A Riesz-Medvedev Stieltjes Integral Method and Nearness Principle for Solvability of Fractional Integral Equations

1. Introduction

In this paper we present, below, solvability results for quadratic type Volterra-Stieltjes integral equations with applications to a very general class of fractional dynamic problems and various singular problems taking the general form.

$$u\left(t\right)=h\left(t\right)+f(t,u\left(t\right)){\int }_0^{t}k(t,s)g(s,u\left(s\right))d_s\Phi (t,s);t\in [0,1].$$

(1)

In this presentation, we are particularly motivated by the newer approach reported in a series of publications [10,11,13,14] by Banas and his followers using the properties of the classical total variation functions. We recall, in 2011, Jozef Banas and Tomasz Zajac [10] presented a new approach to the investigation of functional equations of Volterra-Stieltjes type integral equations of fractional integral order. In particular, they initiated a very natural solvability formulation for functional integral equation of Volterra-Stieljes type, subject to existence of solution $r_0$ to the two auxiliary inequalities below:

$$\parallel h\parallel +K\left(k{F}_1\right)\Phi \left(r\right)\le r;\mathrm{and}kK\Phi \left(r\right)<1.$$

(2)

In the same year, Banas and Rzepka [11] published another set of enlightening results for the solvability of the more general formulation below:

$$u\left(t\right)=h(t,u\left(t\right)),u\left(a\left(t\right)\right)+\left(Gu\right)\left(t\right){\int }_0^{t}k(t,s)\nu (t,s)g\left(u\left(s\right)\right)ds;$$

(3)

subject to solvabilty requirement on a more complicated auxiliary inequalities. viz the solvability requirement that the following auxiliary inequalities,

$${\eta }_{1}r+\overline{h}+\overline{k}\overline{\nu }\phi \left(r\right)\psi \left(r\right)\le r;$$

(4)

has a solution $r_0$ which satisfies

$${\eta }_1+\overline{k}\overline{\nu }{\eta }_2\psi \left(r_0\right)<1.$$

(5)

The aim of this paper is to employ the more general Riesz-Medvedev type of total variation and the principle of nearness of superposition operators to modify this new approach to obtain a more natural and more relaxed but similar constraint on solvability results which yield invaluable applications to fractional dynamics problems. The quadratic fractional integral equations we study herein are the form below:

$$u\left(t\right)=h\left(t\right)+{\displaystyle \frac{f(t,u(t\left)\right)}{\Gamma \left(\alpha \right)}}{\int }_0^t{\displaystyle \frac{k(t,s)}{{(t-s)}^{1-\alpha }}}g(s,u\left(s\right))ds;t\in [0,1].$$

(6)

Though our method is newer and more natural, equations (6) are special cases of (3) investigated by Banas Rzepka [11] which constitutes generalizations and extensions of problems studied in literature [12,28,33] under the broad class called quadratic integral equations which includes a wide class of singular integral equations as special cases. The class of quadratic integral equations are mostly used to formulate models describing numerous events and problems of real world problems like the theory of radiative transfer [17,18,20], the kinetic theory of gases [29], the theory of neutron transfer [17,20] and the theory of vehicular traffic [6,12]. The Chandrasekhar [20] type quadratic integral equations are special cases of such problems often encountered in models formulations for neutron transport theory and vehicular traffic theory [4,6,12,22,33,37].

It should be noted that solvability requirement for the pair of auxiliary inequalities in (36) is an inevitable constraint (inherent in applications of Darbo fixed point theorem) on the hypotheses for existence of solution of associated equations like (3). It is also important to highlight that the inequalities in (36) are more complicated than the inequalities in (2) because the associated integral equation is less complicated than the integral equation (3). Apart from these complications, the inequalities further impose smallness constraints on the data $h,f,k,$ and g of the integral equations. But the novelty of our presentation includes relaxation of these smallness constraints on these data by allowing for alternations of smallness among the data in our formulation herein. In particular, it is only the Young function $\Phi (t,s)$ that is affected by smallness constraints in our presentation.

Under similar constraints by sets of auxiliary inequalities, various authors have attempted improvements in order to relax the requirement of solvability of complicated auxiliary inequalities (2) and (5). In this direction, we mention the following recent contributions: Benkerouche [15], Darwish [22,23,24]. El-Sayed et al [26], Zhou et al [38], Cicho [21], Danville [25], Kelley [28] and various others [1,2] and the references therein.

2. Preliminaries

Our methodology requires applications of the following classical results.

Definition 1. 

[31,37] Let Ω denote a compact metric space and $Z(\Omega )$ denote a class of functions $f:\Omega \times \mathbb{R}\to \mathbb{R}$. The function $f(t,x)$ is said to generate a superposition operator $F:Z_1(\Omega )\to Z_2(\Omega )$ if for each $x\in Z_1(\Omega )$ there corresponds a function $y\in Z_2(\Omega )$ such that

$$\begin{array}{cc}\left(Fx\right)\left(t\right)=y\left(t\right)=f(t,x(t\left)\right)& \\ \mathrm{i}.\mathrm{e}.F\left(x\right)=y\in Z_2(\Omega ),x\in Z_1(\Omega )\end{array}$$

(7)

The theory of superposition operators $\left(Fx\right)\left(t\right)=f(t,x(t\left)\right)$ often called Nemytskii operators (or outer composition operators or substitution operators) still has many open problems [3,35] which are extension of the basic basic problem in the theory of superposition operators. The basic problem of superposition operators concerned with formulations of conditions (called defining conditions) to guarantee validity of definition of a superposition operator $F:Z_1(\Omega )\to Z_2(\Omega )$ between any given pair of function spaces $Z_1(\Omega )$ and $Z_2(\Omega )$. But this theory for superposition operators $F:C[0,1]\to C[0,1]$ on the space of continuous functions on a bounded closed interval $[a,b]$ has been well established [3,31].

In the space $C[0,1]$ of continuous functions defined on the unit interval $[0,1]$ we have the following important properties;

Theorem 1. 

[3] The superposition operator $F:\Omega \to [0,1]$ generated by a function $f:\Omega \times \mathbb{R}\to \mathbb{R}$ maps the space $C[0,1]$ of continuous functions into itself if and only if $f(t,x)$ is continuous on ${\Omega }^{\prime }\times \mathbb{R}$ where ${\Omega }^{\prime }$ denotes the derived set of Ω (i.e. ${\Omega }^{\prime }$ is the set of all limit points of Ω i.e.

$$F(\Omega )\subseteq C(\Omega )\mathrm{iff}f:{\Omega }^{\prime }\times \mathbb{R}\to \mathbb{R}$$

is continuous.

Observe that theorem 1 implies continuity of $f(.,u)$ on ${\Omega }^{\prime }\forall u\in \mathbb{R}$.

Theorem 2. 

[3] Suppose that the superposition $F:C(\Omega )\to C(\Omega )$ generated by $f(t,x)$ maps $C(\Omega )$ into itself. Then F is bounded if and only if $f(x,.)$ is bounded $\forall t\in \Omega {\Omega }^{\prime }$ where ${\Omega }^{\prime }$ denotes the derived set of $\Omega .$

Theorem 3. 

Suppose $F:C[0,1]\to C[0,1]$ is a superposition operator $F\left(u\right)=$ generated by $f:[0,1]\times \mathbb{R}\to \mathbb{R}$. Then the following condonations are equivalent:

(a)

There exists a superposition operator $G:B_r\left(C[0,1]\right)\to B_{k\left(r\right)}\left(C(0,1)\right)$ generated by a function $g:[0,1]\times \mathbb{R}\to \mathbb{R}$ such that $\left|f\right(t,x)-f(t,y\left)\right|\le g(t,z)|x-y|$ where, $z=max\left|x\right|,\left|y\right|$.

(b)

The superposition operator $F\left(u\right)=f(t,u)$ satisfies the Lipschitz condition $\left|F\right(x)-F(y\left)\right|\le k\left(r\right)|x-y|$

(c)

The superposition operator $F\left(u\right)=f(t,u)$ satisfies the Darbo condition $H_C\left(F(\Omega )\right)\le k\left(r\right)H_C(\Omega )$; for all $\Omega \subseteq B_r\left(C[0,1]\right),$ where $H_C$ denotes the Hausdorff measure of no compactness in $C[0,1]$.

Theorem 4 

(cf [5]). A function $\phi (t,s)$ is a convex function if, and only if, $\Phi (t,s)$ has monotonically increasing one-sided partial derivative $\phi (t,s)={\displaystyle \frac{\partial \Phi (t,s)}{\partial s}}$ with respect to s.

Corollary 1 

(cf [5]). Let $f\left(x\right)$ be a twice differentiable function. Then $f\left(x\right)$ is convex if, and only if, $f^{\prime \prime }\left(x\right)\ge 0$ for all x of our interval.

Definition 2. 

A continuous function $\Phi :[0,\infty )\to [0,\infty )$ is called a young function if

(i)

$\Phi \left(0\right)=0$

(ii)

$\Phi \left(t\right)>0\forall t>0$

(iii)

${lim}_{t\to \infty }\Phi \left(t\right)=\infty$

A Young function is often called a gauge function.

Definition 3. 

Given a Young function $\Phi :[0,\infty )\to [0,\infty )$, and a partition $P=\{t_0,t_1,\dots ,t_n\}\in \mathcal{P}\left([\mathcal{0},\mathcal{1}]\right)$ and a function $f:[0,1]\to \mathbb{R}$, the Wiener-Young variation $va{r}_{\Phi }^W(f,P)={\sum }_{j=i}^{\infty }\Phi \left(\right|f\left(t_j\right)-f\left(t_{t-1}\right)\left|\right)$.

Definition 4. 

Given a Young function Φ, a partition $P=\{t_0,t_1,t_2,\dots ,t_n\}$ of $[0,1]$ and a function $f:[0,1]\to \mathbb{R}$. The Riesz-Medvedev variation of f on $[0,1]$ with respect to a partition P is defined by

$$va{r}_{\Phi }^R\left(f\right)=\sum _{j=1}^n\Phi \left({\displaystyle \frac{|f\left(t_j\right)-f\left(t_{j-1}\right)|}{t_j-t_{j-1}}}\right)·(t_j-t_{j-1})$$

Observe that the Riesz-Medvedev variation with respect to a partition P is the Wiener-Young variation if $\Phi \left(\right|f\left(t_j\right)-f\left(t_{j-1}\right))$ is replaced with $\left({\displaystyle \frac{|f\left(t_j\right)-f\left(t_{j-1}\right)|}{t_j-t_{j-1}}}\right)·(t_j-t_{j-1})$.

Definition 5. 

The Riesz-Medvedev total variation of f on $[0,1]$ is given as $sup\{u_{\Phi }^R(f,P):P\in \mathcal{P}\left([0,1]\right)\}$ where $\mathcal{P}\left(\right[\mathcal{0},\mathcal{1}\left]\right)$ denotes the set of all possible finite partitions P of $[0,1]$.

Example 1. 

Let $0\le s\le t\le 1$. Then the function $\Phi (t,s)={\displaystyle \frac{t^{\alpha }-{(t-s)}^{\alpha }}{\alpha }}$ is of bounded Riesz-Medvedev total variation.

We recall [36] a mapping $A:X\to E$ of a subset X of a Banach space E is said to be near a mapping $B:X\to E$ if there exist two constants $\lambda >0$ and ${\eta }_2\in [0,1)$ such that the following property holds:

$$\parallel (Bu-Bv)-\lambda (Au-Av)\parallel \le {\eta }_2\parallel Bu-Bv\parallel$$

(8)

It follows from (8) that when B is the identity mapping i.e. $B=I$ then A is said to be near identity mapping I. Our investigation is limited to the property of nearness of A to identity mapping given below:

$$\parallel (u-v)-\lambda (Au-Av)\parallel \le {\eta }_2\parallel u-v\parallel$$

The nearness principle [16,34] follows from the following generalization of Neuman’s lemma by Campanato

Theorem 5. 

[16,19] Let E be a real Banach space. $T:E\to E$ a nonlinear mapping such that there exist $\lambda >0$ and ${\eta }_2\in [0,1)$ and

$$\parallel (u-v)-\lambda (Tu-Tv)\parallel \le {\eta }_2\parallel u-v\parallel \mathrm{then}Lip(I-\lambda A)\le {\eta }_2\mathrm{and}Lip\left(A^{-1}\right)\le {\displaystyle \frac{\lambda }{1-c}}$$

Where $Lip\left(T\right)$ is the Lipchitz norm of T.

Our solvability procedure is based on application of Rothe fixed point theorem below to prove existence and uniqueness of the quadratic integral equations studied herein.

Theorem 6. 

[27,32] Let E be a Banach space, X a closed unit ball in E with boundary $\partial X$. Let T be a countable compact mapping of X into E such that $T(\partial X)\subset X$, then T has a fixed point.

3. Application to Fractional Dynamics

The results in this section are Corollaries following from our main results above and applications to problems of fractional dynamics and other singular problems.

Corollary 3. 

Let the hypotheses H1, H2, H3, and H5 hold, Suppose $\varphi \left(t\right)$ is the right derivative of the Young’s function $\Psi (t,s)$ then the singular integral equation

$$u\left(t\right)=h\left(t\right)+f(t,u\left(t\right)){\int }_0^{t}k(s,t)\varphi (t,s)g(t,s)ds$$

has a unique solution.

The next theorem illustrates applications of nearness principle to resolution of hitherto difficult solvability of fractional dynamic problems with degenerate perturbation of identity by nonlinearities which are neither small nor contractive.

Theorem 9. 

Let h, f, k, and g satisfy the hypotheses H1 - H6 and ${\int }_0^s\phi (t,\tau )d\tau$ is a Young function in variable s, then the quadratic integral equation

$$u\left(t\right)=f(t,u\left(t\right)){\int }_0^{t}k(t,s)\phi (t,s)g(s,u\left(s\right))ds$$

has a unique solution.

Theorem 10. 

Let the hypotheses (H1-H6) be satisfied for h, f, k and g in the equation

$$u\left(t\right)=h\left(t\right)+f(t,u\left(t\right)){\int }_0^t{\displaystyle \frac{k(t,s)}{{(t-s)}^{1-\alpha }}}g(s,u\left(s\right))ds$$

if the condition below is satisfied

$${\displaystyle \frac{{\tau }_0}{\alpha }}\overline{k}{\eta }_1({\eta }_2+1)\parallel h\parallel <(1-{\Phi }_0\overline{k}{\eta }_1\overline{f})(1-\overline{f})$$

then the quadratic equation $u\left(t\right)$ has a unique solution.

4. Illustrative Examples

Example 3. 

Investigate the solvability of the following quadratic integral equation

$$u\left(t\right)=h\left(t\right)+f(t,u\left(t\right)){\int }_0^t{\displaystyle \frac{t^2-s^2}{1+s}}sinu\left(s\right)ds,0\le t\le 1$$

(34)

where $f\left(t\right)$

Solution

But ${\displaystyle \frac{ds}{1+s}}=dln(1+s)$ So we set $\Phi (t,s)=ln(1+s)$ Since $ln(1+s)$ is a convex increasing function. In fact $ln(1+s)$ is convex since by theorem

$${\displaystyle \frac{d}{ds}}ln(1+s)={\displaystyle \frac{1}{1+s}}>0\forall s>0.$$

The function $sinu$ is near identity with nearness constant i.e.

$$\parallel u_1-u_2-\lambda (sin{u}_1-sin{u}_2)\parallel \le k\parallel u_1-u_2\parallel$$

with $\lambda >0$ and $k\in [0,1)$ Hence, by theorem 8 Also $\overline{f1}=supf(t,u\left(t\right)),\overline{k}=1,{\Phi }_0=ln(1+t_0)$, ${\Phi }_0\le {\eta }_1$, is the Lipchitz constant of f${\eta }_1$,${\eta }_2=1-cos\gamma$$r\in (0,1)$

$$ln(1+{\tau }_0){\eta }_1(2-cos\gamma )\parallel h\parallel <(1-ln(1-{\tau }_0)\eta \overline{f})$$

Hence there exists solution $\left(u\right(t)$ on $C[0,t_0]$ for the equation 34.

Example 4. 

$$u\left(t\right)=h\left(t\right)+f(t,u\left(t\right)){\int }_0^t{\displaystyle \frac{t^2-s^2}{{(t-s)}^{1-\alpha }}}sinu\left(s\right)ds,0\le t\le 1$$

(35)