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Научни трудове на Съюза на учените в България - Пловдив Серия В. Техника и технологии, том XIII., Съюз на учените, сесия 5 - 6 ноември 2015 Scientific Works of the Union of Scientists in Bulgaria-Plovdiv, series C. Technics and Technologies, Vol. XIII., Union of Scientists, ISSN 1311-9419, Session 5 - 6 November 2015.
НЕПРЕКЪСНАТИ РЕШЕНИЯ НА НЕЛИНЕЙНИ ВОЛТЕРА-ХАМЕРСТЕЙН ИНТЕГРАЛНИ УРАВНЕНИЯ Атанаска Георгиева, Лозанка Тренкова
ПУ „Паисий Хилендарски", Факултет по Математика и Информатика, Пловдив
CONTINUOUSSOLUTIONSOFNONLINEARVOLTERRA-HAMMERSTEININTEGRALEQUATIONS AtanaskaGeorgieva, LozankaTrenkova
PU „PaisiiHilendarski", Faculty of Mathematics and Informatics, Plovdiv
Abstract: In this paper we prove the existence of continuous solution for nonlinear Volterra-Hammerstein integral equations using Schauder s fixed point theorem.
Keywords: nonlinear Volterra-Hammerstein integral equations, Schauder fixed point theorem, compact operator
1. Introduction
Many problems which arise in mathematical physics, engineering, biology, economics and etc., lead to mathematical models described by nonlinear integral equations. (see [1],[2],[3]). For instance, the Volterra - Hammerstein integral equations appear in nonlinear physical phenomena such as electro-magnetic fluid dynamics, reformulation of boundary value problems with a nonlinear boundary condition (see [2]).
2. Preliminaries
Let X be a Banach space with norm ||. Let C(X, X) be the space of allcontinuous operator acting in X and K + = [0, .
Let C ([a,b]) = {x: [a,b] ^ № is continuous} denotes the Banach space with the norm
denotes the Banach space with the norm
b
a
We consider the nonlinear Volterra-Hammerstein integral equation
x(t) = f (t) + J V (t, s)g (s, x(s)) ds, (2.1)
where the functions f e C([a,b]), g(.,.):[a,b] x № ^ № and the kernel V(.,.): [a,b] x [a,b] ^ M.
By using Schauder fixed point theorem, we prove the existence of a continuous solution of nonlinear Volterra-Hammerstein integral equation (2.1).
Theorem 1. [1] Let K be a closed and convex subset of a Banach space X. The operator T e C(K, K) and compact. Then T has a fixed point. 3. Existence of continuous solutions
Theorem 2.Let the following conditions are fulfilled: 1. The function g (s, x) satisfies the condition
sup| |gx)|:
~ (s,x)
dx
<
G(s)^(|x|)
for some measure function G(•) and the function is continuous over M + and satisfying the
0(x) r
condition sup——— = L < .
x>0 X
2. The kernel V (t, s) is continuous with respect to t and satisfies thecondition V(t,s)| < V1(t)V2(s) where the functions V(0 e C([a,b]) and function
G()V2() e L ([a,b]) . Moreover, assume that functions V (•) and V2 (•) satisfy the condition
IVII. L VG < 1 .
Then the equation (2.1) has a solution in C([a, b]).
Proof: Let r > 0 be a positive real number that will be fixed later on and define the subset Br of
X by Br = {x e C([a, b]): ||x||^ < r} . It is clear that Br is a closed and convex subset of X .
t
Let T be the operator defined on C([a,b])by Tx(t) = f (t) + J V(t, s)g(s, x(s))ds .Let
a
r < r and we denote the set T(Br) = {Tx: x e Br} .We will prove that T(Br) is a compact subset in Br .By using Arzella Ascoli theorem the compactness of the set T (B ) is ensured if T (B ) is equicontinuous and uniformly bounded.Let x e B . From conditions 1.and 2. for every t e [a,b] we have
Nol < II f IL+J V1(t)V2(t )G(s)^(|x(s)| )s <| f IL+1 ^IL lV2Gl LML
a
Consequently, T(B ) is uniformly bounded.
We will prove the equicontinuous of the set T(B ). Let t', t"e [a, b]and t' < t". From conditions 1.and 2. we obtain
\Tx(t" ) - Tx(t' )| < | f (f ) - f (f )| + Lrx J \ V (t", s) - V (f, s)| G(s)ds +1 V\\» lrrx J V2 (s)G(s)ds
a t'
By applying the dominanted convergence theorem to the right hand side of the previous inequality, one conclude thatlim\Tx(t " ) - Tx(t ' )| = 0 .Consequently, T(Br ) is equicontinuous.
We will prove that Tx e C([a, b]), whenever x e C([a,b]). Let h > 0 . By using conditions 1 .and 2. and by applying the dominanted convergence theorem and since f e C([a, b]) we have
lim\Tx(t + h) - Tx(t)\ < lim| f (t + h) - f (t )| + L ||x|| » J lim | V (t + h, s) - V (t, s)| G( s)ds ■
t +h
+ IIVIL LH» h J V2(s)G(s)ds = 0
' +
hmCi1 v hm0r v ' " .....»j hm01
0
t +h
1
hm0 t
Consequently, Tx e C([a,b]).
Next we prove that T is continuous over Br. Let {xn } be a sequence of Br converging to x in the II» norm. Since (Br ,||-||» ) is complete, then x e Br. For all n e N, s e [a,b], 0, e [0,1] , xn e Br and x e Br we obtain
K xn (s) + (1 -0, ) x(s)| <0, |xn (s)| + (1 -0, )| x(s)| <0,| xj »+ (1 -0, )|| 4 »< r. Since the function (() is continuous over M +, then one concludes that there exists a positiveconstant M( such that ((\0sxn(s) + (1 -0S)x(s)|) <M(.
Moreover, from condition 1.and 2. we have
t
\Txn nt ) - Tx(t )| < J | V (t, s)| | g (s, xn (s)) - g (s, x(s))| ds <
a
<J rç(t )V2(s) | (s\ (s) + (1 -K)x(s) ) (s) - x(s)| ds <
a
< mjlxn -4»\V2(s)G(s)t(\0sxn(s) + (1 -0)x(s)|)ds <
a
<1 VU xn-x » M ivgi
Then lim \Txn - Tx = 0 and hence the operator T is continuous over Br.
nm»
It remains to choose the positive real number r in such a way that T(Br) c Br. Let x e Br, then we have
\Tx(t )| <| f (t )| + J'V (t, s)|| g (s, x(s))| ds <|| f\ L+Vi )V2(s)G(s)((\x(s) )ds
a a
<1 f\ L+l VII „îV2(s)G(s)Lx(s) ds <|| f\ LI |V| » Lr || v2g| i !
<
Hence, the condition T (Br ) ΠBr is satisfied for any positive real number r satisfying
fixed point in Br for all r > r0.
ACKNOWLEDGEMENT. This work is partially supported by project NI15-FMI-004 of the Scienti c Research Fund, Plovdiv University, Bulgaria.
References:
[1] KaroniAbderrazek, On the existence of continuous solutions of nonlinear integral equations, Applied Mathematics Letters, 18, (2005), 299-305
[2] MaleknejadKh., Torabi P., Application of fixed point method for solving nonlinear Volterra-Hammerstein integral equation, UPB Sci.Bull.,Series A, Vol.74, Iss.1, 2012
[3] ORegan D., Meeham M., Existence theory for nonlinear integral and integro-differential equations,Kluwer Academic, Dordrecht, 1998
r
11/11
r0. By Schauder s fixed point theorem, one concludes the T has a
