TRIPLE POSITIVE SOLUTIONS FOR BOUNDARY VALUE PROBLEM OF A NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION. Communicated by Mohammad Asadzadeh


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1 Bulletin of the Iranian Mathematical Society Vol. 33 No. 2 (27), pp . TRIPLE POSITIVE SOLUTIONS FOR BOUNDARY VALUE PROBLEM OF A NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION R. DEHGHANI AND K. GHANBARI* Communicated by Mohammad Asadzadeh Abstract. In this paper we generalize the main results of Bai, Wang and Ge (Electron J. Diff. Eqns. 6 (2), 8) by considering a general type of boundary value problem associated with a nonlinear fractional differential equation. We obtain sufficient conditions for the existence of at least three positive solution with corresponding upper and lower bounds.. Introduction Fractional differential equations have been studied by many authors caused by its applications in solving ordinary differential equations and also many applications in physics, mechanics, chemistry, and engineering. Positive solutions for ordinary differential equations and difference equations also have been considered by many authors, e.g. [, 6, 8]. The major tool in finding positive solutions for both fractional and ordinary differential equations have been fixed point theorems and Leray Schauder theory. In this paper we consider a typical fractional differential equations and improve the results obtained in [, 5] using fixed MSC(2): Primary: 3B5; Secondary: 3B8 Keywords: Fractional differential equation, Boundary value problem, Positive solution, Green s function, Fixedpoint theorem Received: 3 January 27, Revised: 7 June 27, Accepted: 25 June 27 Corresponding author c 27 Iranian Mathematical Society.
2 2 Dehghani and Ghanbari point theorems. We consider a nonlinear fractional differential equation of the following type: { D α x(t) + q(t)f(t, x(t), x (t)) ( < t < ) (.) x() x(), (.2) where x(t) and q(t) are nonnegative in (, ), f : [, ] [, ) R [, ) is continuous, 2 α < 3 is a real number, and D α is the standard RiemannLioville fractional differentiation (see definition 2.2). Furthermore q(t) does not identically vanish on any subinterval of (, ) and q(t) satisfies the following inequality: < [t( t)] α q(t) dt <. The motivation for considering 2 α < 3 is the importance of second order differential equations, which is equivalent to α 2 in this case. We obtain our main result by using the fixed point theorem of a cone preserving operator on an ordered Banach space that will be defined in Section 2. First we obtain an integral representation of the solution by the corresponding Green s function. 2. Background materials and definitions First we present the necessary definitions from fractional calculus theory and cone theory in Banach spaces, e.g. [2, 9]. Definition 2.. The fractional integral of order α > of a function y : (, ) R is given by I α y(t) t (t s) α y(s) ds, provided the right side is pointwise defined on (, ). Definition 2.2. The fractional derivative of order α > of a continuous function y : (, ) R is given by D α y(t) Γ(n α) ( d dt )n t y(s) ds, (t s) α n+
3 Nonlinear fractional differential equations 3 where n [α] +. Remark 2.3. As a basic example we quote for λ >, D α t λ Γ(λ + ) Γ(λ α + ) tλ α, giving in particular D α t α m, m, 2,..., N. Where N is the smallest integer greater than or equal to α. From Definition 2.2 and Remark 2.3, we obtain the following lemma. Lemma 2.. Let α >. If we assume u C(, ) L (, ), then the fractional differential equation D α u(t), has u(t) C t α + C 2 t α C N t α N, C i R, i,..., N as unique solutions. As D α I α u u for all u C(, ) L(, ), from Lemma 2. we deduce the following law of composition. Lemma 2.5. Assume that u C(, ) L (, ) with a fractional derivative of order α > that belongs to u C(, ) L (, ). Then I α D α u(t) u(t) + C t α + C 2t α C Nt α N, for some C i R, i,..., N. Now we find the integral representation of the solution and specify the corresponding Green s function. Lemma 2.6. Given y C[, ] and 2 α < 3, the unique solution of D α u(t) + y(t) ( < t < ) (2.) u() u(), (2.2)
4 Dehghani and Ghanbari is where G(t, s) u(t) G(t, s)y(s) ds, [t( s)] α (t s) α ( s t ) [t( s)] α. ( t s ) (2.3) Proof. We may apply Lemma 2.2 to reduce Eq.(2.) to an equivalent equation u(t) I α y(t) C t α C 2t α 2 C 3t α 3, for some C, C 2, C 3 R. Consequently, the general solution of Eq.(2.) is t (t s) α u(t) y(s) ds C t α C 2t α 2 C 3t α 3. By (2.2), therefore, C 2 C 3 and C ( s) α y(s) ds. Therefore, the unique solution of the differential equation (2.) subject to (2.2) is u(t) t t (t s) α ( s) α t α y(s) ds + y(s) ds [t( s)] α (t s) α [t( s)] α y(s) ds + y(s) ds G(t, s)y(s) ds. The proof is now completed. Lemma 2.7. The function G(t, s) defined by Eq.(2.3) satisfies the following conditions: (i) G(t, s) >, for t, s (, ) (ii) There exists a positive function γ C(, ) such that min G(t, s) γ(s) max G(t, s) γ(s)g(s, s), for < s <. t 3 t t
5 Nonlinear fractional differential equations 5 Proof. From (2.3) it is clear that G(t, s) > for s, t (, ). In the following, we consider the existence of γ(s). For s (, ), G(t, s) is s increasing with respect to t for t, with respect to t for t s ( ( s) α α 2 ), ( ( s) α α 2 ) and decreasing, and for s (, ), G(t, s) is decreasing with respect to t for s t and increasing with respect to t for s t. If we define then g (t, s) [t( s)]α (t s) α, g 2 (t, s) min t 3 G(t, s) min{g (, s), g (, s)}, s (, ) min{g ( 3, s), g 2(, s)}, s (, 3 ) g 2 (, s), s ( 3, ) min{g (, s), g (, s)}, s (, ) g ( 3, s), s (, r) g 2 (, s), s (r, ) min{ ( s)α, [t( s)]α, {[ ( s)]α ( s)α }}, s (, ) {[ 3 ( s)]α ( 3 s)α }, s (, r) ( s) α, α s (r, ) where < r < 3 is the unique solution of the equation [ 3 ( r)]α ( 3 r)α ( r)α α ; in particular r.5 as α 2 and r.26 as α 3. Using the monotonicity of G(t, s), we have max G(t, s) G(s, s) t [s( s)]α, s (, ).
6 6 Dehghani and Ghanbari Consequently, the desired function γ is γ(s) This completes the proof. min{ ( s)α, [ [s( s)] α ( s)]α ( s)α }, s (, [s( s)] α ) [ 3 ( s)]α ( 3 s)α, s ( [s( s)] α, r). s (r, ) (s) α Definition 2.8. Let E be a real Banach space over R. A nonempty convex closed set P E is said to be a cone provided that (i) au P for all u P and all a and (ii) u, u P implies u. Note that every cone P E induces an ordering in E given by x y if y x P. Let β and θ be nonnegative continuous convex functionals on P, ϕ be a nonnegative continuous concave functional on P, and ψ be a nonnegative continuous functional on P. Then for positive real numbers a, b, c and d, we define the following convex sets: P (β, d) {x P β(x) < d}, P (β, ϕ, b, d) {x P b ϕ(x), β(x) d}, P (β, θ, ϕ, b, c, d) {x P b ϕ(x), θ(x) c, β(x) d}, and a closed set R(β, ψ, a, d) {x P a ψ(x), β(x) d}. The following fixed point theorem due to Avery and Peterson is fundamental in the proof of our main results. Theorem 2.9. [2] Let P be a cone in a real Banach space E. Let β and θ be nonnegative continuous convex functionals on P, ϕ be a nonnegative continuous concave functional on P, and ψ be a nonnegative continuous functional on P satisfying ψ(λx) λψ(x) for λ such that for some positive numbers M and d, ϕ(x) ψ(x) and x M β(x) (2.) for all x P (β, d). Suppose that T : P (β, d) P (β, d) is completely continuous and there exist positive numbers a, b, and c with a < b such
7 Nonlinear fractional differential equations 7 that (s) {x P (β, θ, ϕ, b, c, d) ϕ(x) > b}, and ϕ(t x) > b for x P (β, θ, ϕ, b, c, d); (s2) ϕ(t x) > b for x P (β, ϕ, b, d) with θ(t x) > c; (s3) R(β, ψ, a, d) and ψ(t x) < a for x R(β, ψ, a, d) with ψ(x) a. Then T has at least three fixed points x, x 2, x 3 P (β, d) such that β(x i ) d for i, 2, 3; b < ϕ(x ); a < ψ(x 2 ) with ϕ(x 2 ) < b; ψ(x 3 ) < a. 3. Main results In this section, we impose growth conditions on f which allow us to apply Theorem 2. to establish the existence of triple positive solutions of the boundary value problem (.) with initial conditions (.2). Let X C [, ] be endowed with the ordering x y if x(t) y(t) for all t [, ], and the maximum norm Define the cone P X by x max{ max x(t), max t t x (t) }. P {x X x(t) on [, ], and x() x() }. Let the nonnegative continuous concave functional ϕ, the nonnegative continuous convex functionals θ, β, and the nonnegative continuous functional ψ be defined on the cone P by β(x) max t x (t), The following is wellknown. ψ(x) θ(x) max t x(t), ϕ(x) min x(t). t 3 Lemma 3.. [5]. If x P, then max t x(t) 2 max t x (t). By Lemma 3., the functionals defined above satisfy ϕ(x) θ(x) ψ(x), x max{θ(x), β(x)} β(x) (3.)
8 8 Dehghani and Ghanbari for all x P (β, d) P. Therefore, condition (2.) is satisfied. Lemma 3.2. Let T : P P be the operator defined by T x(t) G(t, s)q(s)f(s, x(s), x (s)) ds Then T : P P is completely continuous. Proof. The operator T : P P is continuous in view of nonnegativeness and continuity of G(t, s) and q(t)f(t, x, x ). Let Ω P be bounded, i.e., there exists a positive constant M > such that x M for all x Ω. Let L max q(t)f(t, x(t), t, x M x (t)) +. Then for x Ω, we have T x(t) G(t, s)q(s)f(s, x(s), x ) ds L G(s, s) ds. Hence, T (Ω) is bounded. On the other hand, given ɛ >, we set δ 2 ( ɛ L ) α. Then, for each x Ω, t, t 2 [, ], t < t 2, and t 2 t < δ one has T x(t 2 ) T x(t ) < ɛ. That is to say, T (Ω) is equicontinuous.
9 Nonlinear fractional differential equations 9 In fact, we have T x(t 2 ) T x(t ) G(t 2, s)q(s)f(s, x(s), x (s)) ds G(t, s)q(s)f(s, x(s), x (s)) ds + t [G(t 2, s) G(t, s)]q(s)f(s, x(s), x (s)) ds t 2 [G(t 2, s) G(t, s)]q(s)f(s, x(s), x (s)) ds t2 + [G(t 2, s) G(t, s)]q(s)f(s, x(s), x (s)) ds t < L [ t ( s) α (t α 2 t α ) ds + + ( s) α (t α 2 t α ) ds t 2 t2 t ( s) α (t α 2 t α ) ds In the following, we divide the proof into two cases. CASE. δ t < t 2 <. ] < L (tα 2 t α ). T x(t 2 ) T x(t ) < L (tα 2 t α ) L α δ 2 α (t 2 t ) CASE 2. t < δ, t 2 < 2δ. L (α )δα ɛ. T x(t 2 ) T x(t ) < L (tα 2 t α ) L tα 2 L (2δ)α ɛ. By the means of the ArzelaAscoli theorem, we have T : P P is completely continuous. The proof is now completed.
10 Dehghani and Ghanbari Let M N δ 3 α ( s)α s α 2 q(s) ds, G(s, s)q(s) ds, γ(s)g(s, s)q(s) ds. We assume there exist constants < a < b d 8 such that (A) f(t, u, v) d M, for (t, u, v) [, ] [, d 2 ] [ d, d]; (A2) f(t, u, v) > b δ, for (t, u, v) [, 3 ] [b, b] [ d, d]; (A3) f(t, u, v) < a N, for (t, u, v) [, ] [, a] [ d, d]. Theorem 3.3. Under the assumptions (A)(A3), the boundaryvalue problem (.)(.2) has at least three positive solutions x, x 2, and x 3 satisfying max t x i(t) d, for i, 2, 3; b < min x (t) ; (3.2) t 3 a < max x 2(t), with min x t 2 (t) < b; t 3 max x 3(t) < a. t Proof. The boundary value problem (.) with initial conditions (.2) has a solution x x(t) if and only if x solves the operator equation x(t) T x(t) G(t, s)q(s)f(s, x(s), x (s)) ds where T : P P, is completely continuous (Lemma 3.2). We now show that all the conditions of Theorem 2. are satisfied. If x P (β, d), then β(x) max t x (t) d. By Lemma 3. and max x(t) d t 2, the assumption (A) implies f(t, x(t), x (t)) d M.
11 Nonlinear fractional differential equations Define H(t, s) (α ) [( s)α t α 2 (t s) α 2 ], ( s t ) (α ) ( s)α t α 2, ( t s ) where 2 α < 3. For given s (, ), H(t, s) is decreasing with respect to t for s t and increasing with respect to t for t s. Using monotonicity of H(t, s), we have Consequently, (α ) max H(t, s) H(s, s) t ( s)α s α 2. β(t x) max (T t x) (t) t max (α ) t [( s)α t α 2 (t s) α 2 ]q(s)f(s, x(s), x (s)) ds (α ) + t ( s)α t α 2 q(s)f(s, x(s), x (s)) ds max H(t, s)q(s)f(s, x(s), x (s)) ds t (α ) d M (α ) ( s) α s α 2 q(s)f(s, x(s), x (s)) ds Hence, T : P (β, d) P (β, d). ( s) α s α 2 q(s) ds d M M d. To check the condition (s) of Theorem 2.9, we choose x(t) b, t. It is easy to see that x(t) b P (β, θ, ϕ, b, b, d) and ϕ(x) ϕ(b) > b, and so x {P (β, θ, ϕ, b, b, d) ϕ(x) > b}. Hence, if x P (β, θ, ϕ, b, b, d), then b x(t) b, x (t) d for t 3. From assumption (A2), we have f(t, x(t), x (t)) b δ for t 3, and by Lemma 2. ϕ(t x) min (T x)(t) t 3 b δ 3 γ(s)g(s, s)q(s) ds b δ δ b. γ(s)g(s, s)q(s)f(s, x(s), x (s)) ds
12 2 Dehghani and Ghanbari Thus, ϕ(t x) > b, for all x P (β, θ, ϕ, b, b, d). This shows that the condition (s) of Theorem 2.9 is satisfied. Notice that the condition (s) of Theorem 2.9 implies the condition (s2) of Theorem 2.9. Clearly, as ψ() < a, there holds that R(β, ψ, a, d). Suppose that x R(β, ψ, a, d) with ψ(x) a. Then, by the assumption (A3), ψ(t x) max (T x)(t) max t t < a N max t G(t, s)q(s) ds a N G(t, s)q(s)f(s, x(s), x (s)) ds G(s, s)q(s) ds a. So, the condition (s3) of Theorem 2.9 is satisfied. Therefore, an application of Theorem 2. implies that the boundary value problem (.) with initial conditions (.2) has at least three positive solutions x, x 2 and x 3 satisfying (3.). Example 3.. Consider the boundaryvalue problem where D 5 2 x(t) + f(t, x(t), x (t)) < t <, (3.3) x() x(), (3.) { e f(t, u, v) t u3 + ( v 6 )3 for u 2, e t ( v 6 )3 for u 2. Here we have α 5/2, q(t). By definition of M, N, and δ we have M α ( s)α s α 2 q(s) ds, (α )Γ(α ) π/8.22. Γ(2α ) Similarly we have N π/32.5. Using Lemma 2., r is the root of the equation [ 3 ( r)]α ( 3 r)α ( r)α α,
13 Nonlinear fractional differential equations 3 which is r.53 in this example. Substituting γ(s) given by Lemma 2. and observing that G(s, s) [s( s)]α, we find δ 3 γ(s)g(s, s)q(s) ds.. Choosing a, b 3, d 6, we have f(t, u, v) < a N 2 for t, u, 6 v 6; f(t, u, v) > b δ 3 for t 3, 3 u 2, 6 v 6; f(t, u, v) < d M for t, u 3, 6 v 6. Then all assumptions of Theorem 3. hold. Thus, by Theorem 3., the boundary value problem (3.3) with initial conditions (3.) has at least three positive solutions x, x 2 and x 3 such that max t x i(t) 6, for i, 2, 3 ; 3 < min x (t) ; t 3 < max x 2(t), with min x t 2 (t) < 3 ; t 3 max x 3(t) <. t Concluding Remark. As we mentioned in the abstract, we considered a general boundary values problem for a nonlinear fractional differential equation of the form (.). The particular case α 2 has been studied by [5] for an ordinary differential equation of the form x + q(t)f(t, x(t), x (t)) with the same boundary conditions as (.2). Our paper in fact generalizes the main results of [5] for a more general boundary value problem. Acknowledgment The authors would like to thank the anonymous referee for careful reading of the manuscript and valuable comments.
14 Dehghani and Ghanbari References [] R. P. Agarwal, D. O Regan, and P. J. Y. Wong, Positive Solutions of Differential, Difference, and Integral Equation, Kluwer Academic Publishers, Boston, 999. [2] R. I. Avery and A. C. Peterson, Three positive fixed points of nonlinear operators on ordered Banach spaces, Comput. Math. Appl. 2 (2), [3] A. Babakhani and V. D. Gejji, Existence of positive solutions of nonlinear fractional differential equations, J. Math. Anal. Appl. 278 (23), 3 2. [] Z. Bai and H. Lu, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl. 3 (25), [5] Z. Bai, Y. Wang and W. Ge, Triple positive solutions for a class of twopoint boundaryvalue problems, Electron. J. Diff. Eqns. Vol. 2 (6)(2), pp. 8. [6] J. Henderson and Y. Wang, Positive solutions for nonlinear eigenvalue problems J. Math. Anal. Appl. 28 (997), [7] M. A. Krasnosel skii, Positive Solutions of Operator Equations, Noordhoff, Groningen, 96. [8] W. C. Lian, F. H. Wong and C. C. Yeh, On the existence of positive solutions of nonlinear second order differential equations Proc. Amer. Math. Soc. 2 (996), [9] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley, New York, 993. [] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, New York, 999. [] J. Spanier and K. Oldham, The Fractional Calculus, Theory and Applications of Differentiation and Integration to Arbitrary Order, Academid Press, New York, 97. [2] S. Q. Zhang, Existence of positive solution for some class of a nonlinear fractional differential equations, J. Math. Anal. Appl. 278 (23), [3] S. Q. Zhang, The existence of a positive solution for a nonlinear fractional differential equation, J. Math. Anal. Appl. 252 (2), R. Dehghani and K. Ghanbari Department of Mathematics Sahand University of Technology Tabriz, Iran
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