ON EXISTENCE OF EXTREMAL SOLUTIONS OF NONLINEAR FUNCTIONAL INTEGRAL EQUATIONS IN BANACH ALGEBRAS

An algebraic fixed point theorem involving the three operators in a Banach algebra is proved using the properties of cones and they are further applied to a certain nonlinear integral equations of mixed type x(t) = k(t,x(μ(t))) + [ f (t,x(θ(t)))](q(t) + ∫ σ(t) 0 v(t, s)g(s,x(η(s)))ds) for proving the existence of maximal and minimal solutions. Our results include the earlier fixed point theorems of Dhage (1992 and 1999) as special cases with a different but simple method.


Introduction
It is known that the algebraic methods are useful for proving the existence of extremal solutions for various classes of differential and integral equations under certain monotonicity conditions.The exhaustive account of this subject appears in Amann [1], Deimling [4], and Heikkilä and Lakshmikantham [12] and the references therein.The existence theorems for nonlinear integral equations of mixed type are generally obtained by using the fixed point theorems of Krasnosel'skii [13] and Dhage [6,8].Most recently, the author in [6,7] proved some fixed point theorems involving two operators in a Banach algebra via the use of measure of noncompactness and they are further applied to a certain nonlinear integral equation in Banach algebras for proving the existence of extremal solutions under the mixed algebraic and topological conditions, such as monotonicity and continuity of the nonlinearities involved in the equation.In this paper, we generalize the fixed point theorem of Dhage [6,7] to three operators under the weaker conditions with a different method and apply the newly developed fixed point theorem to a certain nonlinear integral equation in Banach algebras for proving the existence of extremal solutions.
(i) K + K ⊆ K, (ii) λK ⊆ K whenever λ ∈ R and λ ≥ 0, (iii) −K ∩ K = {0}, where 0 is a zero element of X. Further, a cone K is called positive if (iv) K • K ⊂ K, where • is the multiplicative composition in X.We define an order relation ≤ in X as follows.Let x, y ∈ X.Then x ≤ y ⇐⇒ y − x ∈ K. (2.1) Notice that condition (iv) implies that if x ≤ y and z ∈ K, then xz ≤ yz.
y for some real number N > 0. The details of cones and their properties may be found in the monographs like Guo and Lakshmikantham [11] and Heikkilä and Lakshmikantham [12].
Let u,v ∈ X be such that u ≤ v. Then the set is called an order interval in X.Since K is closed, every order interval is closed in X.
Definition 2.1.A mapping T : X → X is called increasing if for all x, y ∈ X, Tx ≤ T y whenever x ≤ y.
A mapping A : X → X is called Ᏸ-Lipschitzian if there exists a continuous nondecreasing function φ : R + → R + satisfying for all x, y ∈ X with φ A (0) = 0. Sometimes the function φ A is called a Ᏸ-function of A on X.In the special case when φ A (r) = αr, α > 0, A is called a Lipschitzian with a Lipschitz constant α.In particular, if α < 1, A is called a contraction with a contraction constant α.Further, if φ A (r) < r for r > 0, then A is called a nonlinear contraction on X.
The following fixed point theorem for the nonlinear contraction is well known and is useful for proving the existence and the uniqueness theorems for the nonlinear differential and integral equations.
Theorem 2.2 (Boyd and Wong [2]).Let A : X → X be a nonlinear contraction.Then A has a unique fixed point x * and the sequence {A n x} of successive iterations of A converges to x * for each x ∈ X.
An operator T : X → X is called compact if T(X) is a compact subset of X.Similarly, T : X → X is called totally bounded if T maps a bounded subset of X into the relatively compact subset of X.Finally, T : X → X is called completely continuous operator if it is continuous and totally bounded operator on X.However, the two notions of compactness and total boundedness of T are equivalent on bounded subsets of X.
The following theorem of [7] is well known and is useful in the theory of nonlinear differential and integral equations in Banach algebras.

B. C. Dhage 273
We note that an operator T on a Banach space X into itself is called positive if range(T) ⊆ K.
Theorem 2.3.Let the order cone K of a real algebra X be positive and normal, and let u,v ∈ X be such that u ≤ v. Let A,B : [u,v] → X be two positive operators such that (a) A is Ᏸ-Lipschitzian and increasing, (b) B is completely continuous and increasing, (c) Mφ A (r) < r for r > 0, where Proof.Clearly [u,v] is convex and closed subset of X which is further bounded in view of the normality of the cone K in X.
Let y ∈ [u,v] be a fixed element.Define an operator A y : [u,v] → X by

.4)
Since A and B are positive and increasing, one has Moreover, condition (e) implies that

.6)
Hence A y defines a mapping (2.7) This shows that A y is a nonlinear contraction on X in view of the hypothesis (c).Therefore, an application of Theorem 2.2 yields that A y has a unique fixed point, say x * , in [u,v] and the sequence {A n y (u)} converges to x * .By the definition of A y , (2.9) It is easy to prove that A n y (u) ≤ A n w (u) for all n ∈ N. Hence we have where z is the unique solution of the equation  (2.12) where c = Aa + αdiam(S) and a ∈ S, a fixed element.Let > 0 be given.Since B is completely continuous, B(S) is totally bounded.Then there is a set Y = {y 1 ,..., y n } in S such that (2.16)This is true for every y ∈ S and hence where z i = N y i .As a result, N(S) is totally bounded.Since N is continuous, it is a compact operator on S.Moreover, N is increasing because A, B, and . Now the limits of these sequences exist (see Heikkilä and Lakshmikantham [12]) and they belong to the order interval [u,v].Denote By the continuity of N, we obtain and since N is increasing, it follows that x * ≤ x * .Assume that x ∈ [u,v] is a fixed point of the operator N. Then by the definition of S, Assume that z ∈ [u,v] is a solution of the operator equation AxBx + Cx = x.Then from the definitions of N, it follows that z is also a fixed point of N. Since u ≤ z ≤ v, we have Nu ≤ Nz = z ≤ Nv.Proceeding in this way, by induction, we get N n u ≤ z ≤ N n v, and so x * ≤ x * .Thus x * is the least solution and x * is the greatest one of the equation AxBx + Cx = x in [u,v].This completes the proof.
Notice that the proof of Theorem 2.3 involves only the elementary ideas of functional analysis and does not involve the use of the advanced notion of measure of noncompactness in Banach spaces.
As a consequence of Theorem 2.3, we obtain Corollary 2.5 in its applicable form to the nonlinear equations.
Corollary 2.5.Let the order cone K of a real algebra X be positive and normal, and let u,v ∈ X be such that u ≤ v. Let A,B : X → K and C : X → X be three increasing operators such that (a) A and B are Lipschitzians with Lipschitz constants α and β, respectively, Then operator equation AxBx + Cx = x has a least solution and a greatest one in [u,v].
When C ≡ 0 and φ(r) = αr, 0 ≤ α < 1, in Theorem 2.4, we get the interesting Corollary 2.6 to [7, Theorem 2.3] which has numerous applications in the theory of nonlinear differential and integral equations.
Corollary 2.6.Let the order cone K of a real Banach algebra X be positive and normal, and let u,v ∈ X be such that u ≤ v. Let A,B : X → K be two increasing operators such that

Functional integral equations
Let R denote the real line.Given a closed and bounded interval J = [0,1] in R, consider the nonlinear functional integral equation (in short FIE) for all t ∈ J, where µ,θ,σ,η : J → J, q : J → R, v : J × J → R, and f ,g,k : The special cases of the FIE (3.1) occur in some natural, physical, and social sciences; see Chandrasekhar [3] and Deimling [4] and the references therein.The FIE (3.1) and some of its special cases have been discussed in Dhage [5,8] and Dhage and O'Regan [9] for the existence results.In this section, we will prove the existence theorem for the extremal solutions of FIE (3.1) by an application of the abstract fixed point theorem embodied in Corollary 2.5.
Let M(J,R) and B(J,R) denote, respectively, the spaces of all measurable and bounded real-valued functions on J.We will seek the solution of FIE (3.1) in the space BM(J,R) of B. C. Dhage 277 bounded and measurable real-valued functions on J. Define a norm Clearly, BM(J,R) is a Banach algebra with this maximum norm.We define an order relation ≤ in BM(J,R) with the help of the cone K in BM(J,R) defined by Clearly, K is a positive and normal cone in BM(J,R).Let L(J,R) denote the space of Lebesgue integrable real-valued functions on J, with a norm • L 1 defined by We need the following definitions in the sequel.
for all t ∈ J.
We consider the following set of assumptions.(H 0 ) The functions µ,θ,σ,η : J → J are continuous.(H 1 ) The function q : J → R + is continuous with (H 3 ) The function k : J × R → R + is continuous and there is a function for all x, y ∈ R. (H 5 ) The function g : The functions f (t,x), g(t,x), and k(t,x) are nondecreasing in x almost everywhere for t ∈ J. (H 7 ) FIE (3.1) has a lower solution u and an upper solution v with u ≤ v. Remark 3.4.Suppose that hypotheses (H 5 ), (H 6 ), and (H 7 ) hold.Then the function h : J → R, defined by is Lebesgue integrable and g(t,x) ≤ h(t) (3.11) for all t ∈ J and for all x ∈ [a,b].
Theorem 3.5.Assume that the hypotheses (H 0 )-(H 7 ) hold.If We will show that the operators A, B, and C satisfy all the conditions of Theorem 2.4 on BM(J,R).Since the functions q, v, f , and g are nonnegative, A and B define the operators A,B : BM(J,R) → K. Let x, y ∈ X be such that x ≤ y.Then we have for all t ∈ J. Hence A is increasing on X.Similarly, it is shown that the operators B and C are also increasing on X.Let x, y ∈ BM(J,R).Then by (H 4 ), (3.16) Taking the maximum over t, This shows that A is a Lipschitzian with a Lipschitz constant α 1 .Similarly, it is shown that C is a Lipschitzian with a Lipschitz constant β 1 .
Next we will show that the operator B is continuous and compact on [u,v].Since g(t,x) is L 1 -Carathéodory, by using the dominated convergence theorem (see Granas et al. [10]), it can be shown that B is continuous on BM(J,R).Let {x n } be a sequence in [u,v].Then by Remark 3.4, Thus all the conditions of Theorem 2.3 are satisfied and hence an application of it yields that the operator equation (3.2) has a minimal solution and a maximal one in [a,b].This further implies that the FIE (3.1) has a minimal solution and maximal one in [a,b].This completes the proof.
We define an order relation ≤ in AC(J,R) with the help of the cone K AC defined by Clearly, the cone K AC is positive and normal in AC(J,R).We need the following definition in the sequel.Now the desired conclusion follows by an application of Theorem 2.4 with Q = |x 0 / f (0,x 0 )| because AC(J,R) ⊂ BM(J,R).The proof is complete.
converging to a point y.Now N y n − N y = AN y n By n − AN(y)By + C N y n − C(N y) ≤ AN y n By n − AN(y)By n + AN(y)By n − AN(y)By + C N y n − C(N y) ≤ AN y n − AN(y) By n + AN(y) By n − By + C N y n − C(N y) ≤ Mφ A N y n − N y + AN y By n − By + φ C N y n − N y .

(2. 11 )
Taking the limit superior as n → ∞ and using the fact that φ A and φ C are continuous functions, we obtain limsup n→∞ N y n − N y ≤ Mφ A limsup n→∞ N y n − N y + AN y limsup n→∞ By n − By + φ C limsup n→∞ N y n − N y .
.20) and so the operator equation x = AxBx + Cx has a solution in [u,v].As a result, x * and x * are solutions of the operator equation AxBx + Cx = x.
(a) A is Lipschitzian with a Lipschitz constant α and is increasing on [u,v], (b) B is continuous and compact on [u,v], (c) αM < 1, where M = B([u,v]) = sup{ B(x) : x ∈ [u,v]}, (d) u ≤ AuBu and AvBv ≤ v. Then operator equation AxBx = x has a least solution and a greatest one in [u,v].
Then operator equation AxBx = x has a least solution and a greatest one in [u,v].Then operator equation AxBx + Cx = x has a least solution and a greatest one in [u,v].
= B(y i ) and δ = ((1 − (αM + β))/c) and Ꮾ δ (w i ) is an open ball centered at w i of radius δ.Therefore, for any y ∈ S, we have a y k ∈ Y such that Since q, p, and k s (t) = k(t,s) are continuous on J, they are uniformly continuous and consequently Bx n (t) − Bx n (τ) −→ 0 as t −→ τ.
n : n ∈ N} is in equicontinuous set in BM(J,R).Hence B([u,v]) is compact by Arzelá-Ascoli theorem for compactness.Thus B is a continuous and compact operator on[u,v].Finally, we have B. C. Dhage 281 Definition 4.1.A function u ∈ AC(J,R) is called a lower solution of FDE (4.2)-(4.7)if Theorem 4.2.Assume that the hypotheses (H 3 )-(H 7 ) hold.Further, suppose that Then FDE (4.2)-(4.7)has a minimal solution and a maximal one on J.