Abstract
This article studies the existence of solutions for a three-point inclusion problem of Langevin equation with two fractional orders. Our main tools of study include a nonlinear alternative of Leray-Schauder type, selection theorem due to Bressan and Colombo for lower semicontinuous multivalued maps, and a fixed point theorem for multivalued map due to Covitz and Nadler. An illustrative example is also presented.
Mathematical Subject Classification 2000: 26A33; 34A12; 34A40.
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1 Introduction
The study of fractional calculus has recently gained a great momentum and has emerged as an interesting and important field of research. The popularity of the subject can easily be witnessed by a huge number of articles and books published in the last few years. It is mainly due to the extensive application of fractional calculus in the mathematical modeling of physical, engineering and biological phenomena. Fractional differential equations appear naturally in a number of fields such as physics, chemistry, biology, economics, control theory, signal and image processing, biophysics, blood flow phenomena, aerodynamics, fitting of experimental data, etc. For recent development in theory and applications of fractional calculus, see the books [1–4]. Some results concerning the initial and boundary value problems of fractional equations and inclusions can be found in a series of articles [5–26] and the references therein.
Langevin equation is found to be an effective tool to describe the evolution of physical phenomena in fluctuating environments [27–29]. Various generalizations of Langevin equation have been proposed to describe dynamical processes in a fractal medium. One such generalization is the generalized Langevin equation [30–35] which incorporates the fractal and memory properties with a dissipative memory kernel into the Langevin equation. In another possible extension, ordinary derivative is replaced by a fractional derivative in the Langevin equation to yield the fractional Langevin equation [36–39]. Recently, Lim et al. [40] have studied a new type of Langevin equation with two different fractional orders. The solution to this new version of fractional Langevin equation gives a fractional Gaussian process parameterized by two indices, which provides a more flexible model for fractal processes as compared with the usual one characterized by a single index. In [41], Lim et al. discussed the fractional oscillator process with two indices. Recently, a Dirichlet boundary value problem for Langevin equation involving two fractional orders has been studied in [42]. Some more recent work on Langevin equation can be found in [43–46]. In a more recent article [47], the authors studied a nonlinear Langevin equation involving two fractional orders α ∈ (0, 1] and β ∈ (1, 2] with three-point boundary conditions.
Motivated by recent work on Langevin equation of fractional order, we study the following inclusion problem of Langevin equation of two fractional orders in different intervals with three-point boundary conditions
where cD is the Caputo fractional derivative, is a compact valued multivalued map, and is the family of all subsets of ℝ.
We present some existence results for the problem (1.1), when the right hand side is convex as well as nonconvex valued. The first result relies on the nonlinear alternative of Leray-Schauder type. In the second result, we shall combine the nonlinear alternative of Leray-Schauder type for single-valued maps with a selection theorem due to Bressan and Colombo for lower semicontinuous multivalued maps with nonempty closed and decomposable values. The third result, dealing with a nonconvex valued right hand side of the fractional inclusion in (1.1), employs a fixed point theorem for multivalued map due to Covitz and Nadler. Here we remark that the single-valued case of (1.1) was discussed in [47].
2 Background material for multivalued analysis
First of all, we recall some basic definitions on multi-valued maps [48, 49].
For a normed space (X, || · ||), let , , , and . A multi-valued map is convex (closed) valued if G(x) is convex (closed) for all x ∈ X. The map G is bounded on bounded sets if is bounded in X for all (i.e. ). G is called upper semi-continuous (u.s.c.) on X if for each x0 ∈ X, the set G(x0) is a nonempty closed subset of X, and if for each open set N of X containing G(x0), there exists an open neighborhood of x0 such that . G is said to be completely continuous if is relatively compact for every . If the multi-valued map G is completely continuous with nonempty compact values, then G is u.s.c. if and only if G has a closed graph, i.e., x n → x*, y n → y*, y n ∈ G(x n ) imply y* ∈ G(x*). G has a fixed point if there is x ∈ X such that x ∈ G(x). The fixed point set of the multivalued operator G will be denoted by FixG. A multivalued map G : [0; 1] → Pcl(ℝ) is said to be measurable if for every y ∈ ℝ, the function
is measurable.
Definition 2.1 A multivalued map is said to be Carathéodory if
(i) t → F (t, x) is measurable for each x ∈ ℝ;
(ii) x → F (t, x) is upper semicontinuous for almost all t ∈ [0, 1].
Further a Carathéodory function F is called L1-Carathéodory if
(iii) for each ρ > 0, there exists φ ρ ∈ L1([0, 1], ℝ+) such that
for all ||x|| ∞ ≤ ρ and for a.e. t ∈ [0, 1].
For each y ∈ C([0, 1], ℝ), define the set of selections of F by
Let X be a nonempty closed subset of a Banach space E and is a multivalued operator with nonempty closed values. G is lower semi-continuous (l.s.c.) if the set {y ∈ X : G(y) ∩ B = ∅} is open for any open set B in E. Let A be a subset of [0, 1] × ℝ. A is ℒ ⊗ ℬ measurable if A belongs to the σ-algebra generated by all sets of the form , where is Lebesgue measurable in [0, 1] and is Borel measurable in ℝ. A subset of L1([0, 1], ℝ) is decomposable if for all and measurable , the function , where stands for the characteristic function of .
Definition 2.2 Let Y be a separable metric space and letbe a multivalued operator. We say N has a property (BC) if N is l.s.c. and has nonempty closed and decomposable values.
Let be a multivalued map with nonempty compact values. Define a multivalued operator associated with F as
which is called the Nemytskii operator associated with F.
Definition 2.3 Letbe a multivalued function with nonempty compact values. We say F is of l.s.c. type if its associated Nemytskii operator ℱ is l.s.c. and has nonempty closed and decomposable values.
Let (X, d) be a metric space induced from the normed space (X, ||·||). Consider given by
where d(A, b) = inf a∈ A d(a; b) and d(a, B) = inf b∈ B d(a; b). Then (P b, cl (X), H d ) is a metric space and (Pcl(X), H d ) is a generalized metric space (see [50]).
Definition 2.4 A multivalued operator N : X → Pcl(X) is called
(a) γ-Lipschitz if and only if there exists γ > 0 such that
(b) a contraction if and only if it is γ-Lipschitz with γ < 1.
The following lemmas will be used in the sequel.
Lemma 2.1[51]Let X be a Banach space. Let F : [0, 1] × ℝ → Pcp,c(X) be an L1-Carathéodory multivalued map and let Θ be a linear continuous mapping from L1([0, 1], X) to C([0, 1], X), then the operator
is a closed graph operator in C([0, 1], X) × C([0, 1], X).
In passing, we remark that if dim X < ∞, then S F (x) ≠ ∅ for any x(·) ∈ C([0, 1], X) with F (·, ·) as in Lemma 2.1.
Lemma 2.2 (Nonlinear alternative for Kakutani maps[52]) Let E be a Banach space, C a closed convex subset of E, U an open subset of C and 0 ∈ U. Suppose that is a upper semicontinuous compact map; here denotes the family of nonempty, compact convex subsets of C. Then either
(i) F has a fixed point in Ū, or
(ii) there is a u ∈ ∂U and λ ∈ (0, 1) with u ∈ λF (u).
Lemma 2.3[53]Let Y be a separable metric space and letbe a multivalued operator satisfying the property (BC). Then N has a continuous selection, that is, there exists a continuous function (single-valued) g : Y → L1([0, 1], ℝ) such that g(x) ∈ N(x) for every x ∈ Y.
Lemma 2.4[54]Let (X, d) be a complete metric space. If N : X → Pcl(X) is a contraction, then FixN ≠ ∅.
For some recent results on multivalued maps, we refer the reader to the articles [55, 56].
3 Existence results
We are concerned with the existence of solutions for the problem (1.1) when the right hand side has convex as well as nonconvex values. Initially, we assume that F is a compact and convex valued multivalued map.
Definition 3.1[47]A function x ∈ C([0, 1], ℝ) is said to be a solution of (1.1), if there exists a function v ∈ L1([0, 1], ℝ) with v(t) ∈ F (t, x(t)) a.e. t ∈ [0, 1] and
Theorem 3.1 Suppose that
(H1) the mapis Carathéodory and has nonempty compact convex values;
(H2) there exist a continuous non-decreasing function ψ : [0, ∞) → (0, ∞) and a positive continuous function p such that
for each (t, u) ∈ [0, T ] × ℝ;
(H3) there exists a number M > 0 such that
withand.
Then the problem (1.1) has at least one solution.
Proof. In view of Definition 3.1, the existence of solutions to (1.1) is equivalent to the existence of solutions to the integral inclusion
Let us introduce the operator as
for v ∈ S F, x . We will show that the operator N satisfies the assumptions of the nonlinear alternative of Leray-Schauder type. The proof consists of several steps. As a first step, we show that N(x) is convex for each x ∈ C([0, 1], ℝ). For that, let h1, h2 ∈ N(x). Then there exist v1, v2 ∈ S F, x such that for each t ∈ [0, 1], we have
Let 0 ≤ ω ≤ 1. Then, for each t ∈ [0, 1], we have
Since S F, x is convex (F has convex values), therefore it follows that ωh1 + (1 - ω)h2 ∈ N(x).
Next, we show that N(x) maps bounded sets into bounded sets in C([0, 1], ℝ). For a positive number r, let B r = {x ∈ C([0, 1], ℝ): ||x|| ∞ ≤ r} be a bounded set in C([0, 1], ℝ). Then, for each h ∈ N(x), x ∈ B r , there exists v ∈ S F, x such that
and
where ||p||∞ = supt ∈ [0, 1]p(t). Using the relations for Beta function B(., .):
we find that
where
Now we show that N maps bounded sets into equicontinuous sets of C([0, 1], ℝ). Let t', t'' ∈[0, 1] with t' < t'' and x ∈ B r , where B r is a bounded set of C([0, 1], ℝ). For each h ∈ N(x), we obtain
Obviously the right hand side of the above inequality tends to zero independently of x ∈ B r as t'' -t' → 0. As N satisfies the above three assumptions, therefore it follows by Ascoli-Arzelá theorem that is completely continuous.
In our next step, we show that N has a closed graph. Let x n → x*, h n ∈ N(x n ) and h n → h*. Then we need to show that h* ∈ N(x*). Associated with h n ∈ N(x n ), there exists such that for each t ∈ [0, 1],
Thus we have to show that there exists such that for each t ∈ [0,1],
Let us consider the continuous linear operator Θ: L1([0, 1], ℝ) → C([0, 1], ℝ) so that
Observe that
Thus, it follows by Lemma 2.1 that Θ ∘ S F is a closed graph operator. Further, we have . Since x n → x*, it follows that
for some .
Finally, we discuss a priori bounds on solutions. Let x be a solution of (1.1). Then there exists v ∈ L1([0, 1], ℝ) with v ∈ S F, x such that, for t ∈ [0, 1], we have
In view of (H2) together with the condition , for each t ∈ [0, 1], we find that
which can alternatively be written as
In view of (H3), there exists M such that ||x|| ∞ ≠ M. Let us set
Note that the operator is is upper semicontinuous and completely continuous. From the choice of U, there is no x ∈ ∂U such that x ∈ μN(x) for some μ ∈ (0, 1). Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 2.2), we deduce that N has a fixed point x ∈Ū which is a solution of the problem (1.1). This completes the proof. □
Next, we study the case where F is not necessarily convex valued. Our approach here is based on the nonlinear alternative of Leray-Schauder type combined with the selection theorem of Bressan and Colombo for l.s.c. maps with decomposable values.
Theorem 3.2 Assume that (H2)-(H3) and the following conditions hold:
(H4) is a nonempty compact-valued multivalued map such that
(a) (t, x) ↦ F(t, x) is ℒ ⊗ ℬ measurable,
(b) x ↦ F(t, x) is lower semicontinuous for each t ∈ [0, 1];
(H5) for each σ > 0, there exists φ σ ∈ C([0, 1], ℝ+) such that
Then the boundary value problem (1.1) has at least one solution on [0, 1].
Proof. It follows from (H4) and (H5) that F is of l.s.c. type. Then from Lemma 2.3, there exists a continuous function f : C([0, 1], ℝ) → L1([0, 1], ℝ) such that f (x) ∈ ℱ(x) for all x ∈ C([0, 1], ℝ).
Consider the problem
Observe that if x ∈ C3([0, 1]) is a solution of (3.2), then x is a solution to the problem (1.1). In order to transform the problem (3.2) into a fixed point problem, we define the operator as
It can easily be shown that is continuous and completely continuous. The remaining part of the proof is similar to that of Theorem 3.1. So we omit it. This completes the proof.
□
Now we prove the existence of solutions for the problem (1.1) with a nonconvex valued right hand side by applying a fixed point theorem for multivalued map due to Covitz and Nadler (Lemma 2.4).
Theorem 3.3 Assume that the following conditions hold:
(H6) F : [0, 1] × ℝ → Pcp(ℝ) is such that F (·, x): [0, 1] → Pcp(ℝ) is measurable for each x ∈ ℝ;
(H7) for almost all t ∈ [0, 1] and with m∈ C([0, 1], ℝ+) and d(0, F (t, 0)) ≤ m(t) for almost all t ∈ [0, 1].
Then the boundary value problem (1.1) has at least one solution on [0, 1] if
Proof. Observe that the set S F, x is nonempty for each x ∈ C([0, 1], ℝ) by the assumption (H6), so F has a measurable selection (see [[57], Theorem III.6]). Now we show that the operator N defined by (3.1) satisfies the assumptions of Lemma 11.2.4. To show that N(x) ∈ Pcl((C[0, 1], ℝ)) for each x ∈ C([0, 1], ℝ), let {u n }n ≥ 0∈ N(x) be such that u n → u (n → ∞) in C([0, 1], ℝ). Then u ∈ C([0, 1], ℝ) and there exists v n ∈ S F, x such that, for each t ∈ [0, 1],
As F has compact values, we pass onto a subsequence to obtain that v n converges to v in L1([0, 1], ℝ). Thus, v ∈ S F, x and for each t ∈ [0, 1],
Hence, u ∈ N(x).
Next we show that there exists γ < 1 such that
Let and h1 ∈ N(x). Then there exists v1(t) ∈ F (t, x(t)) such that, for each t ∈ [0, 1],
By (H7), we have
So, there exists such that
Define by
Since the multivalued operator is measurable ([[57], Proposition III.4])), there exists a function v2(t) which is a measurable selection for V . So and for each t ∈ [0, 1], we have .
For each t ∈ [0, 1], let us define
Thus,
Hence,
Analogously, interchanging the roles of x and , we obtain
Since N is a contraction, it follows by Lemma 11.2.4 that N has a fixed point x which is a solution of (1.1). This completes the proof. □
Example 3.1 Consider the problem
where is a multivalued map given by
For f ∈ F, we have
Thus,
with p(t) = 1, ψ(||x|| ∞ ) = 3/4. Further, using the condition
we find that
Clearly, all the conditions of Theorem 3.1 are satisfied. So there exists at least one solution of the problem (3.3) on [0, 1].
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Acknowledgements
This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant no. 07/31/Gr. The authors, therefore, acknowledge with thanks DSR technical and financial support. The authors also thank the referees for their useful comments.
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Ahmad, B., Nieto, J.J. & Alsaedi, A. A nonlocal three-point inclusion problem of Langevin equation with two different fractional orders. Adv Differ Equ 2012, 54 (2012). https://doi.org/10.1186/1687-1847-2012-54
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DOI: https://doi.org/10.1186/1687-1847-2012-54