THÔNG TIN TÀI LIỆU
Baleanu et al Advances in Difference Equations 2013, 2013:368 http://www.advancesindifferenceequations.com/content/2013/1/368 RESEARCH Open Access Existence and uniqueness of solutions for multi-term nonlinear fractional integro-differential equations Dumitru Baleanu1,2,3* , Sayyedeh Zahra Nazemi4 and Shahram Rezapour4 * Correspondence: dumitru@cankaya.edu.tr Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O Box 80204, Jeddah, 21589, Saudi Arabia Department of Mathematics, Cankaya University, Ogretmenler Cad 14, Balgat, Ankara, 06530, Turkey Full list of author information is available at the end of the article Abstract In this manuscript, by using the fixed point theorems, the existence and the uniqueness of solutions for multi-term nonlinear fractional integro-differential equations are reported Two examples are presented to illustrate our results Keywords: Caputo fractional derivative; fixed point theorem; multi-term nonlinear fractional differential equation Introduction The study of fractional differential equations ranges from the theoretical aspects of existence and uniqueness of solutions to the analytic and numerical methods for finding solutions Fractional differential equations appear naturally in a number of fields such as physics, polymer rheology, regular variational in thermodynamics, biophysics, blood flow phenomena, aerodynamics, electro-dynamics of complex medium, viscoelasticity, Bode’s analysis of feedback amplifiers, capacitor theory, electrical circuits, electron-analytical chemistry, biology, control theory, fitting of experimental data, etc An excellent account in the study of fractional differential equations can be found in [, ] and [] For more details and examples, one can study [–] and [] It is considerable that there are many works about fractional integro-differential equations (see, for example, [–] and []) In , Xinwei and Landong reviewed the existence of solutions for the nonlinear fractional differential equation c Dα u(t) = f t, u(t), c Dβ u(t) ( < t < ) with boundary values u() = u () = or u () = u() = or u() = u() = , where < α ≤ , < β ≤ , and f is continuous on [, ] × R × R [] In , Su and Zhang studied the existence and uniqueness of solutions for the following nonlinear two-point fractional boundary value problem c Dα u(t) = f t, u(t), c Dβ u(t) ( < t < ) with boundary values a u() – a u () = A and b u() + b u () = B, where α, β, , bi (i = , ) satisfy certain conditions [] In , Ahmad and Sivasundaram studied the ©2013 Baleanu et al.; licensee Springer This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Baleanu et al Advances in Difference Equations 2013, 2013:368 http://www.advancesindifferenceequations.com/content/2013/1/368 Page of 17 existence of solutions for the nonlinear fractional integro-differential equation c Dq u(t) = f t, u(t), (φu)(t), (ψu)(t) ( < t < and < q ≤ ) with boundary values u () + au(η ) = , bu () + u(η ) = and < η ≤ η < , where c Dq is the Caputo fractional derivative, a, b ∈ (, ), f : [, ] × X × X × X → X is continuous and t for the mappings γ , λ : [, ] × [, ] → [, ∞) with the property supt∈[,] | λ(t, s) ds| < ∞ t t and supt∈[,] | γ (t, s) ds| < ∞, the maps φ and ψ are defined by (φu)(t) = γ (t, s)u(s) ds t and (ψu)(t) = λ(t, s)u(s) ds Here, X is a Banach space (see []) Main results 2.1 The basic problem In this paper, we study the existence and uniqueness of solutions for the multi-term nonlinear fractional integro-differential equation c Dα u(t) = f t, u(t), (φu)(t), (ψu)(t), c Dβ u(t), c Dβ u(t), , c Dβn u(t) ( < t < ) () with boundary values u() + au() = and u () + bu () = , where < α < , < βi < , α – βi ≥ , a, b = –, f : [, ] × Rn+ → R is continuous, and for the mappings γ , λ : [, ] × [, ] → [, ∞) t t with the property supt∈[,] | λ(t, s) ds| < ∞ and supt∈[,] | γ (t, s) ds| < ∞, the maps φ t t and ψ are defined by (φu)(t) = γ (t, s)u(s) ds and (ψu)(t) = λ(t, s)u(s) ds In this way, we need the following result, which has been proved in [] Lemma . Let α > and n = [α] + Then I α c Dα u(t) = u(t) + c + c t + c t + · · · + cn– t n– , where c , c , , cn– are some real numbers The proof of the following result by using Lemma . is straightforward Lemma . Let y ∈ C[, ], a, b = – and < α < Then the problem c Dα u(t) = y(t) with boundary values u() + au() = and u () + bu () = has the unique solution (α) u(t) = + t (t – s)α– y(s) ds – ab – b( + a)t ( + a)( + b) (α – ) a ( + a) (α) ( – s)α– y(s) ds ( – s)α– y(s) ds 2.2 Some results on solving the problem Let C(I) be the space of all continuous real-valued functions on I = [, ] and X = u : u ∈ C(I) and c Dβi u ∈ C(I) ( < βi < ) for i = , , , n endowed with the norm u = maxt∈I |u(t)| + · ) is a Banach space n c βi i= maxt∈I | D u(t)| It is known that (X, Baleanu et al Advances in Difference Equations 2013, 2013:368 http://www.advancesindifferenceequations.com/content/2013/1/368 Page of 17 Theorem . Assume that there exist κ ∈ (, α – ) and μ(t) ∈ L κ ([, ], (, ∞)) such that f (t, x, y, w, u , u , , un ) – f t, x , y , w , v , v , , ≤ μ(t) x – x + y – y + w – w + |u – v | + |u – v | + · · · + |un – | for all t ∈ [, ] and x, y, w, x , y , w , u , u , , un , v , v , , ∈ R Then problem () has a unique solution whenever ( + |a|)μ∗ – κ | + a| (α) α – κ = ( + γ + λ ) n + –κ |b|( + |a|)μ∗ | + a|| + b| (α – ) α – κ – –κ (α – κ)μ∗ (α – ) (α – βi – κ + ) α – κ – + i= + –κ –κ |b|μ∗ | + b| ( – βi ) (α – ) α – κ – where γ = supt∈I | t γ (t, s) ds|, λ = supt∈I | t –κ –κ –κ < , λ(t, s) ds|, μ∗ = ( (μ(s)) κ ds)κ Proof Define the mapping F : X → X by t (Fu)(t) = – (t – s)α– f s, u(s), (φu)(s), (ψu)(s), c Dβ u(s), c Dβ u(s), , c Dβn u(s) ds (α) a ( + a) ( – s)α– (α) × f s, u(s), (φu)(s), (ψu)(s), c Dβ u(s), c Dβ u(s), , c Dβn u(s) ds + ab – b( + a)t ( + a)( + b) ( – s)α– (α – ) × f s, u(s), (φu)(s), (ψu)(s), c Dβ u(s), c Dβ u(s), , c Dβn u(s) ds For each u, v ∈ X and t ∈ [, ], by using the Hölder inequality, we have (Fu)(t) – (Fv)(t) t = (t – s)α– f s, u(s), (φu)(s), (ψu)(s), c Dβ u(s), c Dβ u(s), , c Dβn u(s) (α) – f s, v(s), (φv)(s), (ψv)(s), c Dβ v(s), c Dβ v(s), , c Dβn v(s) ds – a ( + a) ( – s)α– f s, u(s), (φu)(s), (ψu)(s), c Dβ u(s), c Dβ u(s), , c Dβn u(s) (α) – f s, v(s), (φv)(s), (ψv)(s), c Dβ v(s), c Dβ v(s), , c Dβn v(s) ds + ab – b( + a)t ( + a)( + b) ( – s)α– f s, u(s), (φu)(s), (ψu)(s), c Dβ u(s), c Dβ u(s), (α – ) , c Dβn u(s) – f s, v(s), (φv)(s), (ψv)(s), c Dβ v(s), c Dβ v(s), , c Dβn v(s) ds t ≤ (t – s)α– f s, u(s), (φu)(s), (ψu)(s), c Dβ u(s), c Dβ u(s), , c Dβn u(s) (α) Baleanu et al Advances in Difference Equations 2013, 2013:368 http://www.advancesindifferenceequations.com/content/2013/1/368 Page of 17 – f s, v(s), (φv)(s), (ψv)(s), c Dβ v(s), c Dβ v(s), , c Dβn v(s) ds |a| | + a| + ( – s)α– f s, u(s), (φu)(s), (ψu)(s), c Dβ u(s), c Dβ u(s), , c Dβn u(s) (α) – f s, v(s), (φv)(s), (ψv)(s), c Dβ v(s), c Dβ v(s), , c Dβn v(s) ds |ab – b( + a)t| | + a|| + b| + ( – s)α– f s, u(s), (φu)(s), (ψu)(s), c Dβ u(s), c Dβ u(s), (α – ) , c Dβn u(s) – f s, v(s), (φv)(s), (ψv)(s), c Dβ v(s), c Dβ v(s), , c Dβn v(s) ds t ≤ (t – s)α– μ(s) u(s) – v(s) + (φu)(s) – (φv)(s) + (ψu)(s) – (ψv)(s) (α) + c Dβ u(s) – c Dβ v(s) + c Dβ u(s) – c Dβ v(s) + · · · + c Dβn u(s) – c Dβn v(s) ds + |a| | + a| ( – s)α– μ(s) u(s) – v(s) + (φu)(s) – (φv)(s) + (ψu)(s) – (ψv)(s) (α) + c Dβ u(s) – c Dβ v(s) + c Dβ u(s) – c Dβ v(s) + · · · + c Dβn u(s) – c Dβn v(s) ds + |b|( + |a|) | + a|| + b| ( – s)α– μ(s) u(s) – v(s) + (φu)(s) – (φv)(s) (α – ) + (ψu)(s) – (ψv)(s) + c Dβ u(s) – c Dβ v(s) + c Dβ u(s) – c Dβ v(s) + · · · + c Dβn u(s) – c Dβn v(s) ds ≤ ( + γ + λ ) u – v (α) + + ≤ t |a|( + γ + λ ) u – v | + a| (α) + ( – s)α– μ(s) ds |b|( + |a|)( + γ + λ ) u – v | + a|| + b| (α – ) ( + γ + λ ) u – v (α) + (t – s)α– μ(s) ds t t ds μ(s) κ κ ds –κ ( – s)α– –κ ds ( – s)α– –κ μ(s) κ κ ds –κ ds μ(s) –κ α–κ –κ + κ ds |a|μ∗ ( + γ + λ ) u – v | + a| (α) |b|( + |a|)μ∗ ( + γ + λ ) u – v | + a|| + b| (α – ) ≤ ( + γ + λ ) κ μ ( + γ + λ ) u – v (α) + –κ –κ (t – s)α– |b|( + |a|)( + γ + λ ) u – v | + a|| + b| (α – ) ∗ + ( – s)α– μ(s) ds |a|( + γ + λ ) u – v | + a| (α) × ≤ ( + |a|)μ∗ – κ | + a| (α) α – κ –κ |b|( + |a|)μ∗ | + a|| + b| (α – ) α – κ – –κ α–κ – –κ –κ u–v –κ –κ α–κ –κ Baleanu et al Advances in Difference Equations 2013, 2013:368 http://www.advancesindifferenceequations.com/content/2013/1/368 Page of 17 Also, we have c Dβi (Fu)(t) – c Dβi (Fv)(t) t (t – s)–βi (Fu) (s) ds – ( – βi ) t (t – s)–βi ( – βi ) = = t (t – s)–βi (Fv) (s) ds ( – βi ) (s – τ )α– (α – ) s × f τ , u(τ ), (φu)(τ ), (ψu)(τ ), c Dβ u(τ ), c Dβ u(τ ), , c Dβn u(τ ) dτ b +b – ( – τ )α– (α – ) × f τ , u(τ ), (φu)(τ ), (ψu)(τ ), c Dβ u(τ ), c Dβ u(τ ), , c Dβn u(τ ) dτ ds t – s (t – s)–βi ( – βi ) (s – τ )α– (α – ) × f τ , v(τ ), (φv)(τ ), (ψv)(τ ), c Dβ v(τ ), c Dβ v(τ ), , c Dβn v(τ ) dτ b +b – ( – τ )α– (α – ) × f τ , v(τ ), (φv)(τ ), (ψv)(τ ), c Dβ v(τ ), c Dβ v(τ ), , c Dβn v(τ ) dτ ds t ≤ (t – s)–βi ( – βi ) s (s – τ )α– (α – ) × f τ , u(τ ), (φu)(τ ), (ψu)(τ ), c Dβ u(τ ), c Dβ u(τ ), , c Dβn u(τ ) – f τ , v(τ ), (φv)(τ ), (ψv)(τ ), c Dβ v(τ ), c Dβ v(τ ), , c Dβn v(τ ) dτ ds + |b| | + b| × t (t – s)–βi ( – βi ) ( – τ )α– f τ , u(τ ), (φu)(τ ), (ψu)(τ ), c Dβ u(τ ), c Dβ u(τ ), , c Dβn u(τ ) (α – ) – f τ , v(τ ), (φv)(τ ), (ψv)(τ ), c Dβ v(τ ), c Dβ v(τ ), , c Dβn v(τ ) dτ ds ≤ ( + γ + λ ) u – v ( – βi ) (α – ) + ≤ ≤ (s – τ )α– μ(τ ) dτ ds t (t – s)–βi –κ t (t – s)–βi sα–κ– ds –κ α–κ – –κ α–κ – |b|μ∗ ( + γ + λ ) u – v | + b| ( – βi ) (α – ) ( – τ )α– μ(τ ) dτ ds –κ α–κ – |b|μ∗ ( + γ + λ ) u – v | + b| ( – βi ) (α – ) μ∗ ( + γ + λ ) u – v ( – βi ) (α – ) + s (t – s)–βi |b|( + γ + λ ) u – v | + b| ( – βi ) (α – ) μ∗ ( + γ + λ ) u – v ( – βi ) (α – ) + t –κ (t – s)–βi ds –κ –κ α–κ – t ( – ξ )–βi ξ α–κ– dξ –κ Baleanu et al Advances in Difference Equations 2013, 2013:368 http://www.advancesindifferenceequations.com/content/2013/1/368 Since B(α – κ, – βi ) = c –βi α–κ– ξ dξ ( – ξ ) Page of 17 (α–κ) (–βi ) , (α–βi –κ+) = –κ (α – κ)μ∗ (α – ) (α – βi – κ + ) α – κ – Dβi (Fu)(t) – c Dβi (Fv)(t) ≤ ( + γ + λ ) + we obtain –κ |b|μ∗ | + b| ( – βi ) (α – ) α – κ – –κ –κ u–v for all i = , , , n Hence, we get Fu – Fv ≤ ( + γ + λ ) n –κ + –κ |b|( + |a|)μ∗ | + a|| + b| (α – ) α – κ – –κ (α – κ)μ∗ (α – ) (α – βi – κ + ) α – κ – + i= + ( + |a|)μ∗ – κ | + a| (α) α – κ –κ |b|μ∗ | + b| ( – βi ) (α – ) α – κ – –κ –κ –κ u–v = u–v Since < , F is a contraction mapping, therefore, by using the Banach contraction principle, F has a unique fixed point, which is the unique solution of problem () by using Lemma . Corollary . Assume that there exists L > such that f (t, x, y, w, u , u , , un ) – f t, x , y , w , v , v , , ≤ L x – x + y – y + w – w + |u – v | + |u – v | + · · · + |un – | for all t ∈ [, ] and x, y, w, x , y , w , u , u , , un , v , v , , ∈ R Then problem () has a unique solution whenever ( + γ + λ ) n + i= ( + |a|)( + (α + )|b|)L | + a|| + b| (α + ) |b|L L + (α – βi + ) | + b| ( – βi ) (α) where γ = supt∈I | t γ (t, s) ds|, λ = supt∈I | t < , λ(t, s) ds| Now, we restate the Schauder’s fixed point theorem, which is needed to prove next result (see Theorem .. in []) Theorem . Let E be a closed, convex and bounded subset of a Banach space X, and let F : E → E be a continuous mapping such that F(E) is a relatively compact subset of X Then F has a fixed point in E Baleanu et al Advances in Difference Equations 2013, 2013:368 http://www.advancesindifferenceequations.com/content/2013/1/368 Page of 17 Theorem . Let f : [, ] × Rn+ → R be a continuous function such that there exists a constant l ∈ (, α – ) and a real-valued function m(t) ∈ L l ([, ], (, ∞)) such that f (t, x, y, w, u , u , , un ) ≤ m(t) + d|x|ρ + d |y|ρ + d |w|ρ + d |u |ρ + d |u |ρ + · · · + dn |un |ρn , (∗) where d, d , d , di ≥ and < ρ, ρ , ρ , ρi < for i = , , , n, or f (t, x, y, w, u , u , , un ) ≤ d|x|ρ + d |y|ρ + d |w|ρ + d |u |ρ + d |u |ρ + · · · + dn |un |ρn , where d, d , d , di > and ρ, ρ , ρ , ρi > for i = , , , n Then problem () has a solution Proof First, suppose that f satisfy condition (∗) Define Br = {u ∈ X, u ≤ r}, where r ≥ max (n + )Ad (n + )Ad K= n –l + –ρn –ρ p , (n + )Ad λ –l |b|( + |a|)M | + a|| + b| (α – ) α – l – –l |b|M | + b| ( – βi ) (α – ) α – l – ( + |a|)( + ( + α)|b|) + | + a|| + b| (α + ) n i= –ρ , (n + )Ad –ρ , , (n + )K , –l –l –l (α – l)M (α – ) (α – βi – l + ) α – l – i= + p , (n + )Ad γ , , (n + )Adn ( + |a|)M – l | + a| (α) α – l + A= –ρ –ρ –l , |b| + (α – βi + ) | + b| (α) ( – βi ) and M = ( (m(t)) l ds)l Note that Br is a closed, bounded and convex subset of the Banach space X For each u ∈ Br , we have t (Fu)(t) = – (t – s)α– f s, u(s), (φu)(s), (ψu)(s), c Dβ u(s), c Dβ u(s), , c Dβn u(s) ds (α) a ( + a) ( – s)α– (α) × f s, u(s), (φu)(s), (ψu)(s), c Dβ u(s), c Dβ u(s), , c Dβn u(s) ds + ab – b( + a)t ( + a)( + b) ( – s)α– (α – ) × f s, u(s), (φu)(s), (ψu)(s), c Dβ u(s), c Dβ u(s), , c Dβn u(s) ds t ≤ + (t – s)α– f s, u(s), (φu)(s), (ψu)(s), c Dβ u(s), c Dβ u(s), , c Dβn u(s) ds (α) |a| | + a| ( – s)α– (α) Baleanu et al Advances in Difference Equations 2013, 2013:368 http://www.advancesindifferenceequations.com/content/2013/1/368 Page of 17 × f s, u(s), (φu)(s), (ψu)(s), c Dβ u(s), c Dβ u(s), , c Dβn u(s) ds |b|( + |a|) | + a|| + b| + ( – s)α– (α – ) × f s, u(s), (φu)(s), (ψu)(s), c Dβ u(s), c Dβ u(s), , c Dβn u(s) ds t ≤ (t – s)α– m(s) ds (α) + dr ρ + d γ rρ + d λ rρ + d rρ + d rρ + · · · + dn rρn p |a| | + a| + t p (t – s)α– ds (α) α– ( – s) m(s) ds (α) |a| p p + drρ + d γ rρ + d λ rρ + d rρ + d rρ + · · · + dn rρn | + a| × |b|( + |a|) ( – s)α– ds + (α) | + a|| + b| ( – s)α– m(s) ds (α – ) |b|( + |a|) p p + drρ + d γ rρ + d λ rρ + d rρ + d rρ + · · · + dn rρn | + a|| + b| × ( – s)α– ds (α – ) t (α) ≤ + + + (t – s)α– –l –l t ds m(s) l l ds |a| | + a| (α) ( – s)α– –l –l ds m(s) l l ds |b|( + |a|) | + a|| + b| (α – ) ( – s)α– –l –l ds m(s) l l ds ( + |a|)( + ( + α)|b|) | + a|| + b| (α + ) × drρ + d γ rρ + d λ rρ + d rρ + d rρ + · · · + dn rρn p ≤ p ( + |a|)M – l | + a| (α) α – l + –l + –l |b|( + |a|)M | + a|| + b| (α – ) α – l – –l ( + |a|)( + ( + α)|b|) | + a|| + b| (α + ) × drρ + d γ rρ + d λ rρ + d rρ + d rρ + · · · + dn rρn p p Also, we have c Dβi (Fu)(t) = t (t – s)–βi (Fu) (s) ds ( – βi ) t (t – s)–βi ( – βi ) = s (s – τ )α– (α – ) × f τ , u(τ ), (φu)(τ ), (ψu)(τ ), c Dβ u(τ ), c Dβ u(τ ), , c Dβn u(τ ) dτ – b +b ( – τ )α– (α – ) Baleanu et al Advances in Difference Equations 2013, 2013:368 http://www.advancesindifferenceequations.com/content/2013/1/368 Page of 17 × f τ , u(τ ), (φu)(τ ), (ψu)(τ ), c Dβ u(τ ), c Dβ u(τ ), , c Dβn u(τ ) dτ ds t ≤ c (s – τ )α– f τ , u(τ ), (φu)(τ ), (ψu)(τ ), (α – ) Dβ u(τ ), c Dβ u(τ ), , c Dβn u(τ ) dτ ds |b| | + b| + c s (t – s)–βi ( – βi ) t (t – s)–βi ( – βi ) ( – τ )α– f τ , u(τ ), (φu)(τ ), (ψu)(τ ), (α – ) Dβ u(τ ), c Dβ u(τ ), , c Dβn u(τ ) dτ ds t ≤ ρ + dr + d t × p γ rρ t + d λ r ρ + d r ρ + d r ρ + · · · + d n r ρn p (s – τ )α– dτ ds (α – ) (t – s)–βi ( – βi ) ( – τ )α– m(τ ) dτ ds (α – ) |b| p p drρ + d γ rρ + d λ rρ + d rρ + d rρ + · · · + dn rρn | + b| + t × s × ( – τ )α– dτ ds (α – ) (t – s)–βi ( – βi ) (α – ) ( – βi ) ≤ (s – τ )α– m(τ ) dτ ds (α – ) s (t – s)–βi ( – βi ) |b| | + b| + s (t – s)–βi ( – βi ) t (t – s)–βi (s – τ )α– –l –l m(τ ) l l dτ ds (drρ + d γ rρ + d λ rρ + d rρ + d rρ + · · · + dn rρn ) (α) ( – βi ) p + t × p (t – s)–βi sα– ds + t × (t – s)–βi |b| | + b| (α – ) ( – βi ) ( – τ )α– –l –l dτ m(τ ) l l dτ |b|(drρ + d γ rρ + d λ rρ + d rρ + d rρ + · · · + dn rρn ) | + b| (α) ( – βi ) p + p –l M (α – ) ( – βi ) α – l – ≤ s dτ t (t – s)–βi sα–l– ds (drρ + d γ rρ + d λ rρ + d rρ + d rρ + · · · + dn rρn ) (α) ( – βi ) p + –l t × (t – s)–βi sα– ds p ds Baleanu et al Advances in Difference Equations 2013, 2013:368 http://www.advancesindifferenceequations.com/content/2013/1/368 Page 10 of 17 –l |b|M | + b| (α – ) ( – βi ) α – l – + t (t – s)–βi ds |b|(drρ + d γ rρ + d λ rρ + d rρ + d rρ + · · · + dn rρn ) | + b| (α) ( – βi ) p + p –l M (α – ) ( – βi ) α – l – ≤ –l –l ( – ξ )–βi ξ α–l– dξ (drρ + d γ rρ + d λ rρ + d rρ + d rρ + · · · + dn rρn ) (α) ( – βi ) p + × p ( – ξ )–βi ξ α– dξ –l |b|M | + b| (α – ) ( – βi ) α – l – + |b|(drρ + d γ rρ + d λ rρ + d rρ + d rρ + · · · + dn rρn ) | + b| (α) ( – βi ) p Since B(α – l, – βi ) = βi ) = –βi α– ξ dξ ( – ξ ) c –l + p ( – ξ )–βi ξ α–l– dξ = = (α) (–βi ) , (α–βi +) (α–l) (–βi ) (α–βi –l+) we conclude that –l (α – l)M (α – ) (α – βi – l + ) α – l – Dβi (Fu)(t) ≤ and, on the other hand, B(α, – –l –l + –l |b|M | + b| (α – ) ( – βi ) α – l – + drρ + d γ rρ + d λ rρ + d rρ + d rρ + · · · + dn rρn (α – βi + ) + |b|(drρ + d γ rρ + d λ rρ + d rρ + d rρ + · · · + dn rρn ) | + b| (α) ( – βi ) p p p p for all i = , , , n Thus, Fu ≤ ( + |a|)M – l | + a| (α) α – l n + –l |b|( + |a|)M | + a|| + b| (α – ) α – l – –l (α – l)M (α – ) (α – βi – l + ) α – l – + i= + –l –l |b|M | + b| (α – ) ( – βi ) α – l – –l –l –l + drρ + d γ rρ + d λ rρ + d rρ + d rρ + · · · + dn rρn p × ( + |a|)( + ( + α)|b|) + | + a|| + b| (α + ) ρ = K + dr + d ≤ p p γ rρ r × (n + ) = r n+ +d p λ r ρ n i= |b| + (α – βi + ) | + b| (α) ( – βi ) + d r ρ + d r ρ + · · · + d n r ρn A Baleanu et al Advances in Difference Equations 2013, 2013:368 http://www.advancesindifferenceequations.com/content/2013/1/368 Page 11 of 17 Hence, F maps Br into Br Now, suppose that f satisfy the second condition In this case, choose for j = , , and i = , , , , then the second condition of Theorem . holds Thus, by using Theorem ., the boundary value problem () has a solution Conclusions Fractional nonlinear differential equations, fractional integro-differential equations and their applications represent a topic of high interest in the area of fractional calculus and its applications in various fields of science and engineering [] In this article, based on the Schauder’s fixed point theorem, we have proved some existence results for the multi-term nonlinear fractional integro-differential equation () Competing interests The authors declare that they have no competing interests Authors’ contributions All authors have equal contributions All authors read and approved the final manuscript Author details Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O Box 80204, Jeddah, 21589, Saudi Arabia Department of Mathematics, Cankaya University, Ogretmenler Cad 14, Balgat, Ankara, 06530, Turkey Institute of Space Sciences, Magurele, Bucharest, Romania Department of Mathematics, Azarbaijan Shahid Madani University, Azarshahr, Tabriz, Iran Acknowledgements Research of the second and the third authors was supported by Azarbaijan Shahid Madani University Also, the authors express their gratitude to the referees for their helpful suggestions, which improved the final version of this paper Received: 21 March 2013 Accepted: 28 August 2013 Published: 13 Dec 2013 References Hilfer, R: Applications of Fractional Calculus in Physics World Scientific, Singapore (2000) Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations North-Holland Mathematics Studied, vol 204 Elsevier, Amsterdam (2006) Podlubny, I: Fractional Differential Equations Mathematics in Science and Engineering Academic Press, San Diego (1999) Ahmad, B: Existence of solutions for fractional differential equations of order q ∈ (2, 3) with anti-periodic boundary conditions J Appl Math Comput 34, 385-391 (2009) Baleanu, D, Agarwal, RP, Mohammadi, H, Rezapour, S: Some existence results for a nonlinear fractional differential equation on partially ordered Banach spaces Bound Value Probl 2013, 112 (2013) Baleanu, D, Mohammadi, H, Rezapour, S: Positive solutions of a boundary value problem for nonlinear fractional differential equations Abstr Appl Anal 2012, Article ID 837437 (2012) Baleanu, D, Mohammadi, H, Rezapour, S: Some existence results on nonlinear fractional differential equations Philos Trans R Soc Lond A 371, 20120144 (2013) Baleanu, D, Mohammadi, H, Rezapour, S: On a nonlinear fractional differential equation on partially ordered metric spaces Adv Differ Equ 2013, 83 (2013) Chang, YK, Nieto, JJ: Some new existence results for fractional differential inclusions with boundary conditions Math Comput Model 49, 605-609 (2009) 10 Chen, A, Chen, Y: Existence of solutions to anti-periodic boundary value problem for nonlinear fractional differential equations Differ Equ Dyn Syst 19(3), 237-252 (2011) 11 Davi, JV, Lakshmikantham, V: Nonsmooth analysis and fractional differential equations Nonlinear Anal 69, 2677-2682 (2008) 12 Lakshmikantham, V, Vatsala, AS: Basic theory of fractional differential equations Nonlinear Anal 69, 2677-2682 (2008) 13 Li, CF, Luo, XN, Zhou, Y: Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations Comput Math Appl 59, 1363-1375 (2010) 14 Sun, S, Li, Q, Li, Y: Existence and uniqueness of solutions for a coupled system of multi-term nonlinear fractional differential equations Comput Math Appl (2012) doi:10.1016/j.camwa.2012.01.065 15 Jaradat, O, Al-Omari, A, Momani, SM: Existence of the mild solution for fractional semilinear initial value problems Nonlinear Anal 69, 3153-3159 (2008) 16 Momani, SM, El-Khazali, R: On the existence of extremal solutions of fractional integro-differential equations J Fract Calc 18, 87-92 (2000) 17 Momani, SM: Some existence theorems on fractional integro-differential equations Abhath Al-Yarmouk J 10, 435-444 (2001) 18 Momani, SM, Hadid, SB: On the inequalities of integro-differential fractional equations Int J Appl Math 12(1), 29-37 (2003) Page 16 of 17 Baleanu et al Advances in Difference Equations 2013, 2013:368 http://www.advancesindifferenceequations.com/content/2013/1/368 19 Momani, SM, Hadid, SB: Some comparison results for integro-fractional differential inequalities J Fract Calc 24, 379-387 (2003) 20 Xinwei, S, Landong, L: Existence of solution for boundary value problem of nonlinear fractional differential equation Appl Math J Chin Univ Ser B 22(3), 291-298 (2007) 21 Su, X, Zhang, S: Solutions to boundary value problems for nonlinear differential equations of fractional order Electron J Differ Equ 2009, 26 (2009) 22 Ahmad, B, Sivasundaram, S: On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order Appl Math Comput 217, 480-487 (2010) 23 Agarwal, RP, O’Regan, D, Sahu, DR: Fixed Point Theory for Lipschitzian-Type Mappings with Applications Springer, Berlin (2009) 24 Ahmad, B: Nonlinear fractional differential equations with anti-periodic type fractional boundary conditions Differ Equ Dyn Syst (2012) doi:10.1007/s12591-012-0154-2 25 Ahmad, B, Nieto, JJ: Anti-periodic fractional boundary value problems with nonlinear term depending on lower order Fract Calc Appl Anal 15, 451-462 (2012) 10.1186/1687-1847-2013-368 Cite this article as: Baleanu et al.: Existence and uniqueness of solutions for multi-term nonlinear fractional integro-differential equations Advances in Difference Equations 2013, 2013:368 Page 17 of 17
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