International Journal of Analysis and Applications ISSN 2291-8639 Volume 11, Number 1 (2016), 1-10 http://www.etamaths.com EXISTENCE RESULTS FOR SOME NONLINEAR FUNCTIONAL-INTEGRAL EQUATIONS IN BANACH ALGEBRA WITH APPLICATIONS LAKSHMI NARAYAN MISHRA1,2,∗ H. M. SRIVASTAVA3,4 AND MAUSUMI SEN1 Abstract. In the present manuscript, we prove some results concerning the existence of solutions for some nonlinear functional-integral equations which contains various integral and functional equations that considered in nonlinear analysis and its applications. By utilizing the techniques of noncom- pactness measures, we operate the fixed point theorems such as Darbo’s theorem in Banach algebra concerning the estimate on the solutions. The results obtained in this paper extend and improve essentially some known results in the recent literature. We also provide an example of nonlinear functional-integral equation to show the ability of our main result. 1. Introduction Measures of noncompactness and fixed point theorems are the most valuable and effective imple- ments in the framework of nonlinear analysis, which act as principal role for the solvability of linear and nonlinear integral equations. Recently, the theory of such integral equations is developed effectively and emerge in the fields of mathematical analysis, engineering, mathematical physics and nonlinear functional analysis (see [2, 1, 33, 4, 13, 23, 27, 26, 34, 35, 36, 37, 19, 8, 15, 18, 7] and some references therein). In connection with some of the integro-differential equations, the paper should be further motivated by somehow connecting the work with the works [25, 17, 3, 12, 16, 31, 32]. Maleknejad et al. [29, 30] examined the existence of solutions for the nonlinear functional-integral equations (for short NLFIE) of the form x(t) = g(t,x(t)) + f  t, t∫ 0 u(t,s,x(s))ds,x(α(t))   ,(1.1) and x(t) = f(t,x(α(t)) t∫ 0 u(t,s,x(s))ds,(1.2) respectively, by availing the Darbo fixed-point theorem with suitable combination of measure of non- compactness defined in [5]. Banaś and Sadarangani [11] as well as Maleknejad et al. [28] discussed the existence of solutions for NLFIE (1.3) f  t, t∫ 0 v(t,s,x(s))ds,x(α(t))   ·g  t, a∫ 0 u(t,s,x(s))ds,x(β(t))   . 2010 Mathematics Subject Classification. 45G10, 47H08, 47H10. Key words and phrases. measures of noncompactness; nonlinear functional-integral equation; fixed point theorem; Banach algebra. c©2016 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 1 2 MISHRA, SRIVASTAVA AND SEN Banaś and Rzepka [9, 10] dealt the existence of solutions of NLFIE and nonlinear quadratic Volterra integral equation x(t) = f(t,x(t)) t∫ 0 u(t,s,x(s))ds,(1.4) x(t) = p(t) + f(t,x(t)) t∫ 0 v(t,s,x(s))ds,(1.5) respectively. The popular nonlinear Volterra integral equation and Urysohn integral equation are given as follows x(t) = a(t) + t∫ 0 u(t,s,x(s))ds,(1.6) x(t) = b(t) + a∫ 0 v(t,s,x(s))ds,(1.7) respectively. Dhage [20] discussed the following nonlinear integral equation x(t) = a(t) a∫ 0 v(t,s,x(s))ds +   t∫ 0 u(t,s,x(s))ds   ·   a∫ 0 v(t,s,x(s))ds   .(1.8) Moreover, the familiar quadratic integral equation of Chandrasekhar type [14] has the form x(t) = 1 + x(t) a∫ 0 t t + s φ(s)x(s)ds,(1.9) which is applicable in the theories of radiative transfer, neutron transport and kinetic energy of gases (see [14, 22, 24]). In this paper, we study the existence of solutions of NLFIE x(t) =  q(t) + f(t,x(t),x(θ(t))) + F  t,x(t), t∫ 0 u(t,s,x(á(s)))ds,x(b(t))     ×G  t,x(t), a∫ 0 v(t,s,x(c(s)))ds,x(d(t))   ,(1.10) for t ∈ [0,a]. It is worthwhile mentioning that up to now equations (1.1)-(1.9) are a particular case of equation (1.10). Moreover, NLFIE (1.10) also involve with the functional equation of the first order having the form x(t) = f(t,x(t),x(θ(t))). This paper investigates existence of solutions of NLFIE (1.10) under some relevant results of fixed point theorem for the product of two operators which satisfies the Darbo condition with suitable combination of a measure of noncompactness in the Banach algebra of continuous functions in the interval [0,a]. The existence results are interesting in themselves although their solutions are continuous and stable. 2. Definitions and preliminaries This section is devoted to revise some data which will be required in our further circumstances. Let E is a real Banach space with the norm ‖ ·‖ and zero element θ ′ . Symbolically B(x,r) represents the closed ball centered at x and with radius r, as well as we indicates by Br the ball B(θ ′ ,r). The notation ME appears for the family of all nonempty and bounded subsets of E and notation NE also appears for its subfamily consisting of all relatively compact subsets. Additionally, if X(6= φ) ⊂ E NONLINEAR FUNCTIONAL-INTEGRAL EQUATIONS 3 then the symbols X̄,ConvX in consideration of the closure and convex closure of X, respectively. We exercise the definition on the concept of a measure of noncompactness [5] as follows. Definition 2.1. Let X ∈ME and µ(X) = inf { δ > 0 : X = m⋃ i=1 Xi with diam(Xi) ≤ δ, i = 1, 2, ...m } , where for a fixed number t ∈ [0,a], we denote diam X(t) = sup{|x(t) −y(t)| : x,y ∈ X}. Clearly, 0 ≤ µ(X) < ∞. µ(X) is called the Kuratowski measure of noncompactness. Theorem 2.1. Let X,Y ∈ME and λ ∈ R. Then (i) µ(X) = 0 if and only if X ∈NE; (ii) X ⊆ Y ⇒ µ(X) ≤ µ(Y ); (iii) µ(X̄) = µ(ConvX) = µ(X); (iv) µ(X ∪Y ) = max{µ(X),µ(Y )}; (v) µ(λX) = |λ|µ(X), where λX = {λx : x ∈ X}; (vi) µ(X + Y ) ≤ µ(X) + µ(Y ), where X + Y = {x + y : x ∈ X,y ∈ Y}; (vii) |µ(X) −µ(Y )| ≤ 2dh(X,Y ), where dh(X,Y ) denotes the Hausdorff metric of X and Y , i.e. dh(X,Y ) = max { sup y∈Y d(y,X), sup x∈X d(x,Y ) } , where d(., .) is the distance from an element of E to a set of E. Furthermore, every function µ : ME → [0,∞), satisfying conditions (i)-(vi) of Theorem 2.1, will be called a regular measure of noncompactness in the Banach space E (cf. [9]). Now let us theorize that Ω is a nonempty subset of a Banach space E and S : Ω → E is a continuous operator, which transforms bounded subsets of Ω to bounded ones. Additionally, let µ be a regular measure of noncompactness in E. Definition 2.2. (see [5]) The continuous operator S satisfies the Darbo condition with a constant K ′ with respect to measure µ provided µ(SX) ≤ K ′ µ(X) for each X ∈ME such that X ⊂ Ω. If K ′ < 1, then S is called a contraction with respect to µ. In the continuation, consider the space C[0,a] is consisting of all real functions defined and contin- uous on the interval [0,a]. The space C[0,a] is equipped with standard norm ‖x‖ = sup{|x(t)| : t ∈ [0,a]}. Evidently, the space C[0,a] has also the structure of Banach algebra. Taking into our considerations, we will utilize a regular measure of noncompactness defined in [6] (cf. also [5]). Let us fix a set X ∈ MC[0,a]. For x ∈ X and for a given � > 0 denote by w(x,�) the modulus of continuity of x, i.e., w(x,�) = sup{|x(t) −x(s)| : t,s ∈ [0,a]; |t−s| ≤ �}. Further, put w(X,�) = sup{w(x,�) : x ∈ X}, w0(X) = lim �→0 w(X,�). The function w0(X) is a regular measure of noncompactness in the space C[0,a], which can be shown in [6]. For our purposes we will require the following lemma and theorem [21, 6]. Lemma 2.1. Let D be a bounded, closed and convex subset of E. If operator S : D → D is a strict set contraction, then S has a fixed point in D. 4 MISHRA, SRIVASTAVA AND SEN Theorem 2.2. Let us suppose that Ω is a nonempty, bounded, convex and closed subset of C[0,a] and the operators P and T transform continuously the set Ω into C[0,a], just like that P(Ω) and T(Ω) are bounded. Furthermore, let the operator S = P ·T transform Ω into itself. If the each operators P and T satisfies the Darbo condition on the set Ω with the constants K1 and K2, respectively, then the operator S satisfies the Darbo condition on Ω with the constant ‖P(Ω)‖K2 + ‖T(Ω)‖K1. Remark 2.1. In Theorem 2.2, if ‖P(Ω)‖K2 + ‖T(Ω)‖K1 < 1, then S is a contraction with respect to the measure w0 and has at least one fixed point in the set Ω. Now we will identify solutions of the integral equation (1.10). 3. Main result In this section, we will study the solvability of NLFIE (1.10) for x ∈ C[0,a], under the following hypotheses. (A1) The function q : [0,a] → R is continuous and bounded with k = supt∈[0,a] |q(t)|. (A2) The functions f : [0,a] × R × R → R; F,G : [0,a] × R × R × R → R are continuous and there exists nonnegative constants l,m such that |f(t, 0, 0)| ≤ l, |F(t, 0, 0, 0)| ≤ m, |G(t, 0, 0, 0)| ≤ m. (A3) There exists the continuous functions aj : [0,a] → [0,a], for j = 1, 2, ...8 such that |f(t,x1,x2) −f(t,y1,y2)| ≤ a1(t)|x1 −y1| + a2(t)|x2 −y2|, |F(t,x1,y1,x2) −F(t,x3,y2,x4)| ≤ a3(t)|x1 −x3| + a4(t)|y1 −y2| + a5(t)|x2 −x4|, |G(t,x1,y1,x2) −G(t,x3,y2,x4)| ≤ a6(t)|x1 −x3| + a7(t)|y1 −y2| + a8(t)|x2 −x4|, for all t ∈ [0,a] and x1,x2,x3,x4,y1,y2 ∈ R. (A4) The functions u = u(t,s,x(á(s))) and v = v(t,s,x(c(s))) act continuously from the set [0,a]× [0,a] × R into R. Moreover, the functions θ, á,b,c and d transform continuously the interval [0,a] into itself. (A5) There exists a nonnegative constant K such that K = max j {aj(t) : t ∈ [0,a]}, for j = 1, 2, ...8. (A6) (Sublinear condition) There exists the constants ξ and η such that |u(t,s,x(á(s)))| ≤ ξ + η|x|, |v(t,s,x(c(s)))| ≤ ξ + η|x|, for all t,s ∈ [0,a] and x ∈ R. (A7) 4στ < 1, for σ = 4K + Kaη and τ = k + l + Kaξ + m. Now we can formulate the main result of this paper. Theorem 3.1. Under the assumptions (A1) − (A7), NLFIE (1.10) has at least one solution in the Banach algebra C = C[0,a]. Proof. To prove this result using Theorem 2.2, we consider the operators P and T on the Banach algebra C[0,a] in the following way: (Px)(t) = q(t) + f(t,x(t),x(θ(t))) + F  t,x(t), t∫ 0 u(t,s,x(á(s)))ds,x(b(t))   , (Tx)(t) = G  t,x(t), a∫ 0 v(t,s,x(c(s)))ds,x(d(t))   , NONLINEAR FUNCTIONAL-INTEGRAL EQUATIONS 5 for t ∈ [0,a]. Now, taking into account the assumptions (A1), (A2) and (A4), it is clear that P and T transforms the Banach algebra C[0,a] into itself. Now, the operator S defined on the algebra C[0,a] as follows Sx = (Px) · (Tx). Definitely, S transform C[0,a] into itself. Next, let us fix x ∈ C[0,a], then using our imposed assumptions for t ∈ [0,a], we obtain |(Sx)(t)| = |(Px)(t)|× |(Tx)(t)| = ∣∣∣∣∣∣q(t) + f(t,x(t),x(θ(t))) + F  t,x(t), t∫ 0 u(t,s,x(á(s)))ds,x(b(t))   ∣∣∣∣∣∣ × ∣∣∣∣∣∣G  t,x(t), a∫ 0 v(t,s,x(c(s)))ds,x(d(t))   ∣∣∣∣∣∣ ≤ { k + |f(t,x(t),x(θ(t))) −f(t, 0, 0)| + |f(t, 0, 0)| + ∣∣∣∣∣∣F  t,x(t), t∫ 0 u(t,s,x(á(s)))ds,x(b(t))  −F(t, 0, 0, 0) ∣∣∣∣∣∣ + |F(t, 0, 0, 0)| } ×   ∣∣∣∣∣∣G  t,x(t), a∫ 0 v(t,s,x(c(s)))ds,x(d(t))  −G(t, 0, 0, 0) ∣∣∣∣∣∣ + |G(t, 0, 0, 0)|   ≤  k + a1(t)|x(t)| + a2(t)|x(θ(t))| + l + a3(t)|x(t)| + a4(t) t∫ 0 |u(t,s,x(á(s)))|ds + a5(t)|x(b(t))| + m   ×  a6(t)|x(t)| + a7(t) a∫ 0 |v(t,s,x(c(s)))|ds + a8(t)|x(d(t))| + m   ≤{k + 4K‖x‖ + l + Ka(ξ + η‖x‖) + m} ·{2K‖x‖ + Ka(ξ + η‖x‖) + m} ≤{(4K + Kaη)‖x‖ + k + l + Kaξ + m}2. Let σ = 4K + Kaη and τ = k + l + kaξ + m, then from the above estimate, it follows that ‖Px‖≤ σ‖x‖ + τ,(3.1) ‖Tx‖≤ σ‖x‖ + τ,(3.2) ‖Sx‖≤ (σ‖x‖ + τ)2,(3.3) for x ∈ C[0,a]. From estimate (3.3), we conclude that the operator S maps the ball Br ⊂ C[0,a] into itself for r1 ≤ r ≤ r2, where r1 = 1 − 2στ − √ 1 − 4στ 2σ2 , r2 = 1 − 2στ + √ 1 − 4στ 2σ2 . In the following, we will assume that r = r1. Moreover, let us observe that from estimates (3.1) and (3.2), we obtain ‖PBr‖≤ σr + τ,(3.4) ‖TBr‖≤ σr + τ.(3.5) 6 MISHRA, SRIVASTAVA AND SEN Now, we have to prove that the operator P is continuous on the ball Br. To do this, fix � > 0 and take arbitrary x,y ∈ Br such that ‖x−y‖≤ �. Then for t ∈ [0,a], we have |(Px)(t) − (Py)(t)| ≤ |f(t,x(t),x(θ(t))) −f(t,y(t),y(θ(t)))| + ∣∣∣∣∣∣F  t,x(t), t∫ 0 u(t,s,x(á(s)))ds,x(b(t))  −F  t,y(t), t∫ 0 u(t,s,y(á(s)))ds,y(b(t))   ∣∣∣∣∣∣ ≤ a1(t)|x(t) −y(t)| + a2(t)|x(θ(t)) −y(θ(t))| + ∣∣∣∣∣∣F  t,x(t), t∫ 0 u(t,s,x(á(s)))ds,x(b(t))  −F  t,y(t), t∫ 0 u(t,s,x(á(s)))ds,y(b(t))   ∣∣∣∣∣∣ + ∣∣∣∣∣∣F  t,y(t), t∫ 0 u(t,s,x(á(s)))ds,y(b(t))  −F  t,y(t), t∫ 0 u(t,s,y(á(s)))ds,y(b(t))   ∣∣∣∣∣∣ ≤ a1(t)|x(t) −y(t)| + a2(t)|x(θ(t)) −y(θ(t))| + a3(t)|x(t) −y(t)| + a5(t)|x(θ(t)) −y(θ(t))| + a4(t) t∫ 0 |u(t,s,x(á(s))) −u(t,s,y(á(s)))|ds ≤ 4K‖x−y‖ + Ka w(u,�) ≤ 4K� + Ka w(u,�), where w(u,�) = sup{|u(t,s,x) −u(t,s,y)| : t,s ∈ [0,a]; x,y ∈ [−r,r];‖x−y‖≤ �}. In view of uniformly continuous of the function u = u(t,s,x) on the bounded subset [0,a]×[0,a]×[−r,r], we have that w(u,�) → 0 as � → 0. Thus, from the above inequality the operator P is continuous on Br. Similarly, the operator T is also continuous on Br. Hence, we conclude that S is continuous operator on Br. Next, we prove that the operators P and T satisfies the Darbo condition with respect to the measure w0, defined in Section 2, in the ball Br. To do this, we take a nonempty subset X of Br and x ∈ X. Let � > 0 be fixed and t1, t2 ∈ [0,a] with t2 − t1 ≤ � and we can assume that t1 ≤ t2. Then, taking into account our assumptions, it follows |(Px)(t2) − (Px)(t1)| ≤ |q(t2) −q(t1)| + |f(t2,x(t2),x(θ(t2))) −f(t1,x(t1),x(θ(t1)))| + ∣∣∣∣∣∣F  t2,x(t2), t2∫ 0 u(t2,s,x(á(s)))ds,x(b(t2))   − F  t1,x(t1), t1∫ 0 u(t1,s,x(á(s)))ds,x(b(t1))   ∣∣∣∣∣∣ ≤ w(q,�) + |f(t2,x(t2),x(θ(t2))) −f(t2,x(t1),x(θ(t1)))| + |f(t2,x(t1),x(θ(t1))) −f(t1,x(t1),x(θ(t1)))| + ∣∣∣∣∣∣F  t2,x(t2), t2∫ 0 u(t2,s,x(á(s)))ds,x(b(t2))   (3.6) NONLINEAR FUNCTIONAL-INTEGRAL EQUATIONS 7 − F  t2,x(t1), t1∫ 0 u(t1,s,x(á(s)))ds,x(b(t1))   ∣∣∣∣∣∣ + ∣∣∣∣∣∣F  t2,x(t1), t1∫ 0 u(t1,s,x(á(s)))ds,x(b(t1))   − F  t1,x(t1), t1∫ 0 u(t1,s,x(á(s)))ds,x(b(t1))   ∣∣∣∣∣∣ ≤ w(q,�) + a1(t)|x(t2) −x(t1)| + a2(t)|x(θ(t2)) −x(θ(t1))| + wf (�, ., .) + a3(t)|x(t2) −x(t1)| + a4(t) ∣∣∣∣∣∣ t2∫ 0 u(t2,s,x(á(s)))ds− t1∫ 0 u(t1,s,x(á(s)))ds ∣∣∣∣∣∣ + a5(t)|x(b(t2)) −x(b(t1))| + wF (�, ., ., .) ≤ w(q,�) + 2Kw(x,�) + Kw(x,w(θ,�)) + wf (�, ., .) + K   t1∫ 0 |u(t2,s,x(á(s))) −u(t1,s,x(á(s)))|ds + t2∫ t1 |u(t2,s,x(á(s)))|ds   + Kw(x,w(b,�)) + wF (�, ., ., .) w(Px,�) ≤ w(q,�) + 2Kw(x,�) + Kw(x,w(θ,�)) + wf (�, ., .) + K{wu(�, ., .)a + K′�} + Kw(x,w(b,�)) + wF (�, ., ., .)(3.7) where wf (�, ., .) = sup{|f(t,x1,x2) −f(t′,x1,x2)| : t,t′ ∈ [0,a]; |t− t′| ≤ �; x1,x2 ∈ [−r,r]}, wu(�, ., .) = sup{|u(t,s,x) −u(t′,s,x)| : t,t′ ∈ [0,a]; |t− t′| ≤ �; x ∈ [−r,r]}, wF (�, ., ., .) = sup{|F(t,x1,y1,x2) −F(t′,x1,y1,x2)| : t,t′ ∈ [0,a]; |t− t′| ≤ �; x1,x2 ∈ [−r,r]; y1 ∈ [−K′a,K′a]}, K′ = sup{|u(t,s,x)| : t,s ∈ [0,a]; x ∈ [−r,r]}. Since, the functions q = q(t), f = f(t,x1,x2) and F = F(t,x1,y1,x2) are uniformly continuous on the set [0,a], [0,a]×R×R and [0,a]×R×R×R, respectively, and the function u = u(t,s,x) is also uniformly continuous on the set [0,a]×[0,a]×R. Hence, we deduce that w(q,�) → 0,wf (�, ., .) → 0,wu(�, ., .) → 0 and wF (�, ., ., .) → 0 as � → 0. Thus, from the above estimate (3.6) we conclude (3.8) w0(PX) ≤ 4Kw0(X). Similarly, we can show that (3.9) w0(TX) ≤ 2Kw0(X). Finally, from the estimates (3.4), (3.5), (3.7), (3.8) and keeping in mind Theorem 2.2, we conclude that the operator S satisfies the Darbo condition on Br with respect to the measure w0 with constant 4K(σr + τ) + 2K(σr + τ). Thus, we have 6K(σr + τ) = 6K(σr1 + τ) = 6K { σ ( (1 − 2στ) − √ 1 − 4στ 2σ2 ) + τ } = 3K σ (1 − √ 1 − 4στ). 8 MISHRA, SRIVASTAVA AND SEN Taking into account the assumption (A7), since 1 − √ 1 − 4στ < 1 and 3K σ = 3K 4K + Kaη < 1. Therefore, the operator S is a contraction on Br with respect to measure w0. Thus, S has at least one fixed point in the ball Br, by applying Theorem 2.2 and Remark 2.1. Consequently, the NLFIE (1.10) has at least one solution in the ball Br. � 4. An example Now, we begin with an example of a NLFIE and to illustrate the existence of its solutions by using Theorem 3.1. Example 4.1. Consider the following NLFIE: x(t) = [ te−(t+3) + t 7(1 + t) arctan |x(t)| + t 16 ln(1 + |x(1 − t)|) + 1 12 t∫ 0 { cos(x(1 −s)) 3 + 2t arctan ( |x(1 −s)| 1 + |x(1 −s)| )} ds ] × [ 1 17 1∫ 0 { t sin x( √ s) 3 + (1 + t) ln(1 + |x( √ s)|) } ds ] ,(4.1) where t ∈ [0, 1]. Observe that equation (4.1) is a particular case of equation (1.10). Let us take q : [0, 1] → R; f : [0, 1] ×R×R → R; F,G : [0, 1] ×R×R×R → R and u,v : [0, 1] × [0, 1] ×R → R and comparing (4.1) with equation (1.10), we get q(t) = te−(t+3),f(t,x1,x2) = t 7(1 + t) arctan |x1| + t 16 ln(1 + |x2|), F(t,x1,y1,x2) = 1 12 y1,G(t,x1,y1,x2) = 1 17 y1, u(t,s,x) = cos x 3 + 2t arctan ( |x| 1 + |x| ) ,v(t,s,x) = t sin x 3 + (1 + t) ln(1 + |x|), then we can easily test that the assumptions of Theorem 3.1 are satisfied. In fact, we have that the function q(t) is continuous and bounded on [0, 1] with k = e−4 = 0.0183156... . Thus, the assumption (A1) is satisfied. Moreover, these functions are continuous and satisfies the assumption (A3) with a1 = 1 14 ,a2 = 1 16 ,a3 = a5 = a6 = a8 = 0,a4 = 1 12 ,a7 = 1 17 . In this case, we have K = max { 1 14 , 1 16 , 0, 1 12 , 1 17 } = 1 12 . Further, |f(t, 0, 0)| = 0, |F(t, 0, 0, 0)| = 0, |G(t, 0, 0, 0)| = 0, |u(t,s,x)| ≤ 1 3 + 2|x|, |v(t,s,x)| ≤ 1 3 + 2|x|. It is observed that l = m = 0,ξ = 1 3 ,η = 2 and a = 1. Finally, we see that 4στ = 4(4K + Kaη)(k + l + Kaξ + m) < 1. Hence, all the assumptions from (A1) to (A7) are satisfied. Now, on the basis of result obtained in Theorem 3.1, we deduce that NLFIE (4.1) has at least one solution in Banach algebra C[0, 1]. NONLINEAR FUNCTIONAL-INTEGRAL EQUATIONS 9 Acknowledgments The authors wishes to express their deep gratitude to the anonymous learned referee(s) and the editor for their valuable suggestions and constructive comments, which resulted in the subsequent improvement of this research article. The authors are also grateful to all the editorial board members and reviewers of this esteemed journal. The first author Lakshmi Narayan Mishra is thankful to the Ministry of Human Resource Development, New Delhi, India and Department of Mathematics, National Institute of Technology, Silchar, India for supporting this research article. References [1] R.P. Agarwal, N. Hussain, M.A. Taoudi, Fixed point theorems in ordered Banach spaces and applications to nonlinear integral equations, Abstr. Appl. Anal. 2012 (2012), Article ID 245872. [2] R.P. Agarwal, D. O’Regan, P.J.Y. Wong, Positive Solutions of Differential, Difference and Integral Equations, Kluwer Academic, Dordrecht, 1999. [3] G. Anichini, G. Conti, Existence of solutions of some quadratic integral equations, Opuscula Math. 28 (4) (2008), 433-440. [4] J. Banaś, A. Chlebowicz, On existence of integrable solutions of a functional integral equation under Carathéodory conditions, Nonlinear Anal. 70 (9) (2009), 3172-3179. [5] J. Banaś, K. Goebel, Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathe- matics, vol. 60, Marcel Dekker, New York, 1980. [6] J. Banaś, M. Lecko, Fixed points of the product of operators in Banach algebra, Panamer. Math. J. 12 (2) (2002), 101-109. [7] J. Banaś, M. Mursaleen, Sequence Spaces and Measures of Noncompactness with Applications to Differential and Integral Equations, Springer, New York, 2014. [8] J. Banaś, B. Rzepka, An application of a measure of noncompactness in the study of asymptotic stability, Appl. Math. Lett. 16 (1) (2003), 1-6. [9] J. Banaś, B. Rzepka, On existence and asymptotic stability of solutions of a nonlinear integral equation, J. Math. Anal. Appl. 284 (1) (2003), 165-173. [10] J. Banaś, B. Rzepka, On local attractivity and asymptotic stability of solutions of a quadratic Volterra integral equation, Appl. Math. Comput. 213 (1) (2009), 102-111. [11] J. Banaś, K. Sadarangani, Solutions of some functional-integral equations in Banach algebra, Math. Comput. Modelling 38 (2003), 245-250. [12] V.C. Boffi, G. Spiga, An equation of hammerstein type arising in particle transport theory, J. Math. Phys. 24 (6) (1983), 1625-1629. [13] T.A. Burton, B. Zhang, Fixed point and stability of an integral equation: nonuniqueness, Appl. Math. Lett. 17 (7) (2004), 839-846. [14] S. Chandrasekhar, Radiative Transfer, Oxford Univ Press, London, 1950. [15] C. Corduneanu, Integral Equations and Applications, Cambridge University Press, New York, 1990. [16] G. Darbo, Punti uniti in trasformazioni a codominio non compatto, Rend. Sem. Mat. Univ. Padova 24 (1955), 84-92. [17] M.A. Darwish, K. Sadarangani, Nondecreasing solutions of a quadratic Abel equation with supremum in the kernel, Appl. Math. Comput. 219 (14) (2013), 7830-7836. [18] Deepmala, H.K. Pathak, A study on some problems on existence of solutions for nonlinear functional-integral equations, Acta Math. Sci. 33 (5) (2013), 1305-1313. [19] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, New York, 1985. [20] B. C. Dhage, On α-condensing mappings in Banach algebras, Math. Student 63 (1994), 146-152. [21] D. Guo, V. Lakshmikantham, X.Z. Liu, Nonlinear Integral Equations in Abstract Spaces, Kluwer, Dordrecht, 1996. [22] S. Hu, M. Khavani, W. Zhuang, Integral equations arising in the kinetic theory of gases, Appl. Anal. 34 (1989), 261-266. [23] X.L. Hu, J.R. Yan, The global attractivity and asymptotic stability of solution of a nonlinear integral equation, J. Math. Anal. Appl. 321 (1) (2006), 147-156. [24] C.T. Kelly, Approximation of solutions of some quadratic integral equations in transport theory, J. Integral Equa- tions 4 (3) (1982), 221-237. [25] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North- Holland Mathematical Studies, vol. 204, Elsevier (North-Holland) Science Publishers, Amsterdam, London and New York, 2006. [26] Z. Liu, S.M. Kang, Existence and asymptotic stability of solutions to a functional-integral equation, Taiwanese J. Math. 11 (1) (2007), 187-196. [27] Z. Liu, S.M. Kang, Existence of monotone solutions for a nonlinear quadratic integral equation of Volterra type, Rocky Mountain J. Math. 37 (6) (2007), 1971-1980. [28] K. Maleknejad, R. Mollapourasl, K. Nouri, Study on existence of solutions for some nonlinear functional integral equations, Nonlinear Anal. 69 (8) (2008), 2582-2588. 10 MISHRA, SRIVASTAVA AND SEN [29] K. Maleknejad, K. Nouri, R. Mollapourasl, Existence of solutions for some nonlinear integral equations, Commun. Nonlinear Sci. Numer. Simul. 14 (6) (2009), 2559-2564. [30] K. Maleknejad, K. Nouri, R. Mollapourasl, Investigation on the existence of solutions for some nonlinear functional- integral equations, Nonlinear Anal. 71 (12) (2009), e1575-e1578. [31] L.N. Mishra, R.P. Agarwal, M. Sen, Solvability and asymptotic behavior for some nonlinear quadratic integral equation involving Erdélyi-Kober fractional integrals on the unbounded interval, Progr. Fract. Differ. Appl. in press. [32] L.N. Mishra, M. Sen, On the concept of existence and local attractivity of solutions for some quadratic Volterra integral equation of fractional order, Appl. Math. Comput. 285 (2016) 174-183. [33] L.N. Mishra, M. Sen, R.N. Mohapatra, On existence theorems for some generalized nonlinear functional-integral equations with applications, Filomat, in press. [34] D. O’Regan, Existence results for nonlinear integral equations, J. Math. Anal. Appl. 192 (3) (1995), 705-726. [35] D. O’Regan, M. Meehan, Existence Theory for Nonlinear Integral and Integrodifferential Equations, Kluwer, Dor- drecht, 1998. [36] H.K. Pathak, Deepmala, Remarks on some fixed point theorems of Dhage, Appl. Math. Lett. 25 (11) (2012), 1969-1975. [37] P.P. Zabrejko, A.I. Koshelev, M.A. Krasnosel’skii, S.G. Mikhlin, L.S. Rakovshchik, V.J. Stetsenko, Integral Equa- tions, Noordhoff, Leyden, 1975. 1Department of Mathematics, National Institute of Technology, Silchar 788 010, Cachar, Assam, India 2L. 1627 Awadh Puri Colony, Phase III, Beniganj, Opposite Industrial Training Institute (I.T.I.), Ayodhya Main Road, Faizabad 224 001, Uttar Pradesh, India 3Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada 4China Medical University, Taichung 40402, Taiwan, Republic of China ∗Corresponding author: lakshminarayanmishra04@gmail.com