Existence results for nonlocal boundary value problems of nonlinear fractional q-difference equations

Advances in Difference Equations, Aug 2012

In this paper, we study a nonlinear fractional q-difference equation with nonlocal boundary conditions. The existence of solutions for the problem is shown by applying some well-known tools of fixed-point theory such as Banach’s contraction principle, Krasnoselskii’s fixed-point theorem, and the Leray-Schauder nonlinear alternative. Some illustrating examples are also discussed. MSC:34A08, 39A05, 39A12, 39A13.

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Existence results for nonlocal boundary value problems of nonlinear fractional q-difference equations

Advances in Difference Equations Existence results for nonlocal boundary value problems of nonlinear fractional q-difference equations Bashir Ahmad 1 Sotiris K Ntouyas 0 Ioannis K Purnaras 0 0 Department of Mathematics, University of Ioannina , Ioannina, 451 10 , Greece 1 Department of Mathematics, Faculty of Science, King Abdulaziz University , P.O. Box 80203, Jeddah, 21589 , Saudi Arabia In this paper, we study a nonlinear fractional q-difference equation with nonlocal boundary conditions. The existence of solutions for the problem is shown by applying some well-known tools of fixed-point theory such as Banach's contraction principle, Krasnoselskii's fixed-point theorem, and the Leray-Schauder nonlinear alternative. Some illustrating examples are also discussed. MSC: 34A08; 39A05; 39A12; 39A13 fractional q-difference equations; nonlocal boundary conditions; existence; Leray-Schauder nonlinear alternative; fixed point - cDqαx(t) = f t, x(t) ,  ≤ t ≤ ,  < α ≤ , 2 Preliminaries on fractional q-calculus The q analogue of the power (a – b)n is (a – b)() = , (a – b)(n) = a – bqk , a, b ∈ R, n ∈ N. The q-gamma function is defined by ( – q)(x–) q(x) = ( – q)x– , x ∈ R \ {, –, –, . . .},  < q <  and satisfies q(x + ) = [x]q q(x) (see, []). For  < q < , we define the q-derivative of a real valued function f as Dqf (t) = Dqf () = tl→im Dqf (t). The higher order q-derivatives are given by Dqf (t) = f (t), Dqnf (t) = DqDqn–f (t), n ∈ N. Iqf (x) = f (s) dqs = x( – q)qnf xqn provided that the series converges. f (s) dqs = Iqf (b) – Iqf (a) = ( – q) qn bf bqn – af aqn , DqIqf (x) = f (x), IqDqf (x) = f (x) – f (). For more details of the basic material on q-calculus, see the book []. Definition . ([]) Let α ≥  and f be a function defined on [, ]. The fractional qintegral of the Riemann-Liouville type is (Iqf )(t) = f (t) and f (s) dq(s), Dqαf (t) = D[qα]I[α]–αf (t), q cDqαf (t) = Iq[α]–αD[qα]f (t), Definition . ([]) The fractional q-derivative of the Riemann-Liouville type of order α ≥  is defined by (Dqf )(t) = f (t) and Lemma . Let α, β ≥  and let f be a function defined on [, ]. Then the next formulas hold: (i) (Iqβ Iqαf )(t) = (Iqα+β f )(t), (ii) (DqαIqαf )(t) = f (t). q(k + )  < a < x < b. ⎪⎪⎩ αx() + βDqx() = γx(η), is given by x(t) = g(s) dqs g(s) dqs g(s) dqs g(s) dqs, g(s) dqs Proof In view of Lemmas . and ., integrating equation in (.), we have x(t) = g(s) dqs + ct + c, t ∈ [, ]. Using the boundary conditions of (.) in (.), we have g(s) dqs g(s) dqs, Solving the above system of equations for c, c, we get c = g(s) dqs g(s) dqs c = g(s) dqs g(s) dqs g(s) dqs . Substituting the values of c, c in (.), we obtain (.). In view of Lemma ., we define an operator F : C([, ], R) → C([, ], R) as (Fx)(t) = f s, x(s) dqs f s, x(s) dqs f s, x(s) dqs f s, x(s) dqs Observe that problem (.)-(.) has a solution if the operator equation Fx = x has a fixed point, where F is given by (.). 3 Main results Theorem . Assume that f : [, ] × R → R is continuous and that there exists a qintegrable function L : [, ] → R such that Then the boundary value problem (.)-(.) has a unique solution provided (Fx)(t) ≤ where k is given by (.). A = ≤ M L(s) dqs L(s) dqs L(s) dqs L(s) dqs ≤ M which, in view of (.) and (.), implies that This shows that FBρ ⊂ Bρ . Now, for x, y ∈ C, we obtain (Fx) – (Fy) f s, x(s) – f s, y(s) dqs f s, x(s) – f s, y(s) dqs f s, x(s) – f s, y(s) dqs f s, x(s) – f s, y(s) dqs  ( – qs)(α–) q(α – ) f s, x(s) – f s, y(s) dqs L(s) dqs which, in view of (.), yields (Fx)(t) – (Fy)(t) ≤ k x – y . In case L(t) = L (L is a constant), the condition (.) becomes LA <  and Theorem . takes the form of the following result. Corollary . Assume that f : [, ] × R → R is a continuous function and that Then the boundary value problem (.)-(.) has a unique solution. f (t, x) ≤ μ(t)φ |x| , (t, x) ∈ [, ] × R; (A) there exists a constant r with r ≥ where μ = supt∈[,T] |μ(t)|. L(s) dqs then the boundary value problem (.)-(.) has at least one solution on [, ]. q(α + ) + |γ|δ qη((αα–+)) + |γ|δ qη((αα–+)) + |α|δ q(α + ) + |β|δ q(α) ≤ r. (Px)(t) = f s, x(s) dqs, t ∈ [, ], f s, x(s) dqs f s, x(s) dqs f s, x(s) dqs  ( – qs)(α–) q(α – ) f s, x(s) dqs, t ∈ [, ]. For x, y ∈ Br, we find that (Px + Qy)(t) ≤ (Px)(t) – (Px)(t) t (t – qs)(α–) – (t – qs)(α–) q(α) f s, x(s) dqs f s, x(s) dqs – f s, x(s) dqs f s, x(s) dqs t (t – qs)(α–) – (t – qs)(α–) q(α) In the special case when φ(u) ≡ , we see that there always exists a positive r so that (.) holds true, thus we have the following corollary. ∀(t, x) ∈ [, ] × R, and μ ∈ C [, ], R+ . The next existence result is based on Leray-Schauder nonlinear alternative. Lemma . (Nonlinear alternative for single valued maps []) Let E be a Banach space, C a closed, convex subset of E, (...truncated)


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Bashir Ahmad, Sotiris K Ntouyas, Ioannis K Purnaras. Existence results for nonlocal boundary value problems of nonlinear fractional q-difference equations, Advances in Difference Equations, 2012, pp. 140, Volume 2012, Issue 1, DOI: 10.1186/1687-1847-2012-140