed and convex set C such that Σ(C) ⊂ C. • b) the map Σ is an uppersemicontinous and compact map; • c) the set Σ(q) is an acyclic set, for every q ∈ C.
a)
Let C = B(0, M ) be a ball of C1(J, IRn
); hence y ∈ B(0, M) if and only if
||y|| ≤ M and ||y′|| ≤ M. Let q ∈ C; we have:
|y(t)| ≤ |∫t 0k(t, s)f (s), ∫s 0q(τ )dτ, y(s))ds≤ |∫t 0h(s)(a| ∫s
0 q(τ )dτ| + b|y(s)|)ds| ≤ ha||q|| + bh||y||.
So we have||y|| ≤ haM + bh||y||. Then, it follows||y|| ≤ haM
1− hb. Hence||y|| ≤ M if
ha
1− hb < 1. Since 2ha + hb < 1, then the last inequality holds.
Moreover we have: y′(t) =∫0t∂k ∂t(t, s)f (s, ∫ s 0 q(τ )dτ, y(s))ds + k(t, t)f (t, ∫ s 0 q(τ )dτ, y(t)). Hence
||y′|| ≤ h(aM + b||y||) + h(aM + b||y||) ≤ 2h(aM + b||y||) ≤
2h(aM + bhaM 1− hb) = 2h
aM
1− hb. Thus we have ||y′|| ≤ M if 2ha
1− hb < 1 and since 2ha + hb < 1, we have
Then the condition Σ(B(0, M ))⊂ B(0, M) is satisfied. b)
We want to prove that the map Σ : B(0, M )−→ B(0, M) is compact. Let y∈ B(0, M) and fix ϵ > 0. For any t1, t2∈ [0, 1], we have
|y′(t2)− y′(t1)| = |∫t2 0 ∂k ∂t(t2, s)f (s, ∫ s 0 q(τ )dτ, y(s))ds+ k(t2, t2)f (t2, ∫t2 0 q(τ )dτ, y(t2))− ∫t1 0 ∂k ∂t(t1, s)f (s, ∫ s 0 q(τ )dτ, y(s))ds− k(t1, t1)f (t1, ∫t1 0 q(τ )dτ, y(t1))| = |∫t1 0 ∂k ∂t(t2, s)f (s, ∫ s 0 q(τ )dτ, y(s))ds+ k(t2, t2)f (t2, ∫t2 0 q(τ )dτ, y(t2))− k(t1, t1)f (t1, ∫t1 0 q(τ )dτ, y(t1))− ∫t1 0 ∂k ∂t(t1, s)f (s, ∫ s 0 q(τ )dτ, y(s))ds + ∫ t2 t1 ∂k ∂t(t2, s)f (s, ∫ s 0 q(τ )dτ, y(s))ds| ≤∫t1 0 | ∂k ∂t(t2, s)− ∂k ∂t(t1, s)|(a| ∫ s 0 q(τ )dτ|+b|y(s)|)ds+h|f(t2, ∫ t2 0 q(τ )dτ, y(t2))− f (t1, ∫ t1 0 q(τ )dτ, y(t1))| + h | ∫ t2 t1 f (s, ∫ s 0 q(τ )dτ, y(s))|ds.
By the continuity of the functions q, h, f and k, it follows that
|y′(t2)− y′(t1)| < ϵ, t1, t2∈ [0, 1].
Since|y(t)|, |y′(t)| ≤ M, the the set Σ(B(0, M)) is (relatively) compact. Let us now show that Σ is an upper semi-continuous mapping.
Let qn a sequence converging to q0 in the C1-norm, yn∈ Σ(qn), i.e. let
yn(t) =
∫t
0k(t, s)f (s,
∫t
0qn(s)ds, yn(s))ds, t∈ J = [0, 1].
Assume that yn−→ y0 in the C1−norm. We need to show that y0∈ Σ(q0).
From limn→+∞f (s,
∫t
0qn(s)ds, yn(s))ds = f (t,
∫t
0q0(s)ds, y0(t)), it follows,
from the Dominated Lebesgue convergence Theorem: limn→+∞yn(t) = limn→+∞ ∫t 0k(t, s)f (s, ∫s 0 qn(τ )dτ, yn(s))ds = ∫t 0limn→+∞k(t, s)f (s, ∫s 0qn(τ )dτ, yn(s))ds = ∫t 0k(t, s)f (s, ∫s 0 q0(τ )dτ, y0(s))ds; hence y0(t) = ∫t 0k(t, s)f (s, ∫s 0 q0(τ )dτ, y0(s))ds i.e. y0∈ Σ(q0). c)
Now, we want to show that, for every fixed q∈ B(0, M), the set Σ(q) is an acyclic set.
Consider the integral equation:
y(t) =∫0tk(t, s)f (s,∫0sq(τ )dτ, y(s))ds. (1q)
Put f (s,∫0sq(τ )dτ, y) = g(s, y). Than the equation (1q) can be written in
the following way:
There exists, for every n∈ IN, a Lipschitz function gn : [0, 1]× IRn −→ IRn
such that for every (s, y), (s, y)∈ [0, 1] × IRn
|gn(s, y)− gn(s, y)| ≤ Ln|y − y| and |gn(s, y)− g(s, y)| ≤
1 2nh. Let Hn : B(0, M )−→ C1(J, IRn) be the operator defined as follows:
Hn(y)(t) =
∫t
0k(t, s)gn(s, y(s))ds, t∈ [0, 1].
Clearly Hn is a compact operator for every n∈ IN.
Moreover |Hn(y)(t)− H(y)(t)| ≤ ∫t 0|k(t, s)||gn(s, y(s))− g(s, y(s))|ds ≤ h 2nh = 1 2n < 1 n and |H′ n(y)(t)− H′(y)(t)| ≤ | ∫t 0 ∂k ∂t(t, s)gn(s, y(s))ds− ∫ t 0 ∂k ∂t(t, s)g(s, y(s))ds+ k(t, t)gn(t, y(t))− k(t, t)g(t, y(t))| ≤ ∫t 0h(s)|gn(s, y(s))− g(s, y(s))|ds+ h|gn(t, y(t))− g(s, y(s))| ≤ 2h 2nh = 1 n.
Let now β∈ B(0, M). Consider the equation y−Hn(y) = β; we want to prove
that this equation has at most one solution. Let y be another solution: we have, by the Gronwall Lemma,|y(t) − y(t)| ≤∫0th(s)|gn(s, y(s))− gn(s, y(s))|ds ≤ 0,
so that y(t) = y(t), for every t∈ [0, 1].
Then we can conclude that the set Σ(q) of the solutions of the equation (1q)
is an acyclic set.
4
Examples
Let us consider the following integro-differential equation with implicit deriva-tive: x(t) = ∫ t 0 cos t s2+ 4 sin(x′(s)) + ln(|x(s)| + 1) x2(s) + 1 ds, t∈ J = [0, 1]. (2)
By recalling our main result, we have, for every (t, s)∈ J × J:
k(t, s) = cos t s2+ 4 and so h(s) = 1 s2+ 4 ≤ 1 4; in analogous way we have|∂k
∂t(t, s)| ≤ h(s) ≤
1 4. So we can put: a = 1, b = 1; h = 14.
Finally the crucial inequality 2ha + hb < 1 becomes 2×14×1+14×1 = 34 < 1.
Then we can say that the conditions of our main theorem are satisfied and that the equation (2) admits at least one solution.
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