Nonlinear fractional differential equations with nonlocal fractional integro-differential boundary conditions

Nonlocal boundary value problems of fractional differential equations have been extensively studied in the recent years. In fact, the subject of fractional calculus has been quite attractive and exciting due to its applications in the modeling of many physical and engineering problems. For theoretical and practical development of the subject, we refer to the books [1–5]. Some recent results on fractional boundary value problems can be found in [6–14] and references therein. In [11], the authors studied a boundary value problem of fractional differential equations with fractional separated boundary conditions.

where ${}^{c}D^{\alpha }$ denotes the Caputo fractional derivative of order α, f is a given continuous function, and ${\alpha }_i$, ${\beta }_i$, ${\gamma }_i$ ($i=1,2$) are suitably chosen real constants.

The main aim of the present study is to obtain some existence results for the problem (1.1). As a first step, we transform the given problem to a fixed point problem and show the existence of fixed points for the transformed problem which in turn correspond to the solutions of the actual problem. The methods used to prove the existence results are standard; however, their exposition in the framework of the problem (1.1) is new.

Let us recall some basic definitions of fractional calculus [1, 2].

where $[q]$ denotes the integer part of the real number q.

provided the integral exists.

To define the solution of the boundary value problem (1.1), we need the following lemma, which deals with a linear variant of the problem (1.1).

Substituting the values of $c_0$, $c_1$ in (2.4), we get (2.2). This completes the proof. □

In case ${\beta }_1=0={\beta }_2$, the boundary conditions in (2.1) coincide with (2.6) and consequently the corresponding solutions become identical.

Let $\mathcal{C}=C([0,1],\mathbb{R})$ denote the Banach space of all continuous functions from $[0,1]$ into ℝ endowed with the usual supremum norm.

Observe that the problem (1.1) has solutions if and only if the operator equation $\mathcal{F}x=x$ has fixed points.

where ${\tilde{\mu }}_1=|{\gamma }_1|(|{\mathrm{\Delta }}_1|+|{\mathrm{\Delta }}_2|)$, ${\tilde{\mu }}_2=|{\gamma }_1{\mathrm{\Delta }}_3|+|{\mathrm{\Delta }}_4|$ with ${\mathrm{\Delta }}_i$ ($i=1,2,3,4$) given by (2.3).

Our first result is based on the Leray-Schauder nonlinear alternative [15].

Lemma 3.1 (Nonlinear alternative for single valued maps)

Theorem 3.2 Let $f:[0,1]\times \mathbb{R}\to \mathbb{R}$ be a jointly continuous function. Assume that:

(${\mathrm{A}}_1$) there exist a function $p\in C([0,1],{\mathbb{R}}^{+})$ and a nondecreasing function $\psi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ such that $|f(t,x)|\le p(t)\psi (\parallel x\parallel )$, $\mathrm{\forall }(t,x)\in [0,1]\times \mathbb{R}$;

Then the boundary value problem (1.1) has at least one solution on $[0,1]$.

Obviously, the right-hand side of the above inequality tends to zero independently of $x\in B_r$ as $t^{″}-t^{\prime }\to 0$. As ℱ satisfies the above assumptions, therefore, it follows by the Arzelá-Ascoli theorem that $\mathcal{F}:C([0,1],\mathbb{R})\to C([0,1],\mathbb{R})$ is completely continuous.

Note that the operator $\mathcal{F}:\overline{U}\to C([0,1],\mathbb{R})$ is continuous and completely continuous. From the choice of U, there is no $x\in \partial U$ such that $x=\lambda \mathcal{F}(x)$ for some $\lambda \in (0,1)$. Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 3.1), we deduce that ℱ has a fixed point $x\in \overline{U}$ which is a solution of the problem (1.1). This completes the proof. □

In the special case when $p(t)=1$ and $\psi (|x|)=\kappa |x|+N$ (κ and N are suitable constants) in the statement of Theorem 3.2, we have the following corollary.

Corollary 3.3 Let $f:[0,1]\times \mathbb{R}\to \mathbb{R}$ be a continuous function. Assume that there exist constants $0\le \kappa <1/\omega$, where ω is given by (3.2) and $N>0$ such that $|f(t,x)|\le \kappa |x|+N$ for all $t\in [0,1]$, $x\in C[0,1]$. Then the boundary value problem (1.1) has at least one solution.

Next, we prove an existence and uniqueness result by means of Banach’s contraction mapping principle.

Theorem 3.4 Suppose that $f:[0,1]\times \mathbb{R}\to \mathbb{R}$ is a continuous function and satisfies the following assumption:

(${\mathrm{A}}_3$) $|f(t,x)-f(t,y)|\le L|x-y|$, $\mathrm{\forall }t\in [0,1]$, $L>0$, $x,y\in \mathbb{R}$.

where ω is given by (3.2).

Note that ω depends only on the parameters involved in the problem. As $L\omega <1$, therefore, ℱ is a contraction. Hence, by Banach’s contraction mapping principle, the problem (1.1) has a unique solution on $[0,1]$. □

Now, we prove the existence of solutions of (1.1) by applying Krasnoselskii’s fixed point theorem [16].

Theorem 3.5 (Krasnoselskii’s fixed point theorem)

Let M be a closed, bounded, convex, and nonempty subset of a Banach space X. Let A, B be the operators such that (i) $Ax+By\in M$ whenever $x,y\in M$; (ii) A is compact and continuous; (iii) B is a contraction mapping. Then there exists $z\in M$ such that $z=Az+Bz$.

Theorem 3.6 Let $f:[0,1]\times \mathbb{R}\to \mathbb{R}$ be a jointly continuous function satisfying the assumption (${\mathrm{A}}_3$). In addition we assume that:

(${\mathrm{A}}_4$) $|f(t,x)|\le \mu (t)$, $\mathrm{\forall }(t,x)\in [0,1]\times \mathbb{R}$, and $\mu \in C([0,1],{\mathbb{R}}^{+})$.

Now, we prove the compactness of the operator $\mathcal{P}$.

which is independent of x. Thus, $\mathcal{P}$ is equicontinuous. Hence, by the Arzelá-Ascoli theorem, $\mathcal{P}$ is compact on $B_{\overline{r}}$. Thus, all the assumptions of Theorem 3.5 are satisfied. So, the conclusion of Theorem 3.5 implies that the boundary value problem (1.1) has at least one solution on $[0,1]$. □

Thus, all the conditions of Corollary 3.3 are satisfied and consequently the problem (4.1) has at least one solution.

where α, p, ${\alpha }_i$, ${\beta }_i$, ${\gamma }_i$, ($i=1,2$) η, σ are the same as given in (4.1) and $f(t,x)=\frac{L}{2}(x+{tan}^{-1}x)+\frac{\sqrt{t(t+3)}}{t^4+2}$. Clearly, $|f(t,x)-f(t,y)|\le L|x-y|$ and thus, for $L<1/\omega =1/3.578386$, all the conditions of Theorem 3.4 are satisfied. Hence, the boundary value problem (4.2) has a unique solution on $[0,1]$.