Investigation of the solvability of $n$- term fractional quadratic integral equation in a Banach algebra

1.   Introduction and preliminaries

Several fixed point problems involving product of operators have been investigated in many literature and monographs, for example ^[1,2,3,4,5]. However, the problem considered in ^[6] is more general than those ^[5,7,8,9,10].

Quadratic integral equations(QIEs) have been investigated from different points of view and using different techniques (see ^{[11,12,13,14,15,16,17,18]}). The QIEs can be widely applicable in more applications like the dynamic theory of gases, the theory of radiative exchange, the traffic theory, etc. see ^[4,6,16,19]. For the case of Banach algebras, many recent references have been appeared, for example ^[20,21,22]. For solvability of some QIEs on unbounded interval see ^[19,23,24].

The multi-term fractional quadratic integral equation

has been studied in a Banach algebra ^[25] by using some fixed point theorem ^[26].

The quadratic integral equation of fractional order

has been investigated in ^[27] by applying the nonlinear alternative of Leray-Schauder type.

Motivated by these results and by the monographs that studied $\phi$-fractional quadratic integral equations, in this article, we focus our attention on a nonlinear multi-term quadratic functional integral equation of fractional order

for all $\; t\in J = [0, 1], \;$ where $\; a_i:J\rightarrow \mathbb{R}, \; f_i:J\times \mathbb{R}\rightarrow \mathbb{R}, \; , \; g_i:J\times \mathbb{R}\rightarrow \mathbb{R} \;$ and $\; \eta_i, \; \psi_i: J\rightarrow J.\;$

By a solution of the quadratic functional integral equation of fractional order (1.2), we mean a function $\; x\in C(J, \; \mathbb{R})\;$ that satisfies Eq (1.2), where $\; C(J, \; \mathbb{R})\;$ stands for the space of continuous real-valued functions on $\; J.\;$

For proving the existence results for the quadratic functional integral equation of fractional Eq (1.2). We recall the following fixed point theorem ^[26] which enables us to prove the existence theorem for solutions of the functional integral Eq (1.2).

Theorem 1.1. Let $\; n\;$ be a positive integer, and $\; \mathcal{C}\;$ be a nonempty, closed, convex and bounded subset of a Banach Algebra $\; X.\;$ Assume that the operators $\; A_i:X\rightarrow X\;$ and $\; B_i:\mathcal{C}\rightarrow X, \; \; i = 1, 2, ...., n, \;$ satisfy

(a) For each $\; i\in \{1, 2, ...., n\}, \; \; A_i \;$ is $\; D-$Lipschitzian with a $\; D-$function $\; \phi_i$;

(b) For each $\; i\in \{1, 2, ...., n\}, \; \; B_i \;$ is continuous and $\; B_i(\mathcal{C})\;$ is precompact;

(c) For each $\; y\in \mathcal{C}, \; \; x = \sum\limits_{i = 1}^n A_ix. B_iy\;$ implies that $\; x\in \mathcal{C}$.

Then, the operator equation $\; x = \sum\limits_{i = 1}^n A_ix. B_ix\;$ has a solution provided that

where $\; M_i = \sup\limits_{x\in \mathcal{C}}||B_ix||, \; \; i = 1, 2, ...., n.\;$

2.   Existence results

Equation (1.2) is investigated under the assumptions:

($i^*$) $\; g_i:\; J\times \mathbb{R}\; \rightarrow\; \mathbb{R}, \; \; i = 1, 2, ...., n\;$ satisfy Carathéodory condition (i.e., measurable in $\; t\;$ for all $x \; \in\; \mathbb{R}$ and continuous in $\; x\;$ for almost all $t\; \in\; J$) such that:

and $\; k_i = \sup\limits_{{t\in J}}I_\phi^{\beta_i}\; m_i(t)\;$ for any $\; \beta_i\leq \alpha_i, \; \; i = 1, 2, ...., n\;$ such that $\; k_i\neq 0\; \; \forall\; i\;$

($ii^*$)$\; f_i:\; J\; \times\; \mathbb{R}\; \rightarrow \mathbb{R}, \; \; i = 1, 2, ...., n\;$ are continuous and bounded with $\; h_i = \sup\limits_{{(t, x)\in J\times \mathbb{R}}}\; | f_i(t, x) |, \; \; i = 1, 2, ...., n. \;$

($iii^*$)There exist constants $\; L_i, \; \; i = 1, 2, ...., n\;$ satisfying

for all $\; t\in J\;$ and $\; x, \; y\in \mathbb{R}.\;$

($iv*$)$\eta_i, \psi_i:J \rightarrow J, \; i = 1, 2, 3......, n$ are continuous functions.

($v*$)$\; \phi\;$ is increasing and absolutely continuous function.

($vi*$) $\; a_i:J\rightarrow \mathbb{R}, i = 1, ..., n\;$ are continuous and bound with $\; \mathfrak{k}_i = \sup\limits_{t\in J}\; | a(t) |.\;$

Theorem 2.1. Let the assumptions ($i^*$)-($v^*$) be satisfied. Furthermore, if

then the quadratic integral Eq (1.2) has at least one solution in $\; {C}(J, \mathbb{R}).\;$

Proof. Set $\; X = C(J, \; \mathbb{R}).\;$ Consider the closed ball $\; \overline{\mathbb{B}}_{r}(0)\;$ in $\; X \;$ centered at origin $\; 0\;$ and of radius $\; r, \;$ where $\; r = \sum\limits_{i = 1}^n \left[\frac{h_i k_i}{\Gamma(\alpha_i-\beta_i+1)}+\mathfrak{k}_i\right] \left[1-\sum\limits_{i = 1}^n\frac{b_ih_i }{\Gamma(\alpha_i+1)}\right]^{-1} > 0.\;$

Consider the mapping $\; A_i:X \rightarrow X\;$ and $\; B_i:\overline{\mathbb{B}}_{r}(0)\rightarrow X\; \;$ on $\; {C}(J, \mathbb{R})\;$ defined by:

Then the quadratic integral Eq (1.2) can be written in the form:

We shall show that $\; A_i\;$ and $\; B_i\;$ satisfy all the conditions of Theorem 1.1.

Let us define a subset $\; \mathcal{C}\;$ of $\; {C}(J, \mathbb{R})\;$ by

Obviously, $\; \mathcal{C}\;$ is nonempty, bounded, convex and closed subset of $\; {C}(J, \mathbb{R}).\;$

For all $\; t\in J, \;$ since the assumptions (ii*) and (iii*) are satisfied, the mapping $\; f_i\;$ is well defined and the function $\; A_ix\;$ is continuous and bounded on $\; J.\;$ Again, since each $\; g_i\;$ is continuous in $\; x\;$ and each $\; \psi_i\;$ is continuous function, then the function $\; B_ix\;$ is also, continuous and bounded by $\; m_i'(t) = m_i(t)+b_ir\; , \; m_i'\in L^1(J, \mathbb{R})\;$(in view of assumptions (i*) and (vi*)). Firstly, we show that $\; A_i\;$ is Lipschitz on $\; X.\;$ Let $\; x, \; y\in X\;$ be arbitrary. Then by assumptions (ii*) and (iii*)

For all $\; t\in J.\;$ Taking supremum over $t\;$

for all $\; x, \; y\in X.\;$ This shows that $\; A_i\;$ is a Lipschitz mapping on $\; X\;$ with the Lipschitz constants $\; L_i.\;$ Secondly, we show that $\; B_i\;$ is continuous and compact operator on $\; \overline{\mathbb{B}}_{r}(0).\;$ First we show that each $\; B_i\;$ is continuous on $\; \overline{\mathbb{B}}_{r}(0).\;$ To do this, let us fix arbitrary $\; \epsilon > 0\;$ and let $\; \{x_n\}\;$ be a sequence of functions in $\; \overline{\mathbb{B}}_{r}(0)\;$ converging to $\; x\in \overline{\mathbb{B}}_{r}(0).\;$ Then we get

Thus

Furthermore, let us assume that $\; t\in J.\;$ Then, by Lebesgue dominated convergence theorem, we obtain the estimate:

for all $\; t\in J.\;$ Thus, $\; B_ix_n\rightarrow B_ix\;$ as $\; n\rightarrow \infty\;$ uniformly and hence each $\; B_i\;$ is a continuous operator on $\; \overline{\mathbb{B}}_{r}(0)\;$into $\; \overline{\mathbb{B}}_{r}(0)\;$ has a Cauchy subsequence. Now by (i*) and (vi*)

for all $\; t\in J.\;$ Then $\; ||B_ix_n(t)||\leq \mathfrak{k}_i+\frac{k_i}{\Gamma(\alpha_i-\beta_i+1)}+\frac{b_ir}{\Gamma(\alpha_i+1)}\;$ for all $\; n\in N.\;$ This shows that $\; \{B_ix_n\}\;$ is a uniformly bounded sequence in $\; B_i(\overline{\mathbb{B}}_{r}(0)).\;$

Now, we proceed to show that it is also equicontinuous. Let $\; t_1, \; t_2\in J\;$ (without loss of generality assume that $\; t_1 < t_2$), then we have

Therefore,

Using the uniform continuity of the functions $\; a_i\;$ and $\phi$ on $\; J, \;$ we obtain

As a consequence, $\; |B_ix_n(t_2)- B_ix_n(t_1)| \rightarrow 0 \; \; \; \; \text{as}\; \; t_2\rightarrow t_1.\;$ This shows that $\; \{B_ix_n\}\;$ is an equicontinuous sequence in $\; X.\;$ Now an application of Arzela-Ascoli ^[28] theorem yields that $\; \{B_ix_n\}\;$ has a uniformly convergent subsequence on the the compact subset $\; J\;$ of $\; \mathbb{R}.\;$ without loss of generality, call the subsequence it self. We show that $\; \{B_i x_n\}\;$ is Cauchy sequence in $\; X.\;$ Now $\; |B_ix_n(t)- B_ix(t)| \rightarrow 0 \; \text{as}\; \; n\rightarrow \infty\;$ for all $\; t\in J.\;$ Then for given $\; \epsilon > 0\;$ there exists an $\; n_0\in \mathbb{N}\;$ such that for all $\; m, \; n\geq n_0.\;$ Therefore (or $\; m, \; n\geq n_0\;$), we have

This shows that $\; \{B_ix_n\}\subset F(\overline{\mathbb{B}}_{r}(0))\subset X\;$ is Cauchy sequence. Since $\; X\;$ is complete, $\; \{B_ix_n\}\;$ converges to a point in $\; X.\;$ Hence $\; B_i(\overline{\mathbb{B}}_{r}(0))\;$ is relatively compact and consequently $\; B_i\;$ is a continuous and compact operator on $\; \overline{\mathbb{B}}_{r}(0).\;$

Next, we show that $\; A_ixB_iy = x\Rightarrow x\in \overline{\mathbb{B}}_{r}(0)\;$ for all $\; y\in \overline{\mathbb{B}}_{r}(0).\;$ Then,

Taking the supremum over $\; t.\;$ we obtain $\; ||x||\leq r\;$ for all $\; y\in \overline{\mathbb{B}}_{r}(0), \; r = \sum\limits_{i = 1}^n \left[\frac{h_i k_i}{\Gamma(\alpha_i-\beta_i+1)}+\mathfrak{k}_i\right] \left[1-\sum\limits_{i = 1}^n\frac{b_ih_i }{\Gamma(\alpha_i+1)}\right]^{-1}.\;$ Hence hypothesis (c) of Theorem 1.1 holds. Here,

Therefore we can get, for every $x\; \in\; \mathcal{C}\;$ we have

Then, $\; \mathcal{T}x\; \in\; \mathcal{C} \;$ and hence $T\mathcal{C} \; \subset\; \mathcal{C}.$

Since all conditions of Theorem 1.1 are satisfied, then the operator $\; \mathcal{T} = \sum\limits_{i = 1}^n A_i. B_i\;$ has a fixed point in $\; \mathcal{C}.\;$

3.   Features of the solution

In this section, we shall demonstrate some characteristics for the solutions of the $n$-term quadratic integral Eq (1.2).

3.1.   Uniqueness of the solution

In aim of proving the uniqueness of the solution of (1.2), we replace assumption ($i^*$) by the following assumption:

$\; g_i:\; J\times \mathbb{R}\; \rightarrow\; \mathbb{R}, \; \; i = 1, 2, ...., n\;$ satisfy Carathéodory condition (i.e., measurable in $\; t\;$ for all $x \; \in\; \mathbb{R}$ and continuous in $\; x\;$ for almost all $t\; \in\; J$) and

and $\; \rho_i = \sup\limits_{{t\in J}}|g_i(t, 0)|\;$ for any $\; i = 1, 2, ...., n\;$ such that $\; k_i\neq 0\; \; \forall\; i\;$.

Theorem 3.1. Let the assumptions ($i$) and ($ii*$)–($v*$) be satisfied. Moreover, if

then the quadratic integral Eq (1.2) has a unique solution in $\; {C}(J, \mathbb{R}).\;$

Proof. From the assumption $(i^{*})$ we have

Let $x_{1}, x_{2}$ be two solutions of the integral Eq (1.2), then

Then

Since $\; \sum\limits_{i = 1}^n \bigg(L_i (\mathfrak{k}_i+\frac{\rho_i+c_ir}{\Gamma(\alpha_i+1)})+\frac{h_i c_i }{\Gamma(\alpha_i+1)}\bigg) < 1, \;$ then $\; x_1(t) = x_2(t).\;$

3.2.   Continuous dependency

Firstly, we discuss the continuous dependence of the unique solution of the Eq (1.2) on the delays functions $\; \eta_i\;$ and $\; \psi_i.$

Definition 1. The solutions of the quadratic functional integral Eq (1.2) depends continuously on the delay functions $\eta_i$ and $\; \psi_i\;$ if $\forall\; \epsilon\; > 0, \; \exists \delta, \; \sigma\; > 0$, such that

Theorem 3.2. Let the assumptions of Theorem 3.1 be satisfied, then the solution of the functional integral Eq (1.2) depends continuously on the delay functions $\eta_i$ and $\psi_i$.

Proof. Let $\delta > 0$ and $\sigma > 0$ be given such that $|\eta_i(t)-\eta_i^{*}(t)|\leq\delta$ and $|\psi_i(t)-\psi_i^{*}(t)|\leq\sigma$ $\forall t\; \in J$, then for

Then

Next, we investigate the continuous dependence of the unique solution of the Eq (1.2) on the functions $\; f_i\;$ and $\; g_i.\;$

Definition 2. The solutions of the quadratic functional integral Eq (1.2) depends continuously on the functions $f_i$ and $\; g_i\;$ if $\forall\; \epsilon\; > 0, \; \exists \delta, \; \sigma\; > 0$, such that

Theorem 3.3. Let the assumptions of Theorem 3.1 be satisfied, then the solution of the functional integral equation (1.2) depends continuously on the functions $f_i$ and $g_i$.

Proof. Let $\delta > 0$ and $\sigma > 0$ be given such that $|f_i(t, x)-f_i^{*}(t, x)|\leq\delta$ and $|g_i(t, x)-g_i^{*}(t, x)|\leq\sigma$ $\forall t\; \in J$, then for

Then

4.   Discussion and remarks

● In case $\; a_i = 0, \;$ the operators $\; B_i\;$ have the following form

Then we get the $n$-term quadratic integral equation of fractional order

By a simple calculation we can verify that the operators $\; B_i\;$ satisfy the assumptions of Theorem 1.1 and hence the quadratic integral Eq (4.1) has a solution in $\; \mathcal{C}.\;$ For $\; \phi(t) = t\;$ in Eq (4.1), we obtain the quadratic equation which is studied in ^[25].

● Taking $\; n = 1\;$ we obtain the quadratic integral equation

and putting $\; \phi(t) = t, \;$ then we have the same result studied in ^[27].

● Letting $\; \alpha_i \rightarrow 1, i = 1, 2.$ Then we have the $n$- term quadratic integral equation

and for $\; n = 2, \;$ we have

Therefore we obtain the same result obtained in ^[29] when $\; a_i = 0, i = 1, 2\;$ and $\; \eta_i(t) = \psi_i(t) = t.\;$

● Letting $\; n = 2\;$ we obtain the two term quadratic integral equation

● Taking $\; \phi(t) = t, \; \eta_1(t) = \eta_2(t) = t\; , \; \psi_1(t) = \psi_2(t) = t$ and $\; f_1(t, x) = 1\;$ in (4.3), then we get the hybrid differential equation of fractional order

where $\; ^RD^{\alpha_2}\;$ is the Riemann-Liouville fractional derivative of order $\; \alpha_2\in (0, 1).$

● Taking $\alpha_2 \rightarrow 1, \; g_1(t, x) = 0, \; \phi(t) = t, \; \eta_1(t) = \eta_2(t) = t\; , \; \psi_1(t) = \psi_2(t) = t$ and $\; f_1(t, x) = 1\;$ in (4.3) we have the initial value problem

Example 1. For $n = 2,$ consider the nonlinear quadratic integral equation

where

We can easily verify that the functions $\; f_1, f_2, g_1\;$ and $\; g_2\;$ satisfy all assumptions of Theorem 2.1. Then $h_1 = h_2 = 1, \; b_1 = \frac{1}{2}, \; b_2 = \frac{3}{10},$ which implies that $\; \frac{b_1h_1 }{\Gamma(\alpha_1+1)} +\frac{b_2h_2 }{\Gamma(\alpha_2+1)} = \frac{0.3 }{\Gamma(\frac{1}{2}+1)} +\frac{0.5 }{\Gamma(\frac{1}{4}+1)} = 0.898437 < 1.$ Moreover, we get $\; r > 0.\;$

5.   Conclusions

Many previous papers have discussed quadratic integral equations of fractional order by using different techniques in several classes of functions ^{[3,18,30,31,32,33,34]}.

We have investigated existence and uniqueness of the solutions for a multi-term $\phi$-quadratic integral equation of fractional order in $\; C(J, \mathbb{R}).\;$ The giving problem is converted into an analogous fixed point problem which is solved using typical functional analysis tools to prove our results. As a pursuit of this, sufficient conditions are given for the existence of solutions to that singular quadratic integral equation. Next, the continuous dependence of the solution on some functions has been proved. Finally, some particular cases, remarks and example to validate our results.

Acknowledgments

The researchers would like to thank the Deanship of Scientific Research, Qassim University for funding the publication of this project. The authors are thankful to the referees for remarks and suggestions for the improvement of this paper.

Conflict of interest

All authors declare no conflicts of interest in this paper.