J. Math. Comput. Sci. 7 (2017), No. 2, 264-279 ISSN: 1927-5307
APPROXIMATING SOLUTION FOR GENERALIZED QUADRATIC FUNCTIONAL INTEGRAL EQUATIONS WITH MAXIMA
DNYANESHWAR V. MULE1,2,∗, BHIMRAO R. AHIRRAO3AND VILAS S. PATIL4
1Department of Mathematics, North Maharashtra University, Jalgaon 425001, India 2Department of Mathematics, Mahatma Phule College, Kingaon, Ahmedpur-413515, India
3Department of Mathematics, Z.B. Patil College, Dhule 424002, India
4Department of Mathematics, S.I.C.E. Society’s Degree College of Arts, Science and Commerce,
Ambernath(W)-421505, India
Copyright c2017 Mule, Ahirrao and Patil. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract. In this paper we prove an existence as well as approximation result for a certain nonlinear generalized quadratic functional integral equation with maxima. An algorithm for the solutions is developed and it is shown that the sequence of successive approximations starting with a lower or an upper solution converges monotonically to the solution of the related quadratic functional integral equation with maxima under some suitable mixed hybrid conditions. We base our main results on the Dhage iteration principle embodied in a recent hybrid fixed point theorem of Dhage (2014) in a partially ordered normed linear space.An example illustrating the existence result is also presented.
Keywords:quadratic functional integral equation; approximate positive solution; fixed point theorem.
2010 AMS Subject Classification:45G10.
1. Introduction
∗Corresponding author
E-mail address: [email protected] Received October 11 , 2016; Published March 1, 2017
The quadratic integral equations have been a topic of interest since long time because of their occurrence in the problems of some natural and physical processes of the universe. See Argyros [2], Deimling [6], Chandrasekher [4] and the references therein. The study gained momentum after the formulation of fixed point principles in Banach algebras due to Dhage [10],[12]. The existence results for such equations are generally proved under the mixed Lipschitz and com-pactness type conditions together with a certain growth condition on the nonlinearities of the quadratic integral equations. See Dhage [10],[12] and the references therein. The Lipschitz and compactness hypotheses are considered to be very strong conditions in the theory of nonlinear differential and integral equations which do not yield any algorithm to determine the numerical solutions. Therefore, it is of interest to relax or weaken these conditions in the existence and ap-proximation theory of quadratic integral equations.However, the literature on existence results for a special class of functional differential equations, namely nonlinear quadratic differential equations with maxima under weaker partial Lipschitz and partial compactness type conditions via Dhage iteration method. is not enriched yet, recently, the first authors in [12] ,[13],[14] have studied the existence results of functional differential equations with maxima. Therefore, it is admirable to extend this method to nonlinear quadratic integral equations with maxima. This is the main motivation of the present paper.
In this paper we prove the existence as well as approximations of the positive solutions of a certain generalized quadratic integral equation with maxima via an algorithm based on succes-sive approximations under partially Lipschitz and compactness conditions
Given a closed and bounded intervalJ= [0,T]of the real lineRfor someT >0, we consider the quadratic functional integral equation (in short GQFIE) with maxima
x(t) =k(t,x(t),X(t)) + [f(t,x(t),X(t))]
a(t) +
Z t
0
v(t,s)g(s,x(s),X(s))ds
(1)
for all t ∈J, v:J×R→R and f,g:J×R×R→R, are continuous functions, and X(t) =
max0≤η≤tx(η).
The GQFIE (1) with maxima is new to the literature . In particular, If f(t,x,y) =0 for all t∈Jandx,y∈Rthe GQFIE (1) with maxima reduces to the nonlinear functional equation with maxima
x(t) =k(t,x(t),X(t))), t∈J, (2) and ifk(t,x,y) =0 and f(t,x,y) =1 for allt ∈J andx,y∈R, it is reduced to nonlinear usual Volterra integral equation with maxima
x(t) =a(t) +
Z t
0
v(t,s)g(s,x(s),X(s)))ds, t∈J. (3)
Therefore, the QFIE (1) is general and the results of this paper include the existence and ap-proximations results for above nonlinear functional and Volterra integral equations with maxima as special cases.
The paper is organized as follows. In the following section we give the preliminaries and auxiliary results needed in the subsequent part of the paper.
2. Preliminary Notes / Materials and Methods
Unless otherwise mentioned, throughout this paper that follows, letE denote a partially or-dered real normed linear space with an order relationand the normk · kin which the addition and the scalar multiplication by positive real numbers are preserved by. A few details of a partially ordered normed linear space appear in Dhage [8], Heikkil¨a and Lakshmikantham [15] and the references therein.
We need the following notion and results.
Definition 2.1 : A mapping T :E →E is called isotone or monotone nondecreasing if it preserves the order relation, that is, if xy impliesT xT y for allx,y∈E. Similarly,
T is called monotone nonincreasing ifxy implies TxTy for all x,y∈E. Finally, T is called monotonic or simply monotone if it is either monotone nondecreasing or monotone nonincreasing onE.
Definition 2.2 : A mappingT :E→Eis called partially continuous at a pointa∈Eif forε>0
T called partially continuous onE if it is partially continuous at every point of it. It is clear that ifT is partially continuous onE, then it is continuous on every chainCcontained inE.
Definition 2.3 : A non-empty subsetSof the partially ordered Banach spaceEis called partially bounded if every chainCinSis bounded. An operatorT on a partially normed linear spaceE into itself is called partially bounded ifT (E)is a partially bounded subset ofE. T is called uniformly partially bounded if all chainsCinT (E)are bounded by a unique constant.
Definition 2.4 : A non-empty subsetSof the partially ordered Banach spaceEis called partially compact if every chainCinSis a relatively compact subset ofE. A mappingT :E→Eis called partially compact ifT (E)is a partially relatively compact subset ofE. T is called uniformly partially compact ifT is a uniformly partially bounded and partially compact operator onE. T is called partially totally bounded if for any bounded subsetSofE,T(S)is a partially relatively compact subset ofE. IfT is partially continuous and partially totally bounded, then it is called partially completely continuous onE.
Remark 2.1:Suppose thatT is a nondecreasing operator onEinto itself. ThenT is a partially bounded or partially compact if T (C) is a bounded or relatively compact subset ofE for each chainCinE.
Definition 2.5 : The order relation and the metric d on a non-empty set E are said to be compatible if{xn}n∈Nis a monotone, that is, monotone nondecreasing or monotone
nonincreas-ing sequence inE and if a subsequence{xnk}n∈N of{xn}n∈N converges to x∗ implies that the
original sequence{xn}n∈N converges tox∗. Similarly, given a partially ordered normed linear
space(E,,k · k), the order relationand the normk · kare said to be compatible ifand the metricd defined through the normk · kare compatible.
Clearly, the set Rof real numbers with usual order relation ≤and the norm defined by the absolute value function| · |has this property. Similarly, the finite dimensional Euclidean space Rnwith usual componentwise order relation and the standard norm possesses the compatibility property.
Definition 2.6 : A upper semi-continuous and monotone nondecreasing functionψ:R+→R+
is called a D-function provided ψ(r) = 0 iff r =0. Let (E,,k · k) be a partially ordered
exists aD-functionψ :R+→R+ such that
kTx−T yk ≤ψ(kx−yk) (4)
for all comparable elementsx,y∈E. If ψ(r) =k r,k>0, thenTis called a partially Lipschitz
with a Lipschitz constantk.
Let(E,,k · k)be a partially ordered normed linear algebra. Denote
E+=x∈E|xθ, whereθ is the zero element ofE
and
K ={E+⊂E|uv∈E+ for all u,v∈E+}. (5)
The elements of K are called the positive vectors of the normed linear algebra E. The following lemma follows immediately from the definition of the set K and which is often times used in the applications of hybrid fixed point theory in Banach algebras.
Lemma 2.1 [Dhage[8]]: Ifu1,u2,v1,v2∈K are such thatu1v1 andu2v2, thenu1u2
v1v2.
Definition 2.7 : An operatorT : E→Eis said to be positive if the rangeR(T)ofT is such thatR(T )⊆K .
Theorem 2.1 [Dhage [8]]: Let E,,k · k
be a regular partially ordered complete normed linear algebra such that the order relation and the norm k · k in E are compatible in every compact chain ofE. LetA,B:E→K andC :E→E be three nondecreasing operators such that
(a) A and C is partially bounded and partially nonlinear D-Lipschitz with D-functions
ψA andψC respectively,
(b) B is partially continuous and uniformly partially compact, and
(c) MψA(r)+ψC ,<r,r>0, whereM=sup{kB(C)k :Cis a chain inE},and
(d) there exists an elementx0∈Xsuch thatx0Ax0Bx0+Cx0 or x0Ax0Bx0+Cx0. Then the operator equation
has a solutionx∗inEand the sequence{xn}of successive iterations defined byxn+1=AxnBxn+ Cxn, n=0,1, . . . ,converges monotonically tox∗.
Remark 2.2: The compatibility of the order relation and the norm k · k in every compact chain of E holds if every partially compact subset of E possesses the compatibility property with respect toandk · k. This simple fact has been utilized to prove the main results of this paper.
3. Results and Discussion
The QFIE (1) is considered in the function spaceC(J,R)of continuous real-valued functions defined onJ. We define a normk · kand the order relation ≤ inC(J,R)by
kxk=sup
t∈J
|x(t)| (7)
and
x≤y ⇐⇒ x(t)≤y(t)∀t∈J, (8) respectively.
Clearly,C(J,R) is a Banach algebra with respect to above supremum norm and is also par-tially ordered w.r.t. the above parpar-tially order relation≤. It is known that the partially ordered Banach algebraC(J,R)has some nice properties concerning the compatibility property with re-spect to the normk · kand the order relation≤in certain subsets of of it. The following lemma in this connection follows by an application of Arzel´a-Ascoli theorem.
Lemma 3.1 :Let C(J,R),≤,k · k
be a partially ordered Banach space with the normk · kand the order relation ≤ defined by (7) and (8) respectively. Then k · k and ≤ are compatible in every compact chainCinS.
Proof. LetSbe a partially compact subset ofC(J,R)and let{xn}n∈N be a monotone
nonde-creasing sequence of points inS. Then we have
x1(t)≤x2(t)≤ · · · ≤xn(t)≤ · · ·, (9)
Suppose that a subsequence{xnk}k∈N of{xn}n∈N is convergent and converges to a pointxin
S. Then the subsequence{xnk(t)}k∈N of the monotone real sequence{xn(t)}n∈N is convergent.
By monotone characterization, the whole sequence{xn(t)}n∈N is convergent and converges to
a pointx(t)inRfor eacht∈J. This shows that the sequence{xn}n∈N converges toxpoint-wise
onJ. To show the convergence is uniform, it is enough to show that the sequence{xn(t)}n∈N
is equicontinuous. Since S is partially compact, every chain or totally ordered set and conse-quently{xn}n∈N is an equicontinuous sequence by Arzel´a-Ascoli theorem. Hence {xn}n∈N is
convergent and converges uniformly to x. As a result k · k and ≤ are compatible in S. This completes the proof.
We need the following definition in what follows.
Definition 3.1 : A function p∈C(J,R) is said to be a lower solution of the GQFIE (1) if it satisfies
p(t) =k(t,p(t),P(t)) + [f(t,p(t),P(t))]
a(t) +
Z t
0
v(t,s)g(s,p(s),P(s))ds
for all t ∈J.Similarly, a functionq∈C(J,R)is said to be an upper solution of the GQFIE (1) with maxima if it satisfies the above inequalities with reverse sign.
We consider the following set of assumptions in what follows: (H1) f defines a nonnegative function f :J×R×R→R
(H2) There exists a constantMf >0 such that 0≤ f(t,x,y)≤Mf for allt∈Jandx,y∈R.
(H3) There exists aD-functionψf such that
0≤ f(t,x1,x2)−f(t,y1,y2)≤ψf(max{x1−y1,x1−y1})
for allt∈Jandx1,x2,y1,y2∈R,x1≥y1,x2≥y2. (H4) a defines a continuous functiona:J→R+
(H5) v defines a continuous and nonnegative function onJ×J
(H6) g(t,x,y)is nondecreasing inxandyfor allt∈J.
(H7) There exists a constantMg>0 such thatg(t,x,y)≤Mgfor allt∈Jandx,y∈R.
(H9) There exists aD-functionψk such that
0≤k(t,x1,x2)−k(t,y1,y2)≤ψk(max{x1−y1,x1−y1})
for allt∈Jandx1,x2,y1,y2∈R,x1≥y1,x2≥y2.
(H10) The GQFIE (1) with maxima has a lower solutionp∈C(J,R).
Theorem 3.1 :Assume that hypotheses (H1)-(H10) hold. Furthermore, assume that
(kak+V T Mg)ψf(r) +ψk(r)<r, r>0, (10)
then the GQFIE (1) with maxima has a solutionx∗ defined on J and the sequence{xn}n∈N of
successive approximations defined by
xn+1(t) =k(t,xn(t),Xn(t)) + [f(t,xn(t),Xn(t))]
a(t) +
Z t
0
v(t,s)g(s,xn(s),Xn(s))
ds, t∈J.
(11) for all t ∈J,wherex0=pandXn(t) =max0≤η≤txn(η), converges monotonically tox∗.
Proof. Set E =C(J,R). Then, from Lemma 3.1, it follows that every compact chain inE possesses the compatibility property with respect to the normk · kand the order relation ≤in E.
Define two operatorsA andBonE by
Ax(t) = f(t,x(t),X(t)), t∈J, (12)
and
Bx(t) =a(t) +
Z t
t0
v(t,s)g(s,x(s),X(s))ds, t∈J, (13) and
Cx(t) =k(t,x(t),X(t)), t∈J, (14)
From the continuity of the integral and the hypotheses (H1)-(H10), it follows thatA ,B andC
define the maps A,B :E →K and C :E→E. Now by definitions of the operatorsA, B, andC the GQFIE (1) is equivalent to the operator equation
We shall show that the operatorsA,BandC satisfy all the conditions of Theorem 2.1. This is achieved in the series of following steps.
Step I: A ,B andC are nondecreasing on E.
Letx,y∈E be such thatx≥y.Thenx(t)≥y(t)for allt ∈J. Sinceyis continuous on[a,t], there exists a η∗∈[a,t] such that y(η∗) = max
a≤η≤ty(η). By definition of ≤, one has x(η
∗)≥
y(η∗). Consequently, we obtain
X(t) = max
a≤η≤tx(η) =x(η
∗)≥y(
η∗) = max
a≤η≤ty(η) =Y(t)
for eacht∈J. Then by hypothesis (H2), we obtain
Ax(t) = f(t,x(t),X(t))≥ f(t,y(t),Y(t))) =Ay(t),
and
Cx(t) =k(t,x(t),X(t))≥k(t,y(t),Y(t))) =Cy(t),
for allt ∈J. This shows thatA and C are nondecreasing operators on E into E. Similarly, using hypothesis (H5),
Bx(t) = a(t) +
Z t
t0
v(t,s)g(s,x(s),X(s))ds
≥ a(t) +
Z t
t0
v(t,s)g(s,y(s),Y(s))ds
= By(t)
for allt∈J. Hence, it is follows that the operatorB is also a nondecreasing operator onEinto itself. Thus,A ,BandC are nondecreasing positive operators onEinto itself.
Step II: A andC are partially bounded and partiallyD-Lipschitz on E. Letx∈E be arbitrary. Then by (H2),
|Ax(t)| ≤
f(t,x(t),X(t)) ≤Mf,
Next, letx,y∈E be such thatx≥y. Then, we have
|x(t)−y(t)| ≤ |X(t)−Y(t)|
and that
|X(t)−Y(t)| = X(t)−Y(t)
= max
t0≤η≤t
x(η)− max
t0≤η≤t y(η)
≤ max
t0≤η≤t
[x(η)−y(η)]
= max
t0≤η≤t
|x(η)−y(η)|
≤ kx−yk
for eacht∈J. As a result, by hypothesis (A3),
|Ax(t)−Ay(t)| = f(t,x(t),X(t))−f(t,y(t),Y(t))
≤ ψf(max{|x(t)−y(t)| |X(t)−Y(t)|})
≤ ψf(kx−yk),
for allt ∈J. Taking supremum overt, we obtain
kAx−Ayk ≤ψf(kx−yk)
for allx,y∈Ewithx≥y. Similarly, by hypothesis (A3)
kCx−Cyk ≤ψk(kx−yk)
for allx,y∈E withx≥y. HenceA andC are partially nonlinearD-Lipschitz operators onE which further implies thatA andC are partially continuous onE .
Step III: B is a partially continuous operator on E.
Let {xn}n∈N be a sequence in a chain C of E such that xn →x for all n∈N. Then, by
lim
n→∞B
xn(t) = a(t) +
Z t
t0
v(t,s)g(s,xn(s),Xn(s))ds
≥ a(t) +
Z t
t0
[lim
n→∞
v(t,s)g(s,xn(s),Xn(s))]ds
= a(t) +
Z t
t0
v(t,s)g(s,x(s),X(s))ds
= Bx(t),
for allt ∈J. This shows thatBxnconverges monotonically toBxpointwise onJ.
Next, we will show that {Bxn}n∈N is an equicontinuous sequence of functions in E. Let t1,t2∈Jbe arbitrary witht1<t2.Then
Bxn(t2)−Bxn(t1)
≤ |a(t1)−a(t2)|
+
Z t2
0
v(t1,s)g(s,xn(s),Xn(s))ds− Z t1
0
v(t1,s)g(s,xn(s),Xn(s))ds
≤ |a(t1)−a(t2)|+
Z t2
0
v(t2,s)g(s,xn(s),Xn(s))ds
− Z t1
0
v(t2,s)g(s,xn(s),Xn(s))ds
+
Z t2
0
v(t1,s)g(s,xn(s),Xn(s))ds− Z t1
0
v(t1,s)g(s,xn(s),Xn(s))ds
≤ |a(t1)−a(t2)|+
Z t2
0
|v(t2,s)−v(t1,s)||g(s,xn(s),Xn(s))|ds +
Z t2
t1
|v(t1,s)||g(s,xn(s),Xn(s))|ds ≤
a(t1)−a(t2)|+
Z T
0
|v(t2,s)−v(t1,s)|Mgds
+ V Mg|t2−t1| (16)
Step IV:Bis uniformly partially compact operator on E.
LetC be an arbitrary chain inE. We show thatB(C) is a uniformly bounded and equicon-tinuous set inE. First we show thatB(C)is uniformly bounded. Lety∈B(C)be any element. Then there is an elementx∈Cbe such thaty=Bx. Now, by hypothesis (A5),
|y(t)| ≤ a(t) +
Z t
0
v(t,s)|g(s,x(s),X(s))|ds
≤ kak+V MgT
≤ r
for allt∈J. Taking the supremum overt, we obtainkyk ≤ kBxk ≤rfor ally∈B(C). Hence,
B(C) is a uniformly bounded subset of E. Moreover, kB(C)k ≤ r for all chains C in E. Hence,B is a uniformly partially bounded operator onE. Next, we will show thatB(C)is an equicontinuous set inE. Lett1,t2∈Jbe arbitrary witht1<t2.Then, for anyy∈B(C), one has,
y(t2)−y(t1)
=
Bx(t2)−Bx(t1)
≤ |a(t1)−a(t2)|+
Z t2
0
v(t1,s)g(s,x(s),X(s))ds− Z t1
0
v(t1,s)g(s,x(s),X(s))ds
≤ |a(t1)−a(t2)|+
Z t2
0
v(t2,s)g(s,x(s),X(s))ds− Z t2
0
v(t1,s)g(s,x(s),X(s))ds +
Z t2
0
v(t1,s)g(s,x(s),X(s))ds− Z t1
0
v(t1,s)g(s,x(s),X(s))ds
≤ |a(t1)−a(t2)|+
Z t2
0
|v(t2,s)−v(t1,s)||g(s,x(s),X(s))|ds
+
Z t2
t1
|v(t1,s)||g(s,x(s),X(s))|ds
≤ |a(t1)−a(t2)|+
Z t2
t1
|g(s,x(s),X(s))|ds ≤
a(t1)−a(t2)|+
Z T
0
|v(t2,s)−v(t1,s)|Mgds
+ V Mg|t2−t1|
uniformly for all y∈B(C).Hence B(C) is an equicontinuous subset ofE. Now, B(C) is a uniformly bounded and equicontinuous set of functions inE, so it is compact. Consequently,
Bis a uniformly partially compact operator onEinto itself.
Step V: p satisfies the operator inequality p≤A pBp+Cp.
By hypothesis (H10), the GQFIE (1) has a lower solutionpdefined onJ. Then, we have
p(t) =k(t,p(t),P(t)) + [f(t,p(t),P(t))]
a(t) +
Z t
0
v(t,s)g(s,p(s),P(s))ds
(17)
for allt∈J.From the definitions of the operatorsA ,BandC it follows thatp(t)≤A p(t)Bp(t)+
Cp(t)for allt∈J. Hencep≤A pBp+Cp.
Step VI: TheD-functionsψA andψC satisfy the growth condition MψA(r) +ψC <r, for
r>0.
Finally, theD-functionψA of the operatorA satisfy the inequality given in hypothesis (d)
of Theorem 2.1, viz.,
MψA(r) +ψC ≤(kak+V MgT)ψf(r) +ψk(r)<r
for allr>0.
ThusA, B andC satisfy all the conditions of Theorem 2.1 and we conclude that the oper-ator equation AxBx+Cx=x has a solution. Consequently the GQFIE(1) has a solutionx∗ defined onJ. Furthermore, the sequence{xn}n∈N of successive approximations defined by (11)
converges monotonically tox∗. This completes the proof.
Example
Given a closed and bounded intervalJ= [0,1], consider the GQFIE,
x(t) = 1
4
4+tan−1x(t) +tan−1X(t)
1 t+1+
Z t
0
1 t2+1·
[1+tanhx(s)]
4 ds
+1
4[tan
−1x(t) +tan−1X(t)] (18)
Here,q(t) =t+t1 andv(t,s) = 1
t2+1 which are continuous andkq(t)k= 12and V=1. Similarly, the functionsk,f andgare defined byk(t,x,y) = 14[tan−1x(t) +tan−1y(t)]
f(t,x,y) =1
4
4+tan−1x(t) +tan−1y(t) and
g(t,x,y) =g(t,x) =1+tanhx
4 .
The function f satisfies the hypothesis (H3) withψf(r) = 211+rη2 for each 0<η<r. To see this, we have
0≤ f(t,x1,x2)−f(t,y1,y2)
≤ 1 4·
1 1+η2·(
x1−y1) +1
4· 1 1+η2·(
x2−y2)
≤ 1 2·
1
1+η2·max{x1−y1,x2−y2}
for allx1,y1,x2,y2∈R,x1≥y1andx2>η>y2. Moreover, the function f is nonnegative and
bounded onJ×R×Rwith boundMf =2 and so the hypothesis (H2) is satisfied.
Similarly,the function k satisfies the hypothesis (H9)ψk(r) = 121+rη2 for each 0<η<r. To see this, we have
0≤k(t,x1,x2)−k(t,y1,y2)
≤ 14· 1
1+η2·(x1−y1) + 1 4·
1
1+η2·(x2−y2)
≤ 12· 1
1+η2 ·max{x1−y1,x2−y2}
for all x1,y1,x2,y2 ∈R, x1 ≥y1 and x2 >η >y2. Moreover, the function k is bounded on
J×R×Rwith boundMk= π
4 and so the hypothesis (H9) is satisfied. Also we have
(kak+V T Mg)ψf(r) +ψk(r)<r, r>0,
Again, sincegis nonnegative and bounded on J×R×R with boundMg=12, the hypothesis (H5) holds. Furthermore, g(t,x,y) =g(t,x) is nondecreasing in xandy for allt ∈J, and thus
Thus, condition (10) of Theorem 3.1 is held. Finally, the GQFIE (1) has a lower solution p(t) =0 onJ. Thus all the hypotheses of Theorem 3.1 are satisfied. Hence we apply Theorem 2.1 and conclude that the GQFIE (1) has a solutionx∗ defined onJ and the sequence{xn}n∈N
defined by
xn+1(t) = 1
4
4+tan−1xn(t) +tan−1Xn(t)
1 t+1+
Z t
0
1 t2+1
[1+tanhx(s)]
4 ds
+1
4[tan
−1x
n(t) +tan−1Xn(t)] (19)
for allt ∈J, wherex0=0, converges monotonically tox∗.
Conflict of Interests
The authors declare that there is no conflict of interests.
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