Nonlinear q-fractional differential equations with nonlocal and sub-strip type boundary conditions

This paper is concerned with new boundary value problems of nonlinear qfractional differential equations with nonlocal and sub-strip type boundary conditions. Our results are new in the present setting and rely on the contraction mapping principle and a fixed point theorem due to O’Regan. Some illustrative examples are also presented.


Introduction
In this paper, we introduce a sub-strip type boundary condition of the form which relates the contribution due to a sub-strip of arbitrary length with the value of the unknown function at an arbitrary (nonlocal) point off the sub-strip.Precisely, we consider the following boundary value problem of nonlinear fractional q-difference equations with nonlocal and sub-strip type boundary conditions: B. Ahmad, S. K. Ntouyas, A. Alsaedi and H. Al-Hutami where c D υ q denotes the Caputo fractional q-derivative of order υ, f : [0, 1] × R → R and g : C([0, 1], R) → R are given continuous functions, and b is a real constant.Here we emphasize that the nonlocal conditions are more plausible than the standard initial conditions to describe some physical phenomena.In (1.1), g(x) may be understood as g(x) = ∑ p j=1 α j x(t j ) where α j , j = 1, . . ., p, are given constants and 0 < t 1 < . . .< t p ≤ 1.For more details we refer to the work by Byszewski [1,2].
Recent extensive studies on fractional boundary value problems indicate that it is one of the hot topics of the present-day research.There have appeared numerous articles covering a variety of aspects of these problems.The nonlocal nature of a fractional order differential operator, which takes into account hereditary properties of various material and processes, has helped to improve the mathematical modelling of many real world problems of physical and technical sciences [3,4].For some recent work on the topic, please see [5][6][7][8][9][10][11][12][13] and the references therein.
Fractional q-difference (q-fractional) equations are regarded as the fractional analogue of qdifference equations.Motivated by recent interest in the study of fractional-order differential equations, the topic of q-fractional equations has attracted the attention of many researchers.The details of some recent development of the subject can be found in [14][15][16][17][18][19][20], whereas the background material on q-fractional calculus can be found in a recent text [22].
The paper is organized as follows.In Section 2, we recall some fundamental concepts of fractional q-calculus and establish a lemma for the linear variant of the given problem.Section 3 contains the existence results for the problem (1.1) which are shown by applying Banach's contraction principle and a fixed point theorem due to O'Regan.In Section 4, we consider a new problem with a condition of the form . Finally, some examples illustrating the applicability of our results are presented in Section 5.
For a real parameter q ∈ R + \ {1}, a q-real number denoted by [a] q is defined by The q-analogue of the Pochhammer symbol (q-shifted factorial) is defined as The q-analogue of the exponent (x − y) k is The q-gamma function Γ q (y) is defined as where y ∈ R \ {0, −1, −2, . ..}. Observe that Γ q (y + 1) = [y] q Γ q (y).
Nonlinear q-fractional differential equations 3 Definition 2.1 ([21]).Let f be a function defined on [0, 1].The fractional q-integral of the Riemann-Liouville type of order β ≥ 0 is (I 0 q f )(t) = f (t) and Observe that β = 1 in the Definition 2.1 yields q-integral I q f (t For more details on q-integral and fractional q-integral, see Section 1.3 and Section 4.2 respectively in [22]. Remark 2.2.The q-fractional integration possesses the semigroup property ([22, Proposition 4.3]): Further, it was shown in Lemma 6 of [23] that Before giving the definition of fractional q-derivative, we recall the concept of q-derivative.We know that the q-derivative of a function f (t) is defined as Furthermore,

Definition 2.3 ([22]
).The Caputo fractional q-derivative of order β > 0 is defined by where β is the smallest integer greater than or equal to β.
B. Ahmad, S. K. Ntouyas, A. Alsaedi and H. Al-Hutami is equivalent to an integral equation: where Proof.Applying the operator I υ q on the equation c D υ q x(t) = y(t) and using (2.2), we get where a 0 , a 1 ∈ R are arbitrary constants.Using the given boundary conditions, it is found that a 1 = y 0 , and ( Substituting the values of a 0 , a 1 in (2.7) yields (2.5).Conversely, applying the operator c D υ q on (2.5) and taking into account (2.3), it follows that c D υ q x(t) = y(t).From (2.5), it is easy to verify that the boundary conditions x(0) = y 0 , x(ξ) = b 1 η x(s) d q s are satisfied.This establishes the equivalence between (2.4) and (2.5).

Main results
We denote by C = C([0, 1], R) the Banach space of all continuous functions from [0, 1] → R endowed with a topology of uniform convergence with the norm defined by Also by L 1 ([0, 1], R) we denote the Banach space of measurable functions x : [0, 1] → R which are Lebesgue integrable and normed by In view of Lemma 2.4, we can transform the problem (1.1) into an equivalent fixed point problem: F x = x, where the operator F : C → C is defined by Observe that the existence of a fixed point for the operator F implies the existence of a solution for the problem (1.1).
For convenience we introduce the notations: and Then the boundary value problem (1.1) has a unique solution.
Proof.For x, y ∈ C and for each t ∈ [0, 1], from the definition of F and assumptions (A 1 ) and (A 2 ), we obtain As δ < 1 by (A 3 ), the operator F is a contraction map from the Banach space C into itself.Hence the conclusion of the theorem follows by the contraction mapping principle (Banach fixed point theorem).
Our next existence result relies on a fixed point theorem due to O'Regan in [24].In the sequel, we will use Lemma 3.2 by taking C to be E.For more details of such fixed point theorems, we refer a paper [25] by Petryshyn.
To apply Lemma 3.2, we define F i : C → C, i = 1, 2 by and Theorem 3.3.Suppose that (A 2 ) holds.In addition, we assume that: Nonlinear q-fractional differential equations , where µ 0 and k 0 are defined in (3.2) and (3.3) respectively.
Then the boundary value problem (1.1) has at least one solution on [0, 1].
Proof.By the assumption (A 6 ), there exists a number r 0 > 0 such that We shall show that the operators F 1 and F 2 defined by (3.4) and (3.5) respectively, satisfy all the conditions of Lemma 3.2.
Step 1.The operator F 1 is continuous and completely continuous.Let us consider the set and show that F 1 ( Ωr 0 ) is bounded.For any x ∈ Ωr 0 , we have Thus the operator F 1 ( Ωr 0 ) is uniformly bounded.For any t 1 , t 2 ∈ [0, 1], t 1 < t 2 , we have which is independent of x and tends to zero as t 2 − t 1 → 0. Thus, F 1 is equicontinuous.Hence, by the Arzelà-Ascoli theorem, F 1 ( Ωr 0 ) is a relatively compact set.Now, let x n ⊂ Ωr 0 with B. Ahmad, S. K. Ntouyas, A. Alsaedi and H. Al-Hutami x n − x → 0. Then the limit x n (t) − x(t) → 0 is uniformly valid on [0, 1].From the uniform continuity of f (t, x) on the compact set [0, 1] × Ωr 0 , it follows that f (t, x n (t)) − f (t, x(t)) → 0 is uniformly valid on [0, 1].Hence F 1 x n − F 1 x → 0 as n → ∞ which proves the continuity of F 1 .This completes the proof of Step 1.
Step 3. The set F ( Ωr 0 ) is bounded.The assumptions (A 2 ) and (A 4 ) imply that for any x ∈ Ωr 0 .This, with the boundedness of the set F 1 ( Ωr 0 ) implies that the set F ( Ωr 0 ) is bounded.
Step 4. Finally, it will be shown that the case (C2) in Lemma 3.2 does not hold.On the contrary, we suppose that (C2) holds.Then, we have that there exist κ ∈ (0, 1) and x ∈ ∂Ω r 0 such that x = κF x.So, we have x = r 0 and Using the assumptions (A 2 ) and (A 4 )-(A 6 ), we get Γ q (υ) Γ q (υ) Taking the supremum over t ∈ [0, 1], and using the definition of Ωr 0 , we obtain Thus, we get a contradiction: Thus the operators F 1 and F 2 satisfy all the conditions of Lemma 3.2.Hence, the operator F has at least one fixed point x ∈ Ωr 0 , which is the solution of the problem (1.1).This completes the proof.
Remark 3.4.If we consider the equation of the form ( c D υ q + λ)x(t) = f (t, x(t)), λ ∈ R in the problem (1.1), then the condition (A 3 ) in the statement of Theorem 3.1 modifies to δ := (L + λ)µ 0 + k 0 < 1 whereas the condition (A 6 ) in the statement of Theorem 3.3 takes the form sup We emphasize that the equations similar to one considered in this remark appear in applied problems, for example, see [26,27].

A boundary value problem with flux sub-strip conditions
In this section, we discuss the existence of solutions for a boundary value problem of nonlinear fractional q-difference equations with nonlocal and flux sub-strip type boundary conditions.Precisely, we consider the following boundary value problem where c D υ q denotes the Caputo fractional q-derivative of order υ, f : [0, 1] × R → R and g : C([0, 1], R) → R are given continuous functions, and b, λ are real constants.
For the sequel, we set Now we are in a position to give the existence results for the problem (4.1).We do not provide the proofs for these results as the method of proof is similar to the one employed in the preceding section.Theorem 4.2.Assume that (A 2 ), (A 4 )-(A 6 ) hold with µ 0 and k 0 in place of µ 0 and k 0 , where µ 0 and k 0 are given by (4.2) and (4.3) respectively.Then there exists at least one solution for the problem (4.1) on [0, 1].

Theorem 4 . 1 .
Let the assumptions (A 1 )-(A 3 ) hold with µ 0 and k 0 in place of µ 0 and k 0 , where µ 0 and k 0 are given by (4.2) and (4.3) respectively.Then the boundary value problem (4.1) has a unique solution.
3.2.Let U be an open set in a closed, convex set C of a Banach space E. Assume 0 ∈ U. Also assume that F ( Ū) is bounded and that F : Ū → C is given by F = F 1 + F 2 , in which F 1 : Ū → E is continuous and completely continuous and F 2 : Ū → E is a nonlinear contraction (i.e., there exists a continuous nondecreasing function ϑ :