Extremal solutions for certain type of fractional differential equations with maxima
© Ibrahim; licensee Springer. 2012
Received: 13 September 2011
Accepted: 8 February 2012
Published: 8 February 2012
In this article, we employ the Tarski's fixed point theorem to establish the existence of extremal solutions for fractional differential equations with maxima.
Fractional calculus has become an exciting new mathematical method of solution of diverse problems in mathematics, science, and engineering. Indeed, recent advances of fractional calculus are dominated by modern examples of applications in differential and integral equations and inclusions, physics, signal processing, fluid mechanics, viscoelasticity, mathematical biology, engineering, dynamical systems, control theory, electrical circuits, generalized voltage divider, computer sciences, and electrochemistry (see [1, 2]).
The theory and applications of fractional differential equations received in recent years considerable interest both in pure mathematics and in applications. There exist several different definitions of fractional differentiation. Whereas in mathematical treatises on fractional differential equations the Riemann-Liouville approach to the notion of the fractional derivative is normally used [3–5], the Caputo fractional derivative often appears in applications , Erdèlyi-Kober fractional derivative  and The Weyl-Riesz fractional operators . There are some advantages in studying the extremal solution for fractional differential equations, because some boundary conditions are automatically fulfilled and due to lower order differential requirements (see ).
Differential equations with maximum arise naturally when solving practical and phenomenon problems, in particular, in those which appear in the study of systems with automatic regulation and automatic control of various technical systems. It often occurs that the law of regulation depends on maximum values of some regulated state parameters over certain time intervals. Many studies of the existence of solutions are imposed such as periodicity, asymptotic stability and oscillatory [10–12]. In , the authors discusses the existence of univalent solutions for fractional integral equations with maxima in complex domain, by using technique associated with measures of non-compactness.
In this article, we establish the extreme solutions (maximal and minimal solutions) for fractional differential equation with maxima in sense of Riemann-Liouville fractional operators, by using the Tarski's fixed point theorem. Moreover, we extend the existence of extremal solutions from initial value problems to boundary value problems for infinite quasi-monotone functional systems of fractional differential equations.
The fundamental tool in our work is the following well-known Tarski's fixed point theorem which can be found in :
One of the most frequently used tools in the theory of fractional calculus is furnished by the Riemann-Liouville operators (see ).
and ϕ α (t) = 0, t ≤ 0 and ϕ α → δ(t) as α → 0 where δ(t) is the delta function.
3 Main results
Analogously we say that u j is an upper solution of (1) if the above inequalities are reversed. We say that is a solution of (1) if it is both a lower and an upper solution. A solution u* in A ⊂ S is a maximal solution in the set A if u* ≥ u for any other solution u ∈ A. The minimal solution in A is defined analogously by reversing the inequalities; when both a minimal and a maximal solution in A exist, we call them the extremal solutions in A.
Next we pose our main result
- (i)For each ξ ∈ [γ, λ] S the initial value problem(3)
- (ii)For each and t ∈ J if u(t) ≤ v(t) and then
The function is nondecreasing in [γ, λ] S . Moreover, the function ϕ is nondecreasing in [-η, 0].
Then problem (1) has a maximal solution, u*, and a minimal one, u*, in [γ, λ] S .
Proof. We shall prove the existence of the maximal solution since the existence of the minimal solution follows from the dual arguments.
Next we proceed to prove that Φ satisfies the conditions of Theorem 2.1.
Step 1. Φ: [γ, λ] S → [γ, λ] S is nondecreasing.
Since is arbitrary we conclude that (Φξ1) ≤ (Φξ2).
Step 2. [γ, λ] S is a complete lattice.
Therefor and ξ* = sup B. The existence of inf B is proved by similar manner. Hence is a complete lattice and consequently .
Step 3. X* is the maximal solution of problem (1) in [γ, λ] S .
Then by (5) it follows that for every solution x of the problem (1) satisfies x ≤ x*. This completes the proof of Theorem 3.1.
Next we replace the condition (i) by assuming F in the set of -Carathéodory.
Definition 3.2. A mapping p : J × ℝ → ℝ is said to be Carathéodory if
(C1) t → p (t, u) is measurable for each u ∈ ℝ,
(C2) u → p (t, u) is continuous a.e. for t ∈ J.
A Carathéodory function p (t, u) is called L1 (J, ℝ)-Carathéodory if (C3) for each number r > 0 there exists a function h r ∈ L1(J, ℝ) such that |p(t, u)| ≤ h r (t) a.e t ∈ J for all u ∈ ℝ with |u| = r.
A Carathéodory function p (t, u) is called - Carthéodory if (C4) there exists a function h ∈ L1(J, ℝ) such that |p (t, u)| ≤ h (t) a.e t ∈ J for all u ∈ ℝ where h is called the bounded function of p.
Theorem 3.2. Let F be - Carathéodory. If the assumptions (ii) and (iii) hold then the problem (1) has at least one solution u (t) on J.
where is the space of all continuous real valued functions on J with a supremum norm that is P : B ρ → B ρ . Therefore, P maps B ρ into itself. In fact, P maps the convex closure of P [B ρ ] into itself. Since f is bounded on B ρ , thus P [B ρ ] is equicontinuous and the Schauder fixed point theorem shows that P has at least one fixed point u ∈ A such that Pu = u, which is corresponding to solution of the problem (1). To obtain the maximal and minimal solutions, we use the same arguments in Theorem 3.1.
Moreover condition (i) can replaced by letting F in the set of all functions which are μ - Lipschitz. We have the following definition:
- (i)a μ - Lipschitz if and only if there exists a positive constant μ such that
A contraction if and only if it is μ - Lipschitz with μ < 1.
Theorem 3.3. Let F be μ - Lipschitz. If , then (1) has a unique solution u(t) on J.
Hence by the assumption of the theorem we have that P is a contraction mapping then in view of the Banach fixed point theorem, P has a unique fixed point which is corresponding to the solution of Equation (1). In this case u (t) = u* (t) = u* (t).
It is clear that F is - Carathéodory with any decreasing growth function h ∈ L1(J, ℝ+) such that ||F (t, u)|| ≤ h (t) a.e t ∈ J for all u ∈ ℝ. Therefore in view of Theorem 3.2, the problem (8) has maximal and minimal solutions.
Obviously F does not satisfy the condition (i) of Theorem 3.1, and hence the problem (9) hasn't extremal solutions.
This research has been funded by the University Malaya, under the Grant No. RG208-11AFR.
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