Analysis of Abel-type nonlinear integral equations with weakly singular kernels

Abel-type integral equations are associated with a wide range of physical problems such as heat transfer [1], nonlinear diffusion [2], propagation of nonlinear waves [3], and they can also be applied in the theory of neutron transport and in traffic theory. In the past 70 years, many researchers investigated the existence and uniqueness of nontrivial solutions for a large number of Abel-type integral equations by using various analysis methods (see [4–16] and references therein).

Fractional calculus provides a powerful tool for the description of hereditary properties of various materials and memory processes. In particular, integral equations involving fractional integral operators (which can be regarded as an extension of Abel integral equations) appear naturally in the fields of biophysics, viscoelasticity, electrical circuits, and etc. There are some remarkable monographs that provide the main theoretical tools for the qualitative analysis of fractional order differential equations, and at the same time, show the interconnection as well as the contrast between integer order differential models and fractional order differential models [17–24].

It is remarkable that many researchers pay attention to the study of the existence and attractiveness of solutions for fractional integral equations by using functional analysis methods such as the contraction principle, the Schauder fixed point theorem and a Darboux-type fixed point theorem involving a measure of noncompactness (see [25–33] and references therein).

Many analysis techniques are used to construct the suitable order interval (see Lemma 2, [13]) and the spaces with suitable weighted norms.

are special cases of equation (2), which of course all have trivial solutions.

Thus, the main purpose of this paper is to prove the existence and uniqueness of nontrivial solutions for equation (2). The key difficult comes from the weakly singular kernels ${(t^{\beta }-s^{\beta })}^{1-\alpha }$ and nonlinear terms in equation (2). Although we are motivated by [13], we have to introduce novel techniques and results to overcome the difficult from the weakly singular kernels and nonlinear terms h and g. For example, the first important step is how to construct a suitable order interval to help us to apply the fixed point theorem in such an order interval. More details of the novel techniques and results will be found in the proof. As a byproduct of our method, we improve a gap in the proof of [[13], Theorem 5]. So even for equation (1) (or (5)) we get a new result.

The plan of this paper is as follows. In Section 2, some notation and preparation results are given. Existence and uniqueness results of a nontrivial solution of equation (2) in an order interval are given in Section 3. In Section 4, we find general solutions in closed form of some Erdélyi-Kober-type fractional integral equations, and finally theoretical results with three illustrate examples are presented in Section 5.

Let ℳ be the set $\mathcal{M}:=\{f\in C[0,T]:f(0)=0\}$ with the supremum-norm ${\parallel f\parallel }_{\mathcal{M}}:={sup}_{0<t\le T}\{|f(t)|\}$. Clearly, the set $(\mathcal{M},{\parallel \cdot \parallel }_{\mathcal{M}})$ is a closed subspace of Banach space $(C[0,T],{\parallel \cdot \parallel }_C)$. Thus, $(\mathcal{M},{\parallel \cdot \parallel }_{\mathcal{M}})$ is a Banach space.

where $c_1={max}_{t\in [0,T]}\{g(t)\}$ and $c_2={min}_{t\in [0,T]}\{g(t)\}$. Note that the similar inequality (5) of [13] is incorrect.

Consider the cone $P_{\mathcal{M}}:=\{u\in \mathcal{M}:u(t)\ge 0,t\in [0,T]\}$ in ℳ. The so-called partial ordering induced by the cone $P_{\mathcal{M}}$ is given by $u\le v⟺u(t)\le v(t)$ for all $u,v\in \mathcal{M}$ and all $t\in [0,T]$. In general [34, 35], a set $[f,g]=\{h\in E:f\le h\le g\}$ is called an order interval where E is an ordered Banach space. We know that every order interval $[f,g]$ is closed. Moreover, if ${\parallel f\parallel }_E\le {\parallel g\parallel }_E$ for all $f,g\in E$ with $0\le f\le g$, then every order interval $[f,g]$ is bounded.

Similarly for $K(t,s)=k_1(t^{\beta }-s^{\beta })$ and $n=0$.

We note [34] an important estimate on the function K, which will be used in the sequel.

Proof We only check the case of $K(t,s)=k_1(t^{\beta }-s^{\beta })$ with $n\ge 1$, since the other cases are trivial.

Replacing t by $(t^{\beta }-s^{\beta })$, we obtain the desired result. □

To end this section, we collect the following basic facts, which will be used several times in the next section.

is the well-known Beta function.

The proof is completed. □

In this section, we will use the fixed point method to prove the existence and uniqueness of nontrivial solution for equation (2) in an order interval.

where ${\mathbb{K}}_{\mathrm{low}}$ and ${\mathbb{K}}_{\mathrm{up}}$ are defined in equation (8) or (9).

Remark 3.1 Note that $\beta (n+\alpha -1)q_{\mp }+\gamma p_{\pm }\ge \beta (\alpha -1)q_{\mp }+\beta p_{\pm }=\beta (p_{\pm }-q_{\mp })+\alpha \beta q_{\mp }>0$ and $\beta (n+\alpha -1)+\gamma \ge \beta (\alpha -1)+\beta =\alpha \beta >0$. Next, ${\tau }_{-}\ge {\tau }_{+}$.

The following result is clear.

then $F(t)\le G(t)<M$ for all $t\in [0,T]$. Consequently, the order interval $[F,G]\subset P_{\mathcal{M}}$ is well defined.

which is satisfied, since equation (3) implies $a_{+}\ge a_{-}>0$, $b_{+}\ge b_{-}>0$ (see equation (4)) and clearly $\frac{{\mathbb{K}}_{\mathrm{low}}}{{\mathbb{K}}_{\mathrm{up}}}\le 1$. This case occurs for instance when $h(x)=x^p\tilde{h}(x)$ and $g(x)=x^q\tilde{g}(x)$ with $p>q$ and $0<{inf}_{\mathbb{R}}h(x)\le {sup}_{\mathbb{R}}h(x)<\mathrm{\infty }$, $0<{inf}_{\mathbb{R}}g(x)\le {sup}_{\mathbb{R}}g(x)<\mathrm{\infty }$.

From now on, we suppose that all above assumptions hold: equations (3), (4), (i)-(iii), and (11).

Lemma 3.4 Any solution $x\in P_{\mathcal{M}}$ of equation (2), with $M>x(t)>0$ for all $t\in (0,T]$, satisfies $x\in [F,G]$.

Proof Step 1: We prove that $x\le G$ for a solution x of equation (2).

Since $\frac{\beta (n+\alpha -1)q_{+}+\gamma p_{-}}{\beta (p_{-}-q_{+})}>\frac{\gamma }{\beta }$ implies $\mathbb{B}(\frac{\beta (n+\alpha -1)q_{+}+\gamma p_{-}}{\beta (p_{-}-q_{+})},n+\alpha )<\mathbb{B}(\frac{\gamma }{\beta },n+\alpha )$, estimate (13) is an improvement of equation (12).

Hence we can complete the proof. □

Lemma 3.5 The operator $S_{h,g}$ maps the order interval $[F,G]$ into itself.

Consequently, $S_{h,g}$ is well defined and $S_{h,g}([F,G])\subset [F,G]$. The proof is completed. □

From the Arzela-Ascoli theorem and since $S_{h,g}:[F,G]\to [F,G]$ is nondecreasing, it follows that $S_{h,g}$ is compact, so the Schauder fixed point theorem implies the following existence result [35, 38, 39].

Now we are ready to state the following uniqueness result. But first we note that the above considerations can be repeated for any $0<T_1\le T$, so we get ${\mathbb{K}}_{\mathrm{low}}(T_1)$, ${\mathbb{K}}_{\mathrm{up}}(T_1)$, $A(T_1)$, $B(T_1)$, $F_{T_1}$, and $G_{T_1}$ as continuous functions of $T_1$. Note ${\mathbb{K}}_{\mathrm{low}}(T_1)$ is nonincreasing, ${\mathbb{K}}_{\mathrm{up}}(T_1)$ is nondecreasing, and ${\mathbb{K}}_{\mathrm{low}}(T_1)$, ${\mathbb{K}}_{\mathrm{up}}(T_1)$ can be continuously extended to $T_1=0$. Then ${\mathbb{K}}_{\mathrm{low}}(0)={\mathbb{K}}_{\mathrm{up}}(0)$. We still keep the notation ${\mathbb{K}}_{\mathrm{low}}={\mathbb{K}}_{\mathrm{low}}(T)$, ${\mathbb{K}}_{\mathrm{up}}={\mathbb{K}}_{\mathrm{up}}(T)$, $F=F_T$, and $G=G_T$.

where we set $0^0=1$.