Malaya Journal of Matematik, Vol. 6, No. 3, 56-53, 28 https://doi.org/.26637/mjm63/8 Qualitative behaviour of solutions of hybrid fractional integral equation with linear perturbation of second kind in Banach Space Kavita Sakure * and Samir Dashputre2 Abstract In this paper, we present existence and qualitative behaviour of solution of hybrid fractional integral equation with linear perturbation of second kind by applying measure of noncompactness in Banach space. We established our result in the Banach space of real-valued functions defined, continuous and bounded in the right hand real axis. Keywords Hybrid fractional integral equation, measure of noncompactness, attractivity of solution. AMS Subject Classification 45G, 45M, 47H. Department of Mathematics, Govt. Digvijay Auto. P.G. College, Rajnandgaon, 4944, Chhattisgarh, India. of Mathematics, Govt. College, Arjunda, 49225, Chhattisgarh, India. *Corresponding author: kavitaaage@gamil.com; 2 samir23973@gmail.com Article History: Received 2 March 28; Accepted 27 May 28 2 Department Contents Introduction....................................... 56 2 Basic Definitions and Results.................... 57 3 Attractivity and Positivity of Solutions........... 59 4 Conclusion........................................ 52 5 Acknowledgement................................ 52 References........................................ 52. Introduction Measure of noncompactness and fixed point theorems are the most valuable and effective implements in the framework of nonlinear analysis, which acts as principle role for 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 (e.g.,, 3-5,9,, 5-7, 9). Nonlinear integral equation with bounded intervals has been studied extensively in the literature as regard various aspects of the solutions. This includes existence, uniqueness, stability and extremality of solutions. But the study of nonlinear integral equation with unbounded intervals is relatively new and exploited for the new characteristics of attractivity c 28 MJM. and asymptotic attractivity of solutions. There are two approaches for dealing with theses characteristics of solutions, namely, classical fxed point theorems involving the hypotheses from analysis and topology and the fixed point theorems involving the use of measure of noncompactness. Each one of these approaches has some advantage and disadvantages over the others was discussed in Dhage [2]. In 25, Apell [2] discussed some measure of noncompactness in the application of nonlinear integral equations. Let J = [t,t a] in R be a closed and bounded interval where t R and a R with a > and a given a real number < q <. Consider the hybrid fractional integral equation(hfie) with linear perturbation of second type a(t) = h(t) f (t, a(t)) (t s)q g(t, a(s))ds (.) t where t, s J,h : J R, f : J R R is continuous and g : J R R is locally Holder continuous. In this paper,we are going to discuss two qualitative behavior such as global attractivity and positivity of hybrid fractional integral equation (.) with linear perturbation of second type using measure of noncompactness under certain conditions. We established our result in the Banach space of real-valued functions defined, continuous and bounded on the right hand real half axis R.
Banach Space 57/53 Definition 2.6. A mapping K : X X is called D set Lipschitz if there exists a continuous nondecreasing function φ : R R such that µ(k(x )) φ (µ(x )) for all X Pbd (X) with K(X ) Pbd (X), where φ () =. Sometimes we call the function φ to be a D f unction of K on X. In the special case, when φ (r) = kr, k >, K is called a k set Lipschitz mapping and if k <, then K is called a k set contraction on X. If φ (r) < r for r >, then K is called a nonlinear D set contraction on X. 2. Basic Definitions and Results This section is devoted to conflict to collect some definitions and auxialiary results which will be needed in the further considerations of this paper. At the beginning we present some basic facts concerning the measure of noncompactness. We accept the following definitions of the concept of a measure of noncompactness given in Dhage [7]. Let X be a Banach space, P(X) a class of subset of E with property p. Pcl (X),Pbd (X),Pcl,bd (X),Prcp (X) denote the class of closed, bounded, closed and bounded and relatively compact subsets of X respectively. A function dh : P(X) P(X) R defined by dh (A, B) = max{sup D(a, B), sup D(b, A)} a A (2.) Theorem 2.7. ([4]) Let C be a non-empty, closed, convex and bounded subset of a Banach space X and let K : C C be a continuous and nonlinear D set contraction. Then K has a fixed point. b B Remark 2.8. Let us write Fix(K) by the set all fixed points of the operator K which belongs to C. It can be easily shown that the Fix(K) existing in Theorem 2.7 belongs to family kerµ. In fact if Fix(K) / kerµ, then µ(fix(k)) > and K(Fix(K)) = Fix(K). From nonlinear D set contraction it follows that µ(k(fix(k))) φ (µ(fix(k))) which is a contradiction since φ (r) < r for r >. Hence Fix(K) ker(µ). satisfies all the conditions of a metric on P(X) and is called Hausdorff Pompeiu metric on X, where D(a, B) = inf{ka bk : b B}. It is clear that the space (Pcl (X), dh ) is complete metric space. Definition 2.. A sequence {Xn } of non-empty sets in P p (X) is said to be converge to a set X, called the limiting set if dh (Xn ) as n. a mapping µ : P p (X) R is called continuous if for any sequence {Xn } in P p (X) we have dh (Xn, X) µ(xn ) µ(x) as Let the Banach space BC(R, R) be consisting of all real functions a = a(t) defined, continuous and bounded on R. This space is equipped with the standard supremum norm n kak = sup{ a(t) : t R } Definition 2.2. A mapping µ : P p (X) R is called if X, X2 We will use the Hausdorff or ball measure of noncompactness P p (X) are two sets with A B, then µ(x ) µ(x2 ), where in BC(R, R). A formula for Hausdorff measure of noncom is a order relation by inclusion in P p (X). pactness useful in application is defined as follows. Let us fix Now we define the measure of noncompactness for a a nonempty and bounded subset X of the space BC(R, R) bounded subset of the Banach space X. and a positive number T. For x X and ε denote by ω T (a, ε) the modulus of continuity of the function a on the Definition 2.3. Let X X. A function µ : Pbd (X) R is closed and bounded interval [, T ] defined by called a measure of noncompactness, if it satisfies: ω T (a, ε) = sup{ a(t) a(s) : t, s [, T ], t s ε}. φ 6= µ () Prcp (X), Next, let us 2. µ(x ) = µ(x ), where X is closure of X, ω T (X, ε) = sup{ ω T (a, ε) : a X }, 3. µ(x ) = µ(conv(x )), where Conv(X ) is convex hull of X, ωt (X ) = lim ω T (X, ε). ε 4. µ is nondecreasing, and 5. if {Xn } is a decreasing sequence of sets in Pbd (X) such that limn µ(xn ) =, then the limiting set X = limn = n= Xn is nonempty. ωt It is known that is a measure of noncompactness in the Banach space C([, T ], R) of real valued and continuous functions defined on a closed and bounded interval [, T ] in R which is equivalent to Hausdorff or ball measure Definition 2.4. The family kerµ is said to be the kernel of measure of noncompactness where χ(x ) = ωt (X ) 2 for any bounded subset X of C([, T ], R). We define kerµ = {X Pbd (X) µ(x ) = } Prcp (X). ω (X ) = lim ωt (X ). Definition 2.5. A measure µ is complete or full if the kernel kerµ of µ consists of all possible relatively compact subsets of X. For a fixed number t R let us write X (t) = {a(t) : a X }, The following definition appear in Dhage[2]. 57
Banach Space 58/53 k X (t) k= sup{a(t) : a X }, Definition 2.. We say that solutions of the equation (2.8) are locally attractive if there exists a closed ball B r (a ) in the space BC(R, R) for some a BC(R, R) such that arbitrary solutions a = a(t) and b = b(t) of the equation(2.8) belonging to B r (a ) Ω we have that and k X (t) c k= sup{a(t) c : x X }. Let the functions µ s be defined on the family Pcl,bd (X ) by the formulas µa (X ) = max{ω (X ), lim sup diamx (t)}, lim (a(t) b(t)) =. (2.2) µb (X ) = max{ω (X ), lim sup k X (t) k}, In this case when the limit(2.9) is uniform with respect to the set B r (a ) Ω, i.e., when for each ε > there exists T > such that (2.3) µc (X ) = max{ω (X ), lim sup k X (t) c k}. (2.9) a(t) b(t) ε (2.4) (2.) for all a, b B r (a ) Ω being solution of (2.) and for t T, we will say that solutions of equation(2.8) are uniformally locally attractive on R. Let T > be fixed. Then for any a BC(R, R) define δt (a) = sup{ a(t) a(t) : a X }, δt (X ) = sup{δt (a) : a X } Definition 2.. The solution a = a(t) of equation(2.8) is said to be globally attractive if (2.9) holds for each solution b = b(t) of (2.8) on ω. In other words, we may say that the solutions of the equation (2.9) are globally attractive if for arbitrary solutions a(t) and b(t) of (2.8) on Ω, the condition (2.9) is satisfied. In the case when the condition (2.9) is satisfied uniformly with respect to the set Ω, i.e., if for every ε > there exists T > such that the inequality(2.) is satisfied for all a, b Ω being the solutions of (2.8) and for t T, we will say that solutions of the equation(2.8) are uniformly globally attractive on R. δ (X ) = lim δt (X ) Define functions µad, µbd, µcd : Pbd (X) R by µad (X ) = max{µa (X ), δ (X )}, (2.5) µbd (X ) = max{µb (X ), δ (X )} (2.6) µcd (X ) = max{µc (X ), δ (X )} (2.7) for all X Pcl,bd (X). Remark 2.9. It is shown as in Banas and Goebel [7] that the functions µa, µb, µc, µad, µbd and µcd are measure of noncompactness in the space BC(R, R). The kernels kerµa, kerµb and kerµc of the measures µa, µb and µc consists of nonempty and bounded subsets X of BC(R, R) such that functions from X are locally equicontinuous on R and the thickness of bundle formed by functions from X tends to zero at infinity. The functions from kerµc come closer along a line y(t) = c and the functions from kerµb come closer to line y(t) = c as t increases to through R. A similar situation is true for the kernels kerµad, kerµbd and kerµcd. Moreover, these measure µad, µbd and µcd characterize the ultimate positivity of the functions belonging to the kernels of kerµad, kerµbd and kerµcd. The above property of kerµa, kerµb, kerµc and kerµad, kerµbd, kerµcd permits us to characterize solutions of the integral equations considered in the sequel. In order to introduce further concepts used in this paper, let us assume that X = BC(R, R) and Ω be a subset of X. Let K : X X be an operator and consider the following operator equation in X, Ka(t) = a(t) (2.8) for all t R. We give different characterizations of the solutions for the the operator equation (2.8) on R. The following definitions appear in Dhage[3]. 58 Definition 2.2. A line b(t) = c, where c a real number, is called an attractor for a solution a BC(R, R) to the equation(2.8) if lim [a(t) c] = and the solution a to the equation (2.8) is also called asymptotic to the line b(t) = c and the line is an asymptote for the solution a on R. The following definitions appear in Dhage[2]. Definition 2.3. The solutions of equations (2.8) are said to be globally asymptotic attractive if for any two solutions a = a(t) and b = b(t) of the equation (2.8), the condition (2.9) is satisfied and there is a line which is a common attractor to them on R. When the condition (2.9) is satisfied uniformly, i.e., if for every ε > there exists T > such that the inequality (2.) is satisfied for t T and for all a, b being the solution of (2.8) and having a line as common attractor, we will say that solutions of the equation (2.8) are uniformly globally asymptotically attractive on R. Remark 2.4. The notion of global attractivity of solutions are introduced in Hu and Yan [7] and concept of global and local asymptotic attractivity have been presented in Dhage[3] while concept of uniform global and local attractivity were introduced in Banas and Rzepka [6] and concept of global asymptotic attractivity of solutions are presented in Dhage[2] and local attractivity of a nonlinear quadratic fractional integral equation have been presented in [].
Banach Space 59/53 (K4 ) The function g : J R R is continuous such that there is a continuous map B : J J such that Definition 2.5. A solution a of the equation (2.8) is called locally ultimately positive if there exists a closed ball B r (a ) in BC(R, R) for some a BC(R, R) such that a B r (a ) and lim [ a(t) a(t)] =. g(t, a) B(t) for t, s J. Moreover, we assume that (2.) lim When the limit (2.) is uniform with respect to the solution set of the operator equation (2.8),i.e., when for each ε > there exist T > such that a(t) a(t) ε Theorem 3.. Under the assumptions (K )- (K4 ), the FIE(.) for t = has atleast one solution in the space C(J, R). Moreover solution of FIE(.) are globally uniformly attractive on J. (2.2) for all a being solutions of (2.8) and for t T,we will say that solutions of equation (2.8) are uniformly locally ultimately positive on R. Let the operator K be defined on the space C(J, R) such that Ka(t) = h(t) f (t, a(t)) Definition 2.6. A solution a C(R, R) of the equation (2.8) is called globally ultimate positive if (2.) is satisfied. When the limit (2.) is uniform with respect to the solution set of the operator equation (2.8) in C(R, R), i.e., when for each ε > there exists T > such that (2.2) is satisfied for all x being solutions of (2.8) and for t T, we will say that solutions of equation 2.8 are uniformly globally ultimate positive on R. (t s)q g(t, a(s))ds (3.) By assumptions, the function Ka(t) is continuous for any function of a C(J, R). For arbitrarily fixed t J, Ka(t) = h(t) f (t, a(t)) (t s)q g(t, a(s))ds h(t) f (t, a(t)) f (t, ) f (t, ) Z t (t s)q g(t, a(s)) ds L a(t) h f (t, ) N B(s)ds M a(t) Remark 2.7. The global attractivity and global asymptotic attractivity implies the local attractivity and local asymptotic attractivity, respectively, to the solutions for the operator equation (2.8) on R. Similarly, global ultimate positivity implies local ultimate positivity to the solutions for the operator equation (2.8) on unbounded intervals. The converse of the above statements may not be true. h L x F Nv(t) M x h L x F Nv(t) M x 3. Attractivity and Positivity of Solutions h L F Nv(t) By a solution of FIE(.), we mean a function a C(J, R) that satisfies FIE(.) where C(J, R) is the space of continuous real valued functions on J. Let FIE(.) satisfies the following assumptions: (K ) The function h : J R is continuous. (K ) The function f : J R R is continuous and there is bounded function l : J R with bound L and a positive constant M such that f (t, a ) f (t, a2 ) B(s)ds =. K(x) h L F NV (3.2) for all x C(J, R). This means that the operator K transforms the space C(J, R) into itself. From (3.2), we obtain the operator K transforms continuously the space C(J, R) into the closed ball Br (), where r = h L F NV. Therefore the existence of the solution for FIE(.) is global in nature. We will consider the operator K : Br () Br (). Now we will show that the operator K is continuous on ball Br (). Let ε > be arbitrary and take a, b Br () such that a b ε, then l(t) a a2 M a a2 for t J and for all a, a2 R. Moreover assume that L M. (K2 ) The function t 7 f (t, ) is bounded on J with F = sup{ f (t, ) : t J}. (K3 ) The function (t, s) 7 (t s)q is continuous and there is a positive real number N such that (t s)q N 59 (Ka)(t) (Kb)(t) f (t, a(t)) f (t, b(t)) (t, s)q g(t, a(s)) g(t, b(s)) ds L a(t) b(t) N2B(s)ds M a(t) b(t) L a b 2Nv(t) M x y ε 2Nv(t)
Banach Space 5/53 from assumption (K3 ), there exists T > such that v(t) ε for t T. Thus for t T, we have (Ka)(t) (Kb)(t) 3ε. we may assume that t < t2. Then (K a)(t2 ) (Ka)(t ) h(t2 ) h(t ) f (t2, a(t2 )) f (t, a(t )) 2 (t2 s)q g(t2, a(s))ds (t s)q g(t, a(s)) (3.3) (3.6) ω T (h, ε) f (t2, a(t2 )) f (t2, a(t ) f (t2, a(t )) f (t, a(t )) 2 (t2 s)q g(t2, a(s))ds (3.8) 2 (t2 s)q g(t, a(s))ds 2 (t2 s)q g(t, a(s))ds (3.9) (t s)q g(t, a(s))ds L a(t2 ) a(t ) ω T (h, ε) ωrt ( f, ε) M x(a2 ) a(t ) 2 (t2 s)q g(t2, a(s)) g(t, s, a(s)) ds 2 (t2 s)q g(t, a(s))ds (3.) 2 (t s)q g(t, a(s))ds 2 (t s)q g(t, a(s))ds (3.) (t s)q g(t, a(s))ds Let us assume that t [, T ]. Then (Ka)(t) (Kb)(t) ε (t s)q g(t, a(s)) g(t, b(s)) ds ε N ωrt (g, ε)ds ε NT ωrt (g, ε) (3.4) where ω T (h, ε) N 2 ωrt (g, ε) = sup{ g(t, a) g(t, b) : t [, T ], a, b [r, r], a b ε}. (3.7) (3.5) 2 t (t2 s)q (t s)q g(t, a(s)) ds (t s)q g(t, a(s)) ds N By uniform continuity of the functions g(t, a) on the set [, T ] [r, r], we have ωrt (g, ε) as ε. Now, by (3.4), (3.5) and above established facts we conclude that the operator K maps continuously the closed ball Br () into itself. Further, let us take a nonempty subset X of the ball Br (). Next, fix arbitrarily T > and ε >. Let us choose a X and t,t2 [, T ] with t2 t ε. Without loss of generality (3.3) ωrt (g, ε)ds Z ωt ( N 5 Lω T (a, ε) ωrt ( f, ε) M ω T (a, ε) t2 (t s)q, ε)v ds N GrT ds t Lω T (a, ε) ω T (KX, ε) ω T (h, ε) ωrt ( f, ε) M ω T (a, ε) (3.2) ωrt (g, ε)ds ω T (h, ε) Lω T (x, ε) ωrt ( f, ε) M ω T (a, ε) ωrt (g, ε)ds ωt ( (t s)q, ε)v ds NεGrT (3.4)
Banach Space 5/53 Using measure of noncompactness µa, where µa (KX ) = max{ω (KX ), lim sup KX (t)} Lω (X ) L lim sup X (t), max M ω (X ) M lim sup X (t) L max{ω (X ), lim sup X (t)} M max{ω (X ), lim sup X (t)} Lµa (X ) M µa (X ) (3.9) ω T (h, ε) = sup{ h(t2 ) h(t ) : t,t2 [, T ], t2 t ε}, ωrt ( f, ε) = sup{ f (t2, a) f (t, a) : t,t2 [, T ], t2 t ε, x, y [r, r]}, ωrt (t s)q (t2 s)q : t,t2, s [, T ], t2 t ε, (t s)q, ε = sup since L M, µa (KX ) = φ (µa (X )), Lr Mr ωrt (g, ε) where for r >. Now we apply Theorem 2.7 to deduce that operator K has a fixed point a in the ball Br (). Thus x is solution of the FIE (.). The image of the space C(J, R) under the operator K is contained in the ball Br () because the set Fix(K) of all fixed points of K is contained Br (). The set Fix(K) contain all solutions of the FIE (.) and from Remark 2.8 we conclude that the set Fix(K) belongs to the family kerµa. Now, taking account the description of sets belonging to kerµa, we have that all solutions for the FIE (.) are globally uniformly attractive on J. In order to prove next result concerning the asymptotic positivity of the attractive solutions, we need following hypotheses. (K6 ) The functions f satisfies = sup{ g(t2, a) g(t, b) : t,t2, s [, T ], t2 t ε, a, b [r, r]}, GTr = sup{ g(t, a) : t, s [, T ], x [r, r]}. Thus from the above estimate, we have Lω T (X, ε) ωrt ( f, ε) M ω T (X, ε) N ωrt (g, ε)ds ω T ( (t s)q, ε)v ds NεGrT (3.5) ω T (KX, ε) ω T (h, ε) lim [ f (t, a) f (t, a)] = By the uniform continuity of the functions f and g on the sets [, T ] [r, r], [, T ] [r, r], respectively, we have ω T (h, ε) for all a R. (t s)q, ε) and ω T (g, ε)., ω T ( f, ε), ω T ( Theorem 3.2. If the FIE (3.) satisfies the hypotheses of r It is obvious that GT is finite. Thus, Theorem 3. and (K6 ). Then the FIE (.) has atleast one solution on J and solutions of the FIE(.) are uniformly LωT (X) ωt (3.6) globally attractive and ultimately positive on J. M ωt (X) Proof. Let Br () be a closed ball in the Banach space C(J, R), where the real number r is given as in the proof of Theorem Let t J be arbitrarily fixed. Then 3. and define a map K : C(J, R) C(J, R) by (.). In (Ka)(t) (Kb)(t) f (t, a(t)) f (t, b(t)) (3.7) proof of Theorem 3., we have shown that K is a continuous mapping from the space C(J, R) from the space Br (). In (t s)q particular, K maps Br () into itself. Now we will prove that q K is a nonlinear-set-contraction with respect to measure µad g(t, a(s)) g(t, b(s)) ds of noncompactness in C(J, R). For any a, b R, we have diam(kx)(t) lim sup diam(kx)(t) a b a b a b, L a(t) b(t) (t s)q M a(t) b(t) q therefore g(t, a(s)) g(t, b(s)) ds a b (a b) a b (a b) a a b b for all a, b R. For any a Br (), we have L a(t) b(t) 2v(t)N M a(t) b(t) LdiamX(t) 2v(t)N M diamx(t) L lim sup diamx(t) M lim sup diamx(t) Ka(t) Ka(t) f (t, a(t)) f (t, a(t)) q (t s) Γ δt ( f ) 2Nv(t) (3.8) 5 δt ( f ) 2NVT, g(t, a(s))ds Γ g(t, a(s))ds
Banach Space 52/53 where VT = supt T v(t). Thus we have References [] δt (X ) δt ( f ) 2NVT for all closed X Br (). On taking the limit superior as T, we have [2] lim sup δt (X ) lim sup δt ( f ) 2 lim sup NVT = (3.2) [3] for all closed X Br (). Hence, [4] δ (KX ) = lim δt (X ) = [5] for all closed X Br (). By the measure of noncompactness µa, we have [6] µad (KX ) = max{µad (KX ), δ (KX )} Lµa (X), } max{ M µa (X ) Lµa (X ) = M µa (X ) Lµad (X ) M µad (X ) [7] [8] (3.2) [9] Since L M, therefore we have [] µad (KX ) φ (µad (X )), Lr where φ (r) = Mr for r >. By Theorem (2.7), the operator K has a fixed point a in the ball Br () and x is a solution of FIE (.). The image of the space C(J, R) is contained in Br () under the operator K because the set Fix(K) of all fixed points of K is contained Br (). The set Fix(K) contain all solutions of the FIE (.) and from Remark 2.8 we conclude that the set Fix(K) belongs to the family kerµad. Now, taking account the description of sets belonging to kerµad, we have that all solutions for the FIE (.) are globally uniformly attractive and ultimately positive on J. [] [2] [3] [4] 4. Conclusion The uniformly global attractivity and utimately posivity are the main qualitative behaviour of solution of the nonlinear integral equations and we have shown existence and the above qualitative behaviour of solution of hybrid fractional integral equation with linear perturbation of second kind with the help of measure of noncompactness in our recent paper. [5] [6] [7] 5. Acknowledgement [8] The authors would like to express sincere gratitude to the reviewer to his/her valuable suggestions. 52 Agarwal R.P., ORegan D., Infinite Interval Problems for Differential, Difference and Integral Equations, Kluwer Academic Publishers, Dordrecht, 2. Apell J., Measure of noncompactness, condensing operator and fixed points: an applications-oriented survey, Fixed point theory, 25; 6: 57 229. Balachandran K., Julie D., Asymptotic stability of solutions of nonlinear integral equations, Nonlinear Funct. Anal. Appl., 28; 2: 33 322. Banas J., Cabrera I.J, On existence and asymptotic behaviour of solutions of a functional integral equation, Nonlinear Anal., 27; 66: 2246 2254. Banas J, Sadarangani K., Solutions of some functional integral equations in Banach algebra, Math. Comput. Modell., 23; 38: 245 25. Banas J., Rzepka B., An application of a measure of noncompactness in the study of asymptotic stability, App Math letters, 23; 6; 6. Banas J., Goebel K., Measures of Noncompactness in Banach Spaces, Marcel Dekker, New York, Basel, 98. Bloom F., Closed Asymptotic bound for Solution to a System of Damped Integro-differential Equations of Electromagnetic Theory, J. Math. Ana. Appl.,98; 73; 524 542. Burton T.A., Volterra Integral and Differential Equations, Academic Press, New York, 983. Darwish M.A., On solvability of some quadratic functional integral equation in Banach algebra, Commun. Appl. Anal., 27; ; 44 45. Dhage B.C., Dhage S.B. and Graef J.B., Local attractivity and stability analysis of a nonlinear quadratic fractional integral equation, Applicable Analysis,26; 95(9): 989 23. Dhage B.C., Attractivity and positivity results for nonlinear functional integral equations via measure of noncompactness, Diff. Equations and App., 2; 2(2): 299 38. Dhage B.C., Global attractivity result for nonlinear functional integral equation via Krasnoleskii type fixed point theorem, Nonlinear Analysis, 29; 7: 2485 2493. Dhage B.C., Asymptotic stability of nonlinear functional integral equations via measure of noncompactness, Comm. App. Nonlinear Ana., 28; 5(2): 89. Dhage B.C., A fixed point theorem in Banach algebras with applications to functional integral equations, Kyungpook Math. J., 26; 44: 45 55. Dhage B.C., Lakshmikantham V., On global existence and attractivity results for nonlinear functional integral equations, Nonlinear Anal., 2; 72: 229 2227. Hu X., Yan J., The global attractivity and asymptotic stability of solution of a nonlinear integral equation, J. Math. Anal. Appl., 26; 32: 47 56. Jiang S., Rokhlin V., Second Kind Integral Equation for the Classical Potential Theory on open Surface II,
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