Fractional Calculus of Variations in Terms of a Generalized Fractional Integral with Applications to Physics

and Applied Analysis 3 2. Preliminaries In this section, we present definitions and properties of generalized fractional operators. As particular cases, by choosing appropriate kernels, these operators are reduced to standard fractional integrals and fractional derivatives. Other nonstandard kernels can also be considered as particular cases. For more on the subject of generalized fractional calculus and applications, we refer the reader to 37 . Throughout the text, α denotes a real number between zero and one. Following 45 , we use round brackets for the arguments of functions and square brackets for the arguments of operators. By definition, an operator receives and returns a function. Definition 2.1 generalized fractional integral . The operator K P is given by K P [ f ] x K P [ t −→ f t ] x p ∫x a kα x, t f t dt q ∫b x kα t, x f t dt, 2.1 where P 〈a, x, b, p, q〉 is the parameter set p-set for brevity , x ∈ a, b , p, q are real numbers, and kα x, t is a kernel which may depend on α. The operator K P is referred to as the operator K K-op for simplicity of order α and p-set P , while K P f is called the operation K or K-opn of f of order α and p-set P . Note that if we define G x, t : ⎧ ⎨ ⎩ pkα x, t , if t < x, qkα t, x , if t ≥ x, 2.2 then the operator K P can be written in the form K P [ f ] x K P [ t −→ f t ] x ∫b a G x, t f t dt. 2.3 This is a particular case of one of the oldest and most respectable class of operators, the socalled Fredholm operators 46, 47 . Theorem 2.2 cf. Example 6 of 46 . Let α ∈ 0, 1 and P 〈a, x, b, p, q〉. If kα is a square integrable function on the square Δ a, b × a, b , then K P : L2 a, b → L2 a, b is welldefined, linear, and bounded operator. Theorem 2.3. Let kα be a difference kernel, that is, let kα ∈ L1 a, b with kα x, t kα x − t . Then, K P : L1 a, b → L1 a, b is a well defined bounded and linear operator. Proof. Obviously, the operator is linear. Let α ∈ 0, 1 , P 〈a, t, b, p, q〉, and f ∈ L1 a, b . Define F τ, t : ⎧ ⎨ ⎩ ∣pkα t − τ ∣ · ∣f τ ∣, if τ ≤ t, ∣qkα τ − t ∣ · ∣f τ ∣, if τ > t, 2.4 4 Abstract and Applied Analysis for all τ, t ∈ Δ a, b × a, b . Since F is measurable on the square Δ, we have ∫b