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Chapter 3 Series Involving Zeta Functions. Chapter 4 Evaluations and Series Representations. Chapter 5 Determinants of the Laplacians.
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Chapter 7 Miscellaneous Results. Chapter 1 Introduction and Preliminaries. The q -integral of a function f defined in the interval [ 0 , b ] is given by.
Similarly, as done for derivatives, an operator I q n can be defined, namely,. The fundamental theorem of calculus applies to these operators I q and D q , that is,. Basic properties of the two operators can be found in the book mentioned in [ 13 ]. We now point out three formulas that will be used later D q i denotes the derivative with respect to variable i [ 19 ]. Remark 2. Definition 2. Lemma 2.
Zeta and q-Zeta Functions and Associated Series and Integrals - 1st Edition
Then the next formulas hold :. If the implication. Definition 3. Lemma 3.
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Then , the following equality holds :. Then the following equality holds :. Remark 3. Denote by.
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Theorem 3. It is well known the fixed points of the operator A are solutions of the problem 1. Clearly, the subset D is closed, convex and equicontinuous. The proof will be given in three steps. Step 1: We will show that the operator A maps D into itself. Step 2: We will show that the operator A is weakly sequentially continuous.
Since f satisfies assumptions H1 , we have f t , x n t converge weakly uniformly to f t , x t. Step 3: The implication 2. Thus, A V t is weakly relatively compact since a subset of a reflexive Banach space is weakly relatively compact if and only if it is bounded in the norm topology. This completes the proof.
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Since f satisfies assumptions H1 , we have f t , x n t , converging weakly uniformly to f t , x t. Podlubny I: Fractional Differential Equations. Academic Press, San Diego; Elsevier, Amsterdam; El-Shahed M: Existence of solution for a boundary value problem of fractional order. Zhang S: Existence of solution for a boundary value problem of fractional order. Acta Math. Zhou W, Chu Y: Existence of solutions for fractional differential equations with multi-point boundary conditions.
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Jackson FH: On q -functions and a certain difference operator. Jackson R: On q -definite integrals.
Pure Appl. Springer, New York; Al-Salam WA: Some fractional q -integrals and q -derivatives. Agarwal RP: Certain fractional q -integrals and q -derivatives. Ferreira RAC: Nontrivial solutions for fractional q -difference boundary value problems. Theory Differ. Ferreira RAC: Positive solutions for a class of boundary value problems with fractional q -differences. ISRN Math. Dekker, New York; Nonlinear Anal. Kluwer Academic, Dordrecht; Krzyska S, Kubiaczyk I: On bounded pseudo and weak solutions of a nonlinear differential equation in Banach spaces.
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