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.
MSC 2020:
1. 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, Citation2003; Argyros, Citation1985; Corduneanu, Citation1990; Deimling, Citation1985). 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., Citation2014; Arab, Citation2016; Cabada et al., Citation2018; Deepmala, Citation2013a; Roshan, Citation2017; Tunç, Citation2010a;Tunç Citation2010b; Tunç, Citation2016; Tunç, Citation2020; Tunç and Golmankhaneh, Citation2020; Tunç and Tunç, Citation2018a; Tunç and Tunç, Citation2018b; Tunç and Tunç, Citation2018c; Tunç and Tunç, Citation2019; Tunç, Citation2020 and references therein). In this article, we prove the existence of solution of the following generalized FIE: (1) (1) for
The FIE (Equation1(1) (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 (Equation1(1) (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., Citation2014; Banaś and Sadarangani, Citation2003; Deepmala, Citation2013a; Deepmala, Citation2013b; Maleknejad et al., Citation2008; Maleknejad et al., Citation2009a).
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., Citation2019; Dai et al., Citation2019; Dai et al., Citation2020; Wang et al., Citation2018) and references therein).
2. Preliminaries
Throughout this entire paper, we use the following assumptions:
M: Real Banach space;
norm on a Banach space;
closed ball having y0 as a center with radius
coZ: convex hull of a set Z;
closed convex hull of a set Z;
EM: set of all bounded subsets of a space M;
NM: set of all relatively compact subsets of a space M;
Definition 2.1
(Banas and Goebel, Citation1980). Assume and where
Hence, where is called the Kuratowski MNC.
Theorem 2.1.
Assume and . Then,
if and only if
where
where
where denotes the Hausdorff metric of Y and Z, i.e.
Theorem 2.2
(Banas and Goebel, Citation1980). Let G be a nonempty, bounded, closed and convex subset of M and let be continuous mapping such that there exists a constant , with for any subset of Y of G. Then D has a fixed point in G.
Now, we discuss on which contains set of all real continuous functions defined on the interval with the standard norm
Clearly, has also the structure of Banach algebra. Now, we will focus on a regular MNC defined in Banas and Lecko (Citation2002). We fix a set For and given denote by the modulus of continuity of y, i.e.,
Further,
Thus is a regular MNC in
Theorem 2.3
(Banas and Lecko, Citation2002). Suppose that G is a bounded, convex and closed subset of and Q, H be the operators which transform continuously the set G into such that Q(G) and H(G) are bounded. Again, the operator transforms G into itself. If the operators Q and H satisfy the Darbo’s condition on the set G with the constants D1 and D2, respectively, then the operator D satisfies the Darbo’s condition on G with the constant
If then D will be called contraction with respect to the measure and has a fixed point in the set G.
3. Main result
Now, we will analyze the solvability of the FIE (Equation1(1) (1) ) under the following assumptions:
and are continuous functions and the constants and such that
There exists the continuous functions such that for all and
Moreover, the functions and η convert continuously the interval into itself.
There is a non-negative constant K such that for
There are non-negative constants l and m such that for all and
for and
Theorem 3.1.
Under the assumptions FIE(1) has at least one solution in
Proof.
Let the operators Q and H be defined on M by the formula:
for
From and we see that Q and H transform on M into itself. Now, we put
Clearly, D transforms M into itself. Now, fix Then,
Taking and then we have (2) (2) (3) (3) (4) (4) for
From (Equation4(4) (4) ), we reduce the operator D maps the ball into itself for where
Also, from the estimates (Equation2(2) (2) ) and (Equation3(3) (3) ), it follows that (5) (5) (6) (6)
Next, we show that Q is continuous on the ball To do this, fix and arbitrary such that Then, for we get
where
The function and are uniform continuous on the bounded subset then and as Thus, Q is continuous on Similarly, H is also continuous on Hence, D is a continuous operator on
Now, we will show that the Q and H satisfy the Darbo’s condition in the ball Assume that a non empty subset Z of and Let be fixed and such that and Then, we obtain
and (7) (7) where
In view of our assumptions we deduce that the functions and are uniform continuous on and respectively and the functions and are uniform continuous on Hence, we infer that and as Thus, we get (8) (8)
Similarly, it is obtained that (9) (9)
Finally, it follows that D satisfies the Darbo’s condition on with respect to the measure ω0 with constant Now, we have
Hence, D is a contraction on with respect to Consequently, we conclude that the nonlinear FIE (Equation1(1) (1) ) has at least one solution in ball □
4. Applications
Our proposed functional integral equation contains several integral equations, considered by several authors as a special case.
If and then equation reduces to the following FIE, which was studied in Deepmala and Pathak (2013a).
Taking and then FIE(Equation1(1) (1) ) is converted into the following form which has been studied in Maleknejad et al. (Citation2009a).
On putting and we obtain the following FIE studied in Banaś and Sadarangani (Citation2003) and Maleknejad et al. (Citation2008).
If and then we get the following FIE studied in Banaś and Rzepka (Citation2003).
Taking and then FIE (Equation1(1) (1) ) has the following form studied in Maleknejad et al. (Citation2009b)).
On putting and then, we get following non-linear Volterra integral equation (Corduneanu, Citation1990).
Taking and then we obtain Urysohn integral equation (Corduneanu, Citation1990).
If and then FIE(Equation1(1) (1) ) has the following form
The above integral equation is the famous quadratic integral equation of Chandrasekhar type (Chandrasekhar, Citation1950).
Example 4.1.
Consider the following non-linear FIE: (10) (10) where
The FIE (Equation10(10) (10) ) is a special case of Equationequation (1)(1) (1) . Here
Now, we show that all the assumptions of Theorem 3.1 are satisfied. It is obvious that In this case,
Moreover, it is obvious that
We also have and b = 1.
Finally, we see that
Hence, all assumptions to are satisfied. The FIE (Equation10(10) (10) ) has at least one solution in
5. 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, Citation1994; Hu et al., Citation1989; Kelly, Citation1982).
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