Mathematical structures describe the complex systems which involve multiple elements and interact between one another in various forms. These interactions exist in physics, electromagnetic, mechanics, biology, signal processing, finance, economics, and many more. In order to make sense of the data extracted from such elements, the evolution of the data against time is utilized. The immediate observation would be a system of differential equations. Upon solving such differential equations, the obtained function will have some information that can be used to extract and understand the data at hand and further predict the future information related to the data. A special class of differential equations are boundary value problems (BVP) and nonlinear fractional integro-differential equations [1, 2]. The fundamental investigation on these types of fractional differential equations is pertinent in order to interpret the related data which evolve into such form. Thus to study the solutions of existence and uniqueness of integro-differential equations might benefit data modeling and formulation via fractional integro-differential equations.
Further, BVPs containing fractional derivatives also describe many phenomena in various modeling such as in science and engineering, in particular viscoelasticity, physics, electromagnetism, biology as well as finance, in particular mixed fractional option pricing, and more. The questions linked to the existence of solutions to BVPs for fractional differential equations have been studied by researchers using different methods, here we cite some such as fixed point theorems, the upper and lower solutions, Mawhin’s coincidence degree theory, Laplace transform method, iteration methods, etc. [3–22].
In [23], the existence of solutions for integral boundary value problems of mixed fractional differential equations under resonance was studied, and a very recent study [24] introduced a new method to convert the boundary value problems for impulsive fractional differential equations to integral equations.
Recently much attention has been given to the solvability of such type of differential equations that have left and right fractional derivatives. Further, several works are also devoted to this type of study, for details, see [3, 4, 7, 12–14].
In this study, we consider the existence of solutions for the following type of equation:
$$\begin{aligned} (\mathrm{P})\quad \textstyle\begin{cases} D_{1^{-}}^{\theta }D_{0^{+}}^{\upsilon }x ( t ) =f ( t,x ( t ) ), \quad t\in (0,1), \\ x ( 0 ) =0,\quad\quad D_{0^{+}}^{\upsilon }x ( 1 ) =D_{0^{+}}^{ \upsilon }x ( 0 ) ,\end{cases}\displaystyle \end{aligned}$$
where \(f\in C ([ 0,1] \times \mathbb{R},\mathbb{R} )\), \(0<\theta \), and \(\upsilon <1\) such that \(\theta +\upsilon >1\), while the notations \(D_{1^{-}}^{\theta } \) and \(D_{0^{+}}^{\upsilon }\) refer to the right and left fractional derivatives in the Caputo sense, respectively. Note that problem (P) is at resonance since the homogeneous fractional boundary value problem (BVP)
$$\begin{aligned}& D_{1^{-}}^{\theta }D_{0^{+}}^{\upsilon }x ( t ) =0, \quad t \in (0,1), \\& x ( 0 ) =0,\quad\quad D_{0^{+}}^{\upsilon }x ( 1 ) =D_{0^{+}}^{ \upsilon }x ( 0 ) \end{aligned}$$
has \({x ( t ) =ct^{\upsilon }}\), \(c\in \mathbb{R}\) as nontrivial solutions.
In this study we establish sufficient conditions that will help us to show that there is at least one solution for problem (P). Many difficulties will occur when we deal with the presence of mixed type fractional derivatives having order less than one, and there are only a few studies related to this case. Moreover, the current literature on the study of BVP at resonance having mixed type fractional-order derivatives is not satisfactory and the topic has not been extensively studied so far. There are some initial attempts such as the following.
In [9], the authors studied, by means of Mawhin’s coincidence degree, the existence of solutions in multipoint Riemann–Liouville sense fractional BVP on the half-line:
$$\begin{aligned}& D_{0^{+}}^{\alpha }u ( t ) =f \bigl( t,u ( t ) , D_{0^{+}}^{\alpha }u ( t ) \bigr) , \quad t>0, 1< \alpha < 2, \\& I_{0^{+}}^{2-\alpha }u ( 0 ) =0, \quad\quad \lim_{t\rightarrow \infty }D_{0^{+}}^{\alpha -1}u ( t ) =\sum_{i=1}^{m-2} \beta _{i}D_{0^{+}}^{\alpha -1}u ( \xi _{i} ), \end{aligned}$$
where \(0<\xi _{1}<\cdots<\xi _{m-2}<\infty \), \(\beta _{i}>0\), and \(i=1,\ldots, m-1\).
In [4], the authors investigated the existence and uniqueness of solution by the use of some fixed point theorems for the following type BVP:
$$\begin{aligned}& {}^{C}D_{1^{-}}^{\alpha RL}D_{0^{+}}^{\beta }u ( t ) + \lambda I_{0^{+}}^{p}I_{1^{-}}^{q}h \bigl( t,u ( t ) \bigr) =f \bigl( t,u ( t ) \bigr) , \quad t\in (0,1) \\& u ( 0 ) =u ( \xi ) =0, \quad\quad u ( 1 ) =\delta u ( \mu ), \end{aligned}$$
where \(1<\alpha <2\), \(0<\beta <1\), \(0<\xi <\mu <1\).
Similarly, under certain conditions on f in [14], the authors studied and proved, by using Krasnoselskii’s fixed point theorem, the existence of solutions for the following type nonlinear BVPs:
$$\begin{aligned}& {}^{C}D_{1^{-}}^{\alpha }D_{0^{+}}^{\beta }u ( t ) =f \bigl( t ,u ( t ) \bigr), \quad t \in (0,1), \\& u ( 0 ) =u^{\prime } ( 0 ) =u ( 1 ) =0, \end{aligned}$$
which involve the right Caputo and the left Riemann–Liouville fractional derivatives, respectively.
In [19, 20], some partial treatments were provided for the following hybrid type nonlinear fractional integro-differential equations:
$$\begin{aligned}& \bigl(D^{\alpha }u \bigr) (t)+\lambda u(t) = f \biggl(t,u(t), \int _{t_{0}}^{t} \frac{(t-s)^{\alpha -1}}{\Gamma (\alpha )}g \bigl(s,u(s) \bigr)\,ds \biggr), \\& u(t_{0}) = B_{0} \in \mathbb{R}, \end{aligned}$$
(1)
where f, g are continuous functions and \(\lambda \in \mathbb{R}^{+}\) for all \(t\in J=[a,b]\). Thus we check for a solution of Eq. (1) subject to \(u \in C^{1}(J,\mathbb{R})\).
Next we recall the following definitions and auxiliary lemmas related to fractional calculus theory, for details, see [17, 22, 25].
Definition 1
The left and right Riemann–Liouville fractional integrals with order \(\theta >0\) on \([ a,b ] \) of a function y are defined respectively by
$$\begin{aligned}& I_{a^{+}}^{\theta }y(t) =\frac{1}{\Gamma (\theta )} \int _{a}^{t}(t-s)^{\theta -1}y(s)\,ds, \quad t>a, \\& I_{b^{-}}^{\theta }y(t) =\frac{1}{\Gamma (\theta )} \int _{t}^{b}(s-t)^{\theta -1}y(s)\,ds,\quad t< b. \end{aligned}$$
Definition 2
The left and right Caputo derivatives \(D_{a^{+}}^{\alpha }\) and \(D_{b^{-}}^{\alpha }\) with order \(\alpha >0\) on \([ a,b ] \) of the function \(y\in AC^{n} [ a,b ] \) are defined by
$$\begin{aligned}& D_{a^{+}}^{\theta }y ( t ) =\frac{1}{\Gamma ( n-\theta ) } \int _{a}^{t} ( t-s ) ^{n-\theta -1}y^{ ( n ) } ( s ) \,ds, \quad t>a, \\& D_{b^{-}}^{\theta }y ( t ) =\frac{ ( -1 ) ^{n}}{\Gamma ( n-\theta ) } \int _{t}^{b} ( s-t ) ^{n-\theta -1}y^{ ( n ) } ( s ) \,ds,\quad t< b, \end{aligned}$$
respectively, where \(n= [ \theta ] +1\), and \([ \theta ] \) is the integer part of θ.
In the next lemma we present some properties associated with fractional integrals and derivatives in the Caputo sense.
Lemma 3
The homogenous equation (fractional differential)
$$ D_{a^{+}}^{\theta }g(t)=0 $$
has a solution
$$ g(t)=\sum_{i=0}^{n-1}c_{i} ( t-a ) ^{i}, $$
and similarly,
$$ D_{b^{-}}^{\theta }g(t)=0 $$
has a solution
$$ g(t)=\sum_{i=0}^{n-1}a_{i} ( b-t ) ^{i}, $$
where \(a_{i},c_{i}\in \mathbb{R}\), \(i=1,\ldots,n\), and \(n=[\theta ]+1\) if \(\theta \notin \mathbb{N}= \{ 0,1,\ldots \} \) and \(n=\theta \) if \(\theta \in \mathbb{N}\).
In addition, the following properties are correct:
$$\begin{aligned}& D_{a^{+}}^{\theta }I_{a^{+}}^{\theta }y ( t ) = y ( t ), \qquad D_{b^{-}}^{\theta }I_{b^{-}}^{\theta }y(t)=y(t), \\& D_{a^{+}}^{\theta } ( t-a ) ^{\gamma -1}= \frac{\Gamma ( \gamma ) }{\Gamma ( \gamma -\theta ) } ( t-a ) ^{\gamma -\theta -1}, \end{aligned}$$
and
$$ D_{b^{-}}^{\theta } ( b-t ) ^{\gamma -1}=\frac{\Gamma ( \gamma ) }{\Gamma ( \gamma -\theta ) } ( b-t ) ^{\gamma -\theta -1} , \quad \gamma > [ \theta ] +1. $$
Next we need the following definitions and a theorem for the development of our results.
Let X and Y be two Banach spaces (real), and let us define a linear operator \(L:\operatorname{dom}L\subset X\rightarrow Y\). Then we have the following definition.
Definition 4
A linear operator L is called Fredholm operator with index zero if ImL is a closed subset in Y and \(\dim \ker L=\operatorname{co}\dim \operatorname{Im}L<\infty \).
Now if we define \(P:X\rightarrow X\) and \(Q:Y\rightarrow Y\) as continuous projections such that \(\operatorname{Im}P=\ker L\), \(\ker Q=\operatorname{Im}L\). Then
$$ X=\ker P \oplus \ker L,\qquad Y= \operatorname{Im}Q \oplus \operatorname{Im}L, $$
which leads to
$$ L| _{\ker P \cap \operatorname{dom}L} : \operatorname{dom}L\cap \ker P\rightarrow \operatorname{Im}L $$
is invertible, and we denote its inverse by \(K_{P}\).
Definition 5
Let \(\Omega \subset X\) be a bounded open subset and \(\operatorname{dom}L\cap \Omega \neq \emptyset \). Then the map N: \(X\rightarrow Y\) is called L-compact on Ω̅ if the map \(QN ( \overline{\Omega } ) \) is bounded and further \(K_{P} ( I-QN ) :\overline{\Omega }\rightarrow X\) is compact.
Note that since ImQ is isomorphic to kerL, that is, \(J:\operatorname{Im}Q\rightarrow \ker L\) isomorphism, the equation \(Lx=Nx\) is equivalent to
$$ x=(P+JQN)x+KP(I-Q)Nx. $$
The next theorem is given in [21].
Theorem 6
Let L be a Fredholm operator with index zero and N be L-compact on Ω̅. Further, the following conditions are satisfied:
-
(1)
\(Lx\neq \lambda Nx\), \(\forall ( x,\lambda ) \in [ ( \operatorname{dom}L \backslash \ker L ) \cap \partial \Omega ] \times ( 0,1 ) \);
-
(2)
\(Nx\notin \operatorname{Im}L\), \(\forall x\in \ker L\cap \partial \Omega \);
-
(3)
If \(Q:Y\rightarrow Y\) is a projection and \(\deg ( QN| _{\ker L},\Omega \cap \ker L,0 ) \neq 0\) such that \(\operatorname{Im}L=\ker Q\),
then there is at least one solution for equation \(Lx=Nx\) in \(\operatorname{dom}L\cap \overline{\Omega }\).