On the existence of solutions of some non-linear functional integral equations in Banach algebra with applications

Abstract In this article, we establish some results for the existence of solution of nonlinear functional integral equations by using Darbo’s fixed point theorem in Banach algebra. As an application, we discuss some examples of nonlinear functional integral equations and existence of solutions.


Introduction
Integral equations are an important branch of mathematical analysis, science and equations of such types are applicable in many physical problems such as in the vehicular traffic, the biology, control theory and mathematical physics(see Abdou, 2003;Argyros, 1985;Corduneanu, 1990;Deimling, 1985).Recently the theory of FIE and various kind of functional differential equations are developed effectively and emerged in the field of analysis, engineering, applied mathematics, and nonlinear functional analysis (see Aghajani et al., 2014;Arab, 2016;Cabada et al., 2018;Deepmala, 2013a;Roshan, 2017;Tunc¸, 2010a;Tunc ¸2010b;Tunc¸, 2016;Tunc¸, 2020;Tunc¸and Golmankhaneh, 2020;Tunc¸and Tunc ¸, 2018a;Tunc ¸and Tunc¸, 2018b;Tunca nd Tunc¸, 2018c;Tunc¸and Tunc ¸, 2019;Tunc¸, 2020 and references therein).In this article, we prove the existence of solution of the following generalized FIE: for f 2 ½0, b: The FIE (1) consists of many special types of FIEs, those are very useful in real-world problems of physics, biology, differential equations, etc.Here, our aim to examine the difficulty of the existence of the solutions of FIE (1) using the techniques of MNC and Darbo's fixed point theorem in [0, b].Many authors have taken out some successful attempts to solve many FIE by utilizing Darbo's condition which is an important tool to study these equations (Aghajani et al., 2014;Bana s and Sadarangani, 2003;Deepmala, 2013a;Deepmala, 2013b;Maleknejad et al., 2008;Maleknejad et al., 2009a).
Recently, there are some developments in the field of the travelling wave solutions as well as its applications, so it will enhance the readability and comprehension of the manuscript (for details, see (Chen et al., 2019;Dai et al., 2019;Dai et al., 2020;Wang et al., 2018) and references therein).

Preliminaries
Throughout this entire paper, we use the following assumptions: M: Real Banach space; jj:jj : norm on a Banach space; Bðy 0 , rÞ : closed ball having y 0 as a center with radius r; coZ : convex hull of a set Z; co Z : closed convex hull of a set Z; E M : set of all bounded subsets of a space M; N M : set of all relatively compact subsets of a space M; Definition 2.1 (Banas and Goebel, 1980).Assume Y 2 E M and here dð:Þ is the distance from an element M to a set of M.
Theorem 2.2 (Banas and Goebel, 1980).Let G be a nonempty, bounded, closed and convex subset of M and let D : G ! G be continuous mapping such that there exists a constant k 2 ð0, 1Þ, with lðDYÞ klðYÞ for any subset of Y of G. Then D has a fixed point in G. Now, we discuss on C½0, b, which contains set of all real continuous functions defined on the interval ½0, b with the standard norm Clearly, C½0, b has also the structure of Banach algebra.Now, we will focus on a regular MNC defined in Banas and Lecko (2002).We fix a set Y 2 M C½0, b : For y 2 Y and given >0 denote by xðy, Þ the modulus of continuity of y, i.e., Thus x 0 ðYÞ is a regular MNC in C½0, b: Theorem 2.3 (Banas and Lecko, 2002).Suppose that G is a bounded, convex and closed subset of C½0, b and Q, H be the operators which transform continuously the set G into C½0, b such that Q(G) and H(G) are bounded.Again, the operator D ¼ Q:H transforms G into itself.If the operators Q and H satisfy the Darbo's condition on the set G with the constants D 1 and D 2 , respectively, then the operator D satisfies the Darbo's condition on G with the constant , then D will be called contraction with respect to the measure x 0 and has a fixed point in the set G.

Main result
Now, we will analyze the solvability of the FIE (1) under the following assumptions: are continuous functions and 9 the constants u and w !0 such that jf ðf, 0, 0Þj u, jFðf, 0, 0, 0Þj w, jLðf, 0, 0, 0Þj w: s,yð/ðsÞÞ,u¼uðf,s,yðaðsÞÞ,p¼pðf,s, yðcðsÞÞ, q¼qðf,s,yðnðsÞÞ :½0,bÂ½0,bÂR!R:Moreover, the functions /,a,c,n,d and g convert continu- for all f, s 2 ½0, b and y 2 R: ðB 6 Þ 4cq<1 for c ¼ 3K þ2Kbm and q ¼ uþ2Kbl þw: Theorem 3.1.Under the assumptions ðB 1 ÞÀðB 6 Þ FIE(1) has at least one solution in M ¼ C½0, b: Proof.Let the operators Q and H be defined on M by the formula: for f 2 ½0, b: From ðB 1 Þ and ðB 3 Þ, we see that Q and H transform on M into itself.Now, we put Dy ¼ ðQyÞðHyÞ: (2) for y 2 M: From (4), we reduce the operator D maps the ball B r & M into itself for r 1 r r 2 , where Also, from the estimates ( 2) and ( 3), it follows that jjQB r jj c r þ q, (5) Next, we show that Q is continuous on the ball B r : To do this, fix >0 and arbitrary y, z 2 B r such that jjyÀzjj : Then, for f 2 ½0, b, we get qðf, s, yðnðsÞÞÞds, yðgðfÞÞÞÀLðf, 0, 0, 0Þj þ jLðf, 0, 0, 0ÞjÞ Now, we will show that the Q and H satisfy the Darbo's condition in the ball B r : Assume that a non empty subset Z of B r y 2 Z: Let >0 be fixed and f 1 , f 2 2 ½0, b such that f 1 f 2 and f 1 Àf 2 : Then, we obtain Àf ðf, zðfÞ, zðhðfÞÞÀF f, where In view of our assumptions we deduce that the func- respectively and the functions r ¼ rðf, s, yÞ and u ¼ uðf, s, yÞ are uniform continuous on ½0, b Â ½0, b Â R: Hence, we infer that x f ð:::Þ !0, x r ð:::Þ !0, x u ð:::Þ !0 and x F ð:::Þ !0 as !0: Thus, we get x 0 ðQYÞ 3Kx 0 ðYÞ: Similarly, it is obtained that x 0 ðHYÞ Kx 0 ðHÞ: Finally, it follows that D satisfies the Darbo's condition on B r with respect to the measure x 0 with con- Hence, D is a contraction on B r with respect to x 0 : Consequently, we conclude that the nonlinear FIE (1) has at least one solution in ball B r : w

Applications
Our proposed functional integral equation contains several integral equations, considered by several authors as a special case.
then equation reduces to the following FIE, which was studied in Deepmala and Pathak (2013a). 1) is converted into the following form which has been studied in Maleknejad et al. (2009a).
The above integral equation is the famous quadratic integral equation of Chandrasekhar type (Chandrasekhar, 1950).

Conclusion
Integral equations represent an important field in the area of applied mathematics and a powerful tool for modeling diverse problems arising in all areas of scientific research.Our result contains outcome of several research papers as a particular case.These result may be further extended for the developments in the field of the traveling wave solutions as well as its applications (Dhage, 1994;Hu et al., 1989;Kelly, 1982).