Monotone Solution of Cauchy Type Weighted Nonlocal Fractional Differential Equation

Fractional derivatives now used to model a wide variety of real life problems and it has applications in physics, finance, biology, hydrology, etc., as fractional order models can be found to be more adequate than integer order models in some real world problems because fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes. Nonlinear fractional differential equation with weighted initial data has been studied by several authors. The weighted Cauchy-type problem


Introduction
Fractional derivatives now used to model a wide variety of real life problems and it has applications in physics, finance, biology, hydrology, etc., as fractional order models can be found to be more adequate than integer order models in some real world problems because fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes.
Nonlinear fractional differential equation with weighted initial data has been studied by several authors. The weighted Cauchy-type problem studied by Khaled et al. [13]. The solution of the periodic boundary value problem for a fractional differential equation involving a Riemann-Liouville fractional derivative D α (u(t)) = f (t, u(t)) t 1−α u(t)| t=0 = t 1−α u(t)| t=T (2) studied by Weia et al. [14]. Also the existence of solutions of fractional equations of Volterra type with the Riemann-Liouville derivative, studied by Jankowski [15] etc. And the weighted non-local fractional differential equation studied by Holambe et al. [26,27,28] and the references therein. Problems in nonlinear fractional differential equation were studied by various researchers.
The importance of non-local problems appears to have been first noted in the literature by Bitsadze Samarski [12]. By Byszewski [1,2], the non-local condition can be more useful than the standard initial condition to describe some physical phenomena. Now here we consider the weighted non-local fractional differential equation where c D α is Caputo fractional derivatives of order 0 < α ≤ 1 and 0 < t ≤ T < ∞,0 < τ i < T < ∞.

Auxiliary Results
We need certain definitions,lemmas and theorems in the sequel. We will use extensively studied fractional derivative and integral with its properties by [6,7,10,11,16,17,18,19,20,23,24,25] Let E denote a partially ordered real normed linear space with an order relation ≼ and the norm ∥ · ∥. It is known that E is regular if {x n } n∈N is a non-decreasing (resp. non-increasing) sequence in E such that x n → x * as n → ∞, then x n ≼ x * (resp. x n ≽ x * ) for all n ∈ N. Clearly, the partially ordered Banach space C(J, R) is regular and the conditions guaranteeing the regularity of any partially ordered normed linear space E may be found in Heikkilä and Lakshmikantham [8] and the references therein.
Definition 5 [3]. The order relation ≼ and the metric d on a non-empty set E are said to be compatible if {x n } n∈N is a monotone, that is, monotone nondecreasing or monotone nonincreasing sequence in E and if a subsequence {x n k } n∈N of {x n } n∈N converges to x * implies that the original sequence {x n } n∈N converges to x * . Similarly, given a partially ordered normed linear space (E, ≼, ∥·∥), the order relation ≼ and the norm ∥ · ∥ are said to be compatible if ≼ and the metric d defined through the norm ∥ · ∥ are compatible.
Definition 6 [21]. An upper semi-continuous and monotone nondecreasing function ψ : R + → R + is called a D-function provided ψ(r) = 0 iff r = 0. Let (E, ≼, ∥ · ∥) be a partially ordered normed linear space. A mapping T : E → E is called partially nonlinear D-Lipschitz if there exists a D-function ψ : for all comparable elements x, y ∈ E. If ψ(r) = k r, k > 0, then T is called a partially Lipschitz with a Lipschitz constant k.
Let (E, ≼, ∥ · ∥) be a partially ordered normed linear algebra. Denote The elements of K are called the positive vectors of the normed linear algebra E. The following lemma follows immediately from the definition of the set K and which is often times used in the applications of hybrid fixed point theory in Banach algebras.
The method may be stated as "the monotonic convergence of the sequence of successive approximations to the solutions of a nonlinear equation beginning with a lower or an upper solution of the equation as its initial or first approximation" which is a powerful tool in the existence theory of nonlinear analysis. It is clear that the method is different from the usual Picard's successive iteration method and embodied in the following applicable hybrid fixed point theorems proved in [3] which forms a useful key tool for our work contained in this paper. A few other hybrid fixed point theorems involving the method may be found in [3,4,5].
be a regular partially ordered complete normed linear algebra such that the order relation ≼ and the norm ∥ · ∥ in E are compatible in every compact chain of E. Let

IJARM Volume 11
Then the operator equation has a solution x * in E and the sequence {x n } of successive iterations defined by x n+1 = Ax n Bx n , n = 0, 1, . . . , converges monotonically to x * .
Remark. The compatibility of the order relation ≼ and the norm ∥ · ∥ in every compact chain of E holds if every partially compact subset of E possesses the compatibility property with respect to ≼ and ∥ · ∥. Note that a subset S of the partially ordered Banach space C(J, R) is called partially compact if every chain C in S is compact. This simple fact has been utilized to prove the main results of this paper.

Main Results
The equaivalent integral form of the problem [5] is considered in the function space C(J, R) of continuous real-valued functions defined on J. We define a norm ∥ · ∥ and the order relation and for all t ∈ J respectively. Clearly, C(J, R) is a Banach algebra with respect to above supremum norm and is also partially ordered w.r.t. the above partially order relation ≤. It is known that the partially ordered Banach algebra C(J, R) has some nice properties concerning the compatibility property with respect to the norm ∥ · ∥ and the order relation ≤ in certain subsets of it. The following lemma in this connection follows by an application of Arzelá-Ascoli theorem.
be a partially ordered Banach space with the norm ∥ · ∥ and the order relation ≤ defined by (9) and (10) respectively. Then ∥·∥ and ≤ are compatible in every partially compact subset of C(J, R).
We need the following definition in what follows.
Similarly, a function u u ∈ C(J, R) is said to be an upper solution of the problem (5) if it satisfies the above inequalities with reverse sign.

International Journal of Advanced Research in Mathematics Vol. 11 21
A Caratheódory function f is called L 2 -Carathéodory if (iii) there exists a function h ∈ L 2 (J, R) such that We consider the following set of assumptions in what follows: for all t ∈ J and x, y ∈ R,x ≤ y.
(A 5 ) The problem (5) has a lower solution u l ∈ C(J, R).
The following lemma is useful in what follows.

Lemma 13. For any
Proof. For the nonlocal weighted fractional differential equation (5) by the Caputo fractional derivative of order γ > 0

IJARM Volume 11
By the definition of Reimann-Liouville fractional integral Integrating from 0 to t,we get Operating Reimann-Liouville integral I α on bothside Taking differentiation on both side put t = τ i in the equation (11) International Journal of Advanced Research in Mathematics Vol. 11 23 then the FDE(5) has a solution x * defined on J and the sequence {x n } n∈N∪{0} of successive approximations defined by 24 IJARM Volume 11 Proof. Set E = C(J, R). Then, from Lemma 10 it follows that every compact chain in E possesses the compatibility property with respect to the norm ∥ · ∥ and the order relation ≤ in E.

Define two operators A and B on E by
From the continuity of the integral and the hypotheses (A 1 )-(A 5 ), it follows that A and B define the maps A, B : E → K. Now by definitions of the operators A and B, the FDE (5) is equivalent to the operator equation We shall show that the operators A, B satisfy all the conditions of Theorem 9. This is achieved in the series of following steps.
Step I: A, B are nondecreasing on E.
Let x, y ∈ E be such that x ≥ y. Then by hypothesis (A 2 ),(A 3 ) and (A 4 ), we obtain Ax(t) = g(t, x(t)) ≥ g(t, y(t)) = Ay(t), and for all t ∈ J. This shows that A and B are nondecreasing operators on E into E. Thus, A, B are nondecreasing positive operators on E into itself.
Step II: A is partially bounded and partially D-Lipschitz on E.
Let x ∈ E be arbitrary. Then by (A 2 ), for all t ∈ J. Taking supremum over t, we obtain ∥Ax∥ ≤ M g and so, A is bounded. This further implies that A is partially bounded on E.
Next, let x, y ∈ E be such that x ≤ y. Then, by hypothesis (A 3 ), for all t ∈ J. Taking supremum over t, we obtain for all x, y ∈ E with x ≤ y. Hence A is partially nonlinear D-Lipschitz operators on E which further implies that they are also a partially continuous on E into itself.
Step III: B is a partially continuous operator on E. Let {x n } n∈N be a sequence in a chain C of E such that x n → x for all n ∈ N. Then, by dominated convergence theorem, we have for all t ∈ J. This shows that Bx n converges monotonically to Bx pointwise on J.

IJARM Volume 11
Since the functions α is continuous on compact interval J so uniformly continuous there. Therefore, from the above inequality (15) it follows that |Bx n (t 2 ) − Bx n (t 1 )| → 0 as n → ∞ uniformly for all n ∈ N. This shows that the convergence Bx n → Bx is uniform and hence B is partially continuous on E.

IJARM Volume 11
Step IV: B is uniformly partially compact operator on E.
Let C be an arbitrary chain in E. We show that B(C) is a uniformly bounded and equicontinuous set in E. First we show that B(C) is uniformly bounded. Let y ∈ B(C) be any element. Then there is an element x ∈ C be such that y = Bx. Now, by hypothesis , for all t ∈ J. Taking the supremum over t, we obtain ∥y∥ ≤ ∥Bx∥ ≤ r for all y ∈ B(C). Hence, B(C) is a uniformly bounded subset of E. Moreover, ∥B(C)∥ ≤ r for all chains C in E. Hence, B is a uniformly partially bounded operator on E.
Next, we will show that B(C) is an equicontinuous set in E. Let t 1 , t 2 ∈ J with t 1 < t 2 . Then, for any y ∈ B(C), one has IJARM Volume 11 uniformly for all y ∈ B(C). Hence B(C) is an equicontinuous subset of E. Now, B(C) is a uniformly bounded and equicontinuous set of functions in E, so it is compact. Consequently, B is a uniformly partially compact operator on E into itself.
Step V: u l satisfies the operator inequality u l ≤ Au l Bu l . By hypothesis (A 5 ), the FDE 5 has a lower solution u l defined on J. Then, we have for all t ∈ J. From the definitions of the operators A, B and C it follows that u l (t) ≤ Au l (t) Bu l (t) for all t ∈ J. Hence u l ≤ Au l Bu l .
Step VI: The D-functions ψ A satisfy the growth condition M ψ A (r) < r, for r > 0. Finally, the D-function ψ A of the operator A satisfy the inequality given in hypothesis of Theorem 9, viz., ) ψ f (r) < r for all r > 0.
Thus A and B satisfy all the conditions of Theorem 9 and we conclude that the operator equation Ax Bx = x has a solution. Consequently the FDE (5) has a solution x * defined on J. Furthermore, the sequence {x n } n∈N of successive approximations defined by (13) converges monotonically to x * . This completes the proof.
The conclusion of Theorems 14 also remains true if we replace the hypothesis (A 5 ) with the following one: (A ′ 5 ) The FDE (5) has an upper solution u u ∈ C(J, R).
The proof of Theorem 14 under this new hypothesis is similar and can be obtained by closely observing the same arguments with appropriate modifications.