On the Asymptotic Behavior of Solutions of Nonlinear Differential Equations of Integer and Also of Non-integer Order

We present conditions under which all solutions of the fractional differential equation with the Caputo derivative c D α a x(t) = f (t, x(t)), a > 1, α ∈ (1, 2), (1) are asymptotic to at + b as t → ∞ for some real numbers a, b. This paper is in final form, has been submitted to the Proceedings of 9th Col-loquium on the Qualitative Theory of Differential Equations and no version of it will be submitted for publication for publication elsewhere.

The subject of initial and boundary value problems of fractional order differential equations has recently emerged as an important area of investigation due to its extensive applications in various disciplines of science and engineering such as mechanics, electricity, chemistry, biology, economics, control theory, signal and image processing, polymer rheology, regular variation in thermodynamics, biophysics, aerodynamics, viscoelasticity and damping, electro-dynamics of complex medium, wave propagation, blood flow phenomena, etc.( [1,9,21,23,25,26,29]).Many researchers have contributed to the development of the existence theory for nonlinear fractional boundary value problems, for instance, see ( [2]- [6], [12,13,18,19,28,30]) and the references cited therein.
The present work is motivated by a recent paper [7] where a nonlocal strip condition of the form is considered.In the present study, we have introduced Riemann-Liouville type fractional integral boundary conditions involving nonintersecting finite many strips of arbitrary length.Such boundary conditions can be interpreted in the sense that a controller at the right-end of the interval under consideration is influenced by a discrete distribution of finite many nonintersecting sensors (strips) of arbitrary length expressed in terms of Riemann-Liouville type integral boundary conditions.The results concerning the single valued case of (1.1) are reported in the paper [8].
We establish the new existence results for the problem (1.1), when the right hand side of the inclusion is convex as well as non-convex 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, while in the third result, we shall use the fixed point theorem for contraction multivalued maps due to Covitz and Nadler.The methods employed in the present work are well known, however their exposition in the framework of problem (1.1) is new.
The paper is organized as follows: Section 2 contains some preliminary concepts and results about multivalued maps while the main results are presented in Section 3.

Fractional Calculus
First of all, we recall some basic definitions of fractional calculus [21,25,26] and then obtain an auxiliary result.EJQTDE, 2013 No. 20, p. 2 Definition 2.1 For an at least n-times differentiable function g : [0, ∞) → R, the Caputo derivative of fractional order q is defined as where [q] denotes the integer part of the real number q. Definition 2.2 The Riemann-Liouville fractional integral of order q is defined as provided the integral exists.
Lemma 2.3 [21] For q > 0, the general solution of the fractional differential equation D q x(t) = 0 is given by where In view of Lemma 2.3, it follows that for some In the following, AC n−1 ([0, T ], R) will denote the space of functions x : [0, T ] → R that are (n − 1)−times absolutely continuously differentiable functions.

Basic Material for Multivalued Maps
Here we outline some basic concepts of multivalued analysis.[15,20,27].
Let C([0, T ]) denote a Banach space of continuous functions from [0, T ] into R with the norm x = sup t∈[0,T ] |x(t)|.Let L 1 ([0, T ], R) be the Banach space of measurable functions x : [0, T ] → R which are Lebesgue integrable and normed by For a normed space (X, • ), let A multi-valued map G : X → P(X) : (v) is said to be completely continuous if G(B) is relatively compact for every B ∈ P b (X); (vi) is said to be measurable if for every y ∈ R, the function The fixed point set of the multivalued operator G will be denoted by Fix G.
For each x ∈ C([0, T ], R), define the set of selections of F by We define the graph of G to be the set Gr (G) = {(x, y) ∈ X × Y, y ∈ G(x)} and recall two useful results regarding closed graphs and upper-semicontinuity.
Conversely, if G is completely continuous and has a closed graph, then it is upper semi-continuous.

Lemma 2.8 ([24]
) Let X be a Banach space.Let F : [0, T ] × R → P cp,c (X) be an L 1 − Carathéodory multivalued map and let Θ be a linear continuous mapping from We recall the well-known nonlinear alternative of Leray-Schauder for multivalued maps.
Lemma 2.9 (Nonlinear alternative for Kakutani maps) [17].Let E be a Banach space, C a closed convex subset of E, U an open subset of C and 0 ∈ U. Suppose that F : U → P c,cv (C) is a upper semicontinuous compact map.Then either (i) F has a fixed point in U, or (ii) there is a u ∈ ∂U and λ ∈ (0, 1) with u ∈ λF (u).Definition 2.10 Let A be a subset of [0, T ] × R. A is L ⊗ B measurable if A belongs to the σ−algebra generated by all sets of the form J × D, where J is Lebesgue measurable in [0, T ] and D is Borel measurable in R. R) is decomposable if for all u, v ∈ A and measurable J ⊂ [0, T ] = J, the function uχ J + vχ J−J ∈ A, where χ J stands for the characteristic function of J .EJQTDE, 2013 No. 20, p. 6 Lemma 2.12 ([10]) Let Y be a separable metric space and let N : Y → P(L 1 ([0, T ], R)) be a lower semi-continuous (l.s.c.) multivalued operator with nonempty closed and decomposable values.Then N has a continuous selection, that is, there exists a continuous function (single-valued) h : Let (X, d) be a metric space induced from the normed space (X; • ).Consider H d : P(X) × P(X) → R ∪ {∞} given by  [22]).Definition 2.13 A multivalued operator N : X → P cl (X) is called (a) γ−Lipschitz if and only if there exists γ > 0 such that 3 Main Results

The Carathéodory Case
In this section, 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.For the forthcoming analysis, we set where Ω is given by (3.1).
Then the BVP (1.1) has at least one solution.
Proof.Let us introduce the operator 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, T ], R).For that, let h 1 , h 2 ∈ N(x).Then there exist v 1 , v 2 ∈ S F,x such that for each t ∈ [0, T ], we have Next, we show that N(x) maps bounded sets into bounded sets in C([0, T ], R).For a positive number ρ, let Now we show that N maps bounded sets into equicontinuous sets of C([0, T ], R).Let t ′ , t ′′ ∈ [0, T ] with t ′ < t ′′ and x ∈ B ρ , where B ρ , as above, is a bounded set of C([0, T ], R).For each h ∈ N(x), we obtain Obviously the right hand side of the above inequality tends to zero independently of x ∈ B ρ as t ′′ − t ′ → 0. As N satisfies the above three assumptions, therefore it follows by Ascoli-Arzelá theorem that N : C([0, T ], R) → P(C([0, T ], R)) 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 v n ∈ S F,xn such that for each t ∈ [0, T ], Thus we have to show that there exists v * ∈ S F,x * such that for each t ∈ [0, T ], Let us consider the continuous linear operator Θ : Thus, it follows by Lemma 2.8 that Θ • S F is a closed graph operator.Further, we have h n (t) ∈ Θ(S F,xn ).Since for some v * ∈ S F,x * .Finally, we discuss a priori bounds on solutions.Let x be a solution of (1.1).Then, EJQTDE, 2013 No. 20, p. 11 using the computations proving that N(x) maps bounded sets into bounded sets, we have Consequently, in view of (3.1), we get In view of (H 3 ), there exists M such that x = M. Let us set Note that the operator N : U → P(C([0, T ], R)) 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 [17], we deduce that N has a fixed point x ∈ U which is a solution of the problem (1.1).This completes the proof.
With the given values of the parameters involved, we find that EJQTDE, 2013 No. 20, p. 12 and Ω = T q Γ(q + 1) + T q+n−1 |λ|Γ(q + 1) Using the above values in the condition (H 3 ) : we find that M > M 1 ≃ 1.055229.Clearly, all the conditions of Theorem 3.1 are satisfied.Hence the conclusion of Theorem 3.1 applies to the problem (3.2).

The Lower Semi-Continuous Case
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 lower semi-continuous maps with decomposable values.
Theorem 3.3 Assume that (H 2 ) − (H 3 ) and the following conditions hold: Proof.It follows from (H 2 ) and (H 4 ) that F is of l.s.c.type ( [16]).Then from Lemma 2.12, there exists a continuous function f : ) is a solution of (3.3), then x is a solution to the problem (1.1).In order to transform the problem (3.3) into a fixed point problem, we define the operator N as It can easily be shown that N 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.

The Lipschitz Case
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 [14].
Theorem 3.4 Assume that the following conditions hold: Then the boundary value problem (1.1) has at least one solution on [0, T ] if Proof.We transform the boundary value problem (1.1) into a fixed point problem.
Consider the operator N : C([0, T ], R) → P(C([0, T ], R)) defined at the begin of the proof of Theorem 3.1.We show that the operator N, satisfies the assumptions of Lemma 2.14.The proof will be given in two steps.
Step 1. N(x) is nonempty and closed for every v ∈ S F,x .Note that since the set-valued map F (•, x(•)) is measurable with the measurable selection theorem (e.g., [11,Theorem III.6]) it admits a measurable selection v : I → R.Moreover, by the assumption (H 6 ), we have As F has compact values, we pass onto a subsequence to obtain that v n converges to v in L 1 ([0, T ], R).Thus, v ∈ S F,x and for each t ∈ [0, T ], Hence, u ∈ N(x).

( 2 . 3 )
EJQTDE, 2013 No. 20, p. 3 where and if for each open set N of X containing G(x 0 ), there exists an open neighborhood N 0 of x 0 such that G(N 0 ) ⊆ N; (iv) 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; a∈A d(a, B), sup b∈B d(A, b)}, where d(A, b) = inf a∈A d(a; b) and d(a, B) = inf b∈B d(a; b).

Example 3 . 2
Let us consider the following 4−strip nonlocal boundary value problem: b) x −→ F (t, x) is lower semicontinuous for each t ∈ [0, T ]; Then the boundary value problem (1.1) has at least one solution on [0, T ].