A Note on the Lower and Upper Solutions of Hybrid-Type Iterative Fractional Differential Equations

In this paper, we introduce and investigate a hybrid type of nonlinear Riemann Liouville fractional integro-differential equations. We develop and extend previous work on such non-fractional equations, using operator theoretical techniques, and find the approximate solutions. We prove the existence as well as the uniqueness of the corresponding approximate solutions by using hybrid fixed point theorems and provide upper and lower bounds to these solutions. Furthermore, some examples are provided, in which the general claims in the main theorems are demonstrated.

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“…Here, we recall some related concepts which will be useful in our study; details can be found in [8][9][10][11][12].…”

Section: Preliminariesmentioning

confidence: 99%

On Hybrid Type Nonlinear Fractional Integrodifferential Equations

2020

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“…Next, we introduce some related concepts which will be useful in our study, details can be found for example in [4,5,6,7,8] Denition 2.1 [2,3] Let α ∈ R + . The operator I α is dened on L 1 [a, b] by…”

Section: Preliminariesmentioning

confidence: 99%

On the Solutions of Nonlinear Hybrid Fractional Integrodifferential Equations

Damag¹,

Kılıçman²,

Al-Arioi³

2020

Preprint

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In the present work we study the existence of solutions for hybrid nonlinear fractional integrodifferential equations. We developed an algorithm by using the operator theoretical techniques in order to obtain the approximate solutions. The main results depend on the Dhage iteration method that were incorporated with the modern hybrid fixed point theorems. The approximate solutions were obtained by using Lipschitz conditions and weaker form of mixed partial continuity. Further, we provide some examples to explain the hypotheses and the related results.

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“…During the last few decades diversity of positive solutions of different BVPs for fractional order nonlinear differential equation (FONLDE for short) has extensively considered by using various techniques, for instance see the articles of Agarwal et al [2,3], Afshari et al [1], Asaduzzaman and Ali [5], Bai [8], Chen et al [12], Cu et al [13], Devi et al [17], Sun et al [34], and Torres [36] as well as for lower and upper solutions to the integro-differential and iterative hybrid type fractional differential equations see, Damag et al [14] and Damag et al [15] and for positive solutions of nonlinear dissipative type equations, see Asaduzzaman et al [6].…”

Section: Introductionmentioning

confidence: 99%

Presence and diversity of positive solutions for a Caputo-type fractional order nonlinear differential equation with an advanced argument

Asaduzzaman¹,

Kılıçman²,

Ali³

2020

J. Math. Computer Sci.

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This article aims to construct the presence and diversity principles of minimum one or two positive solutions for a Caputotype fractional-order nonlinear differential equation (CFONLDE for short) with an advanced argument under three-point boundary value conditions (BVCs for short). Guo-Krasnoselskii's fixed point theorem and Fixed-point index theory in cone spaces are used to analyze this article. First, the Green's function of the corresponding boundary value problem for a linear fractional differential equation with an advanced argument has been established. Next, several essential properties of that Green's function have been proved. Finally, in cone spaces, some novel presence and diversity principles of minimum of one or two positive solutions for a CFONLDE with an advanced argument are obtained. To support the analytic proof, some particular examples are included.

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A Note on the Lower and Upper Solutions of Hybrid-Type Iterative Fractional Differential Equations

Cited by 6 publications

References 6 publications

On Hybrid Type Nonlinear Fractional Integrodifferential Equations

On Hybrid Type Nonlinear Fractional Integrodifferential Equations

On the Solutions of Nonlinear Hybrid Fractional Integrodifferential Equations

Presence and diversity of positive solutions for a Caputo-type fractional order nonlinear differential equation with an advanced argument

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